research article turn-based war chess model and its search...

Research ArticleTurn-Based War Chess Model and Its SearchAlgorithm per Turn

Hai Nan12 Bin Fang1 Guixin Wang2 Weibin Yang3 Emily Sarah Carruthers4 and Yi Liu5

1College of Computer Science Chongqing University Chongqing 400044 China2Department of Software Engineering Chongqing Institute of Engineering Chongqing 400056 China3College of Automation Chongqing University Chongqing 400044 China4College of International Education Chongqing University Chongqing 400044 China5PetroChina Chongqing Marketing Jiangnan Company Chongqing 400060 China

Correspondence should be addressed to Bin Fang fbcqueducn

Received 20 August 2015 Revised 25 December 2015 Accepted 10 January 2016

Academic Editor Michela Mortara

Copyright copy 2016 Hai Nan et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

War chess gaming has so far received insufficient attention but is a significant component of turn-based strategy games (TBS) andis studied in this paper First a common gamemodel is proposed through various existing war chess types Based on the model wepropose a theory frame involving combinational optimization on the one hand and game tree search on the other hand We alsodiscuss a key problem namely that the number of the branching factors of each turn in the game tree is huge Then we proposetwo algorithms for searching in one turn to solve the problem (1) enumeration by order (2) enumeration by recursion The maindifference between these two is the permutation method used the former uses the dictionary sequence method while the latteruses the recursive permutation method Finally we prove that both of these algorithms are optimal and we analyze the differencebetween their efficiencies An important factor is the total time taken for the unit to expand until it achieves its reachable positionThe factor which is the total number of expansions that each unit makes in its reachable position is set The conclusion proposedis in terms of this factor Enumeration by recursion is better than enumeration by order in all situations

1 Introduction

Artificial intelligence (AI) is one of the most importantresearch fields in computer science and its related algorithmstechnologies and research results are being widely used invarious industries such as military psychological intelligentmachines and business intelligence Computer games knownas ldquoartificial intelligencersquos drosophilardquo are an importantpart of artificial intelligence research With the increasingdevelopment of computer hardware and research methodsartificial intelligence research in traditional board games hasseen some preliminary results Alus et al [1] have proven thatGo-MokursquosAI provided itmoves first is bound towin againstany (optimal) opponent by the use of threat-space search andproof-number search TheMonte Carlo Tree Search (MCTS)method based on UCT (UCB for tree search) has improvedthe strength of 9 times 9 Go close to the level of a professionalKudan [2]

Computer game based on artificial intelligence is a sort ofdeterministic turn-based zero-sum game containing certaininformation Man-machine games can be classified into twocategories two-player game andmultiplayer game accordingto the number of game players Most traditional chessessuch as the game of Go and Chess belong to the two-playergame category to which120572-120573 search based onmin-max searchand its enhancement algorithms such as Fail-Soft 120572-120573 [3]Aspiration Search [4] NullMove Pruning [4] Principal Vari-ation Search [5] and MTD(f) [5] are usually applied On thecontrary Multiplayer Checkers Hearts Sergeant Major andso forth belong to the multiplayer game category [6] whichruns according to a fixed order of actions with participantsfighting each other and competing to be the sole winner of thegame Its search algorithm involves 119872119886119909

119899 search [7] Para-noid [6] and so forth The 120572-120573 search previously mentionedbased onmin-max search is a special case based on the119872119886119909

119899

search and shadow pruning algorithms [7] Man-machine

Hindawi Publishing CorporationInternational Journal of Computer Games TechnologyVolume 2016 Article ID 5216861 14 pageshttpdxdoiorg10115520165216861

2 International Journal of Computer Games Technology

(a) Wargaming (b) SLG game ldquoBattle Commanderrdquo

Figure 1 Wargaming and SLG game

Figure 2 SLG game ldquoHeroes of Might amp Magicrdquo

games can also be classified into two categories classic boardgames and new board games according to the game con-tent Classic board games involve Go chess backgammoncheckers and so forth which are widespread and have a longhistory While other board games such as Hex [8] Lines ofAction [9] and Scotland Yard [10] are ancient games withthe rapid development of modern board games and mobileclient applications they have been accepted bymore andmoreplayers until their prevalence is comparable to that of theclassic board game The machine game algorithms of theboard games listed above are all based on120572-120573 search and theirenhancement algorithms The MCTS algorithm has devel-oped rapidly in recent years being used increasingly in theseboard games and getting increasingly satisfactory results [8ndash10]

However not all board games can be solved with theexisting algorithms Turn-based strategy games (TBS) as well

as turn-based battle simulation games (SLG) (hereinafter col-lectively referred to as turn-based strategy games) originatedfrom the wargames [11] that swept the world in the mid-19thcentury (Figure 1(a) shows an example of a wargame) Withthe introduction of computer technology this new type ofgame turn-based strategy game has flourished (Figure 1(b)shows a famous TBS game called ldquoBattle Commanderrdquo andFigure 2 shows the popular SLG game ldquoHeroes of Might ampMagicrdquo) Now TBS games have become the second mostfamous type of game after RPGs (role-playing games) Withthe blossoming of mobile games TBS games will have greaterpotential for development in the areas of touch-screen oper-ation lightweight fragmented time and so on The contentof a TBS game generally comprises two levels strategic coor-dination and tactical battle control The latter level whoserules are similar to those of board games for examplemovingpieces on the board beating a specified enemy target for

International Journal of Computer Games Technology 3

victory and turn-based orders is called the turn-based warchess game (TBW) The artificial intelligence in TBW is animportant component of TBS games The AI of modern TBSgames is generally not so intelligent of which the fundamen-tal reason is that the AI in its local battle (TBW) is not sointelligent How to improve the TBWrsquos artificial intelligencethus improving the vitality of the entire TBS game industryis an urgent problem that until now has been overlooked

