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Lecture 4: Informed/Heuristic Search
Lecture 4: Informed/Heuristic Search
Organizational items
Organizational items
Outline
Outline
Limitations of uninformed search
Limitations of uninformed search
Recall tree search
Recall tree search
Recall tree search
Recall tree search
Best-first search
Best-first search
Heuristic function
Heuristic function
Heuristic functions for 8-puzzle
Heuristic functions for 8-puzzle
Greedy best-first search
Greedy best-first search
Romania with step costs in km
Romania with step costs in km
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Greedy best-first search example
Optimal Path
Optimal Path
Properties of greedy best-first search
Properties of greedy best-first search
A* Search
A* Search
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
A* search example
Admissible heuristics
Admissible heuristics
Optimality of A* (proof)
Optimality of A* (proof)
Optimality of A* (proof)
Optimality of A* (proof)
Optimality for graphs
Optimality for graphs
A* is optimal with consistent heuristics
A* is optimal with consistent heuristics
Contours of A* Search
Contours of A* Search
Contours of A* Search
Contours of A* Search
Properties of A*
Properties of A*
Comments on A*
Comments on A*
Memory-bounded heuristic search
Memory-bounded heuristic search
Recursive Best-First Search (RBFS)
Recursive Best-First Search (RBFS)
Recursive Best First Search: Example
Recursive Best First Search: Example
RBFS example
RBFS example
RBFS example
RBFS example
RBFS properties
RBFS properties
(Simplified) Memory-bounded A* (SMA*)
(Simplified) Memory-bounded A* (SMA*)
Heuristic functions
Heuristic functions
Notion of dominance
Notion of dominance
Effective branching factor
Effective branching factor
Effectiveness of different heuristics
Effectiveness of different heuristics
Inventing heuristics via relaxed problems
Inventing heuristics via relaxed problems
More on heuristics
More on heuristics
Pattern databases
Pattern databases
Summary
Summary

: Informed-Heuristic Search. : Padhraic Smyth. : Informed-Heuristic Search.ppt. zip-: 406 .

Informed-Heuristic Search

Informed-Heuristic Search.ppt
1 Lecture 4: Informed/Heuristic Search

Lecture 4: Informed/Heuristic Search

ICS 271 Fall 2007

2 Organizational items

Organizational items

Homework 1 Due Thursday in class Problem 3.12 in text: 2nd part, IDS: some printings say constant step cost, others say variable step costs Assume either one, state clearly in your solution

3 Outline

Outline

Limitations of uninformed search methods Informed (or heuristic) search uses problem-specific heuristics to improve efficiency Best-first A* RBFS SMA* Techniques for generating heuristics Can provide significant speed-ups in practice e.g., on 8-puzzle But can still have worst-case exponential time complexity Reading: Chapter 4, Sections 4.1 and 4.2

4 Limitations of uninformed search

Limitations of uninformed search

8-puzzle Avg. solution cost is about 22 steps branching factor ~ 3 Exhaustive search to depth 22: 3.1 x 1010 states E.g., d=12, IDS expands 3.6 million states on average [24 puzzle has 1024 states (much worse)]

5 Recall tree search

Recall tree search

6 Recall tree search

Recall tree search

This strategy is what differentiates different search algorithms

7 Best-first search

Best-first search

Idea: use an evaluation function f(n) for each node estimate of "desirability Expand most desirable unexpanded node Implementation: Order the nodes in fringe by f(n) (by desirability, lowest f(n) first) Special cases: uniform cost search (from last lecture): f(n) = g(n) = path to n greedy best-first search A* search Note: evaluation function is an estimate of node quality => More accurate name for best first search would be seemingly best-first search

8 Heuristic function

Heuristic function

Heuristic: Definition: using rules of thumb to find answers Heuristic function h(n) Estimate of (optimal) cost from n to goal h(n) = 0 if n is a goal node Example: straight line distance from n to Bucharest Note that this is not the true state-space distance It is an estimate actual state-space distance can be higher Provides problem-specific knowledge to the search algorithm

9 Heuristic functions for 8-puzzle

Heuristic functions for 8-puzzle

8-puzzle Avg. solution cost is about 22 steps branching factor ~ 3 Exhaustive search to depth 22: 3.1 x 1010 states. A good heuristic function can reduce the search process. Two commonly used heuristics h1 = the number of misplaced tiles h1(s)=8 h2 = the sum of the distances of the tiles from their goal positions (Manhattan distance). h2(s)=3+1+2+2+2+3+3+2=18

