DESIGN AND ANALYSIS OF ALGORITHMS

UNIT I INTRODUCTION TO DESIGN AND ANALYSIS OF ALGORITHMS

1. Define an algorithmA: Algorithm is defined as a collection of unambiguous instructions occurring in some specific sequence and such an algorithm should produce output for given set of input in finite amount of time.

2. List out the criteria that all algorithms must satisfy.A: All algorithms must satisfy the following criteria:

Input: zero or more quantities are externally supplied. Output: at least one quantity is produced. Definiteness: Each instruction is clear and unambiguous Finiteness: If we trace the instructions of an algorithm, then for all cases, the

algorithm terminates after a finite number of steps. Effectiveness: Every instruction must be very basic and feasible. Once an algorithm is given for a problem and decided to be correct, an important

step is to determine how much in the way of resources such as time or space, the algorithm will require.

3. What is meant by computational procedure?A: Algorithms that are definite (clear, unambiguous) and effective (feasible) are known as computational procedures.

4. Define program.A: A program is the expression of an algorithm in a programming language.

5. Define an Abstract Data Type.A: An ADT (Abstract Data Type) is a mathematical model which gives a set of utilities available to the user but never states the details of its implementation.

6. List out the steps in designing an algorithm.A: The various steps in designing an algorithm are

Understanding the problem. Ascertaining the capabilities of a computational device Choosing between exact and approximate problem solving Deciding on appropriate data structures Algorithm Design techniques Methods of specifying an algorithm Proving an algorithm’s correctness Analyzing an algorithm Coding an Algorithm.

7. Give the reason for using approximation algorithm.A: The reasons are

There are important problems that simply cannot be solved exactly. e.g.: square root √n.

Available algorithms for solving a problem exactly can be unacceptably slow because of the problem’s intrinsic complexity. E.g.: Traveling Salesman Problem.

An approximation Algorithm can be a part of a more sophisticated algorithm that solves a problem exactly.

8. Define algorithm design technique or algorithmic strategiesA: An algorithm design technique is a general approach to solving problems algorithmically that is applicable to a variety of problems from different areas of computing. These are important because

They provide guidance for designing algorithms for new problems They make it possible to classify algorithms according to an underlying design

idea. E.g. Divide and conquer, Dynamic programming, Greedy method, Backtracking.

9. Define the methods of specifying an algorithm.A: The various methods are

i. Using natural language which is not clearii. Pseudo code: mixture of natural language and programming language - like

constructs. This method is more precise.iii. Flowchart: expresses an algorithm by a collection of various geometric shapes

containing descriptions of the algorithm steps.

10. What are the two kinds of algorithm efficiency?A: Time complexity: the time taken by an algorithm to run, determines whether the algorithm is slow or fast. Space complexity: the amount of space/memory taken by an algorithm, determines the extra memory the algorithm needs.

11. List out the factors to be followed while analyzing an algorithm.A: While analyzing an algorithm we should consider the following factors:

i. Time efficiency of an algorithmii. Space efficiency of an algorithm

iii. Simplicity of an algorithmiv. Generality of an algorithmv. Range of input.

12. What is meant by Time efficiency?A: Time efficiency indicates how fast the algorithm runs.

13. What is Space efficiency?

A: Space efficiency indicates how much extra memory the algorithm needs.

14. List out the areas the study of algorithms includes.A: The various areas are:

Sorting Searching String processing Graph problems Combinatorial problems Geometric problems Numerical problems

15. Define validation.A: Once an algorithm is devised, it is necessary to show that it computes the correct answer for all possible legal inputs. This process is called as algorithm validation.

16. What is meant by analysis of algorithms?A: Analysis of algorithms or Performance analysis refers to the task of determining how much computing time and storage an algorithm requires. An important result of this study is that

i. It allows us to make quantitative judgements about the value of one algorithm over another.

ii. It allows us to predict whether the software will meet any efficiency constraints that exist.

17. What are the steps in analyzing an algorithm?A: A general framework for analyzing the efficiency of an algorithm is:

Measuring input size Measuring running time Computing Orders of growth of algorithms Measuring Time and Space complexities Computing Worst case, best case and average case efficiencies.

18. How can you compute the running time of an algorithm?A: From an algorithm:

We first identify the BASIC OPERATION: important operation (core logic) of an algorithm.

