1Announcements Project 1: Search It’s live! Due 9/14. Start early and ask questions. It’s longer than most! Need a partner? Come up after class or try Piazza Sections: can go to any, but have priority in your ownCS 188: Artificial IntelligenceFall 2011Lecture 3: A* Search9/1/2011Dan Klein – UC BerkeleyMultiple slides from Stuart Russell or Andrew Moore Today A* Search Graph Search Heuristic DesignUCS GreedyA*Recap: Search Search problem: States (configurations of the world) Successor function: a function from states to lists of (state, action, cost) triples; drawn as a graph Start state and goal test Search tree: Nodes: represent plans for reaching states Plans have costs (sum of action costs) Search Algorithm: Systematically builds a search tree Chooses an ordering of the fringe (unexplored nodes) Optimal: finds least-cost plansExample: Pancake ProblemCost: Number of pancakes flippedExample: Pancake Problem324332224State space graph with costs as weights343422General Tree SearchAction: flip top twoCost: 2Action: flip all fourCost: 4Path to reach goal:Flip four, flip threeTotal cost: 7Uniform Cost Search Strategy: expand lowest path cost The good: UCS is complete and optimal! The bad: Explores options in every “direction” No information about goal locationStartGoal…c ≤ 3c ≤ 2c ≤ 1[demo: countours UCS]Example: Heuristic FunctionHeuristic: the largest pancake that is still out of place4302333443444h(x)Example: Heuristic Functionh(x)Best First (Greedy) Strategy: expand a node that you think is closest to a goal state Heuristic: estimate of distance to nearest goal for each state A common case: Best-first takes you straight to the (wrong) goal Worst-case: like a badly-guided DFS…b…b[demo: countours greedy]Combining UCS and Greedy Uniform-cost orders by path cost, or backward cost g(n) Greedy orders by goal proximity, or forward cost h(n) A* Search orders by the sum: f(n) = g(n) + h(n)S a dbGh=5h=6h=215112h=6h=0ch=73eh=11Example: Teg Grenager3 Should we stop when we enqueue a goal? No: only stop when we dequeue a goalWhen should A* terminate?SBAG2322h = 1h = 2h = 0h = 3Is A* Optimal?AGS13h = 6h = 05h = 7 What went wrong? Actual bad goal cost < estimated good goal cost We need estimates to be less than actual costs!Admissible Heuristics A heuristic h is admissible (optimistic) if:where is the true cost to a nearest goal Examples: Coming up with admissible heuristics is most of what’s involved in using A* in practice.415Optimality of A*: Blocking…Notation: g(n) = cost to node n h(n) = estimated cost from n to the nearest goal (heuristic) f(n) = g(n) + h(n) =estimated total cost via n G*: a lowest cost goal node G: another goal nodeOptimality of A*: BlockingProof: What could go wrong? We’d have to have to pop a suboptimal goal G off the fringe before G* This can’t happen: Imagine a suboptimal goal G is on the queue Some node n which is a subpath of G* must also be on the fringe (why?) n will be popped before G…Properties of A*…b…bUniform-Cost A*4UCS vs A* Contours Uniform-cost expanded in all directions A* expands mainly toward the goal, but does hedge its bets to ensure optimalityStartGoalStartGoal[demo: countours UCS / A*]Creating Admissible Heuristics Most of the work in solving hard search problems optimally is in coming up with admissible heuristics Often, admissible heuristics are solutions to relaxed problems, where new actions are available Inadmissible heuristics are often useful too (why?)15366Example: 8 Puzzle What are the states? How many states? What are the actions? What states can I reach from the start state? What should the costs be?8 Puzzle I Heuristic: Number of tiles misplaced Why is it admissible? h(start) = This is a relaxed-problem heuristic8Average nodes expanded when optimal path has length……4 steps …8 steps …12 stepsUCS 112 6,300 3.6 x 106TILES13 39 2278 Puzzle II What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles? Total Manhattan distance Why admissible? h(start) =3 + 1 + 2 + …= 18Average nodes expanded when optimal path has length……4 steps …8 steps …12 stepsTILES13 39 227MANHATTAN12 25 738 Puzzle III How about using the actual cost as a heuristic? Would it be admissible? Would we save on nodes expanded? What’s wrong with it? With A*: a trade-off between quality of estimate and work per node!5Trivial Heuristics, Dominance Dominance: ha≥ hcif Heuristics form a semi-lattice: Max of admissible heuristics is admissible Trivial heuristics Bottom of lattice is the zero heuristic (what does this give us?) Top of lattice is the exact heuristicOther A* Applications Pathing / routing problems Resource planning problems Robot motion planning Language analysis Machine translation Speech recognition …[demo: plan tiny UCS / A*]Tree Search: Extra Work! Failure to detect repeated states can cause exponentially more work. Why?Graph Search In BFS, for example, we shouldn’t bother expanding the circled nodes (why?)Sabdpacephfrqq cGaqephfrqq cGaGraph Search Idea: never expand a state twice How to implement: Tree search + set of expanded states (“closed set”) Expand the search tree node-by-node, but… Before expanding a node, check to make sure its state is new If not new, skip it Important: store the closed set as a set, not a list Can graph search wreck completeness? Why/why not? How about optimality?Warning: 3e book has a more complex, but also correct, variantA* Graph Search Gone Wrong?SABCG11123h=2h=1h=4h=1h=0S (0+2)A (1+4) B (1+1)C (2+1)G (5+0)C (3+1)G (6+0)State space graph Search tree6Consistency of Heuristics Stronger than admissibility Definition:cost(A to C) + h(C) ≥ h(A)cost(A to C) ≥ h(A) - h(C)real cost ≥ cost implied by heuristic Consequences: The f value along a path never decreases A* graph search is optimal3ACGh=4h=11Optimality of A* Graph SearchProof: New possible problem: some n on path to G* isn’t in queue when we need it, because some worse n’ for the same state dequeued and expanded first (disaster!) Take the highest such n in tree Let p be the ancestor of n that was on the queue when n’ was
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