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Introduction to Artificial Intelligence Problem Solving through Search (Goal-based agents) BFS, DFS, Greedy Search

Janyl Jumadinova September 12-16, 2016 Example:

On holiday in Romania; currently in Arad. Flight leaves tomorrow from Formulate goal: be in Bucharest Formulate problem: states: various cities actions: drive between cities Find solution: sequence of cities, e.g., Arad, , Fagaras, Bucharest

2/38 Example: Romania

3/38 Tree search algorithms

Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states)

4/38 Tree search example

5/38 Tree search example

6/38 Tree search example

7/38 Implementation: states vs. nodes

A state is a (representation of) a physical configuration A node is a data structure constituting part of a search tree includes parent, children, depth, path cost g(x) States do not have parents, children, depth, or path cost!

8/38 Breadth-first search Expand shallowest unexpanded node

I FIFO queue, i.e., new successors go at end A

B C

D E F G

9/38 Breadth-first search Expand shallowest unexpanded node

I FIFO queue, i.e., new successors go at end A

B C

D E F G

10/38 Breadth-first search Expand shallowest unexpanded node

I FIFO queue, i.e., new successors go at end A

B C

D E F G

11/38 Breadth-first search Expand shallowest unexpanded node

I FIFO queue, i.e., new successors go at end

A

B C

D E F G

12/38 Breadth-first search

Arad

13/38 Breadth-first search

Arad

Zerind Sibiu Timisoara

14/38 Breadth-first search

Arad

Zerind Sibiu Timisoara

Arad

15/38 Breadth-first search

Arad

Zerind Sibiu Timisoara

Rimnicu Arad Oradea Arad Oradea Fagaras Vilcea Arad Lugoj

16/38 Breadth-first search

17/38 Depth-first search Expand deepest unexpanded node I LIFO queue, i.e., put successors at front

Arad

18/38 Depth-first search Expand deepest unexpanded node I LIFO queue, i.e., put successors at front

Arad

Zerind Sibiu Timisoara

19/38 Depth-first search Expand deepest unexpanded node I LIFO queue, i.e., put successors at front

Arad

Zerind Sibiu Timisoara

Arad Oradea

20/38 Depth-first search Expand deepest unexpanded node I LIFO queue, i.e., put successors at front

Arad

Zerind Sibiu Timisoara

Arad Oradea

Zerind Sibiu Timisoara

21/38 ⇒ Expand most desirable unexpanded node

Implementation: a queue sorted in decreasing order of desirability

Special cases: greedy search, A∗ search

Best-first search

Idea: use an evaluation function for each node – estimate of “desirability”

22/38 Implementation: a queue sorted in decreasing order of desirability

Special cases: greedy search, A∗ search

Best-first search

Idea: use an evaluation function for each node – estimate of “desirability”

⇒ Expand most desirable unexpanded node

22/38 Special cases: greedy search, A∗ search

Best-first search

Idea: use an evaluation function for each node – estimate of “desirability”

⇒ Expand most desirable unexpanded node

Implementation: a queue sorted in decreasing order of desirability

22/38 Best-first search

Idea: use an evaluation function for each node – estimate of “desirability”

⇒ Expand most desirable unexpanded node

Implementation: a queue sorted in decreasing order of desirability

Special cases: greedy search, A∗ search

22/38 Romania with step costs in km

Straight−line distance Oradea 71 to Bucharest Neamt Arad 366 87 Bucharest 0 Zerind 75 151 160 Iasi Dobreta 242 Arad 140 Eforie 161 92 Fagaras 178 Sibiu Fagaras 99 77 118 Hirsova 80 151 Iasi 226 Rimnicu Vilcea Timisoara Lugoj 244 142 Mehadia 241 111 211 Neamt 234 Lugoj 97 Pitesti Oradea 380 70 98 Pitesti 98 146 85 Hirsova Mehadia 101 Urziceni Rimnicu Vilcea 193 75 138 86 Sibiu 253 Bucharest Timisoara 120 329 Dobreta 90 Urziceni 80 Craiova Eforie Vaslui 199 Giurgiu Zerind 374

A* slide 23/38 I Greedy search expands the node that appears to be closest to goal

Greedy search

Evaluation function h(n)(heuristic) = estimate of cost from n to the closest goal

Example: h(n) = straight-line distance from n to Bucharest

24/38 Greedy search

Evaluation function h(n)(heuristic) = estimate of cost from n to the closest goal

Example: h(n) = straight-line distance from n to Bucharest

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

24/38 Greedy Search

Arad 366

25/38 Greedy Search

Arad

Sibiu Timisoara Zerind 253 329 374

26/38 Greedy Search

Arad

Sibiu Timisoara Zerind 329 374

Arad Fagaras Oradea Rimnicu Vilcea 366 176 380 193

27/38 Greedy Search

Arad

Sibiu Timisoara Zerind 329 374

Arad Fagaras Oradea Rimnicu Vilcea 366 380 193

Sibiu Bucharest 253 0

28/38 Greedy search

I Can get stuck in loops, e.g., Iasi → Neamt → Iasi → Neamt → Complete in finite space with repeated-state checking

I A good heuristic can give dramatic improvement in search cost

29/38 http://aispace.org/search/

Explore Best first search (explanation from AIspace: http://www.aispace.org/deduction/help/tut3.shtml)

30/38 I Evaluation function f (n) = g(n) + h(n)

I g(n) = cost so far to reach n

I h(n) = estimated cost to goal from n

I f (n) = estimated total cost of path through n to goal Romania with step costs

A* search

Idea: avoid expanding paths that are already expensive

31/38 A* search

Idea: avoid expanding paths that are already expensive

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

I g(n) = cost so far to reach n

I h(n) = estimated cost to goal from n

I f (n) = estimated total cost of path through n to goal Romania with step costs

31/38 Example: Romania

32/38 A* Search

Arad 366=0+366

33/38 A* Search

Arad

Sibiu Timisoara Zerind 393=140+253 447=118+329 449=75+374

34/38 A* Search

Arad

Sibiu Timisoara Zerind 447=118+329 449=75+374

Arad Fagaras Oradea Rimnicu Vilcea 646=280+366415=239+176 671=291+380 413=220+193

35/38 A* Search

Arad

Sibiu Timisoara Zerind 447=118+329 449=75+374

Arad Fagaras Oradea Rimnicu Vilcea 646=280+366 415=239+176 671=291+380

Craiova Pitesti Sibiu 526=366+160 417=317+100 553=300+253

36/38 A* Search

Arad

Sibiu Timisoara Zerind 447=118+329 449=75+374

Arad Fagaras Oradea Rimnicu Vilcea 646=280+366 671=291+380

Sibiu Bucharest Craiova Pitesti Sibiu 591=338+253 450=450+0 526=366+160 417=317+100 553=300+253

37/38 A* Search

Arad

Sibiu Timisoara Zerind 447=118+329 449=75+374

Arad Fagaras Oradea Rimnicu Vilcea 646=280+366 671=291+380

Sibiu Bucharest Craiova Pitesti Sibiu 591=338+253 450=450+0 526=366+160 553=300+253

Bucharest Craiova Rimnicu Vilcea 418=418+0 615=455+160 607=414+193

38/38