Generally the operation which is more time consuming is a basic operation in the algorithm. The concept of basic operations can be well understood with the help of following example:

Problem Statement Input Size Basic OperationSearching a key

element from the list of n elements

List of n elementsComparison of key with every element

of listPerforming matrix

multiplicationTwo matrices with

order n x nActual

multiplication of the

row and column elements in matrix

Computing GCD of two numbers Two numbers Division

Then we compute total number of time taken by this basic operation. The running time of basic operation can be obtained approximately using the following the formula:

T(n) ≈ Cop C(n)Where

T(n) is Running time of BASIC OPERATIONCop is time taken by the basic operation to executeC(n) is Number of times the operation needs to be executed.

19. Define worst case efficiency of an algorithm?A: The worst case efficiency of an algorithm is its efficiency for the worst case input of size n, which is an input of size n for which the algorithm runs the longest among all possible inputs of that size.

20. Define best case efficiency of an algorithm?A: The best case efficiency of an algorithm is its efficiency for the best case input of size n, which is an input of size n for which the algorithm runs the fastest among all possible inputs of that size.

21. Define average case efficiency of an algorithm?A: The average case efficiency of an algorithm describes an algorithm’s behavior on a “typical” or “random” input. They make assumptions about possible inputs of size n, to analyze an algorithm’s average case efficiency.

22. Why space complexity of a program is necessary?A: The reasons are

i. If the program is to run on a multi user computer system, then it is important to specify the amount of memory to be allocated to the program.

ii. To know in advance whether or not sufficient memory is available in the computer system to run the program.

iii. A problem might have several possible solutions with different space requirements.

iv. To estimate the size of the largest problem that a program can solve.

23. Why Time complexity of a program is necessary?i. Some computer systems require the user to provide an upper limit on the amount

of time the program will run. Once this upper limit is reached, the program is aborted.

ii. The program might need to provide a satisfactory real-time response.

iii. If there is alternative ways to solve a problem, then the decision on which to use will be based primarily on the Expected Performance difference among these solutions.

24. Define Order of Growth.A: Measuring the performance of an algorithm in relation with the input size n is called order of growth. For example, the order of growth for varying input size of n is as given below:

n log n n log n n2 2n

1 0 0 1 22 1 2 4 44 2 8 16 168 3 24 64 25616 4 64 256 6553632 5 160 1024 4294967296

25. What is time space trade-off?A: Time space trade-off is basically a situation where either space efficiency (memory utilization) can be achieved at the cost of time or time efficiency (performance efficiency) can be achieved at the cost of memory.

26. List out the techniques that exploits space for time trade-offs?A: The techniques that exploits space for time trade-offs are

Input enhancement Pre-structuring

27. Define Input enhancement.A: It is a main technique that exploits space for time trade-offs. The idea is to preprocess the problem’s input in whole or in part and store the additional information obtained in order to accelerate solving the problem afterward. E.g. counting methods for sorting, Boyer Moore Algorithm for string matching.

28. Define Pre-structuring.A: It is a main technique that exploits space for time trade-offs. It uses extra space to facilitate a faster access and or more flexible access to the data. E.g. hashing, B trees.

29. What is meant by asymptotic notations?A: To choose the best algorithm, we need to check efficiency of each algorithm. The efficiency can be measured by computing the time complexity of each algorithm. This is done using notations called asymptotic notations. In short, asymptotic notation is a shorthand way to represent the time complexity. The standard notations used are O[Big Oh], Ω[Omega] and Ө[Theta].

30. Define Big Oh Notation.A: Let f and g be two functions defined from a set of natural numbers to the set of non-negative real numbers, that is, f, g: N→R≥0. It is said that f(n) = O( g(n) ), if there exists

two positive constants c € R and n0 € N such that f(n) ≤ c g(n), for all n≥ n0. Here, the function f(n) is bound above by the constant multiple of g(n).

31. Define Omega Notation.A: Let f and g be two functions defined from a set of natural numbers to the set of non-negative real numbers, that is, f, g: N→R≥0. It is said that f(n) = Ω( g(n) ), if there exists two positive constants c € R and n0 € N such that f(n) ≥ c g(n), for all n≥ n0. Here, the function f(n) is bound below by the constant multiple of g(n).

32. Define Theta Notation.A: Let f and g be two functions defined from a set of natural numbers to the set of non-negative real numbers, that is, f, g: N→R≥0. It is said that f(n) = Ө( g(n) ), if there exists three positive constants c1, c2 € R and n0 € N such that c1 g(n) ≤ f(n) ≤ c2 g(n), for all n≥ n0. Here, the function f(n) is bound lower by the constant multiple of g(n). Here the function f(n) is bound both above and below by another function g(n).

