Introduction to Artificial Intelligence Blind Search Ruth Bergman Fall 2004

Search Strategy Partial search tree for route finding from Arad to Bucharest. Arad (a) The initial state (search node) (b) After expanding Arad (c) After expanding Sibiu Arad SibiuTimisoaraZerind Arad SibiuTimisoaraZerind AradFagarasOradea Rimnicu Vilcea goal test choosing one option Which node to expand? Which nodes to store in memory?

Search Trees Search node – book-keeping data structure used to represent the search tree –State is a configuration of the world (defstruct node state ;;; state (domain dependent) ancestor ;;; the ancestor node in the search graph f ;;; the cost of the node action ;;; the specific action that led to this node from its ) ;;; ancestor

Depth-first Search Searching Strategies Expand deepest node first. DFS (state path) if goalp(state) return path else for c in succ(state) return DFS(c, path | state)

DFS implementation in Lisp (defun dfs (node) (cond ((goalp (node-state node)) node) (t (let ((children (successor-fn node))) (dfs2 children))))) (defun dfs2 (states) ;; input list of nodes (cond ((null states) nil) ((dfs (car states))) ((dfs2 (cdr states)))))

Criteria –Completeness: if there is a solution will the algorithm find it? –Time complexity: how much time does the algorithm take to arrive at a solution, if one exists? –Space complexity: how much space does the algorithm require? –Optimality: is the solution optimal? Search Strategy Properties

In-Completeness of DFS DFS is not complete –fails in infinite-depth spaces, spaces with loops Variants –limit depth of search –avoid re-visiting nodes. –avoid repeated states along path => complete in finite spaces

DFS with depth limit (defun dfs (node depth) (cond ((goalp (node-state node)) node) ((zerop depth) nil) ((let ((children (successor-fn node))) (dfs2 children (- depth 1))))) (defun dfs2 (states depth) (cond ((null states) nil) ((dfs (car states) depth)) ((dfs2 (cdr states) depth))))

Properties –Complete: No Guaranteed to stop Complete only if exists solution at level l<d (where d is the maximum depth) –Time complexity: O(b^d) Best case l Worst case (b^(d+1)-1)/(b-1) Where b is the branching factor improved performance when there are many solutions –Space complexity: O(bd) i.e., linear space –Optimal: No DFS with depth limit Performance Searching Strategies

DFS with no revisits avoid nodes that have already been expanded. => exponential space complexity. – Not practical.

DFS with no repeated states (defun dfs (node path) (cond ((goalp (node-state node)) path) ((let ((children (successor-fn node))) (dfs2 children (cons (node-state node) path))))) (defun dfs2 (states path) (cond ((null states) nil) ((member (car states) path) nil) ((dfs (car states) path)) ((dfs2 (cdr states) path)))) => Complete in finite spaces

Backtracking Search Searching Strategies When states are expanded by applying operators The algorithm expands one child at a time (by applying one operator) If search fails, backtrack and expand other children Backtracking search results in even lower memory requirements than DFS DFS node discovery Backtracking search node discovery

Advantages –Low space complexity –Good chance of success when there are many solutions. –Complete if there is a solution shorter than the depth limit. Disadvantages –Without the depth limit search may continue down an infinite branch. –Solutions longer than the depth limit will not be found. –The solution found may not be the shortest solution. DFS Summary Searching Strategies

Breadth-first Search Searching Strategies Expand node with minimal depth. avoid revisiting nodes. Since every node is in memory, the additional cost is negligible.

BFS implementation in Lisp (defun bfs (queue) (let* ((node (car queue)) (state (node-state node))) (cond ((null queue) nil) ((goalp state) node) ((let ((children (successor-fn node))) (bfs (append (cdr queue) children)))))))

BFS Performance Searching Strategies Properties –Complete: Yes (if b is finite) –Time complexity: 1+b+b^2+…+b^l = O(b^l) –Space complexity: O(b^l) (keeps every node in memory) –Optimal: Yes (if cost=1 per step); not optimal in general where b is branching factor and l is the depth of the shortest solution

Uniform cost Search A GS C B SS SS A AA B BB C CC GGG Expand least-cost unexpanded node –the breadth-first search is just uniform cost search with f(n)=DEPTH(n) –Node discovery –Stop at first expanded goal node Searching Strategies

