1 CMPUT 412 Search Algorithms Csaba Szepesvári University of Alberta TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A A AA

2 Contents  Search problems on graphs  BFS  DFS, IDFS  A*  Outlook

3 Search problems on graphs  Single-pair shortest path problem: G=(V,E) g 2 V – goal state s 2 V – start state  Note: Graph could be implicit! (nodes generated as needed)  Examples: Shortest path on a map Solving logical puzzles Moving troops in a RTS game Finding a plan (AI)  Variants: All pairs shortest path problem Single-source shortest path problem Stochastic shortest path Infinite graphs (node=state, transition=move=action) Minimum cost optimal control (~reinforcement learning)..

4 Breadth-first search function BFS queue.add (s) while (!queue.empty()) n  queue.pop() if (goal(n)) return n; foreach (n,m) 2 E queue.add(m);  Space complexity ~ number of nodes at the deepest level; O(b d ) or O(|V|+|E|)  Time complexity: Same  Guarantees In finite graphs finds an optimal solution  Extension: costs; expand node with least cost from start

5 Depth-first search function dfs(n, &visited) if (goal(n)) return n; *visited.add(n) foreach (m,n) 2 E *if (!visited.has(m)) dfs(m,visited)  Space complexity: #elements visited  Time complexity: O(|V|+|E|), O(b d )  Guarantees: Finds the goal (without (*) only in loop-free graphs) Might not find the shortest path  Extensions Use a hash-table to store the list of visited nodes (or no list) + iterative deepening Advantage to BFS: Search can be directed

6 A* (Hart, Nilsson, Raphael, ’68) function A*(s) closed := ; q.insert([s]) while (!q.empty()) path  q.pop_first() n  path.last() if closed.has(n) continue if goal(n) return path closed.add(n) foreach (n,m) 2 E queue.add( list(p,m), c(p)+c(n,m)+h(m) ) return failure  Special best-first search algorithm  Idea: Prioritize node expansion by total estimated cost of the path through the node  Total cost = cost-so-far +cost-to-go (often: f=g+h)  h:V  R, h = h(n) “heuristic function” ~estimated cost-to-go  Guarantees Always finds a solution If h is optimistic, A* find the optimal soln  when no closed list is used -- or --  when h is monotonic Given h, optimally efficient  Time-complexity: O(2 L ), where L is the length of the optimal path |h-h * | = O(log h * )  poly(h)  Space-complexity: Might be the same  Extensions: IDA*, MA*, RBFS

7 Search terminology  If an algorithm A always.... finds a solution then A is complete.. finds an optimal solution then A is admissible.. does the least work amongst all admissible algorithm then A is optimally efficient

8 Outlook  Dijkstra A* with h(n)=0  Bellman-Ford: Computes cost-to-go with “value iteration” for all nodes in |V| iterations.  Pruning (beam search, ® - ¯,..)  Cost-to-go can be learnt (LRTA*, RL)  Problems can be stochastic (RL)  Monte-Carlo search; UCT (directed depth- first search, with changing cost-to-go estimates)  success in go