33. Define conditional asymptotic notation.A: Many algorithms are easier to analyse if we impose conditions on them initially. Imposing such conditions, when we specify asymptotic value, it is called conditional asymptotic notation. Let f, g: N→R≥0, it is said that f(n) = O (g(n) | A(n)), read as g(n) when A(n), if there exists two positive constants c € R and n0 € N such that A(n) →[f(n) ≤ c g(n)], for all n≥ n0.

34. What are the properties of order of growth?A: If f1(n) is O(g1(n)) and f2(n) is O(g2(n)), then f1(n) + f2(n) € O(max(g1(n)+ g2(n))).

1. polynomials of degree m € Ө(nm) that means maximum degree is considered from the polynomial for order of growth Ө(nm)

2. O(1) < O (log n)< O( n) < O(n2)< O(n3)<O(2n)3. exponential functions an have different orders of growth for different values of a.

35. Define recurrence equation.A: It is an equation that defines a sequence recursively. It is normally in the following form:

T(n) = T(n-1) + n for n>0___________ 1T(0) = 0 _________________ 2

1 is called recurrence relation2 is called the initial condition.The general solution to the recursive function specifies some formula.Eg: f(n) = 2f(n-1) + 1 for n>1 f(1) = 1Then by solving this recurrence relation we get, f(n) = 2n – 1.

36. What is meant by divide and conquer?A: A technique for designing algorithms that decompose instance into smaller sub-instance of the same problem, solving the sub-instances independently and

combining the sub solutions to obtain the solution of original instance is called divide and conquer. It uses the top-down approach to obtain solutions of problems.

37. Define divide and conquer algorithm.A: Formally, divide and conquer algorithm consists of the following major phases: Divide: Breaking the problem( if the input size is too large) into several sub-problems that are similar to the original problem but smaller in size.Conquer: conquer recursively to solve the sub-problems.Combine: Take the solutions of sub-problems and “merge” these solutions into a solution for the original problem.

38. Give the recurrence relation for divide and conquer algorithm.A: T(N) = g(n) if n is small

aT(N/b)+f(n) Otherwise

39. Give the examples of divide and conquer algorithm.A: i Sorting: merge sort and quick sort

ii Binary tree traversalsiii Binary Searchiv Multiplication of large integers

40. What is meant by binary search?A: Binary Search is an efficient searching method. While searching the elements using this method the most essential thing is that the elements must be a pre-sorted one.

41. Give the divide and conquer steps of binary search.A: Let A[m] be the middle element of array A and K be the search key. Then there are three conditions that needs to be tested while searching the array using this method:

i. If K= A[m] then desired element is foundii. Otherwise divide the array into two sub-arrays about half as large. If

K< A[m] then search the left sub-listiii. Otherwise if K> A[m] then search the right sub-list

42. Give the applications of binary search.A: Finding a name in the phone book, searching a desired record from the database, for solving nonlinear equation with one unknown.

43. Give the worst case analysis of binary search algorithm.A: 1) The search key is not in the sorted list

2) The search key is in the last position. Here the key comparison is done with half of the size of array. Therefore in worst case T(n) = Θ (log n).

44. Give the average case analysis of binary search algorithm.A: The number of key comparisons on average case is slightly smaller than that in the worst case. Therefore T(n) = Θ (log n).

45. Give the best case analysis of binary search algorithm.A: The number of comparisons on best case is only one. Therefore T(n) = Θ (1)

46. Find the element 10 in the following set using binary search algorithm.10 15 20 25 30 35 40

A: Finding the middle element of the given list of elementsMid = (1+7)/2 = 4. Therefore it becomes 10 15 20 25 30 35 40Now A[mid] = A[4] = 25 and search key is 10.

We know that 10 < A[4]. Hence searching in the left sub-list, Again Mid = (1+4)/2 = 2. Therefore it becomes

10 15 20 25A[2] = 15 and we know that 10 < A[2]. Hence searching in the left sub-list, Again

Mid = (1+1)/2 = 1; A[1] = 10 and 10 = 10. Hence the element is found.

47. Give the best case, worst case, average case analysis of finding maximum and minimum.

A: For finding maximum and minimum element, 3n/2 -2 is the best case, worst case, average case number of comparisons when n is a power of 2. Hence neglecting the order of magnitude, we can declare the time complexity as O(n).

48. Give the divide and conquer steps of merge sort.A: i. Divide the array into two sub-arrays each with n/2 items

ii. Conquer each sub-array by sorting it unless the array is sufficiently small, use recursion to do this.

iii.Combine the solutions to the sub-arrays by merging them into a single sorted array.