Currently the study of artificial intelligence in turn-basedstrategy games is mainly aimed at its macro aspect and theresearch object is primarily the overall macro logistics suchas the overall planning of resources construction produc-tion and other policy elements The main research con-tents involve planning uncertainty decisions spatial reason-ing resource management cooperation and self-adaptationHowever studies on artificial intelligence for a specific type ofcombat in TBS are scarce and the attention paid to research-ing the TBW unitsrsquo moves attacks and presentation of thegame round transformation whose AI is precisely the worstof all parts of the AI in a large number of TBS games is notenough At present the research related to TBWrsquos behaviorinvolves spatial reasoning techniques Bergsma and Spronck[12] divided the AI of TBS (NINTENDOrsquos Advanced Wars)into tactical and strategic modules The tactical moduleessentially has to decide where to move units and what toattack It accomplishes this by computing influence mapsassigning a value to each map tile to indicate the desirabilityfor a unit to move towards the tile This value is computed byan artificial neural network However they unfortunately donot provide any detail on how such amechanismwouldworkPaskaradevan and Denzinger [13] presented a shout-aheadarchitecture based on two rule sets one making decisionswithout communicated intentions and one with these inten-tions Reinforcement learning is used to learn rule weights(that influence decisionmaking) while evolutionary learningis used to evolve good rule sets Meanwhile based on thearchitecture Wiens et al [14] presented improvements thatadd knowledge about terrain to the learning and that alsoevaluate unit behaviors on several scenario maps to learnmore general rules However both approaches are essentiallybased on rules for the artificial intelligence resulting in a lackof flexibility of intelligent behaviors a lack of generality asthey depend on a gamersquos custom settings and moreover alack of reasoning for more than one future turn similar tocommon chess games

At present research on TBWrsquos AI from the perspective ofthemultiround chess gamemethod is scarce because a TBWrsquosplayer needs to operate all his pieces during each roundwhich is an essential difference with other ordinary chessgames Thus the number of situations generated by permu-tation grows explosively such that from this perspective theTBWrsquos AI can hardly be solved during regular playtime by thegame approach described previously

This paper attempts to study TBWrsquos AI from the perspec-tive of the chess game method This is because the TBWrsquosrules have many similarities with other chess games and thedecision made every turn in a TBW can be made wisely as inother chess games In this paper we propose two enumerationmethods in a single round dictionary sequence enumeration

B

1

1

2

2

3 4

43

A

Figure 3 An example of TBWHere are four red units and four blueunits belonging to two players respectively on a square board Theunits are divided into sword men and archers The number writtenin the bottom right of each unit is the unit index White tilts meantheir terrain can be entered However ochre ones marked by ldquo119860rdquoillustrate hilly areas no unit can enter The dark green tilts markedby letter ldquo119861rdquo illustrate lakes or rivers which also cannot be enteredbut archers can remotely attack the other side of the tilts

and recursive enumeration which is the fundamental prob-lem in our new framework The improvement in TBWrsquos AIcan not only bring more challenges to game players butalso bear a new series of game elements such as smart AIteammates which will provide players with a new gamingexperience

A TBW game is essentially the compound of combina-tional optimization laterally and game tree search vertically(Section 32) which can be regarded as a programmingproblem ofmultiagent collaboration in stages and can be seenas a tree search problemwith huge branching factorThus theexpansion and development of the traditional systems hiddenbehind TBW games will make the research more meaningfulthan the game itself

This paper first summarizes the general game model forTBW and illustrates its key feature that is that the branchingfactor is huge in comparison with traditional chess gamesThen it puts forward two types of search algorithms for asingle round from different research angles the dictionarysequence enumeration method (Algorithm 2) and the recur-sive enumeration method (Algorithm 5) Ensuring invari-ability of the number of branches Algorithm 5 has less exten-sion operation of pieces than Algorithm 2 under a variety ofconditionsThe experiments also confirmed this conclusion

2 Game Module of Turn-Based War Chess

21 Rules TBW is played on a square hexagonal or octago-nal tile-basedmap Each tile is a composite that can consist ofterrain types such as rivers forests andmountains or built upareas such as bridges castles and villages (Figure 3) Each tileimposes a movement cost on the units that enter them Thismovement cost is based on both the type of terrain and the


(a)

A B

(b)

B

(c)

B

1 2

(d)

Figure 4 Green tilts illustrate the movement range of a swordsman whose movement point is 2 The movement cost of each tilt is 1 (a) Noobstacle (b) Tilt 119860 is an obstacle and thus tilt 119861 is out of the movement range (c) The swordsman cannot pass the enemy to reach tilt 119861 (d)The swordsman can pass units of the same side to reach tilt 119861

type of unit Each tile is occupied by only one unit at the sametime

Each player in a TWB game controls an army consistingof many units All units can either move or attack an enemyunit Each unit has an allotted number of movement pointsthat it uses to move across the tiles Because different tileshave different movement costs the distance that a unit cantravel often varies All of the tiles the unit can travel tocompose a union of them called movement range (Figure 4)including the tile occupied by the unit itself The movementrange can generally be calculated by some algorithm such asbreadth first search [18]

In addition to the movement point each unit has its ownhealth point (Hp) and attack power (ATK) which are numer-ical values and are various among the different units Likemovement range a unitrsquos attack range is another union of tilesto which the unit can attack from its current tile (Figure 5)Commonly a unitrsquos attack range is determined by its attacktechnique Melee units such as swordsmen generally onlyattack adjacent units and thus their attack range looks like

that shown in Figure 5(a) Ranged attacking units such asarchers can attack enemies as far as two or more tiles away(Figure 5(b)) Special unitsrsquo attack range is also a special oneIf a unit attacks another unit it forfeits all of its movementpoints and cannot take any further actions that turn there-fore if a unit needs to be moved to a different tile it mustperform the move action prior to performing an attackaction A unit also has the option not to take any attack actionafter its movement or even not to take any action and stay onits current tile

Each unit attacked by its enemy must deduct its Hp bythe attacking unitrsquos ATK which indicates the damage Whena unitrsquos Hp is deducted to or below 0 this indicates that it isdead and must be removed from the board immediately Thetilt it occupied becomes empty and can be reached by otherfollowing units

A game of TBW consists of a sequence of turns On eachturn every player gets their own turn to perform all of theactions for each of their sidersquos units This is unlike ordinaryboard games such as chess where turns are only for selecting


(a) Swordsmenrsquos attack range (b) Archerrsquos attack range

Figure 5 Unitsrsquo attack range

a pawn tomoveThe opposing side does not get to perform itsactions until the current side has finished A player wins thegame if all of the units or the leader units of the other playerhave died

22 Setup and Notation TBW is composed of the board andpieces (units)The board is considered as an undirected graph119866(119881 119864) where 119881 is the set of vertices (tilts) and 119864 is the setof edges that connect the neighboring tilts Units are dividedinto two parties 119860 (Alexrsquos) and 119861 (Billiersquos) according to whichplayer they belong toThe sizes of the two parties are denotedas 119899119860and 119899119861 respectively and the indexes of the units in the

two parties are 1 2 119899119860and 1 2 119899

119861 respectively Let

119899 = 119899119860

+ 119899119861be the total number of units An assignment

119872 [1 119899] rarr 119881 places the units in unique tilts forall119894 119895 isin [1 119899]119895 = 119894 119872(119894) isin 119881 119872(119895) isin 119881 119872(119894) = 119872(119895) For each unitthere is a movement range 119877 sube 119881 where 119903 = |119877| and anattack range 119877atk where 119903atk = |119877atk|

