informed search methods - computer science and...

Artificial Intelligence

Informed search methods

2

In which we see how information about

the state space can prevent algorithms

from blundering about in the dark.

3

Uninformed vs. Informed Search

• Uninformed search strategies

– Find solutions to problems by systematically

generating new states and testing for goal

– Most are incredibly inefficient

• Informed search strategies

– Use problem-specific knowledge

– Find solutions more efficiently

4

Informed Search

• Informed search strategy

– One that uses problem-specific knowledge beyond the problem definition itself

• General approach is called “Best First”search

– Node selected for expansion based on an evaluation function

• Measuring a distance to goal

– Node with lowest evaluation is selected

5

Techniques

• Best-first search

– Expand “minimum cost” nodes first

– Greedy search

• Minimize estimated cost to reach goal

– A* search

• Estimated total costs through node n to goal:

(actual cost to reach n) + (estimated cost from n to goal)

Best-First Search Methods

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7

Best-First Search

• Try to expand node that is “closest” to goal

– Use evaluation function to estimate how desirable

• Open set

– Priority queue implementation

– Insert expanded nodes in decreasing order of

desirability (most desirable first)

• Special cases

– Greedy search: expand node closest to goal

– A* search: expand node on least-cost solution path

8

Greedy Search

• Simplest best-first search strategy


• Heuristic function h(n)

– Estimate of cost from node n to goal

• Require h(n) = 0 if n is goal

– e.g., For a road map, hSLD(n) = straight-line distance from city n to Bucharest

• Greedy search expands the node that appears to be closest to the goal

– Takes the “biggest bite” out of remaining cost to reach goal• Each step tries to get as close to goal as possible

– Though, will not consider if its action will be the best in the long run

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Romania Step Costs in km

Oradea

Zerind

Arad

Timisoara

Lugoj

Mehadia

DebretaCraiova

Sibiu

Rimnicu Vilcea

Fagaras

Pitesti

Bucharest

Giurgiu Eforie

Urziceni

Hirsova

Vaslui

Iasi

Neamt

75

71

118

111

70

75

151

140 99

80

146

97

138

211

101

90

85

86

98

142

92

87

120

10

Straight-Line Distance to BucharestArad 366

Bucharest 0

Craiova 160

Dobreta 242

Eforie 161

Fagaras 176

Giurgiu 77

Hirsova 151

Iasi 226

Lugoj 244

Mehadia 241

Neamt 234

Oradea 380

Pitesti 100

Rimnicu Vilcea 193

Sibiu 253

Timisoara 329

Urziceni 80

Vaslui 199

Zerind 374

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Greedy Search Example: Romania

Arad

366

Open = { Arad(366) }

12


Zerind Timisoara

Arad

Sibu

366

374 253 329

Open = { Sibu(253), Timisoara(329), Zerind(374) }

13


Rimnicu

VilceaArad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

374 253 329

366 380 176 193

Open = { Fagaras(176), RimnicuVilcea(193), Timisoara(329), Arad(366), Zerind(374), Oradea(380) }

14


BucharestSibiu

Rimnicu

VilceaArad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

374 253 329

366 380 176 193

253 0

Open = { Bucharest(0), Fagaras(176), RimnicuVilcea(193), Sibu(253), Timisoara(329), Arad(366), Zerind(374), Oradea(380) }

15


BucharestSibiu

Rimnicu

VilceaArad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

374 253 329

366 380 176 193

253 0

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Properties of Greedy Search

• Not complete (can be bad)

– Can get stuck in loops

• Go from Iasi to Fagaras:

Iasi Neamt Iasi Neamt (only 1 link)

– Complete if check for repeated states

• Time (can be bad)

– O(bm), m=maximum depth (worst case, like DFS)

• Space (can be bad)

– O(bm), keeps all nodes in memory (worst case)

• Not optimal (can be bad)

– Heuristic is an estimate

But a good heuristic can give dramatic improvement!

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A* Search• Minimizes total estimated path cost

• Avoids expanding paths already expensive

• Evaluation function f(n) = g(n) + h(n)

– Actual cost to reach node n so far g(n)

– Estimated cost from n to goal h(n)

– Estimated total cost through n to goal

f(n) = g(n) + h(n)

– A* search uses admissible heuristic

– i.e., h(n) h*(n) where h*(n) is true cost from n

to goal

– e.g., hSLD(n) never overestimates actual distance

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A* Search Example: Romania

Arad

366

g(n): cost so far

f(n)=g(n)+h(n) = cost so far + estimated cost to goal

Open = { Arad(366) }

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Zerind Timisoara

Arad

Sibu

366

449 393 447

75

140 118

g(n): cost so far


Open = { Sibu(393),

Timisoara(447), Zerind(449) }

f(Zerind)=75 + 374=449

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Arad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

449 393 447

646 671 415 413

Rimnicu

Vilcea

75

140 118

140 151 99 80

g(n): cost so far


Open = { RimnicuVilcea(413),

Fagaras(415), Timisoara(447),

Zerind(449), Arad(646),

Oradea(671) }

f(Fagaras)=(140+99) + 176=415

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Arad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