49. How can you merge two sorted lists?A: 1) Choose the smaller element of the two lists

2) Remove it from the list and put it into a new list3) Repeat the previous steps until all elements are arranged.

50. Define merge sort treeA: Merge sort can be visualized by means of binary tree where each node of the tree represents a recursive call and each external node represents individual elements of given array. Such a tree is called merge sort tree.

51. Give the running time of merge sort.A: T(n) = Θ(n log n)

UNIT 2

1) What is meant by greedy algorithm?A: Greedy algorithm is an algorithmic design technique mainly used to solve optimization problems. They take decisions on the basis of information at hand without worrying about the effect these decisions may have in the future. This approach suggests constructing a solution through a sequence of steps, each expanding a partially constructed solution obtained so far, until a complete solution to the problem is reached. On each step, the choice made must be

Feasible: it should satisfy problem’s constraints Locally optimal: Among all feasible solutions the best choice is made Irrevocable: Once the choice is made then it should not be changed on the

Sub-sequent steps of the algorithm.

2) What is meant by feasible solution?A: Solutions that satisfy problem’s constraints

3) What is meant by locally optimal solution?A: The best choice among all feasible solutions is called a locally optimal solution

4) What is meant by irrevocable solution? A: Once the choice is made then it should not be changed on the sub-sequent steps of the algorithm.

5) What are the functions of greedy algorithm?A: The greedy approach consists of 4 functions

A function that checks whether a given set of candidates give a solution. A function that checks the feasibility of a set of candidates A selection function which chooses some candidate which has not yet

been chosen An objective function which gives the value of a solution.

6) Write down the advantages and disadvantages of greedy algorithm? A: Advantage:

Greedy algorithms work fast when they work simple algorithms,easy to implement.

Disadvantage: Greedy algorithms don’t always work.Hard to prove correct.

7) What is meant by container loading?A: In this problem, the ship is to be loaded with containers. All containers are the same size. Different containers may have different weights. At each stage, each container is loaded into the ship. The total weight of all the containers must be less than or equal to the capacity of ship.

8) Give the greedy solution for container loading problemA: Select the container that has least weight, then the one with the next smallest weight, and so on until either all containers have been loaded or there isn’t capacity for the next one. That is, until total weight of all the containers must be less than or equal to the capacity of ship.

9) Give the analysis for container loading problemA: Sorting the container takes O(n log n). The remaining part of the algorithm takes O(n) time. The overall time complexity is O(n log n).

10) Define Knapsack problem.A: Given n items, each with profit value v and weight wi and a knapsack that can hold at most weight W, the objective is to find the most valuable subset of n items of given weights and values that fit into a knapsack of a given capacity.

11) Define fractional Knapsack problem.A: Given i items, each with profit value v and weight wi and a knapsack that can hold at most weight W, the objective is to find the most valuable subset of n items of given weights and values that fit into a knapsack of a given capacity. Here the items can be broken into smaller pieces so that only a fraction of xi of items I where 0≤ xi≤1 may be chosen.

12) Write the algorithm for greedy Knapsack.A: 1) Sort items in decreasing order of profit per weight (pi/wi)

2) While still room in the knapsack do3) Consider next item in sorted list4) Take as much as possible.

13) Give the analysis for greedy Knapsack.A: If the items are already sorted into decreasing order of (p i/wi) then the while loop takes a time in O(n). The overall time complexity including the sort is O(n log n).

14) Define subset paradigm.A: If the “inputs” to a problem using greedy approach is considered in an order based on some selection procedure, i.e. some optimization measure for selection is used, then it is said to work in a subset paradigm. Here at every stage, the input is checked whether it leads to an optimal solution. If the inclusion of input into partial solution yields an infeasible solution, discard the input; otherwise, add it to the partial solution.

15) What are the various applications of Greedy method?A: Knapsack problem Container loading Single source shortest path

Prim’s algorithm for minimum spanning treeKruskal’s algorithm for minimum spanning treeJob sequencing with deadlines

16) What is the basic operation in container loading problem?A: The basic operation is:

for (i ← 1 to num AND container[i].wt <= W) do

solution[container[i].id] = 1;W = W – container[i].wt;

17) Give time complexities for a successful and unsuccessful binary searchA: Successful Binary Search

Best case: Θ(1)Average case: Θ(log n)Worst case: Θ(log n)

Unsuccessful Binary SearchBest case: Θ(log n)Average case: Θ(log n)Worst case: Θ(log n)

18) What is Minimum Spanning Tree?A: A Minimum Spanning Tree of a weighted connected graph G is a spanning tree with minimum or smallest weight.

