1	Informed search algorithms Chapter 4
2	Material Chapter 4 Section 1 - 3 Excludes memory-bounded heuristic search
3	Outline Best-first search Greedy best-first search A^* search Heuristics Local search algorithms Hill-climbing search Simulated annealing search Local beam search Genetic algorithms
4	Review: Tree search A search strategy is defined by picking the order of node expansion
5	Best-first search Idea: use an evaluation function f(n) for each node estimate of "desirability" Expand most desirable unexpanded node Implementation: Order the nodes in fringe in decreasing order of desirability Special cases: greedy best-first search A^* search
6	Romania with step costs in km
7	Greedy best-first search Evaluation function f(n) = h(n) (heuristic) = estimate of cost from n to goal e.g., h_SLD(n) = straight-line distance from n to Bucharest Greedy best-first search expands the node that appears to be closest to goal
8	Greedy best-first search example
9	Greedy best-first search example
10	Greedy best-first search example
11	Greedy best-first search example
12	Properties of greedy best-first search Complete? No – can get stuck in loops, e.g., Iasi à Neamt à Iasi à Neamt à Time? O(b^m), but a good heuristic can give dramatic improvement Space? O(b^m) -- keeps all nodes in memory Optimal? No
13	A^* search Idea: avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost from n to goal f(n) = estimated total cost of path through n to goal
14	A^* search example
15	A^* search example
16	A^* search example
17	A^* search example
18	A^* search example
19	A^* search example
20	Admissible heuristics A heuristic h(n) is admissible if for every node n, h(n) ≤ h^(n), where h^(n) is the true cost to reach the goal state from n. An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic Example: h_SLD(n) (never overestimates the actual road distance) Theorem: If h(n) is admissible, A^* using TREE-SEARCH is optimal
21	Optimality of A^* (proof) Suppose some suboptimal goal G₂ has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. f(G₂) = g(G₂) since h(G₂) = 0 g(G₂) > g(G) since G₂ is suboptimal f(G) = g(G) since h(G) = 0 f(G₂) > f(G) from above
22	Optimality of A^* (proof) Suppose some suboptimal goal G₂ has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. f(G₂) > f(G) from above h(n) ≤ h^(n) since h is admissible g(n) + h(n) ≤ g(n) + h^(n) f(n) ≤ f(G) Hence f(G₂) > f(n), and A^* will never select G₂ for expansion
23	Consistent heuristics A heuristic is consistent if for every node n, every successor n' of n generated by any action a, h(n) ≤ c(n,a,n') + h(n') If h is consistent, we have f(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') ≥ g(n) + h(n) = f(n) i.e., f(n) is non-decreasing along any path. Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal
24	Optimality of A^* A^* expands nodes in order of increasing f value Gradually adds "f-contours" of nodes Contour i has all nodes with f=f_i, where f_i < f_i+1
25	Properties of A^* Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) ) Time? Exponential Space? Keeps all nodes in memory Optimal? Yes
26	Admissible heuristics E.g., for the 8-puzzle: h₁(n) = number of misplaced tiles h₂(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) h₁(S) = ? h₂(S) = ?
27	Admissible heuristics E.g., for the 8-puzzle: h₁(n) = number of misplaced tiles h₂(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) h₁(S) = ? 8 h₂(S) = ? 3+1+2+2+2+3+3+2 = 18
28	Dominance If h₂(n) ≥ h₁(n) for all n (both admissible) then h₂ dominates h₁ h₂ is better for search Typical search costs (average number of nodes expanded): d=12 IDS = 3,644,035 nodes A^(h₁) = 227 nodes A^(h₂) = 73 nodes d=24 IDS = too many nodes A^(h₁) = 39,135 nodes A^(h₂) = 1,641 nodes
29	Relaxed problems A problem with fewer restrictions on the actions is called a relaxed problem The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h₁(n) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h₂(n) gives the shortest solution
30	Local search algorithms In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution State space = set of "complete" configurations Find configuration satisfying constraints, e.g., n-queens In such cases, we can use local search algorithms keep a single "current" state, try to improve it
31	Example: n-queens Put n queens on an n × n board with no two queens on the same row, column, or diagonal
32	Hill-climbing search "Like climbing Everest in thick fog with amnesia"
33	Hill-climbing search Problem: depending on initial state, can get stuck in local maxima
34	Hill-climbing search: 8-queens problem h = number of pairs of queens that are attacking each other, either directly or indirectly h = 17 for the above state
35	Hill-climbing search: 8-queens problem
36	Simulated annealing search Idea: escape local maxima by allowing some "bad" moves but gradually decrease their frequency
37	Properties of simulated annealing search One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1 Widely used in VLSI layout, airline scheduling, etc
38	Genetic algorithms A successor state is generated by combining two parent states Start with k randomly generated states (population) A state is represented as a string over a finite alphabet (often a string of 0s and 1s) Evaluation function (fitness function). Higher values for better states. Produce the next generation of states by selection, crossover, and mutation
39	Genetic algorithms Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28) 24/(24+23+20+11) = 31% 23/(24+23+20+11) = 29% etc
40	Genetic algorithms