informed_search.pdf

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Chapter 6Search, Games and Problem Solving

Informed Search

Using Search Heuristics

Heuristics are guidelines of where it is best to lookfor a solution

may find a solution faster than uninformedsearch, but without guarantee

may result in a solution not being found

closely linked with the need to make real-timedecisions with limited resources

in practice, a good solution found quickly ispreferred over an optimal solution, but expen-

sive to find

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Heuristic evaluation function, f(s)

used to mathematically model a heuristic

goal is to find with little effort a solution withminimal total cost

how good is an estimate of the distance fromthe node to the goal

iff(m) < f(n), then m is more likely to be

on an optimal path to the goal nHeuristic search methods (informed search)

uses heuristics to guide the search

The 8-puzzle problem

typically tree depth of 20average branching factor of 3

exhaustive search: 320 3.5 billion states

there are only 9! = 362880 possible (unique)states

use heuristics to avoid repeated states

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Admissibility of heuristics

should not overestimate the cost of changinga given state to the goal state

a search heuristic is admissible when it is al-ways guaranteed to find the minimal cost pathto the goal

Heuristics for 8-puzzle

h1(n): count how many tiles are in the wrongplace

h2(n): consider how far each tile has to moveto its correct state

sum Manhattan distances of each tile from

its correct position Manhattan distance = sum of horizontal &

vertical moves to take from one to anotherposition

h2 is more informed than h1 search using h2 will always perform better

than using h1

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h3(n): consider that there is extra difficultyinvolved if two tiles have to move past eachother, because tiles can not jump over each

other k(n) denotes the number of direct swaps

that need to be made between adjacent tilesto move them into correct sequence

h3(n) =h2(n) + 2k(n)

h3 is more informed than h2(n)

Relaxed problems

has fewer constraints

8-puzzle: a tile can move to an adjacent square

regardless of whether that square is empty ornot

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Many non-dominated heurisics: What then?

h(n) = max{h1(n), h2(n), , hK(n)}

if all hk

are admissible, will h(n) be admissi-ble?

hdominates all hk

Monotonicity

search method is monotone if it always reaches

a given node by the shortest possible pathif it reaches a given node at different depths,

it is not monotone

monotone search method must be admissible,provided there is only one goal state

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Example: Modified traveling salesman problem

find best route between two cities: A to F

Figure 1: Modified Traveling Salesman Problem

Search space

Figure 2: The Search Space

Solution using DFS? And BFS?

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Hill Climbing

Informed search

How does it work?

Disadvantages of hill climbing?

Figure 3: Hill Climbing Illustrated

Steepest ascent hill climbingfollow the path to the highest next point

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h2(n) = #tiles out of place + node depth

Figure 5: 8-tiles, with h2(n); Number of tiles out of place + depth

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Beam Search

Best-first search within a given beam width

beam width: limits number of nodes expandedApplied to 8-puzzle?

A Search

f(n) =g(n) +h(n) g(n) = actual cost of path from start node to

n

h(n) = an underestimate of the distance fromnto the goal

f(n) = path-based evaluation functionexpand the node with the smallest f

Optimality

if h(n) is always an underestimate, then A

is gauranteed to find the shortest path to thegoal

A is optimal, ifhis admissible, and complete

if non-admissible heuristic is used, the algo-rithm is called A

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Completeness

A is complete if the graph it searches is lo-cally finite and if every arc has a non-zero cost

locally finite if graph has finite branching fac-tor

Uniform Cost Search

Also called branch and bound search

Basically A search with h(n) = 0

Complete and optimal, provided that cost of apath increases monotonically

Dijkstras algorithm

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Greedy Search

A search withg(n) = 0

Always selects the path with the lowest estimateof the distance to the goal

Worst case: may never find a solution

Not optimal, and may follow extremely costly paths

Example of non-optimal path: if first step on short-est path toward goal is longer than other paths

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Iterative-Deepening A

As memory requirements grow exponentially with

depth of the goal in the search tree, even thoughheuristics reduce the branching factor

IDA combines iterative-deepening with A:

optimal and complete

has the low memory requirements of depth-first search

Executes a series of depth first searches, but usesa cut-off cost value as the depth limit, and notthe actual depth of the tree:

In the first search, the cost cut-off value is= f(n0) =g(n0) +h(n0) =h(n0)

The cost of an optimal path may just be equalto

It can not be less, as h(n0) h(n0), whereh(n0) is the true cost; remember we work with

underestimates of the true cost to the goal

Expand nodes in a depth-first fashion, back-tracking whenever f(n) exceeds

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Real-Time A

Let nbe the current node, and mis a child node

ofng(m) is the cost (distance) from n to m, and not

the cost from the root node

Will backtrack to n, i.e. not further expand fromm, if the cost of backtracking plus the estimatedcost to solve the problem fromnis less than theestimated cost to solve the problem from m, i.e.

g(m) +h(n)< h(m)

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Recursive Best-First Search (RBFS)

Slightly more memory than IDA, but generates

fewer nodes

kn n

n1

n2

n3

k

n m

Figure 6: RBFS Illustrated

The node expansion process (refer to figure 6)

Noden

is expandedThe f values are computed for the childrenn1, n2, n3

Then the f values of n and all of its ances-tors in the search tree are recomputed called

backing-up of f values

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Computing backed-up values:

backed-up value of a successor, ni, of justexpanded node, ni, is just f(ni)

backed-up value of any ancestor node, n, is

f(n) = minni

f(ni)

If the a successor ofn, sayn1, has the smallestfvalue over all unexpanded nodes, thenn1is

futher expanded

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Let n

be an unexpanded node in the search

tree, where n

is not a successor ofn

If f(n

) is the smallest fvalue over all unex-panded nodes, then find the common ancestor

ofn

If k is the common ancestor, and kn is thesuccessor ofk leading to n, then

delete the entire search tree below kn, andlet

f(kn) = minqi

f(qi)

over all unexpanded nodes,qiof the subtreeofkn

expand node n

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Island Driven Search

Assumes a list of subgoals exists, i.e. n0, n1, , ng

First find path fromn0ton1, thenn1ton2, then...ng1 tong

Is island driven search admissible?

Hierarchical Search

Similar to island-driven search, but no explicit setof islands are available

Assume the availability of macro-operators thatcan make large steps in an implicit search spaceof islands

Perform a metal-level search, to produce a path ofmacro-operators from from base-level start nodeto base-level goal node

Perform base-level searches on islandsAdmissible?

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