18.What is Weight of the tree?A: A weight of the tree is defined as the sum of weights of all its edges.

19.What is Prim’s Algorithm?A: This algorithm is for obtaining minimum spanning tree by selecting the adjacent vertices of already selected vertices.

20.What is Kruskal’s Algorithm?A: This algorithm is for obtaining minimum spanning tree but it is not necessary to choose adjacent vertices of already selected vertices.

21.What is Single Source Shortest Paths Algorithm?A: Dijsktra’s algorithm is a popular algorithm for finding shortest path.This algorithm is called single source shortest algorithm.In this algorithm,for a given vertex called

source the shortest path to all other vertices is obtained.The main focus is not to find only one single path but to find the shortest paths from any vertex to all other remaining vertices.

22.What is the use of Dijkstra’s algorithm?A: It is used to solve the single –source shortest –paths problem:for a given vertex called the source in a weighted connected graph,find the shortes path to all its other vertices.

UNIT 3

1) What is meant by dynamic programming?A: Dynamic Programming is an algorithm design technique that can be used when the solution to a problem can be viewed as the result of a sequence of decisions. For example, KNAPSACK: The solution to the knapsack problem can be viewed as the result of a sequence of decisions. We have to decide the values of x i, 1≤i≤n. First, a decision is made on x1, then on x2, then on x3 and so on. An optimal sequence of decisions maximizes the objective function ∑pi xi. An optimal sequence of decisions is obtained by the principle of optimality.

2) Define the principle of optimalityA: Dynamic Programming works on the principle of optimality. The principle of optimality states that an optimal sequence of decisions has the property that whatever the initial state and decision are, the remaining decisions must constitute an optimal sequence of decisions with regard to the state resulting from the first decision.

3) Compare and contrast between Greedy algorithm and Dynamic programmingA:

GREEDY METHOD DYNAMIC PROGRAMMINGGreedy method is used for obtaining optimum solution

Dynamic programming is also used for obtaining optimum solution

An optimum solution is picked from a set of feasible solutions

There is no special set of feasible solutions in this method

The optimum selection is without revising previously generated solutions.

Considers all possible sequences in order to obtain the optimum solution

Only one decision sequence is ever generated

Many decision sequences may be generated

There is no as such guarantee of getting optimum solution

It is guaranteed that the dynamic programming will generate optimum solution using principle of optimality

4) Give the applications/examples of dynamic programmingA: i Multistage graphs

ii Shortest path problemsiii Longest common sub-sequence problemiv Knapsack Problem.

5) Compare and contrast between Divide and Conquer and Dynamic programmingA:

DIVIDE AND CONQUER DYNAMIC PROGRAMMINGThe problem is divided into small sub-problems. These are solved independently. Finally all the solutions of the sub-problems are combined to get the whole solution to the given problem

Many decision sequences are generated and all the overlapping sub-instances are considered.

Less efficient because of rework on solution

More efficient than divide and conquer strategy.

Uses top down approach of problem solving( recursive method)

Uses bottom up approach of problem solving( iterative method)

Divide and Conquer splits its input at specific deterministic points

It splits its input at ever possible split points rather than at a particular point. After trying all split points, it determines which split point is optimal

Has logarithmic time complexity Has polynomial time complexity

6) Define multistage graphs A: A multistage graph G = (V, E) is a directed graph in which the vertices are partitioned into k>= 2 disjoint sets Vi, 1 ≤ i ≤ k. The objective is to find a minimum cost path from source (s) to sink (t). Each set Vi defines a stage in graph. That is, every path from s to t starts in stage1, goes to stage2, then to stage3 and so on until terminating in stage k. Hence we obtain minimum path length for each vertex at each

stage.eg: draw a multistage graph

7) What is meant by all pairs shortest path problem?A: The all pairs shortest path problem is to determine a matrix A such that A(i, j) is the length of a shortest path from i to j. It is computed by solving ‘n’ single source problems as follows:

Often a path originates from vertex i and goes through some intermediate vertices and terminates at vertex j.

If k is an intermediate vertex on this shortest path, then the sub-paths from i to k and k to j must be shortest paths from i to k and k to j, respectively

Otherwise i to j path is not of minimum length Therefore the principle of optimality holds.

8) Give any two properties of dynamic programming approachA: 1) Optimal substructure: The dynamic programming technique makes use of principle of optimality to find the optimal solution from sub-problems.

2) Overlapping sub-problems: The dynamic programming is a technique in which the problem is divided into sub-problems. The solutions of sub-problems are shared to get the final solution to the problem. It avoids repetition of work and we can get the solution more efficiently.