Uniform cost Search Properties of Uniform-Cost Search –Complete: Yes, if step cost >= e (epsilon) –Time complexity: # of nodes with f <= C* C* = cost of optimal solution Worst case O(b^(C*/e)) O(b^l) if step costs are equal –Space complexity: # of nodes with f <= C*, O(b^l) –Optimal: Yes, if step cost >= e (epsilon) Searching Strategies

Combine the best of both worlds –Depth first search has linear memory requirements –Breadth first search gives an optimal solution. Iterative Deepening Search executes depth first search with depth limit 1, then 2, 3, etc. until a solution is found. The algorithm has no memory between searches. Iterative Deepening Search Searching Strategies

Limit=0 Limit=1 Limit=2 Limit=3 Iterative Deepening Search … Searching Strategies

Properties –Complete: Yes –Time complexity: (l+1)*b^0+l*b+(l- 1)*b^2+…+1*b^l = O(b^l) –Space complexity: O(bl) –Optimal: Yes, if step cost = 1 Can be modified to explore uniform-cost tree Iterative Deepening Search Searching Strategies

Numerical demonstration: Let b=10, l=5. –BFS resource use (memory and # nodes expanded) = 111,111 –Iterative Deepening resource use Memory requirement: 10*5 = 50 # expanded nodes = 123,456 => re-searching cost is small compared with the cost of expanding the leaves Iterative Deepening Search - Discussion Searching Strategies

Simultaneously search both forward from the initial state and backward from the goal, and stop when the two searches meet in the middle. Bidirectional Search StartGoal Searching Strategies

Properties –Complete: Yes (using a complete search procedure for each half) –Time complexity: O(b^(l/2)) –Space complexity: O(b^(l/2)) –Optimal: Yes, if step cost = 1 Can be modified to explore uniform-cost tree Bidirectional Search Performance Searching Strategies

Bidirectional Search Discussion Numerical Example (b=10, l = 6) –Bi-directional search finds solution at d=3 for both forward and backward search. Assuming BFS in each half 22,200 nodes are generated. –Compare with 11,111,100 for a standard BFS. Implementation issues: –Operators are reversible. –There may be many possible goal states. –Check if a node appears in the “other” search tree. –What’s the best search strategy in each half.

–b is the branching factor; –l is the depth of solution; –m is the maximum depth of the search tree; –d is the depth limit. Comparison Search Strategies Criterion Breadth- First Uniform- Cost Depth- First Depth- Limited Iterative Deepening Bi- directional (if applicable) Timeb^l b^mb^db^lb^(l/2) Spaceb^l bmbdblb^(l/2) Optimal?Yes No Yes Complete?Yes No Yes, if d>=l Yes Searching Strategies

Herbert A reactive-based robot that collects soda cans, sometimes.

Alternate implementations of Search Algorithms Iteration Unnamed functions (lambda expressions) Mapping functions

DFS implementation in Lisp (defun dfs (state) (cond ((goalp state) (list state)) (t (do* ((children (new-states state) (cdr children))) (solution (dfs (car children)) (dfs (car children))) ((or solution (null children)) (if solution (cons state solution) nil))))))

DFS with depth limit (defun dfs-d (state depth) (cond ((goalp state) (list state)) ((zerop depth) nil) (t (do* ((children (new-states state) (cdr children))) (solution (dfs (car children) (1- depth)) (dfs (car children) (1- depth))) ((or solution (null children)) (if solution (cons state solution) nil))))))

DFS with no repeated states (defun dfs-d-g (state depth path) (cond ((goalp state) (list state)) ((zerop depth) nil) (t (do* ((children (new-states state) (cdr children))) (solution (if (member (car children) path) nil (dfs (car children) (1- depth) (cons state path)) …)) ((or solution (null children)) (if solution (cons state solution) nil)))))) => Complete in finite spaces

BFS implementation in Lisp (defun bfs (state) (let ((queue (list (list state nil)))) (do* ((state (caar queue) …) (children (new-states state) …)) ((or (null queue) (goalp state)) (if (null queue) nil (car state)) (setq queue (append (cdr queue) (mapcar #'(lambda (state) (cons state (car queue))) children)))))))