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- 1. A.I.: Informed Search Algorithms Chapter III: Part Deux
- 2. Outline • Best-first search • Greedy best-first search • A* search • Heuristics
- 3. Overview • Informed Search: uses problem-specific knowledge. • General approach: best-first search; an instance of TREE- SEARCH (or GRAPH-SEARCH) – where a search strategy is defined by picking the order of node expansion. • With best-first, node is selected for expansion based on evaluation function f(n). • Evaluation function is a cost estimate; expand lowest cost node first (same as uniform-cost search but we replace g with f).
- 4. Overview (cont’d) • The choice of f determines the search strategy (one can show that best-first tree search includes DFS as a special case). • Often, for best-first algorithms, f is defined in terms of a heuristic function, h(n). h(n) = estimated cost of the cheapest path from the state at node n to a goal state. (for goal state: h(n)=0) • Heuristic functions are the most common form in which additional knowledge of the problem is passed to the search algorithm.
- 5. Overview (cont’d) • Best-First Search algorithms constitute a large family of algorithms, with different evaluation functions. – Each has a heuristic function h(n) • Example: in route planning the estimate of the cost of the cheapest path might be the straight line distance between two cities. Recall: • g(n) = cost from the initial state to the current state n. • h(n) = estimated cost of the cheapest path from node n to a goal node. • f(n) = evaluation function to select a node for expansion (usually the lowest cost node).
- 6. Best-First Search • Idea: use an evaluation function f(n) for each node – f(n) provides an estimate for the total cost. Expand the node n with smallest f(n). • Implementation: Order the nodes in the frontier increasing order of cost. • Special cases: – Greedy best-first search – A* search
- 7. Greedy best-first search • Evaluation function f(n) = h(n) (heuristic), the estimate of cost from n to goal. • We use the straight-line distance heuristic: hSLD(n) = straight- line distance from n to Bucharest. • Note that the heuristic values cannot be computed from the problem description itself! • In addition, we require extrinsic knowledge to understand that hSLD is correlated with the actual road distances, making it a useful heuristic. • Greedy best-first search expands the node that appears to be closest to goal.
- 8. Romania with step costs in km
- 9. Greedy best-first search example
- 10. Greedy best-first search example
- 11. Greedy best-first search example
- 12. Greedy best-first search example
- 13. Greedy best-first search • GBFS is incomplete! • Why? • Graph-Search version is, however, complete in finite spaces.
- 14. Properties of greedy best-first search • Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt • Time? O(bm), (in worst case) but a good heuristic can give dramatic improvement (m is max depth of search space). • Space? O(bm) -- keeps all nodes in memory. • Optimal? No (not guaranteed to render lowest cost solution).
- 15. A* Search • Most widely-known form of best-first search. • It evaluates nodes by combining g(n), the cost to reach the node, and h(n), the cost to get from the node to the goal: f(n) = g(n) + h(n) (estimated cost of cheapest solution through n). • A reasonable strategy: try node with the lowest g(n) + h(n) value! • Provided heuristic meets some basic conditions, A* is both complete and optimal.
- 22. 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: hSLD(n) (never overestimates the actual road distance) • Theorem: If h(n) is admissible, A* using TREE- SEARCH is optimal.
- 23. Optimality of A* (proof) • Suppose some suboptimal goal G2 has been generated and is in the frontier. Let n be an unexpanded node in the frontier 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
- 24. 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) ≤ g(n) + h*(n) < f(G) < f(G2) Hence f(G2) > f(n), and A* will never select G2 for expansion.
- 25. Consistent Heuristics • A heuristic is consistent (or monotonic) 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.
- 26. 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=fi, where fi < fi+1. • That is to say, nodes inside a given contour have f-costs less than or equal to contour value.
- 27. Properties of A* • Complete: Yes (unless there are infinitely many nodes with f ≤ f(G) ). • Time: Exponential. • Space: Keeps all nodes in memory, so also exponential. • Optimal: Yes (provided h admissible or consistent). • Optimally Efficient: Yes (no algorithm with the same heuristic is guaranteed to expand fewer nodes). • NB: Every consistent heuristic is also admissible (Pearl). Q: What about the converse?
- 28. Admissible Heuristics E.g., for the 8-puzzle: • h1(n) = number of misplaced tiles • h2(n) = total Manhattan distance (i.e. 1-norm) (i.e., no. of squares from desired location of each tile) Q: Why are these admissible heuristics? • h1(S) = ? • h2(S) = ?
- 29. 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
- 30. Dominance • If h2(n) ≥ h1(n) for all n (both admissible), then h2 dominates h1 . • Essentially, domination translates directly into efficiency: “h2 is better for search. • A* using h2 will never expand more nodes than A* using h1. • 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 (IDS=iterative deepening search)
- 31. Memory Bounded Heuristic Search: Recursive BFS (best-first) • How can we solve the memory problem for A* search? • Idea: Try something like depth-first search, but let’s not forget everything about the branches we have partially explored. • We remember the best f-value we have found so far in the branch we are deleting.
- 32. Memory Bounded Heuristic Search: Recursive BFS • RBFS changes its mind very often in practice. This is because f=g+h become more accurate (less optimistic) as we approach the goal. Hence, higher level nodes have smaller f-values and will be explored first. • Problem: We should keep • in memory whatever we can. Best alternative over frontier nodes, which are not children: i.e. do I want to back up?
- 33. Simple Memory-Bounded A* • This is like A*, but when memory is full we delete the worst node (largest f-value). • Like RBFS, we remember the best descendent in the branch we delete. • If there is a tie (equal f-values) we delete the oldest nodes first. • Simple-MBA* finds the optimal reachable solution given the memory constraint (reachable means path from root to goal fits in memory). • Can also use iterative deepening with A* (IDA*). • Time can still be exponential.
- 34. 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. (why?) • 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.
- 35. Summary • Informed search methods may have access to a heuristic function h(n) that estimates the cost of a solution from n. • The generic best-first search algorithm selects a node for expansion according to an evaluation function. • Greedy best-first search expands nodes with minimal h(n). It is not optimal, but is often efficient. • A* search expands nodes with minimal f(n)=g(n)+h(n). • A* s complete and optimal, provided that h(n) is admissible (for TREE-SEARCH) or consistent (for GRAPH-SEARCH). • The space complexity of A* is still prohibitive. • The performance of heuristic search algorithms depends on the quality of the h(n) function. • One can sometimes construct good heuristics by relaxing the problem definition.