9) What does dynamic programming have in common with divide and conquer? What is the principle difference between the two?

A: Both divide and conquer and dynamic programming solves the problem by breaking it into number of sub-problems. In both these methods the solutions from sub-problems are collected to get the final solution to the problem. But in divide and conquer the sub-problems are solved independently whereas in dynamic programming the sub-problems share the solutions among themselves. The dynamic programming works on principle of optimality. 10) Give the commonly used designing steps for dynamic programming algorithm.A: Dynamic programming design involves 4 major steps

Characterize the structure of optimal solution. Recursively define the value of an optimal solution. By using bottom up technique compute the value of optimal solution. Compute an optimal solution from computed information.

11) Define OBST A: A binary search tree is organized so as to minimize the number of nodes visited in all searches, given that it is known how often each word occurs. To increase the search efficiency more frequently used words are arranged nearer to the root and less frequently used words away from the root, with the help of probability value of each word. This type of arrangement is called optimal binary search tree.

12) Give the components of dynamic programming.A: A dynamic programming solution has three components: i. Formulate the answer as a recurrence relation or recursive algorithm.

ii. Show that the number of different instances of the recurrence is bounded by a polynomial.

iii.Specify an order of evaluation for the recurrence so that the solution is obtained the way we desire.

UNIT 4

1) Define exhaustive search.A: Exhaustive search is a basic search routine in which all possible solutions to a given problem are produced. In this search, it is checked if the problem is solved or not and the search is continued until a correct solution is generated. It produces entire solution space for the problem.

2) Differentiate backtracking over exhaustive searchA: Backtracking can be considered as an improvement over exhaustive search. Unlike exhaustive search, they construct candidate solutions, one component at a time and evaluate the partially constructed solutions; if no potential values of the remaining components can lead to a solution, the remaining components are not generated at all.

3) Define backtrackingA: Backtracking is a general algorithm for finding all (or some) solutions to some computational problem, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.

4) Give categories of the problems in backtrackingA: In backtracking the problem can be categorized into three categories

i Decision problem: finds “whether there is any feasible solution?”ii Optimization problem: finds “whether there exists best solution?”iii Enumeration problem: finds all the possible feasible solutions

5) Compare and contrast between Branch and Bound and BacktrackingA:

BACKTRACKING BRANCH AND BOUNDSolution for backtracking is traced using Depth First Search

In this method, it is not necessary to use DFS for obtaining the solution, even the Breadth First Search or Best First Search can be applied

Typically decision problems can be solved using this method

Typically optimization problems can be solved using this method

While finding solutions to the problem bad choices can be made

It proceeds on better solutions, so there cannot be any bad solutions

The state space tree is searched until the solution is obtained

The state space tree needs to be searched completely as there may be chances of getting an optimum solution anywhere in state space tree

Applns are m-coloring, knapsack Applns are job sequencing,TSP

6) What is the basic idea of backtracking?

A: The principle idea of backtracking is to construct solutions one component at a time and evaluate such partially constructed candidates as follows: If a partially constructed solution can be developed further without

violating problem’s constraints, it is done by taking the ‘first’ remaining legitimate option for the next component.

If there is no legitimate option for the next component, no alternatives for remaining components need to be considered. In this case, the algorithm backtracks to replace the last component of the partially constructed solution with its next option.

7) Define promising node and non-promising nodeA: A node in a state space tree is said to be promising if it corresponds to a partially constructed solution that may lead to a complete solution; otherwise it is called non-promising.

8) What is a state space tree?A: A state space tree is a rooted tree whose nodes represent partially constructed solution to the given problem. In backtracking method, the state space tree is built for finding the solution. This tree is built using Depth First Search fashion.

9) What are the types of constraints used in backtracking?A: Explicit constraints are rules that restrict each xi to take on values only from a given set. Explicit constraints depend on the particular instance I of problem being solved. All tuples that satisfy the explicit constraints define a possible solution space for I

Examples of explicit constraints xi ≥ 0, or all nonnegative real numbers xi = 0, 1 li ≤ xi ≤ ui

Implicit constraints are rules that determine which of the tuples in the solution space of I satisfy the criterion function. Implicit constraints describe the way in which the xi must relate to each other.

10) Give the formal definition of n queens problem.A: Consider a n x n chess board on which we have to place n queens such that no two queens can attack each other by being in the same row or in the same column or on the same diagonal. The diagonal conflicts can be checked by the following formula:

P1 and P2 are two positions that are on same diagonal ifi + j = k + l or i – j = k - l

11) How can you represent the solution for 8 queen problem?

