1 problem solving using search brute-force search heuristic search competitive search

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Problem Solving Using Search

Brute-force searchHeuristic searchCompetitive search

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Copyright P. Doerschuk

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Many problems can only be solved by searching

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Importance of efficient search algorithms

Tic tac toe state space of 9!:

9 possible first moves, followed by 8 possible second moves, followed by 7 possible third moves, etc., or 9x8x7x…x1 moves or 9!

Can use exhaustive searchChess

10120 possible paths

can’t use exhaustive search

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State Space Search

use a state space graph to represent the problem each node represents a partial solution state each arc represents transition between states

can be directed or undirected

tree - a graph in which 2 nodes have at most one path between them (no loops or cycles)

state space search - find a solution path from the start state to a goal state

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traveling salesperson

Fig 3.7O(N!) ; actually (N-1)!

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Tree-based search

tree representation is often used to search for a path from the initial state to the goal state (Ex: eights puzzle)

number of possible states: 9!

1 2 3 1 2 3

4 8 5 4 5 6

7 6 7 8

Initial state Goal state

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Eights puzzle state tree

1 2 34 8 57 6

1 2 3 1 2 3 1 2 34 8 5 4 5 4 8 5

7 6 7 8 6 7 6

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Graph search vs tree search

General graph search algorithm must detect and eliminate loops; this is done by ensuring that a state is only examined once

trees don’t need to do this because they don’t have loops

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Unguided (blind) search

does not use heuristic information during the search process

depth first search examines all descendants of a node before the nodes at the same level are examined

breadth-first search examines all nodes at a given level before the nodes of the descendants

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Unguided (blind) search cont.

Breadth first algorithm:1. create a queue and add the first node to it2. Loop:

If the queue is empty, quit.Remove the first node from the queue.If the node contains the goal state, exit with the node as the solution.For each child of the current node:

Add the new state (no duplicates) to the back of the queue.

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Breadth-first search

ABCDEFGHIJKL

QUEUE:

A

BC

CDE

DEFG

EFGHI

FGHIJK

GHIJKLM

HIJKLMNO

IJKLMNO

JKLMNO

KLMNO

LMNO

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depth first algorithm:1. create a queue and add the first node

to it2. Loop:


Add the new state (no duplicates) to the front of the queue.

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Depth-first search

ABDHIEJKCFL

QUEUE:

A

BC

DEC

HIEC

IEC

EC

JKC

KC

C

FG

LMG

...

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Breadth first vs depth first

Breadth firstguaranteed to find the shortest pathspace inefficient

• all unexpanded nodes for each level are kept in the queue

• if average branching factor is B, have Bn states on level n (root is level 0)

• if solution path is long or B is large, memory requirements large

depth firstnot guaranteed to find shortest pathspace efficient

• at each level, keeps only the children of a single node in the queue (B*n) states at level n

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Depth-first search with iterative deepening perform depth-first search up to a maxDepth control loop continually deepens depth-first

search more efficient than breadth-first and depth-first

on large search spaces Russell and Norvig 1995

time behavior is the same as depth-first and breadth-first search: O(Bn)

Note: all unguided search techniques have worst case exponential time behavior

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Using state space to represent reasoning with predicate calculus

Use and/or graphs to represent implications of the form pqr

p q

r

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Two methods of searching graph: data-driven: find path from known true

facts to goal goal directed: start with the proposition

to be proved (the goal) and search backwards along arcs to find support for the goal among the true propositions

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Examples: p. 110, fig 3.21p. 112, fig 3.23financial advisor fig 3.24sentence parser p. 116, fig 3.25

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Guided (heuristic) search

uses additional information, often in the form of estimates, to guide the search

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Guided (heuristic) search cont.

hill-climbing each node has an associated value or

cost; move to the node with the highest value (or lowest cost)

uses no memory and backtracking can only find local maximum (minimum)

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Ex: Traveling Salesperson

Fig. 3.9

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Best-first algorithm:1. create a queue and add the first node to it2. Loop:


if the child has not yet been visited or the child is reached by a path of lower cost than when visited previously, add the child node to the queue, in order of cost, eliminating any other higher cost paths to this child

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Guided (heuristic) search cont

Heuristic evaluation function may be the sum of two components: f(n) = g(n) + h(n)

g(n) measures cost from the start state to state n

h(n) estimates cost from state n to goal state

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Guided (heuristic) search cont

for the eight puzzle, g(n) = the number of moves from start to the

current state. H1(n) = the number of tiles that are out of

place; H2(n) = the sum of the Manhattan distances

(horizontal and vertical distances) for the tiles that are out of place

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A* searchuse evaluation function

f(n)=g(n)+h(n)where h(n) <=h*(n) //h*(n) is actual minimum cost from n to

goal

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Shortest path problem

Find the shortest path from a start node to a goal node

must keep track of paths and costs of paths

reject paths with loopsfor the shortest path problem, g(n) is the

total distance between the initial state and the state n, and h(n) is the estimated distance from state n to the goal state

A* also eliminates the more costly of redundant paths

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Shortest path problem cont.

A* shortest path algorithm1. create a queue and add the first node to it as a zero-length

path.2. Loop:

If the queue is empty, quit and announce failure.Remove the first path from the queue. If it ends at the goal

state, exit with this path as the solution. Otherwise, create new paths by extending the first path to all the neighbors of the terminal node.

Reject all new paths with loops.If two or more paths reach a common node, delete all those

paths except the one that reaches the common node with the minimum cost.

Sort the entire queue by the sum of the path length and a lower bound estimate of the cost remaining, with least-cost paths in front.

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Example: TSP

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Ex: robot path planning

P. 94-99, Winston 3rd Edition

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Competitive (Game) Search

Used for competitive gamesuses a move generator, a position

evaluator and a look-ahead strategygame states are represented as a

treeinitial state is the rootthe player (computer) is the

maximizer, and the opponent is the minimizer; 'minimax' search

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'minimax' search cont.

example: tic-tac-toeevaluation function measures the

number of rows/columns/diagonals open to the player versus the opponent:

E(s) = H(s) = (rX + cX + dX) - (ro + co + do)

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'minimax' search cont.

alpha beta pruning prunes branches which are not needed

lower bound alpha is the largest current value of all the MAX ancestors of the node

upper bound beta is the smallest current value of all its MIN ancestors

X tries to maximize alpha at each MAX level; O tries to minimize beta at each MIN level

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Summary

Search methods unguided

depth first, breadth first

guidedhill climbingbest firstA*

competitiveminimax

1 problem solving using search brute-force search heuristic search competitive search

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