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Heuristic SearchHeuristic Search

Chapter 4

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OutlineOutline

Heuristic function Greedy Best-first search Admissible heuristic and A* Properties of A* Algorithm IDA*

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Heuristic Search

Heuristic: A rule of thumb generally based on expert experience, common sense to guide problem-solving process

In search, use a heuristic function that estimates how far we are from a goal.

How do we use heuristics?

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Romania with step costs in km

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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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Robot NavigationRobot Navigation

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Robot NavigationRobot Navigation

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f(N) = h(N), with h(N) = Manhattan distance to the goal

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Properties of greedy best-first search

Complete? No – can get stuck in loops; Yes, if we can avoid repeated statesTime? O(bm), but a good heuristic can give dramatic improvementSpace? O(bm) -- keeps all nodes in memoryOptimal? No

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A* SearchA* SearchA* search combines Uniform-cost and Greedy Best-first SearchEvaluation function: f(N) = g(N) + h(N) where: g(N) is the cost of the best path found so far

to N h(N) is an admissible heuristic f(N) is the estimated cost of cheapest solution

through N

0 < c(N,N’) (no negative cost steps).Order the nodes in the fringe in increasing values of f(N)

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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 heuristicAdmissible heuristic

Let h*(N) be the cost of the optimal path from N to a goal node

Heuristic h(N) is admissible if:

0 h(N) h*(N)

An admissible heuristic is always optimistic

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8-Puzzle8-Puzzle

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4 5 6

7 8

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N goal• h1(N) = number of misplaced tiles = 6 is admissible

• h2(N) = sum of distances of each tile to goal = 13 is admissible• What heuristics are overestimates?

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Completeness & Optimality of Completeness & Optimality of A*A*

Claim: If there is a path from the initial to

a goal node, A* using TREE-SEARCH terminates by finding the best path, hence is: complete 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.

We can show f(n) < f(G2), so A* would not have selected G2.

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Optimality of A* (proof)Cont’d:

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) Cont’d:

f(G2) > f(G) from above

h(n) ≤ h*(n) since h is admissibleg(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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Exampel: Graph Search returns a suboptimal solution

h=7

2 15

4

h=6

h=1 h=0

S AB

G

h is admissible

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Exampel: Graph Search returns a suboptimal solution

h=7

2 15

4

h=6

h=1 h=0

S AB

G

h is admissible

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Exampel: Graph Search returns a suboptimal solution

h=7

2 15

4

h=6

h=1 h=0

S AB

G

h is admissible

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Exampel: Graph Search returns a suboptimal solution

h=7

2 15

4

h=6

h=1 h=0

S AB

G

h is admissible

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Exampel: Graph Search returns a suboptimal solution

h=7

2 15

4

h=6

h=1 h=0

S AB

G

h is admissible

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Consistent HeuristicConsistent Heuristic

The admissible heuristic h is consistent (or satisfies the monotone restriction) if for every node N and every successor N’ of N:

h(N) c(N,N’) + h(N’)

(triangular inequality)A consisteny heuristic is admissible.

N

N’ h(N)

h(N’)

c(N,N’)

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Exampel: Graph Search returns a suboptimal solution

h=7

2 15

4

h=6

h=1 h=0

S AB

G

h is admissible but not consistent; e.g. h(S)=7 c(S,A) + h(A) = 5 ?No.

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8-Puzzle8-Puzzle

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7 8

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N goal

• h1(N) = number of misplaced tiles

• h2(N) = sum of distances of each tile to goal

are both consistent.

But do you see why?

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ClaimsClaimsIf h is consistent, then the function f alongany path is non-decreasing:

f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) h(N) c(N,N’) + h(N’) f(N) f(N’)

If h is consistent, then whenever A* expands a node it has already found an optimal path to the state associated with this node

N

N’ h(N)

h(N’)

c(N,N’)

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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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Avoiding Repeated States in Avoiding Repeated States in A*A*

If the heuristic h is consistent, then:Let CLOSED be the list of states associated with expanded nodesWhen a new node N is generated: If its state is in CLOSED, then discard N If it has the same state as another

node in the fringe, then discard the node with the largest f

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HeuristicHeuristic AccuracyAccuracy

h(N) = 0 for all nodes is admissible and consistent. Hence, breadth-first and uniform-cost are particular A* !!!Let h1 and h2 be two admissible and consistent heuristics such that for all nodes N: h1(N) h2(N).

Then, every node expanded by A* using h2 is also expanded by A* using h1.h2 is more informed than h1 h2 dominates h1

Which heuristic for 8-puzzle is better?

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Complexity of A*Complexity of A*

Time: exponential

Space: can keep all nodes in memory

If we want save space, use IDA*

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Iterative Deepening A* Iterative Deepening A* (IDA*)(IDA*)

Use f(N) = g(N) + h(N) with admissible and consistent hEach iteration is depth-first with cutoff on the value of f of expanded nodes

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

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8-Puzzle8-Puzzle

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f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

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About HeuristicsAbout Heuristics

Heuristics are intended to orient the search along promising pathsThe time spent computing heuristics must be recovered by a better searchAfter all, a heuristic function could consist of solving the problem; then it would perfectly guide the searchDeciding which node to expand is sometimes called meta-reasoningHeuristics may not always look like numbers and may involve large amount of knowledge

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When to Use Search When to Use Search Techniques?Techniques?

The search space is small, and There are no other available techniques,

or It is not worth the effort to develop a

more efficient technique

The search space is large, and There is no other available techniques,

and There exist “good” heuristics

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SummarySummary

Heuristic function Greedy Best-first search Admissible heuristic and A* A* is complete and optimal Consistent heuristic and repeated states Heuristic accuracy IDA*