A: All solutions to the 8 queen problem can be represented as an 8-tuple (x1,x2,…,x8) where xi is the column on which queen i is placed.

For example: The Solution is expressed as an 8-tuple, (4,6,8,2,7,1,3,5)

Q

Q

Q

Q

Q

Q

Q

Q

12) Give the explicit and implicit constraint for 8 queen problem.A: Explicit constraint using 8-tuple formulation are Si = 1,2,3,4,5,6,7,8 , 1≤ i ≤ 8. Therefore the solution space consists of 88 8-tuples

Implicit constraints are No two xi can be the same, or all the queens must be in different columns

· All solutions are permutations of the 8-tuple (1, 2, 3, 4, 5, 6, 7, 8)· Reduces the size of solution space from 88 to 8! tuples

No two queens can be on the same diagonal

13) Give the condition that says two queens are on the same diagonal?A: The diagonal conflicts can be checked by the following formula:

P1 and P2 are two positions that are on same diagonal ifi + j = k + l or i – j = k - l

14) Define Subset sum ProblemA: Given n positive numbers wi, 1 ≤ i ≤ n, and m, find all subsets of wi whose sums are m

For example, n = 4, w = (11, 13, 24, 7), and m = 31, the desired subsets are (11, 13, 7) and (24, 7)

The solution vectors can also be represented by the indices of the numbers as (1, 2, 4) and (3, 4)

All solutions are k-tuples, 1 ≤k ≤ n

15) Give the explicit and implicit constraint for Subset sum Problem

A: Explicit constraintsxi = j | j is an integer and 1 ≤ j ≤ n

Implicit constraints No two xi can be the same ∑wxi = m xi < x i+1, 1 ≤ i ≤ k (total order in indices): Helps in avoiding the generation

of multiple instances of same subset; (1, 2, 4) and (1, 4, 2) are the same subset

16) Define Problem stateA: Problem state is each node in the depth-first search tree

17) Define Solution statesA: Solution states are the problem states s for which the path from the root node to s defines a tuple in the solution space

In variable tuple size formulation tree, all nodes are solution states In fixed tuple size formulation tree, only the leaf nodes are solution states Partitioned into disjoint sub-solution spaces at each internal node

18) Define Answer statesA: Answer states are those solution states s for which the path from root node to s defines a tuple that is a member of the set of solutions. These states satisfy implicit constraints.

19) Define Static trees and Dynamic treesA: Static trees are ones for which tree organizations are independent of the problem instance being solved

Fixed tuple size formulation Dynamic trees are ones for which tree organizations are dependent of the problem instance being solved

20) Define Live nodeA: Live node is a generated node for which all of the children have not been generated.

21) Define Dead nodeA: Dead node is a generated node that is not to be expanded any further

All the children of a dead node are already generated Live nodes are killed using a bounding function to make them dead nodes

22) Define m colorability ProblemA: Let G be a graph and m be a given positive integer. The problem is to find whether the nodes of G can be colored in such a way that no two adjacent nodes have the same color yet only m colors are used. This is called as m colorability decision problem. E.g.: If d is the degree of the given graph, then it can be colored with d+1 colors. The

m colorability optimization problem asks for the smallest integer m for which the graph G can be colored. This integer is referred to as the chromatic number of the graph.

23) Define graph coloring problemA: same answer as in question 22 with e.g.

24) Define Knapsack ProblemA: Given n items, each with profit value v and weight w i and a knapsack that can hold at most weight W, the objective is to find The xi’s constitute a zero-one-valued vector. The solution space for this problem consists of the 2n distinct ways to assign zero or one values to the xi’s.

25.Define Hamiltonian circuit problemA: The Hamiltonian is defined as a cycle that passes through all the vertices of the grapg exactly once.It s a sequence of n+1 adjacent vertice Vi0,Vi1,……..,Vi(n-1).V i0 where the first vertex of the sequence is asme as the last one while all the other n-1 vertices are distinct.

26. Define graph. A graph G consists of a nonempty set V which is a nodes of the graph, a set E which is the set of edges of the graph, and a mapping from the set for edge E to ta set of pairs of elements of V. It can also be represented as G=(V,E)

27. List out the types of graph Undirected graph Directed graph Weighted graph Connected graph Dense graph Sparse graph Multigraph

28. Define weighted graph. A graph in which weights are assigned to every edge is called a weighted graph

29. Define undirected graph A graph in which every edge is undirected is called a undirected graph.

30. Define directed graph. A graph in which every edge is directed is called a undirected graph

31. Define connected graph. A graph is said to be connected if for every pair of its vertices u and v there is a path from u to v.

