Informed search A* algorithm

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Informed search A* algorithm. 2013/10/17. Outline. Informed = use problem-specific knowledge Which search strategies? Best-first search and its variants Heuristic functions? How to invent them Local search and optimization Hill climbing, simulated annealing, beam search,…
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Informed searchA* algorithm2013/10/17Outline
  • Informed = use problem-specific knowledge
  • Which search strategies?
  • Best-first search and its variants
  • Heuristic functions?
  • How to invent them
  • Local search and optimization
  • Hill climbing, simulated annealing, beam search,…
  • Evolution algorithms
  • Previously: tree-searchfunction TREE-SEARCH(problem,fringe) return a solution or failurefringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe)loop doif EMPTY?(fringe) then return failurenode REMOVE-FIRST(fringe)if GOAL-TEST[problem] applied to STATE[node] succeedsthen return SOLUTION(node)fringe INSERT-ALL(EXPAND(node, problem), fringe)A strategy is defined by picking the order of node expansionBest-first search
  • General approach of informed search:
  • Best-first search: node is selected for expansion based on an evaluation functionf(n)
  • Idea: evaluation function measures distance to the goal.
  • Choose node which appears best
  • Implementation:
  • fringe is queue sorted in decreasing order of desirability.
  • Special cases: greedy search, A* search
  • A heuristic function
  • [dictionary]“A rule of thumb, simplification, or educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood.”
  • h(n)= estimated cost of the cheapest path from node n to goal node.
  • If n is goal thenh(n)=0
  • More information later.Romania with step costs in km
  • hSLD=straight-line distance heuristic.
  • In this example f(n)=h(n)
  • Expand node that is closest to goal
  • = Greedy best-first searchGreedy search exampleArad (366)
  • Assume that we want to use greedy search to solve the problem of travelling from Arad to Bucharest.
  • The initial state=Arad
  • Greedy search exampleAradZerind(374)Sibiu(253)Timisoara(329)
  • The first expansion step produces:
  • Sibiu, Timisoara and Zerind
  • Greedy best-first will select Sibiu.
  • Greedy search exampleAradSibiuArad(366)Rimnicu Vilcea(193)Fagaras(176)Oradea(380)
  • If Sibiu is expanded we get:
  • Arad, Fagaras, Oradea and Rimnicu Vilcea
  • Greedy best-first search will select: Fagaras
  • Greedy search exampleAradSibiuFagarasSibiu(253)Bucharest(0)
  • If Fagaras is expanded we get:
  • Sibiu and Bucharest
  • Goal reached !!
  • Yet not optimal (see Arad, Sibiu, Rimnicu Vilcea, Pitesti)
  • Greedy search, evaluation
  • Completeness: NO (cfr. DF-search)
  • Check on repeated states
  • Minimizing h(n) can result in false starts, e.g. Iasi to Fagaras.
  • Greedy search, evaluation
  • Completeness: NO (cfr. DF-search)
  • Time complexity?
  • Cfr. Worst-case DF-search
  • (with m is maximum depth of search space)
  • Good heuristic can give dramatic improvement.
  • Greedy search, evaluation
  • Completeness: NO (cfr. DF-search)
  • Time complexity:
  • Space complexity:
  • Keeps all nodes in memory
  • Greedy search, evaluation
  • Completeness: NO (cfr. DF-search)
  • Time complexity:
  • Space complexity:
  • Optimality? NO
  • Same as DF-search
  • A* search
  • Best-known form of best-first search.
  • Idea: avoid expanding paths that are already expensive.
  • Evaluation function f(n)=g(n) + h(n)
  • g(n) the cost (so far) to reach the node.
  • h(n) estimated cost to get from the node to the goal.
  • f(n) estimated total cost of path through n to goal.
