Informed search algorithms
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Transcript of Informed search algorithms
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Informed search algorithms
Chapter 4
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Outline• Best-first search• Greedy best-first search• A* search• Heuristics
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Review: Tree searchfunction Tree-SEARCH(problem,fringe) return a solution or failure
closed an empty setfringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe))loop doif EMPTY?(fringe) then return failurenode REMOVE-FIRST(fringe)if GOAL-TEST[problem] applied to STATE[node] succeedsthen return SOLUTION(node)if STATE[node] is not in closed thenadd STATE[node] to closedfringe INSERT-ALL(EXPAND(node, problem), fringe)
• A search strategy is defined by picking the order of node expansion
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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 in decreasing order of desirability
• Special cases:– greedy best-first search– A* search
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Romania with step costs in km
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Greedy best-first search
• Evaluation function f(n) = h(n) (heuristic) = estimate of cost from n to goal• e.g., hSLD(n) = straight-line distance from n
to Bucharest• Greedy best-first search expands the node
that appears to be closest to goal
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
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Properties of greedy best-first search
• Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt
• Time? O(bm), but a good heuristic can give dramatic improvement
• Space? O(bm) -- keeps all nodes in memory• Optimal? No
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A* search
• Idea: avoid expanding paths that are already expensive
• 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
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A* search example
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A* search example
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A* search example
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A* search example
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A* search example
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A* search example
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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) (never overestimates the actual road distance)
• Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal
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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
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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
••
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Consistent heuristics• A heuristic is consistent or monotonous if
– for every node n, every successor n' of n generated by any action a, satisfies h(n) ≤ c(n,a,n') + h(n')
– h(G) = 0 for each goal state G
• 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.
• Theorem: Every consistent heuristic is also admissible
• Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal
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Consistent heuristics – PATHMAX
• Any admissible heuristic h can be made into a consistent heuristic hc using the pathmax equation
• hc(n’) = max( h(n’), h(n)-c(n,a,n’) )
• The pathmax equation simply enforces the triangle inequality
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Optimality of A*
• 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
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Properties of A*
• Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )
• Time? Exponential• Space? Keeps all nodes in memory• Optimal? Yes
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Admissible heuristicsE.g., for the 8-puzzle:• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance(i.e., no. of squares from desired location of each tile)
• h1(S) = ? • h2(S) = ? •
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Admissible heuristicsE.g., for the 8-puzzle:• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance(i.e., no. of squares from desired location of each tile)
• h1(S) = ? 8• h2(S) = ? 3+1+2+2+2+3+3+2 = 18
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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):
• d=12 IDS = 3,644,035 nodesA*(h1) = 227 nodes A*(h2) = 73 nodes
• d=24 IDS = too many nodesA*(h1) = 39,135 nodes A*(h2) = 1,641 nodes
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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
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Pattern databases• The idea with pattern
databases is to store the exact solution costs for every subproblem instance.
• with 15 puzzle a speedup of 1,000 vs Manhattan heuristic
31 moves needed to solve red tiles. 22 moves needed to solve blue tiles
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Iterative Deepening A* (IDA*) Idea: Reduce memory requirement of A*
by applying cutoff on values of f Consistent heuristic h Algorithm IDA*:
1. Initialize cutoff to f(initial-node)2. Repeat:
a. Perform depth-first search by expanding all nodes N such that f(N) cutoff
b. Reset cutoff to smallest value f of non-expanded (leaf) nodes
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Advantages/Drawbacks of IDA*
Advantages:• Still complete and optimal• Requires less memory than A*• Avoid the overhead to sort the fringe
Drawbacks:• Can’t avoid revisiting states not on the current
path• Available memory is poorly used
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Recursive Best First Search
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SMA*(Simplified Memory-bounded A*)
Works like A* until memory is full Then SMA* drops the node in the fringe with the largest f
value and “backs up” this value to its parent When all children of a node N have been dropped, the
smallest backed up value replaces f(N) In this way, the root of an erased subtree remembers the
best path in that subtree SMA* will regenerate this subtree only if all other nodes
in the fringe have greater f values SMA* generates the best solution path that fits in
memory SMA* can’t completely avoid revisiting states, but it can
do a better job at this that IDA*