Problem Solving by Searching Informed Methods

Problem Solving by Searching Search Methods : informed (Heuristic) search

Using problem specific knowledge to aid searching ,[object Object],[object Object],Search everywhere!!

Using problem specific knowledge to aid searching ,[object Object],[object Object],[object Object],Search only in this subtree!! A B C E D F G H I J K L O M N

More formally, why heuristic functions work? ,[object Object],[object Object],[object Object]

Heuristic Functions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Approach 1: f Measures the Value of the Current State ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Approach 2: f Measures the Cost to the Goal ,[object Object],[object Object],[object Object],[object Object]

Approach 3: f measures the total cost of the solution path (Admissible Heuristic Functions) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Traditional informed search strategies ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Informed Search Strategies Best First Search

An implementation of Best First Search ,[object Object],[object Object],[object Object],[object Object]

Informed Search Strategies Greedy Search eval-fn : f(n) = h(n)

Greedy Search 99 211 80 Start Goal 97 101 75 118 111 f(n) = h ( n ) = straight-line distance heuristic 140 A B D C E F I G H State Heuristic: h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0

Greedy Search: Tree Search Start A

Greedy Search: Tree Search Start 75 118 140 [374] [329] [253] A B C E

Greedy Search: Tree Search 99 80 Start 75 118 140 [374] [329] [253] [193] [366] [178] A B C E F G A

Greedy Search: Tree Search 99 211 80 Start Goal 75 118 140 [374] [329] [253] [193] [366] [178] [0] [253] A B C E F I G A E

Greedy Search: Tree Search 99 211 80 Start Goal 75 118 140 [374] [329] [253] [193] [366] [178] [0] [253] Path cost(A-E-F-I) = 253 + 178 + 0 = 431 dist(A-E-F-I) = 140 + 99 + 211 = 450 A B C E F I G A E

Greedy Search: Optimal ? 99 211 80 Start Goal 97 101 75 118 111 f(n) = h ( n ) = straight-line distance heuristic dist(A-E-G-H-I) =140+80+97+101= 418 140 A B D C E F I G H State Heuristic: h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0

Greedy Search: Complete ? 99 211 80 Start Goal 97 101 75 118 111 f(n) = h ( n ) = straight-line distance heuristic 140 A B D C E F I G H State Heuristic: h(n) A 366 B 374 ** C 250 D 244 E 253 F 178 G 193 H 98 I 0

Greedy Search: Tree Search Start 75 118 140 [374] [250] [253] A B C E

Greedy Search: Tree Search 111 Start 75 118 140 [374] [250] [253] [244] A B C E D

Greedy Search: Tree Search 111 Start 75 118 140 [374] [250] [253] [244] [250] Infinite Branch ! A B C E D C

Greedy Search: Tree Search 111 Start 75 118 140 [374] [250] [253] [244] [250] [244] Infinite Branch ! A B C E D C D

Greedy Search: Time and Space Complexity ? 99 211 80 Start Goal 97 101 75 118 111 140 ,[object Object],[object Object],[object Object],[object Object],A B D C E F I G H

Informed Search Strategies A* Search eval-fn : f(n)=g(n)+h(n)

A* (A Star) ,[object Object],[object Object],[object Object]

A* (A Star) g(n) h(n) f(n) = g(n)+h(n) n

A* Search f(n) = g(n) + h ( n ) g(n): is the exact cost to reach node n from the initial state. State Heuristic: h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0 A B D C E F I 99 211 G H 80 Start Goal 97 101 75 118 111 140

A* Search: Tree Search Start A

A* Search: Tree Search Start 75 118 140 [393] [449] [447] A B C E

A* Search: Tree Search 99 80 Start 75 118 140 [393] [449] [447] [417] [413] A B C E F G

A* Search: Tree Search 99 80 Start 75 118 140 [393] [449] [447] [417] [413] 97 [415] A B C E F G H

A* Search: Tree Search 99 80 Start 97 101 75 118 140 [393] [449] [447] [417] [413] [415] Goal [418] A B C E F I G H

A* Search: Tree Search 99 80 Start 97 101 75 118 140 [393] [449] [447] [417] [413] [415] Goal [418] [450] A B C E F I G H I

A* with f() not Admissible h() overestimates the cost to reach the goal state

A* Search: h not admissible ! 99 211 80 Start Goal 97 101 75 118 111 f(n) = g(n) + h ( n ) – (H-I) Overestimated g(n): is the exact cost to reach node n from the initial state. 140 A B D C E F I G H State Heuristic: h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 138 I 0

A* Search: Tree Search 99 80 Start 75 118 140 [393] [449] [447] [417] [413] 97 [455] A B C E F G H

A* Search: Tree Search 99 80 Start 97 75 118 140 [393] [449] [447] [417] [413] [455] Goal [450] A B C E F G H I

A* Search: Tree Search 99 80 Start 97 75 118 140 [393] [449] [447] [417] [413] [455] Goal [450] [473] A B C E F G H I D

A* Search: Tree Search 99 80 Start 97 75 118 140 [393] [449] [447] [417] [413] [455] Goal [450] [473] A* not optimal !!! A B C E F G H I D

A* Algorithm A* with systematic checking for repeated states …

A* Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

A* Search: Analysis 99 211 80 Start Goal 97 101 75 118 111 140 ,[object Object],[object Object],[object Object],[object Object],A B D C E F I G H

Informed Search Strategies Iterative Deepening A*

Iterative Deepening A*:IDA* ,[object Object],[object Object]

Consistent Heuristic ,[object Object],[object Object],N N’ h(N) h(N’) c(N,N’)

IDA* Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object]

8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=4 4 6

8-Puzzle Cutoff=4 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6

8-Puzzle Cutoff=4 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6 5

8-Puzzle Cutoff=4 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6 5 5

8-Puzzle Cutoff=4 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6 5 5 6

8-Puzzle Cutoff=5 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 6

8-Puzzle Cutoff=5 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6

8-Puzzle Cutoff=5 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6 5

8-Puzzle Cutoff=5 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6 5 7

8-Puzzle Cutoff=5 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6 5 7 5

8-Puzzle Cutoff=5 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 4 4 6 6 5 7 5 5

When to Use Search Techniques ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Conclusions ,[object Object],[object Object]

Problem Solving by Searching Informed Methods

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Problem Solving by Searching Informed Methods