Let 119874119903119889119862 be a sequence of elements in set 119862 such that

119874119903119889119862

119894is the 119894th element of this sequence We denote 119899 = |119862|

and thus 119874119903119889119862

119894isin 119862 and forall119894 119895 isin [1 119899] 119895 = 119894 119874119903119889

119862

119894= 119874119903119889119862

119895

Let 119876119862 be a set of all sequences of the elements in set 119862such that 119876119862 = 119874119903119889

119862

Thus |119876119862| = 119875119899

119899

Without loss of generality let119874119903119889119860 be an action sequence

of units in Alexrsquos turn such that 119874119903119889119860

119894expresses the index of

the unit doing the 119894th action where 119894 isin [1 119899119860]

23 Game Tree Search We try to use game tree search theoryto research the AI of TBW Game tree search is the mostpopular model for researching common chess games In thegame tree (Figure 6) nodes express states of the game boardBranches derived from nodes express selections of the movemethodThe root node is the current state and the leaf nodesare end states whose depths are specifically expanded fromthe root Both sides take turns Even layer nodes belong tothe current player (squares) while odd layer nodes belongto the other side (circles) If the leaf node is not able to give

a win-lose-draw final state an evaluation on a leaf node isneeded to select the expected better method from the currentstate this is the function of game tree search Game treesearch is based on min-max search which is used to findthe best outcome for the player and the best path leading tothis outcome (Principal Variation) and eventually to find thecorresponding move method in the root state (Root Move)that is the best move for the playerrsquos turn [19]

It is not difficult to see that the evaluation and search algo-rithmare themost important parts of the game tree For TBWthe evaluation factor of the state generally involves powerspositions spaces and motilities of units The most commonalgorithms of game tree search are Alpha-Beta search [20]and Monte Carlo Tree Search [21] which can also bealthoughnot directly applied toTBWrsquos searchThis is becausethe branching factor of the search tree for TBW is huge andthe common algorithms applied to TBWrsquos search cause atimeout

3 Features and Complexity Analysis

31 Complexity Analysis A game of TBW consists of asequence of turns During each turn every player gets theirown turn to perform all of the actions for each of theirsidersquos units which is the most important feature of TBWThesequence of actions is vitalThis is because the units cannot beoverlappedmoreover a different sequence of actionswill alsohave a different state when a unit of another side is eliminated(Figure 7) Thus during each sidersquos turn all of the plans ofactions for its units are calculated by a permutation methodThe amount of plans is estimated fromboth theworst and bestsituations (eg in the case of Alexrsquos turn)

Step 1 Determine the sequence of actions the total numberis 119875119899119860119899119860

= 119899119860

Step 2 Calculate the number of all plans of action in aspecified action sequence


Ply 0

Opponent Ply 1

Ply 2

Ply 3

Ply 4

Current player

Current player

Current player

Opponent

Figure 6 Game tree expanding to ply 4

2

1

(a)

(b) (c) (d)

(e) (f) (g)

1

1

1

1

1

111

1

2

2

2 2

2

2

2

222

2 2

2

Figure 7 Effect of actions sequence (a)The initial state of red sidersquos turn (bndashd) Red swordsman number 1 acts and eliminates blue swordsmannumber 1 followed by red swordsman number 2 (endashg) Red swordsman number 2 acts and eliminates blue swordsman number 1 followedby red swordsman number 1

Let 119877119894be the movement range of unit number 119894 such that

119903119894

= |119877119894| For simplicity we assume that 119903

1= 1199032

= sdot sdot sdot =

119903119899119860 = 119903 In the worst case themovement ranges of all ofAlexrsquos

units are independent without overlapping each other that isforall119894 119895 isin [1 119899

119860] 119895 = 119894 119877

119894cap 119877119895= Moreover in the attack

phase the amount of enemies that fall into each ofAlexrsquos unitsreaches maximum For example on a four-connected boarda melee unit has at most four adjacent tilts around it whichare full of enemiesThen the number of attack plans is atmostfive (including a plan not to attack any enemy) that is 119903atk+1

According to the multiplication principle the number ofstates expanding under a specified actions sequence is

[119903 (119903atk + 1)]119899119860

(1)

According to Step 1 the number of actions sequences is119875119899119860

119899119860

= 119899119860 and thus in the worst situation the number of

plans is

119878worst = [119903 (119903atk + 1)]119899119860

times 119899119860 (2)


SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(a)

SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(b)

Figure 8The growth trend of states 119878 following 119899119860under differentmovement points (a)Movement point 2 movement cost 1 (b)movement

point 5 movement cost 1

In the best situation the movement ranges of all unitsoverlap completely such that 119877

1= 1198772= sdot sdot sdot = 119877

119899119860 = 119877 More-

over there are no enemies in the attack range of every unitThus the amount of states can be calculated by the arrange-ment number 119875119899119860

119903such that we can select 119899

119860from 119903 positions

to make all of the arrangements of the units Therefore thenumber of plans in the best situation is

119878best =119903

(119903 minus 119899119860)

(3)

Above all the total number of plans under all actionsequences denoted by 119878 is

119878best lt 119878 le 119878worst (4)

In the following examples we calculate the actual valuesof the total plans 119878 For ldquoFire Emblemrdquo a typical ordinaryTBW game both sides have five units and in the openbattlefield themovement range of each unit can reach atmost61 tilts (in that map each tilt is adjacent to four other tilts themovement point is 5 the movement cost of each tilt is 1 andthere is no obstacle) Thus 119878best asymp 710 million and 119878worst asymp

317 trillion Assuming that the average computing time forsearching a plan is 200 nanoseconds searching all plans forone sidersquos turn will then take from 24 minutes to approxi-mately two years Note that in the formula 119899

119860is a key factor

such that as it increases the number of plans will dramaticallyexpand (Figure 8) For a large-scale TBW such as ldquoBattleCommanderrdquo whose units may amount to no less than adozen or dozens the search will be more difficult

Table 1 Branching factors comparison between TBW game andother board games

Branchingfactor Comment

Chess [15] asymp30 Maximum branching factor is 40Go [16] asymp100

Amazons [17] asymp1500 There are 2176 branches in thefirst turn

TBW 710 millionsim317trillion

Suppose that the movementpoint is 5 the movement cost is 1and the amount of units is 5 foreach side