449 393 447

646 671 415 413

Craiova Sibiu

526 417 553

Rimnicu

Vilcea

Pitesti

75

140 118

140 151 99 80

146 97

80

g(n): cost so far


Open = { Fagaras(415), Pitesti(417),

Timisoara(447), Zerind(449),

Craiova(526), Sibu(553), Arad(646),

Oradea(671) }

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BucharestSibiu

Arad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

449 393 447

646 671 415 413

591 450

Craiova Sibiu

526 417 553

Rimnicu

Vilcea

Pitesti

75

140 118

140 151 99 80

99 211 146 97

80

g(n): cost so far


Open = { Pitesti(417), Timisoara(447),

Bucharest(450), Zerind(449),

Craiova(526), Sibu(553), Sibu(591),

Arad(646), Oradea(671) }

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BucharestSibiu

Arad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

449 393 447

646 671 415 413

591 450

Craiova Sibiu

526 417 553

Craiova Bucharest

607 615 418

Rimnicu

Vilcea

Pitesti

75

140 118

140 151 99 80

99 211 146 97

80

97 138 101

g(n): cost so far


Rimnicu

Vilcea

Open = { Bucharest(418), Timisoara(447),

Bucharest(450), Zerind(449), Craiova(526),

Sibu(553), Sibu(591), RimnicuVilcea(607),

Craiova(615), Arad(646), Oradea(671) }

24


BucharestSibiu

Arad Oradea

Zerind Timisoara

Arad

Sibu

Fagaras

366

449 393 447

646 671 415 413

591 450

Craiova Sibiu

526 417 553

Craiova Bucharest

607 615 418

Rimnicu

Vilcea

Pitesti

75

140 118

140 151 99 80

99 211 146 97

80

97 138 101

g(n): cost so far


Rimnicu

Vilcea

Recall that Greedy path was

[Arad, Sibu, Fagaras, Bucharest]

= 450

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Admissible Heuristics (Not over-estimate path cost)

• 8-puzzle

• h1(n) = number of misplaced tiles

– Must move each misplaced tile at least once

• h2(n) = total Manhattan distance (city block)

– Number of squares from desired tile location (Horiz+Vert)

5 4

6 1 8

7 3 2

1 2

8

3

4

7 6 5

Start State Goal State

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Admissible Heuristics

5 4

6 1 8

7 3 2

1 2

8

3

4

7 6 5

Start State Goal State

h1(Start) = 7 (# tiles misplaced)

h2(Start) = 2+3+3+2+4+2+0+2=18

(total Manhattan distance)[Typical solution about 20 steps, though varies with

initial state]

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Heuristic Dominance

• If h2(n) h1(n) for all n (both admissible)

– Then h2 dominates h1, and h2 is better for search

• As the larger h2 is closer to the optimal/true

total cost h*, then A* using h2 will expand

fewer nodes (on average)

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Repeated states in A*• Contrary to previous algorithms, avoiding

repeated states can actually harm A*

• Consider:

A(0+7=7) B(2+5=7) C(4+1=5) G(4+4+0=8)

• Goal is discovered before finding shortest path

to it!

• Depending on algorithm construction, could

return sub-optimal path

– At best, devolves to BFS-like performance

A

B

C G

2

4

1

4

7

5

1 0

Heuristic

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Solutions to Repeated states in A*• Don’t bother

– Some problems have no possibility of repeated states (tree-like

structure)

• Modified closed set

– Store the estimated path cost for each state as well

– If a repeat has lower value, allow it (and update closed set)

– Still technically BFS-like performance (exponential) but effect

reduced in half on average

• Consistent heuristic!

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Consistent Heuristics– Consistency: The difference in heuristic values between two

connected states is always smaller than (or equal to) the actual

cost:

𝑐𝑜𝑠𝑡 𝑥, 𝑦 ≥ ℎ 𝑥 − ℎ 𝑦

– Consistency ensures that the total estimated path cost always

increases (or stays the same) as we get closer to a goal state

• For this reason, sometimes also called a monotonic heuristic

• Guarantees that shortest path is always expanded first

– With a consistent heuristic, using a standard closed set for

repeated state avoidance is guaranteed to be both correct and

optimal

– Any consistent heuristic that has value 0 at goal states is also

admissible. However, many admissible heuristics are not

consistent!

– Though many common heuristics are consistent as well as

admissible

For two connected

states x and y

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Properties of A*• Complete (good)

– Unless infinitely many nodes

• Time and space (not good)

– Still exponential in worst case (keeps all nodes in memory)

– Strongly depends on choice of heuristic

• Optimal (good)

– Expands fewest nodes

• Gaming: Sim City traffic

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Summary

• Informed search strategies

• Best-first search

– Expand minimum cost nodes first

– Greedy search


– A* search

• Estimated total costs through n to goal

• (actual cost to reach n) + (estimated cost from n to goal)

• Admissible heuristic never overestimates actual distance to goal

informed search methods - computer science and...

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