32. Define multigraph A multigraph allows multiple edges between the same vertices.

33. Define complete graph A graph with every pair of its vertices connected by an edge is called complete graph.

34. Define connected components of a graph. A connected component is the maximal subgraph of a given graph.

35. How can you represent the graph? Graphs can be represented in two ways:

Adjacency matrix Adjacency linked lists

36. List out the types of graph traversal. Breadth first search Depth first search

37. Differentiate between DFS and BFS

DFS BFSData structures stack queueNo of vertex ordering 2 orderings 1 orderingsEdge types(Undirected graphs) Tree and back edge Tree and back edgesApplications Connectivity, acyclicity,

articulation pointsConnectivity, acyclicity, minimum edge paths

Efficiency for adjacent matrix Θ(|n²|) Θ(|n²|)Efficiency for adjacent linked lists

Θ(|V|+|E|) Θ(|V|+|E|)

38. How can you check the graph is connected?

Start a DFS traversal at an arbitrary vertex and check, after the algorithm halts, whether all the graphs vertices will have been visited.

If they have, the graph is connecte3d: otherwise it is not connected.

39. Define spanning tree. A spanning tree of a connected graph is its connected acyclic subgraph that contains all the vertices of the graph.

40. Define DFS spanning treeSpanning tree obtained using a depth first search is called depth first spanning tree.

41. Define BFS spanning treeSpanning tree obtained using a breadth first search is called breadth first spanning tree.

42. Define biconnected graph.A graph G is biconnected if and only if it contains no articulation points.

43. Define biconnected components.A maximal biconnected subgraph is called biconnected components.

UNIT 5

1. What are the additional features required in branch-and-bound when compared to backtracking?

Compared to backtracking, branch-and-bound requires: A way to provide, for every node of a state space tree, a bound on the

best value of the objective function on any solution that can be obtained by adding further components to the partial solution represented by the node.

The value of the best solution seen so far.

2. List out the types of branch-and-bound technique. First in first out Least cost or max profit

3. Give the applications of branch and bound technique. Assignment problem Knapsack problem Traveling salesperson problem

4. What is the assignment problem?Assigning ‘n’ people to ‘n’ jobs so that the total cost of the assignment is as small as possible. The instance of the problem is specified as a n by n cost matrix C so that the problem can be stated as: select one element in each row of the matrix so that no two selected items are in the same column and the sum is the smallest possible.

5. Give the formula used to find the upper bound for knapsack problem.A simple way to find the upper bound ‘ub’ is to add ‘v’, the total value of the items already selected, the product of the remaining capacity of the Knapsack W-w and the best per unit payoff among the remaining items, which vi+1/wi+1

ub = v + (W-w)( vi+1/wi+1)

6. What are the strengths of backtracking and branch-and-bound?The strengths are as follows:

It is typically applied to difficult combinatorial problems for which no efficient algorithm exists to find exact solutions.

It holds hope for solving some instances of nontrivial sizes in an acceptable amount of time.

7. Define decision algorithmAny problem for which answer is either yes or no is called decision problem. The algorithm for decision problem is called decision algorithm.

8. Define optimization algorithmAny problem that involves the identification of optimal cost (minimum or maximum) is called optimization problem. The algorithm for optimization problem is called optimization algorithm.

9. Define P.Problems that can be solved in polynomial time. (“P” stands for polynomial). E.g. searching of key element, sorting of elements, all pairs shortest path.

10. Define NPIt stands for non-deterministic polynomial time. (“NP” stands for non-polynomial). E.g. Traveling salesperson problem, graph coloring problem, Hamiltonian circuit problem.

11. Define deterministic and non-deterministic algorithmThe algorithm in which every operation is uniquely defined is called deterministic algorithm. The algorithm in which every operation may not have unique result, rather there can be specified set of possibilities for every operation. Such an algorithm is called non-deterministic algorithm. Non- deterministic means that no particular rule is followed to make the guess.This is a two stage algorithm

Non-deterministic (guessing) stage: generate an arbitrary string that can be thought of as a candidate solution.

Deterministic (verification) stage: It takes as input the candidate solution and the instance to the problem and returns yes if the candidate solution represents the actual solution.

12. Name the functions used in a non-deterministic algorithm.1) Choose- Arbitrarily chooses one of the element from given input set.2) Fail- Indicates the unsuccessful completion.3) Success- Indicates successful completion.

13. Give the relationship between P, NP, NP complete and NP hard problems.

Relationship between P, NP, NP complete

P class problems

NP Hard problems

NP Complete Problems