32 Features and Comparison Compared with TBW gamesother board games (such as chess checkers etc) only requireselecting a unit to perform an action in a single round whichnot only results in fewer single-round action plans but alsomakes the number of plans linear with increasing numbers ofunits (for the chess type played by adding pieces such as Goand Go-Moku the number of plans is linear with increasingamounts of empty grids on the board)The number of single-round action plans corresponds to the size of the game treebranching factor Table 1 shows a comparison between TBWgames and some other ordinary board games that have morebranching factors A large branching factor and a rapidlyexpanding number of units are the key features by which theTWB games are distinguished from other board games

A TBW game is essentially the compound of combina-tional optimization laterally and game tree search vertically


1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4

middot middot middot


Figure 9 Search tree of TBW game (a) red sidersquos turn (b) blue sidersquos turn

(Figure 9) Vertically it can be seen as a tree search problemwith a huge branching factor Laterally the relationshipbetween layers is a series of phased combination optimiza-tions which is like a programming problem of multiagentcollaboration Therefore the new game model generatedby the expansion of the explosive branches needs to beresearched by new algorithms

Because the large number of states in a single round is thekey problembywhich the TBWgames are distinguished fromother board games the optimization search and pruning ofa single round have become the most important issues andprocesses for solving TBW games That the search of a single

round can be efficiently completed guarantees that the entiregame tree can be extended In the following we propose twosingle-round search algorithms and compare them

4 Single-Round Search Algorithms

41 Algorithm 2Dictionary Sequence EnumerationAlgorithmEach side of a TBW game (hereafter unless otherwise statedreferring specifically to Alexrsquos side) wants to achieve a singleturn search Based on Section 31 we need to first determinethe sequence of actions of 119899

119860units and then enumerate all of

the action plans of the units in each sequence


Input the original permutation sequence119874119903119889 = 119874119903119889

11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1

sdot sdot sdot 119874119903119889119896minus1

119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1

(2) if 119895 doesnrsquot exist then(3) exit and next permutation sequence doesnrsquot exist(4) else(5) find 119896 = max119894 | 119874119903119889

119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)

(7) reverse the sub-sequence 119874119903119889119895+1


119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119896119874119903119889119899sdot sdot sdot 119874119903119889

119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1

is thenext permutation sequence

(9) end if

Algorithm 1 next permutation(Ord)

(1) initialize a sequence 119874119903119889 which is the first sequence in dictionary sequences(2) while 119874119903119889 exists do(3) call Search(1)(4) 119874119903119889 larr 119899119890119909119905 119901119890119903119898119906119905119886119905119894119900119899(119874119903119889)

(5) end while

Algorithm 2 Dictionary sequence enumeration algorithm

411 Action Sequence of Units Algorithm Determining anaction sequence of 119899

119860units requires a permutation algorithm

There are some famous permutation algorithms such as therecursivemethod based on exchange the orthoposition trademethod the descending carry method and the dictionarysequence method [22ndash25] Their execution strategies aredifferent their time and space complexities vary and theyhave been used in different problems We first apply thedictionary sequencemethod whose time complexity is lowerThe idea of all permutation generation from 119899 elements (eg1 2 119899) is that with the beginning of the first sequence(123 sdot sdot sdot 119899) a series of subsequent larger sequences are gen-erated lexicographically until reaching the reverse order(119899 sdot sdot sdot 321) The algorithm called next permutation whichgenerates the next sequence from an original one is illus-trated as in Algorithm 1

For example 754938621 is a sequence of numbers 1ndash9Thenext sequence obtained by this algorithm is 754961238

412 Algorithm 2 Dictionary Sequence Enumeration Algo-rithm Enumerate all of the plans of unitsrsquo actions in a partic-ular order Because the search depth is limited (equal to thenumber of units) depth-first search is an effective methodBecause the depth is not great realizing the depth-first searchby the use of recursion requires smaller space overheadwhich leads to the sequential enumeration algorithm withpermutation and recursion as in Algorithm 2

Here Search(119894) is the algorithm for enumerating all of theaction plans of the 119894th unit (see Algorithm 3)

42 Algorithm 5 Recursive Enumeration AlgorithmAlgorithm 2 comes from a simple idea that always startsenumeration from the first unit in every search for the next

sequence However compared with the previous sequencethe front parts of units whose orders are not changed are notrequired to be enumerated again which creates redundantcomputing and reduces efficiency For example when thesearch of sequence 119874119903119889

1 1198741199031198892 119874119903119889

119894 119874119903119889

119895

119874119903119889119899is finished if the next sequential order is adjusted only

from the 119894th to the 119895th unit then in the recursive enumerationphases the units from the first one to the 119894 minus 1th can directlyinherit the enumeration results of the previous sequence andwe only need to enumerate the units from the 119894th to the lastone recursively On the basis of this feature we switch to therecursive permutation algorithm to achieve the arrangementso that the recursive algorithm combines with the recursivedepth-first search algorithm for the purpose of removing theredundant computation which is the improved algorithmcalled the recursive enumeration algorithm illustrated as inAlgorithm 4

In Algorithm 4 119899 is the size of our sequence (lines (1)(6)) With respect to the predefined procedure we generatethe permutations from the 119894th to the last unit in the sequenceby calling the function recursive permutation(119894) The latter isrealized using the subpermutations from the 119894 + 1th to thelast unit in the sequence which are generated by calling thefunction recursive permutation(119894 + 1) recursively (lines (5)ndash(11)) The index 119895 points to the unit swapped with the 119894thunit (line (7)) in every recursive call after which the two unitsmust resume their orders (line (9)) for the next step

By initializing the sequenceOrd and running the functionrecursive permutation(1) we can obtain a full permutation ofall the elements

Based on the above the improved single-round searchalgorithm called the recursive enumeration algorithm isdescribed as in Algorithm 5


(1) if 119894 gt 119899 then(2) return(3) else(4) for each action plan of the 119894th unit(5) execute the current plan(6) call 119878119890119886119903119888ℎ(119894 + 1)

(7) cancel this plan and rollback to the previous state(8) end for(9) end if

Algorithm 3 Search(i)

(1) if 119894 ge 119899 then(2) output the generated sequence 119874119903119889

1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1

(11) end while(12) end if

Algorithm 4 recursive permutation(i) (enumerate sequences from 119894th to the last element)

The framework of this new algorithm is similar to that ofthe recursive permutation algorithm where 119899 is the numberof units In the new algorithm all the action plans of the 119894thunit which involve selecting targets for attack are enumer-ated and executed separately (lines (7)-(8)) after the requiredswap process Then after solving the subproblem using therecursive callPlans Search(119894+1) a rollback of the current planis necessary and the state needs to be resumed (line (10))