10 Greedy best-first search

Greedy best-first search

Special case of best-first search Uses h(n) = heuristic function as its evaluation function Expand the node that appears closest to goal

11 Romania with step costs in km

Romania with step costs in km

12 Greedy best-first search example

Greedy best-first search example

13 Greedy best-first search example

Greedy best-first search example

14 Greedy best-first search example

Greedy best-first search example

15 Greedy best-first search example

Greedy best-first search example

16 Optimal Path

Optimal Path

17 Properties of greedy best-first search

Properties of greedy best-first search

Complete? Not unless it keeps track of all states visited Otherwise can get stuck in loops (just like DFS) Optimal? No we just saw a counter-example Time? O(bm), can generate all nodes at depth m before finding solution m = maximum depth of search space Space? O(bm) again, worst case, can generate all nodes at depth m before finding solution

18 A* Search

A* Search

Expand node based on estimate of total path cost through node Evaluation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost from n to goal f(n) = estimated total cost of path through n to goal Efficiency of search will depend on quality of heuristic h(n)

19 A* search example

A* search example

20 A* search example

A* search example

21 A* search example

A* search example

22 A* search example

A* search example

23 A* search example

A* search example

24 A* search example

A* search example

25 Admissible heuristics

Admissible heuristics

A heuristic h(n) is admissible if for every node n, h(n) ? h*(n), where h*(n) is the true cost to reach the goal state from n. An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic Example: hSLD(n) is admissible never overestimates the actual road distance Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal

26 Optimality of A* (proof)

Optimality of A* (proof)

Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. f(G2) = g(G2) since h(G2) = 0 g(G2) > g(G) since G2 is suboptimal f(G) = g(G) since h(G) = 0 f(G2) > f(G) from above

27 Optimality of A* (proof)

Optimality of A* (proof)

Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. f(G2) > f(G) from above h(n) ? h*(n) since h is admissible g(n) + h(n) ? g(n) + h*(n) f(n) ? f(G) Hence f(G2) > f(n), and A* will never select G2 for expansion

28 Optimality for graphs

Optimality for graphs

Admissibility is not sufficient for graph search In graph search, the optimal path to a repeated state could be discarded if it is not the first one generated Can fix problem by requiring consistency property for h(n) A heuristic is consistent if for every successor n' of a node n generated by any action a, h(n) ? c(n,a,n') + h(n') (aka monotonic) admissible heuristics are generally consistent

29 A* is optimal with consistent heuristics

A* is optimal with consistent heuristics

If h is consistent, we have f(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') ? g(n) + h(n) = f(n) i.e., f(n) is non-decreasing along any path. Thus, first goal-state selected for expansion must be optimal Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

30 Contours of A* Search

Contours of A* Search

A* expands nodes in order of increasing f value Gradually adds "f-contours" of nodes Contour i has all nodes with f=fi, where fi < fi+1

31 Contours of A* Search

Contours of A* Search

With uniform-cost (h(n) = 0, contours will be circular With good heuristics, contours will be focused around optimal path A* will expand all nodes with cost f(n) < C*

32 Properties of A*

Properties of A*

Complete? Yes (unless there are infinitely many nodes with f ? f(G) ) Optimal? Yes Also optimally efficient: No other optimal algorithm will expand fewer nodes, for a given heuristic Time? Exponential in worst case Space? Exponential in worst case

33 Comments on A*

Comments on A*

A* expands all nodes with f(n) < C* This can still be exponentially large Exponential growth will occur unless error in h(n) grows no faster than log(true path cost) In practice, error is usually proportional to true path cost (not log) So exponential growth is common

34 Memory-bounded heuristic search

Memory-bounded heuristic search

In practice A* runs out of memory before it runs out of time How can we solve the memory problem for A* search? Idea: Try something like depth first search, but lets not forget everything about the branches we have partially explored.

35 Recursive Best-First Search (RBFS)

Recursive Best-First Search (RBFS)

Similar to DFS, but keeps track of the f-value of the best alternative path available from any ancestor of the current node If current node exceeds f-limit -> backtrack to alternative path As it backtracks, replace f-value of each node along the path with the best f(n) value of its children This allows it to return to this subtree, if it turns out to look better than alternatives

36 Recursive Best First Search: Example

Recursive Best First Search: Example

Path until Rumnicu Vilcea is already expanded Above node; f-limit for every recursive call is shown on top. Below node: f(n) The path is followed until Pitesti which has a f-value worse than the f-limit.

