1. State Space Search Algorithms CSE 472 Introduction to Artificial Intelligence Autumn 2003

2. Depth First Search a b de c f gh Maintain stack of nodes to visit Properties –Complete? –Time Complexity? –Space Complexity? Not for infinite spaces O(b^d) O(d)

3. Breadth First Search a b c def gh Maintain FIFO queue of nodes to visit Properties –Complete? –Time Complexity? –Space Complexity? Yes O(b^d)

4. Iterative Deepening Search DFS with limit; incrementally grow limit Properties –Complete? –Time Complexity? –Space Complexity? Yes O(b^d) O(d) b de c f gh a b c a

5. Dijkstra’s Shortest Path Algorithm Like breadth-first search, but uses a priority queue instead of a FIFO queue Generalizes BFS to case where arcs can have different lengths –Example: maze where roads between intersections are different lengths –Example: complete CSE major doing as little homework as possible, where each course has estimated number of hours of homework

6. Pseudocode for Dijkstra Initialize the cost of each vertex to  cost[s] = 0; heap.insert(s); While (! heap.empty()) n = heap.deleteMin() if (n is a goal vertex) then return “Success!” For (each vertex a which is adjacent to n along edge e) if (cost[n] + edge_cost[e] < cost[a]) then cost [a] = cost[n] + edge_cost[e] previous_on_path_to[a] = n; if (a is in the heap) then heap.decreaseKey(a) else heap.insert(a)

7. Edsger Wybe Dijkstra (1930-2002) Invented concept of structured programming 1972 Turing Award “In their capacity as a tool, computers will be but a ripple on the surface of our culture. In their capacity as intellectual challenge, they are without precedent in the cultural history of mankind.”

8. Map of Manhattan How would you find a path from S to G? 52 nd St 51 st St 50 th St 10 th Ave 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave 3 rd Ave 2 nd Ave S G

9. Best-First Search The Manhattan distance (  x+  y) is an estimate of the distance to the goal –It is a heuristic function Best-First Search –Order nodes in priority queue to minimize estimated distance to the goal h(n) Compare: Dijkstra –Order nodes in priority queue to minimize distance from the start

10. Best First in Action How would you find a path from S to G? 52 nd St 51 st St 50 th St 10 th Ave 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave 3 rd Ave 2 nd Ave S G

11. Problem 1: Led Astray Eventually will expand vertex to get back on the right track 52 nd St 51 st St 50 th St 10 th Ave 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave 3 rd Ave 2 nd Ave S G

12. Problem 2: Optimality With Best-First Search, are you guaranteed a shortest path is found when –goal is first seen? –when goal is removed from priority queue (as with Dijkstra?)

13. Sub-Optimal Solution No! Goal is by definition at distance 0: will be removed from priority queue immediately, even if a shorter path exists! 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) h=2 h=1 h=4 h=5

14. Synergy? Dijkstra / Breadth First guaranteed to find optimal solution Best First often visits far fewer vertices, but may not provide optimal solution –Can we get the best of both?

15. A* (“A star”) Order vertices in priority queue to minimize (distance from start) + (estimated distance to goal) f(n) = g(n) + h(n) f(n) = priority of a node g(n) = true distance from start h(n) = heuristic distance to goal

16. Optimality Suppose the estimated distance (h) is always less than or equal to the true distance to the goal –heuristic is a lower bound on true distance –heuristic is admissible Then: when the goal is removed from the priority queue, we are guaranteed to have found a shortest path!

17. A* in Action 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St vertexg(n)h(n)f(n) 52 nd & 9 th 055

18. A* in Action 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St vertexg(n)h(n)f(n) 52 nd & 4 th 527 51 st & 9 th 145

19. A* in Action 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St vertexg(n)h(n)f(n) 52 nd & 4 th 527 51 st & 8 th 235 50 th & 9 th 257

20. A* in Action 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St vertexg(n)h(n)f(n) 52 nd & 4 th 527 51 st & 7 th 325 50 th & 9 th 257 50 th & 8 th 347

21. A* in Action 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St vertexg(n)h(n)f(n) 52 nd & 4 th 527 51 st & 6 th 415 50 th & 9 th 257 50 th & 8 th 347 50 th & 7 th 437

22. A* in Action 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St vertexg(n)h(n)f(n) 52 nd & 4 th 527 51 st & 5 th 505 50 th & 9 th 257 50 th & 8 th 347 50 th & 7 th 437

23. A* in Action 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St vertexg(n)h(n)f(n) 52 nd & 4 th 527 50 th & 9 th 257 50 th & 8 th 347 50 th & 7 th 437 DONE!

24. What Would Dijkstra Have Done? 52 nd St 51 st St 9 th Ave 8 th Ave 7 th Ave6 th Ave5 th Ave4 th Ave S G (5 blocks) 50 th St 49 th St 48 th St 47 th St

25. Importance of Heuristics D IDS A*(h1)A*(h2) 2 10 6 6 4 11213 12 6 68020 18 8 638439 25 10 4712793 39 12 364404 227 73 143473941 539 113 18 3056 363 24 391351641 h1 = number of tiles in the wrong place h2 = sum of distances of tiles from correct location 7 2 3 8 5 4 1 6

26. Maze Runner Demo