To enumerate the actions plans of all the units thesequence 119874119903119889 is initialized and then the function PlansSearch(1) runs

From step (3) of Algorithm 5 before enumerating theaction plans of the unit we do not need to generate all of thesequences that is for each unit determination of its orderand enumeration of its actions are carried out simultaneously

43 Comparison First we compare the time complexities ofthe two algorithms

The time consumption of the recursive enumerationalgorithm lies in an 119899 times loop and an 119899minus1 times recursionsuch that the time complexity is119874(119899(119899minus1)(119899minus2) sdot sdot sdot 1) = 119874(119899)

[23] It is the same as the time complexity of the dictionarysequence enumeration algorithm [23] Moreover the statessearched by the two algorithms are also the same

Theorem 1 The states searched by Algorithms 2 and 5 are thesame

Proof Suppose 119878(119874119903119889) is the set of the states in the sequence119874119903119889 and Pre

119886are the sequences beginning with 119886 in 119876

119860

According to Algorithm 2 it first determines the order of asequence 119874119903119889 and then enumerates all of the states 119878

1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)

According to the outermost layer of the recursion inAlgorithm 5 we can obtain all of the states 119878

2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)

Because cup119886isin119860

Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782

The difference between Algorithms 2 and 5 reflects theefficiency of their enumerations In the searching process animportant atomic operation (ops1) expands each unitrsquos actionplan on each position itmoves toThis is because (1) the statestaken by search are mainly composed of every unit movingto every position and (2) every unit arriving at every positionand then attacking or choosing other options for action is atime-consuming operation in the searching process Supposethe number of ops1 in Algorithms 2 and 5 is 119867

1and 119867

2

respectively For simplicity we make the following assump-tions

Assumption 2 Assume that every unitrsquos movement rangedoes not overlap anotherrsquos the sizes ofwhich are all equal that


(1) if 119894 gt 119899 then(2) return(3) else(4) 119895 larr 119894

(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)

(7) for each action plan of the 119894th unit(8) execute the current plan(9) call 119875119897119886119899119904 119878119890119886119903119888ℎ(119894 + 1)

(10) cancel this plan and rollback to the previous state(11) end for(12) swap(119874119903119889

119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1


Algorithm 5 Recursive enumeration algorithm 119875119897119886119899119904 119878119890119886119903119888ℎ(119894) (search action plans from 119894th to the last unit)

is |1198771| = |119877

2| = sdot sdot sdot = |119877

119899| = 119903 and 119877

1cap 1198772cap sdot sdot sdot cap 119877

119899=

Moreover every unit cannot attack after moving (ie none ofthe enemies are inside the attack range)

In the following we calculate 1198671and 119867

2 respectively

In Algorithm 2 in each identified sequence ops1 corre-sponds to the nodes of the search tree formed by enumeratingstates (except the root node which represents no action)Thedepth of the tree is 119899 and each of the branching factors is 119903then the number of nodes is 119903119899 + 119903

119899minus1

+ sdot sdot sdot + 119903 Moreover thenumber of all sequences is 119875119899

119899= 119899 and therefore

1198671= 119899 (119903

119899

+ 119903119899minus1

+ sdot sdot sdot + 119903) (7)

InAlgorithm 5 suppose that the number of ops1 of 119899unitsis ℎ119899 The first unit performing an action according to the

order of the current sequence is 119886 According to Algorithm 5every time 119886moves to a tilt it willmake a new state combiningthe following 119899 minus 1 units such that the number of the ops1is 1 + ℎ

119899minus1 Because the number of tilts 119886 can move to is 119903

and the recursion operates 119899 times we can deduce that ℎ119899=

119899119903(1 + ℎ119899minus1

) where ℎ1= 119903 thus

1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)

It is easy to see that the number of ops1 of Algorithm 5 issmaller than that of Algorithm 2 Table 2 lists the experi-mental results showing 119867

1under Assumption 2 119867

2under

a general condition and their differences

Conclusion On the premise that the search states are exactlythe same Algorithm 5 is better than Algorithm 2 regardingthe consumption of ops1 and actual running time

5 Experimental Evaluation

In this section we present our experimental evaluation of theperformance of Algorithms 2 and 5 under all types of con-ditions and their comparison Because they are both single-round search algorithms we set only one sidersquos units on theboard ignoring the other sidersquos whose interference is equiva-lent to narrowing the range of unitsrsquo movement Experimentsare grouped based on the following conditions the numberof units the unitrsquos movement point and the dispersion ofunits The number of units is set to 3 and 4 (setting to 2 istoo simple with a lack of universality while setting to 5 leadsto timeout) The movement point is set to 2 3 and 4 and themovement cost of each tilt is set to 1 The dispersion is set tothe most dispersive ones and the most centralized ones Themost dispersive cases mean that the movement ranges of allof the units are independent without overlapping each othercorresponding to the worst case in Section 31 The mostcentralized cases mean that all of the units are put together(Figure 10) which maximizes the overlap degree and cor-responds to the best case in Section 31 The experimentalgroups set above cover all of the actual situations The boardused in the experiments is completely open without anyboundary and barrier The case of a board with boundariesand barriers can be classified into caseswhere a smallermove-ment point of units is set The experimental tool is a PC withIntel Core i7-2600240GHzCPUand 400GBmemory andthe program was written with Visual C++ 2005 with opti-mized running time

From Table 2 we can see that in all cases the number ofops1 of Algorithm 5 is less than that of Algorithm 2 for differ-ent levels Assuming that the number of units is invariable theoptimization level of Algorithm 5will become low by increas-ing the movement point which can be deduced from (8)and (9) under Assumption 2 ops1 which shows the reducedpercentage of using ops1 in Algorithm 5 instead of inAlgorithm 2 is

ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)


Table 2 (a) For three units the comparison of ops1 under different movement points and different dispersions (b) For four units thecomparison of ops1 under different movement points and different dispersions

(a)

Movement point 2 Movement point 3 Movement point 4Dispersed Compact Dispersed Compact Dispersed Compact

Algorithm 2 14274 9804 97650 79260 423858 371652Algorithm 5 14235 9771 97575 79191 423735 371535119863ops1lowast 027 034 0077 0087 0029 0031

ops1lowastlowast 030 0080 0030

(b)




lowast

119863ops1 shows the reduced percentage of using ops1 in Algorithm 5 instead of in Algorithm 2lowastlowast

ops1 shows the estimated value of119863ops1 under Assumption 2

(a) (b)