37 RBFS example

RBFS example

Unwind recursion and store best f-value for current best leaf Pitesti result, f [best] ? RBFS(problem, best, min(f_limit, alternative)) best is now Fagaras. Call RBFS for new best best value is now 450

38 RBFS example

RBFS example

Unwind recursion and store best f-value for current best leaf Fagaras result, f [best] ? RBFS(problem, best, min(f_limit, alternative)) best is now Rimnicu Viclea (again). Call RBFS for new best Subtree is again expanded. Best alternative subtree is now through Timisoara. Solution is found since because 447 > 418.

39 RBFS properties

RBFS properties

Like A*, optimal if h(n) is admissible Time complexity difficult to characterize Depends on accuracy if h(n) and how often best path changes. Can end up switching back and fortyh Space complexity is O(bd) Other extreme to A* - uses too little memory.

40 (Simplified) Memory-bounded A* (SMA*)

(Simplified) Memory-bounded A* (SMA*)

This is like A*, but when memory is full we delete the worst node (largest f-value). Like RBFS, we remember the best descendant in the branch we delete. If there is a tie (equal f-values) we delete the oldest nodes first. simplified-MA* finds the optimal reachable solution given the memory constraint. Time can still be exponential.

41 Heuristic functions

Heuristic functions

8-puzzle Avg. solution cost is about 22 steps branching factor ~ 3 Exhaustive search to depth 22: 3.1 x 1010 states. A good heuristic function can reduce the search process. Two commonly used heuristics h1 = the number of misplaced tiles h1(s)=8 h2 = the sum of the distances of the tiles from their goal positions (manhattan distance). h2(s)=3+1+2+2+2+3+3+2=18

42 Notion of dominance

Notion of dominance

If h2(n) ? h1(n) for all n (both admissible) then h2 dominates h1 h2 is better for search Typical search costs (average number of nodes expanded) for 8-puzzle problem d=12 IDS = 3,644,035 nodes A*(h1) = 227 nodes A*(h2) = 73 nodes d=24 IDS = too many nodes A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes

43 Effective branching factor

Effective branching factor

Effective branching factor b* Is the branching factor that a uniform tree of depth d would have in order to contain N+1 nodes. Measure is fairly constant for sufficiently hard problems. Can thus provide a good guide to the heuristics overall usefulness.

44 Effectiveness of different heuristics

Effectiveness of different heuristics

Results averaged over random instances of the 8-puzzle

45 Inventing heuristics via relaxed problems

Inventing heuristics via relaxed problems

A problem with fewer restrictions on the actions is called a relaxed problem The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution Can be a useful way to generate heuristics E.g., ABSOLVER (Prieditis, 1993) discovered the first useful heuristic for the Rubiks cube puzzle

46 More on heuristics

More on heuristics

h(n) = max{ h1(n), h2(n),hk(n) } Assume all h functions are admissible Always choose the least optimistic heuristic (most accurate) at each node Could also learn a convex combination of features Weighted sum of h(n)s, where weights sum to 1 Weights learned via repeated puzzle-solving Could try to learn a heuristic function based on features E.g., x1(n) = number of misplaced tiles E.g., x2(n) = number of goal-adjacent-pairs that are currently adjacent h(n) = w1 x1(n) + w2 x2(n) Weights could be learned again via repeated puzzle-solving Try to identify which features are predictive of path cost

47 Pattern databases

Pattern databases

Admissible heuristics can also be derived from the solution cost of a subproblem of a given problem. This cost is a lower bound on the cost of the real problem. Pattern databases store the exact solution to for every possible subproblem instance. The complete heuristic is constructed using the patterns in the DB

48 Summary

Summary

Uninformed search methods have their limits Informed (or heuristic) search uses problem-specific heuristics to improve efficiency Best-first A* RBFS SMA* Techniques for generating heuristics Can provide significant speed-ups in practice e.g., on 8-puzzle But can still have worst-case exponential time complexity Next lecture: local search techniques Hill-climbing, genetic algorithms, simulated annealing, etc

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