Understanding AI Search Problems: Techniques and Graph Algorithms

1. COMP 3715 Spring 2005

2. Searching as an AI problem • Many AI problem framed as search • General idea • You know the individual steps towards solving a problem • You need to find out which (combination of) steps lead to the solution • So you need to search for the right combination

3. Searching as an AI problem • Example: • Playing chess: search for the right moves to move to a winner location • Puzzle: the order to move pieces so that the puzzle is solved • Walking in a maze: making the right moves to get out of the maze

4. Searching as an AI problem • States: the various situations • Certain position of chess pieces; current location in the maze • Actions: Allowable action at each state • Which chess piece can move where; where can I move in the maze • Initial/Start state: the starting location • Starting position of chess; the entrance of a maze • Final/Goal state: the goal • Position that checkmate; the exit of a maze

5. Searching as an AI problem • Searching can be generalized as a graph • Graph • Vertices/Nodes: labels • Edges: connecting two vertices • Directed vs. Not directed • Cycles • Acyclic graph: graph without a cycle • Trees

6. Searching as an AI problem • Searching as a graph problem • States : vertices • Directed Edges : if a action move from one state to another • Start state, goal state: the corresponding vertices • Search problem: find a path from the start vertex to the end vertex

7. Finding a path in a graph • General idea of a graph search • Start at the initial vertex • Visit the nodes that can be reached by the vertices already visited • Continue until the goal is find or until there is no more vertices to visit • Algorithms differ with • Order of visit • Data structure needed

8. Finding a path in a graph • Depth first search • Going “deeper” whenever possible • Always pick a vertex that is just visited to visit next • Need a stack • Breadth first search • Going “wider” whenever possible • Pick a vertex based on “first come first visited” principle • Need a queue

9. Depth first search -- example B C Stack: {I} I D F A Not yet visited E Current G Visited

10. Depth first search -- example B C Stack: {I, A} I D F A Not yet visited E Current G Visited

11. Depth first search -- example B C Stack: {I, A, C} I D F A Not yet visited E Current G Visited

12. Depth first search -- example B C Stack: {I, A, C, D} I D F A Not yet visited E Current G Visited

13. Depth first search -- example B C Stack: {I, A, C, D} I D F A Not yet visited E Current G Visited A is not put on the stack because it has been visited

14. Depth first search -- example B C Stack: {I, A, C} I D F A Not yet visited E Current G Visited D has no other vertex to go to, so is popped

15. Depth first search -- example B C Stack: {I, A, C, F} I D F A Not yet visited E Current G Visited

16. Depth first search -- example B C Stack: {I, A, C, F, G} I D F A Not yet visited E Current G Visited Goal is reached

17. Depth first search • Start with an initial vertex • Insert the vertex on the stack • Make the initial vertex as visited • At each step, check if the vertex at the top of the stack has vertex not yet visited • If exist, then pick one of the vertices to put on top of stack • If not, pop the current vertex out of the stack • Repeat until the goal is reached or the stack is empty

18. Breadth first search -- example B C Queue: {I} I D F A Not yet visited E Current G Visited

19. Breadth first search -- example B C Queue: {A, B} I D F A Not yet visited E Current G Visited

20. Breadth first search -- example B C Queue: {B, C} I D F A Not yet visited E Current G Visited A is not put on the queue because it is visited

21. Breadth first search -- example B C Queue: {C, E} I D F A Not yet visited E Current G Visited

22. Breadth first search -- example B C Queue: {E, D, F} I D F A Not yet visited E Current G Visited

23. Breadth first search -- example B C Queue: {D, F, G} I D F A Not yet visited E Current G Visited A is not put on the queue because it is already there

24. Breadth first search -- example B C Queue: {F, G} I D F A Not yet visited E Current G Visited G is not put on the queue because it is already there

25. Breadth first search -- example B C Queue: {F, G} I D F A Not yet visited E Current G Visited Goal is found

26. Breadth first search • Use a (first-in-first-out) queue instead of a stack • The initial vertex is put on the queue • At each step • Look at the top of the queue • Look at all the vertices that the top of the queue points to • Add them to the end of the queue if they have not been visited and are not on the queue already • Remove the top of the queue • Repeat until goal is reached or queue empty

27. Search – locating the path • For depth first search • Path can be listed by popping from the stack • For breadth first search • We need to keep track of each vertex’s “parent” (which vertex leads to the vertex adding onto the queue)

28. Depth first/Breadth first search: strength and limitations • Both • Easy to implement (try it yourself!) • Exhaustive – guarantee to find the solution • Depth first search • Very dependent on which vertex to choose • Breadth first search • More likely to find a good path • The queue can grow very big

29. Depth first/Breadth first search: strength and limitations • In an Artificial Intelligence context • Graphs are too big -- Can’t generate all the vertices • Exhaustive search takes too long • Need faster solutions (e.g. robot can’t wait a day before it moves)

30. Heuristic search