Figure 10 (a) Compact placement of three units (b) Compact placement of four units

In (10) the numerator is the infinitesimal of higher orderof the denominator that is

ops1 asymp1

2119903minus2

(11)

Table 2 lists the values of ops1 when themovement pointis 2 3 and 4 which are consistent with the experimentalresults

Under the same conditions of the movement point andthe number of units the value of 119863ops1 with compact unitsis more than that with dispersed units This is because themore the units are compact the stronger the interference theunits will cause to each other which is equivalent to a narrowmovement range of 119903 According to (11) therefore 119863ops1will increase correspondingly Moreover under the sameconditions of the movement point and the degree of unitsrsquo

dispersion 119863ops1 will also increase with the increase of thenumber of units In summary the experiments show thatregardless of whether Assumption 2 is satisfied Algorithm 5always performs better than Algorithm 2 on the numberof ops1 which coincides with ops1 from (11) Because thedegree of optimization is not very prominent the runningtimes of these two algorithms are almost the same

6 Conclusions

Based on a modest study of turn-based war chess games(TBW) a common gaming model and its formal descriptionare first proposed By comparisonwith other chess typemod-els the most important feature of TBW has been discussedthe player needs to complete actions for all of his units in aturn which leads to a huge branching factor Then a game


tree theory framework to solve this model is proposedFinally two algorithms for single-round search from themostcomplex part of the framework are proposed Algorithm 2is the dictionary sequence enumeration algorithm andAlgorithm 5 is the recursive enumeration algorithm Finallybased on theoretical derivations and experimental resultsrespectively the completeness of these algorithms is provenAlso the performance comparison shows that under allconditions the number of ops1 of Algorithm 5 decreases to acertain extent compared to that of Algorithm 2

Although these two algorithms are designed from clas-sical algorithms they can be used to solve the single-roundsearch problem completely and effectively Moreover theresearch angles of the two algorithms are completely differentwhich provide two specific frameworks for a further study onTBW

(1) The dictionary sequence enumeration algorithm isimplemented in two steps The first step consists ofthe generation of sequences and the second stepconsists of the enumeration of action plans underthese sequencesTherefore this algorithm is based onsequences Different permutation algorithms can beused to generate different orders of sequences whichmay be more suitable for new demands For instancethe orthoposition trade method [23] can minimizethe difference of each pair of adjacent sequencesThus more action plans from the former sequencecan be reused for the next which can improve theefficiency

(2) The recursive enumeration algorithm is also imple-mented in two steps The first step consists of theenumeration of action plans of the current unit andthe second step consists of the generation of thesequences of the next units Therefore this algorithmis based on action plans Pruning bad action plans inthe depth-first search process can easily cut off all thefollowing action sequences and action plans of laterunits which will lead to a significant improvement ofefficiency

In the current era of digital entertainment TBW gameshave broad application prospects They also have a profoundtheoretical research value However in this study TBW the-ory has been discussed partiallyThe gamemodel frameworkwe proposed is composed of the combinatorial optimizationproblem on one hand and the game tree search problem onthe other hand Thus our future research will mainly startwith the following two points

(1) Introduce the multiagent collaborative planningapproach to efficiently prune the huge branches of thegame tree Moreover by introducing the independentdetection approach [26] we can separate the inde-pendent units that have no effect on each other intodifferent groups with the purpose of decreasing thenumber of units in each group

(2) Introduce the Monte Carlo Tree Search method tosimulate the deep nodes The single-round search

algorithms proposed in this paper are complete algo-rithms and can be used to verify the performance ofthe new algorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Key Basic ResearchProgram of China (973 Program 2013CB329103 of2013CB329100) the National Natural Science Foundationsof China (NSFC-61173129 61472053 and 91420102) theFundamental Research Funds for the Central University(CDJXS11182240 and CDJXS11180004) and the InnovationTeam Construction Project of Chongqing Institute ofEngineering (2014xcxtd05)

References

[1] L V Alus H J van den Herik and M P H Huntjens ldquoGo-Moku solved by new search techniquesrdquo Computational Intelli-gence vol 12 no 1 pp 7ndash23 1996

[2] C Deng Research on search algorithms in computer go [MSthesis] Kunming University of Science and Technology 2013

[3] H C Neto R M S Julia G S Caexeta and A R A BarcelosldquoLS-VisionDraughts improving the performance of an agentfor checkers by integrating computational intelligence rein-forcement learning and a powerful search methodrdquo AppliedIntelligence vol 41 no 2 pp 525ndash550 2014

[4] Y J Liu The research and implementation of computer gameswhich based on the alpha-beta algorithm [MS thesis] DalianJiaotong University 2012

[5] J Jokic Izrada sahovskog engine-a [PhD thesis] University ofRijeka 2014

[6] N R Sturtevant and R E Korf ldquoOn pruning techniques formulti-player gamesrdquo in Proceedings of the 17th National Confer-ence on Artificial Intelligence and 12th Conference on InnovativeApplications of Artificial Intelligence (AAAI-IAAI rsquo00) vol 49pp 201ndash207 Austin Tex USA July-August 2000

[7] C Luckhart and K Irani B ldquoAn algorithmic solution of N-person gamesrdquo in Proceedings of the 5th National Conference onArtificial Intelligence (AAAI rsquo86) vol 1 pp 158ndash162 Philadel-phia Pa USA August 1986

[8] C Browne ldquoA problem case for UCTrdquo IEEE Transactions onComputational Intelligence andAI in Games vol 5 no 1 pp 69ndash74 2013

[9] M H M Winands Y Bjornsson and J-T Saito ldquoMonte Carlotree search in lines of actionrdquo IEEE Transactions on Computa-tional Intelligence and AI in Games vol 2 no 4 pp 239ndash2502010

[10] P Nijssen and M H M Winands ldquoMonte carlo tree search forthe hide-and-seek game Scotland Yardrdquo IEEE Transactions onComputational Intelligence and AI in Games vol 4 no 4 pp282ndash294 2012

[11] G Wang H Liu and N Zhu ldquoA survey of war games technol-ogyrdquo Ordnance Industry Automation vol 31 no 8 pp 38ndash412012


[12] M H Bergsma and P Spronck ldquoAdaptive spatial reasoning forturn-based strategy gamesrdquo in Proceedings of the 4th ArtificialIntelligence and Interactive Digital Entertainment Conference(AIIDE rsquo08) pp 161ndash166 Palo Alto Calif USA October 2008

[13] S Paskaradevan and J Denzinger ldquoA hybrid cooperative behav-ior learning method for a rule-based shout-ahead architecturerdquoin Proceedings of the IEEEWICACM International Conferenceson Web Intelligence and Intelligent Agent Technology (WI-IATrsquo12) vol 2 pp 266ndash273 IEEE Computer Society December2012

[14] S Wiens J Denzinger and S Paskaradevan ldquoCreating largenumbers of game AIs by learning behavior for cooperatingunitsrdquo in Proceedings of the IEEE Conference on ComputationalIntelligence in Games (CIG rsquo13) pp 1ndash8 IEEE Ontario CanadaAugust 2013

[15] G Badhrinathan A Agarwal and R Anand Kumar ldquoImple-mentation of distributed chess engine using PaaSrdquo in Pro-ceedings of the International Conference on Cloud ComputingTechnologies Applications andManagement (ICCCTAM rsquo12) pp38ndash42 IEEE Dubai United Arab Emirates December 2012

[16] X Xu and C Xu ldquoSummarization of fundamental and method-ology of computer gamesrdquo Progress of Artificial Intelligence inChina pp 748ndash759 2009

[17] J Song and M Muller ldquoAn enhanced solver for the game ofAmazonsrdquo IEEETransactions onComputational Intelligence andAI in Games vol 7 no 1 pp 16ndash27 2015

[18] D Gesang Y Wang and X Guo ldquoAI arithmetic design andpractical path of a war-chess gamerdquo Journal of Tibet University(Natural Science Edition) vol 26 no 2 pp 102ndash106 2011

[19] X Xu and J Wang ldquoKey technologies analysis of Chinese chesscomputer gamerdquoMini-Micro Systems vol 27 no 6 pp 961ndash9692006

[20] B Bosansky V Lisy J Cermak R Vıtek and M PechoucekldquoUsing double-oracle method and serialized alpha-beta searchfor pruning in simultaneous move gamesrdquo in Proceedings of the23rd International Joint Conference on Artificial Intelligence pp48ndash54 AAAI Press Beijing China August 2013

[21] A Guez D Silver and P Dayan ldquoScalable and efficient Bayes-adaptive reinforcement learning based on Monte-Carlo treesearchrdquo Journal of Artificial Intelligence Research vol 48 pp841ndash883 2013

[22] R Sedgewick ldquoPermutation generation methodsrdquo ACM Com-puting Surveys vol 9 no 2 pp 137ndash164 1977

[23] D E KnuthTheArt of Computer Programming Vol 4A Combi-natorial Algorithms Part 1 The Peoplersquos Posts and Telecommu-nications Press Beijing China 2012

[24] B R Heap ldquoPermutations by interchangesrdquoTheComputer Jour-nal vol 6 no 3 pp 293ndash298 1963

[25] F M Ives ldquoPermutation enumeration four new permutationalgorithmsrdquo Communications of the ACM vol 19 no 2 pp 68ndash72 1976

[26] G Sharon R Stern M Goldenberg and A Felner ldquoTheincreasing cost tree search for optimalmulti-agent pathfindingrdquoArtificial Intelligence vol 195 pp 470ndash495 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



(a) Wargaming (b) SLG game ldquoBattle Commanderrdquo

Figure 1 Wargaming and SLG game

Figure 2 SLG game ldquoHeroes of Might amp Magicrdquo

games can also be classified into two categories classic boardgames and new board games according to the game con-tent Classic board games involve Go chess backgammoncheckers and so forth which are widespread and have a longhistory While other board games such as Hex [8] Lines ofAction [9] and Scotland Yard [10] are ancient games withthe rapid development of modern board games and mobileclient applications they have been accepted bymore andmoreplayers until their prevalence is comparable to that of theclassic board game The machine game algorithms of theboard games listed above are all based on120572-120573 search and theirenhancement algorithms The MCTS algorithm has devel-oped rapidly in recent years being used increasingly in theseboard games and getting increasingly satisfactory results [8ndash10]

However not all board games can be solved with theexisting algorithms Turn-based strategy games (TBS) as well

as turn-based battle simulation games (SLG) (hereinafter col-lectively referred to as turn-based strategy games) originatedfrom the wargames [11] that swept the world in the mid-19thcentury (Figure 1(a) shows an example of a wargame) Withthe introduction of computer technology this new type ofgame turn-based strategy game has flourished (Figure 1(b)shows a famous TBS game called ldquoBattle Commanderrdquo andFigure 2 shows the popular SLG game ldquoHeroes of Might ampMagicrdquo) Now TBS games have become the second mostfamous type of game after RPGs (role-playing games) Withthe blossoming of mobile games TBS games will have greaterpotential for development in the areas of touch-screen oper-ation lightweight fragmented time and so on The contentof a TBS game generally comprises two levels strategic coor-dination and tactical battle control The latter level whoserules are similar to those of board games for examplemovingpieces on the board beating a specified enemy target for






B

1

1

2

2

3 4

43

A








(a)

A B

(b)

B

(c)

B

1 2

(d)















119899 = 119899119860




119874119903119889119862


and thus 119874119903119889119862


119862

119894= 119874119903119889119862

119895


119862

Thus |119876119862| = 119875119899

119899











= 119899119860



Ply 0

Opponent Ply 1

Ply 2

Ply 3

Ply 4

Current player

Current player

Current player

Opponent


2

1

(a)

(b) (c) (d)

(e) (f) (g)

1

1

1

1

1

111

1

2

2

2 2

2

2

2

222

2 2

2



119903119894


1= 1199032

= sdot sdot sdot =



119860] 119895 = 119894 119877




[119903 (119903atk + 1)]119899119860

(1)


119899119860


plans is

119878worst = [119903 (119903atk + 1)]119899119860

times 119899119860 (2)


SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(a)

SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(b)




1= 1198772= sdot sdot sdot = 119877

119899119860 = 119877 More-





119878best =119903

(119903 minus 119899119860)

(3)


119878best lt 119878 le 119878worst (4)














1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4













11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














B

1

1

2

2

3 4

43

A








(a)

A B

(b)

B

(c)

B

1 2

(d)















119899 = 119899119860




119874119903119889119862


and thus 119874119903119889119862


119862

119894= 119874119903119889119862

119895


119862

Thus |119876119862| = 119875119899

119899











= 119899119860



Ply 0

Opponent Ply 1

Ply 2

Ply 3

Ply 4

Current player

Current player

Current player

Opponent


2

1

(a)

(b) (c) (d)

(e) (f) (g)

1

1

1

1

1

111

1

2

2

2 2

2

2

2

222

2 2

2



119903119894


1= 1199032

= sdot sdot sdot =



119860] 119895 = 119894 119877




[119903 (119903atk + 1)]119899119860

(1)


119899119860


plans is

119878worst = [119903 (119903atk + 1)]119899119860

times 119899119860 (2)


SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(a)

SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(b)




1= 1198772= sdot sdot sdot = 119877

119899119860 = 119877 More-





119878best =119903

(119903 minus 119899119860)

(3)


119878best lt 119878 le 119878worst (4)














1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4













11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

A B

(b)

B

(c)

B

1 2

(d)















119899 = 119899119860




119874119903119889119862


and thus 119874119903119889119862


119862

119894= 119874119903119889119862

119895


119862

Thus |119876119862| = 119875119899

119899











= 119899119860



Ply 0

Opponent Ply 1

Ply 2

Ply 3

Ply 4

Current player

Current player

Current player

Opponent


2

1

(a)

(b) (c) (d)

(e) (f) (g)

1

1

1

1

1

111

1

2

2

2 2

2

2

2

222

2 2

2



119903119894


1= 1199032

= sdot sdot sdot =



119860] 119895 = 119894 119877




[119903 (119903atk + 1)]119899119860

(1)


119899119860


plans is

119878worst = [119903 (119903atk + 1)]119899119860

times 119899119860 (2)


SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(a)

SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(b)




1= 1198772= sdot sdot sdot = 119877

119899119860 = 119877 More-





119878best =119903

(119903 minus 119899119860)

(3)


119878best lt 119878 le 119878worst (4)














1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4













11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119899 = 119899119860




119874119903119889119862


and thus 119874119903119889119862


119862

119894= 119874119903119889119862

119895


119862

Thus |119876119862| = 119875119899

119899











= 119899119860



Ply 0

Opponent Ply 1

Ply 2

Ply 3

Ply 4

Current player

Current player

Current player

Opponent


2

1

(a)

(b) (c) (d)

(e) (f) (g)

1

1

1

1

1

111

1

2

2

2 2

2

2

2

222

2 2

2



119903119894


1= 1199032

= sdot sdot sdot =



119860] 119895 = 119894 119877




[119903 (119903atk + 1)]119899119860

(1)


119899119860


plans is

119878worst = [119903 (119903atk + 1)]119899119860

times 119899119860 (2)


SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(a)

SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(b)




1= 1198772= sdot sdot sdot = 119877

119899119860 = 119877 More-





119878best =119903

(119903 minus 119899119860)

(3)


119878best lt 119878 le 119878worst (4)














1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4













11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ply 0

Opponent Ply 1

Ply 2

Ply 3

Ply 4

Current player

Current player

Current player

Opponent


2

1

(a)

(b) (c) (d)

(e) (f) (g)

1

1

1

1

1

111

1

2

2

2 2

2

2

2

222

2 2

2



119903119894


1= 1199032

= sdot sdot sdot =



119860] 119895 = 119894 119877




[119903 (119903atk + 1)]119899119860

(1)


119899119860


plans is

119878worst = [119903 (119903atk + 1)]119899119860

times 119899119860 (2)


SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(a)

SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(b)




1= 1198772= sdot sdot sdot = 119877

119899119860 = 119877 More-





119878best =119903

(119903 minus 119899119860)

(3)


119878best lt 119878 le 119878worst (4)














1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4













11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(a)

SbestSworst

100

105

1010

1015

Num

ber o

f sta

tes

2 3 4 51nA

(b)




1= 1198772= sdot sdot sdot = 119877

119899119860 = 119877 More-





119878best =119903

(119903 minus 119899119860)

(3)


119878best lt 119878 le 119878worst (4)














1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4













11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

4

B BA

B

1 B

BB

AA1 1

1

1

11

1

1

1

(a)

A B

A B

A

A

(b)

2

2

2

2

2

22

2

2

2

3

3 3

3

3

33

3

3

3

4

4

4

4

44

4

4

4













11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











11198741199031198892sdot sdot sdot 119874119903119889

119895minus1119874119903119889119895119874119903119889119895+1


119874119903119889119896119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(1) find 119895 = max119894 | 119874119903119889119894lt 119874119903119889

119894+1


119894gt 119874119903119889

119895

(6) swap(119874119903119889119895 119874119903119889

119896)



119874119903119889119895119874119903119889119896+1

sdot sdot sdot 119874119903119889119899

(8) Output 1198741199031198891015840

= 11987411990311988911198741199031198892sdot sdot sdot 119874119903119889


119896+1119874119903119889119895119874119903119889119896minus1

sdot sdot sdot 119874119903119889119895+1


(9) end if



(5) end while










1 1198741199031198892 119874119903119889

119894 119874119903119889

119895











1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














1 1198741199031198892 119874119903119889

119899

(3) return(4) else(5) 119895 larr 119894

(6) while 119895 le 119899 do(7) swap(119874119903119889

119894 119874119903119889

119895)

(8) call 119903119890119888119906119903119904119894V119890 119901119890119903119898119906119905119886119905119894119900119899(119894 + 1)

(9) swap(119874119903119889119894 119874119903119889

119895)

(10) 119895 larr 119895 + 1












119860


1under

this sequence

1198781= ⋃

119874119903119889119860isin119876119860

119878 (119874119903119889119860

) (5)


2

1198782= ⋃

119886isin119860

119878 (Pre119886) (6)


Pre119886= 119876119860 and cup

119874119903119889119860isin119876119860119878(119874119903119889

119860

) = 119878(119876119860

)therefore 119878

1= 1198782


1and 119867

2





(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(5) while 119895 le 119899 do(6) swap(119874119903119889

119894 119874119903119889

119895)



119894 119874119903119889

119895)

(13) 119895 larr 119895 + 1



is |1198771| = |119877


119899| = 119903 and 119877


119899=



2 respectively


119899minus1



1198671= 119899 (119903

119899

+ 119903119899minus1






119899119903(1 + ℎ119899minus1


1198672= ℎ119899= 119899

119899minus1

sum

119894=0

119903119899minus119894

119894 (8)

Accordingly

1198671minus 1198672= 119899

119899minus1

sum

119894=2

119894 minus 1

119894119903119899minus119894

(9)



2under






ops1 =1198671minus 1198672

1198671

=sum119899minus1

119894=2((119894 minus 1) 119894) 119903

119899minus119894

sum119899minus1

119894=0119903119899minus119894

(10)



(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(a)




(b)




lowast



(a) (b)



ops1 asymp1

2119903minus2

(11)




6 Conclusions













Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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