  • A* search
  • A* search uses an admissible heuristic
  • A heuristic is admissible if it never overestimates the cost to reach the goal
  • Are optimistic
  • Formally: 1. h(n) <= h*(n) where h*(n) is the true cost from n 2. h(n) >= 0 so h(G)=0 for any goal G.e.g. hSLD(n) never overestimates the actual road distanceRomania exampleA* search example
  • Find Bucharest starting at Arad
  • f(Arad) = c(??,Arad)+h(Arad)=0+366=366
  • A* search example
  • Expand Arrad and determine f(n) for each node
  • f(Sibiu)=c(Arad,Sibiu)+h(Sibiu)=140+253=393
  • f(Timisoara)=c(Arad,Timisoara)+h(Timisoara)=118+329=447
  • f(Zerind)=c(Arad,Zerind)+h(Zerind)=75+374=449
  • Best choice is Sibiu
  • A* search example
  • Expand Sibiu and determine f(n) for each node
  • f(Arad)=c(Sibiu,Arad)+h(Arad)=280+366=646
  • f(Fagaras)=c(Sibiu,Fagaras)+h(Fagaras)=239+179=415
  • f(Oradea)=c(Sibiu,Oradea)+h(Oradea)=291+380=671
  • f(RimnicuVilcea)=c(Sibiu,RimnicuVilcea)+
  • h(RimnicuVilcea)=220+192=413
  • Best choice is RimnicuVilcea
  • A* search example
  • Expand Rimnicu Vilcea and determine f(n) for each node
  • f(Craiova)=c(Rimnicu Vilcea, Craiova)+h(Craiova)=360+160=526
  • f(Pitesti)=c(Rimnicu Vilcea, Pitesti)+h(Pitesti)=317+100=417
  • f(Sibiu)=c(Rimnicu Vilcea,Sibiu)+h(Sibiu)=300+253=553
  • Best choice is Fagaras
  • A* search example
  • Expand Fagaras and determine f(n) for each node
  • f(Sibiu)=c(Fagaras, Sibiu)+h(Sibiu)=338+253=591
  • f(Bucharest)=c(Fagaras,Bucharest)+h(Bucharest)=450+0=450
  • Best choice is Pitesti !!!
  • A* search example
  • Expand Pitesti and determine f(n) for each node
  • f(Bucharest)=c(Pitesti,Bucharest)+h(Bucharest)=418+0=418
  • Best choice is Bucharest !!!
  • Optimal solution (only if h(n) is admissable)
  • Note values along optimal path !!
  • Optimality of A*(standard proof)
  • Suppose suboptimal goal G2 in the queue.
  • Let n be an unexpanded node on a shortest to optimal goal G.
  • f(G2 ) = g(G2 ) since h(G2 )=0> g(G) since G2 is suboptimal>= f(n) since h is admissibleSince f(G2) > f(n), A* will never select G2 for expansionBUT … graph search
  • Discards new paths to repeated state.
  • Previous proof breaks down
  • Solution:
  • Add extra bookkeeping i.e. remove more expensive of two paths.
  • Ensure that optimal path to any repeated state is always first followed.
  • Extra requirement on h(n): consistency (monotonicity)
  • Consistency
  • A heuristic is consistent if
  • If h is consistent, we have
  • i.e. f(n) is nondecreasing along any path.Optimality of A*(more usefull)
  • A* expands nodes in order of increasing f value
  • Contours can be drawn in state space
  • Uniform-cost search adds circles.
  • F-contours are gradually
  • Added: 1) nodes with f(n)<C*2) Some nodes on the goalContour (f(n)=C*).Contour I has allNodes with f=fi, wherefi < fi+1.A* search, evaluation
  • Completeness: YES
  • Since bands of increasing f are added
  • Unless there are infinitly many nodes with f<f(G)
  • A* search, evaluation
  • Completeness: YES
  • Time complexity:
  • Number of nodes expanded is still exponential in the length of the solution.
  • A* search, evaluation
  • Completeness: YES
  • Time complexity: (exponential with path length)
  • Space complexity:
  • It keeps all generated nodes in memory
  • Hence space is the major problem not time
  • A* search, evaluation
  • Completeness: YES
  • Time complexity: (exponential with path length)
  • Space complexity:(all nodes are stored)
  • Optimality: YES
  • Cannot expand fi+1 until fi is finished.
  • A* expands all nodes with f(n)< C*
  • A* expands some nodes with f(n)=C*
  • A* expands no nodes with f(n)>C*
  • Heuristic functions
  • E.g for the 8-puzzle
  • Avg. solution cost is about 22 steps (branching factor +/- 3)
  • Exhaustive search to depth 22: 3.1 x 1010 states.
  • A good heuristic function can reduce the search process.
  • Heuristic functions
  • E.g for the 8-puzzle knows two commonly used heuristics
  • h1 = the number of misplaced tiles
  • h1(s)=8
  • h2 = the sum of the distances of the tiles from their goal positions (Manhattan distance).
  • h2(s)=3+1+2+2+2+3+3+2=18
  • Heuristic quality
  • Effective branching factor b*
  • Is the branching factor that a uniform tree of depth d would have in order to contain N+1 nodes.
  • Measure is fairly constant for sufficiently hard problems.
  • Can thus provide a good guide to the heuristic’s overall usefulness.
  • A good value of b* is 1.
  • Heuristic quality and dominance
  • 1200 random problems with solution lengths from 2 to 24.
  • If h2(n) >= h1(n) for all n (both admissible)
  • then h2 dominates h1 and is better for searchInventing admissible heuristics
  • Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem:
  • Relaxed 8-puzzle for h1 : a tile can move anywhere
  • As a result, h1(n) gives the shortest solution
  • Relaxed 8-puzzle for h2 : a tile can move to any adjacent square.
  • As a result, h2(n) gives the shortest solution.The optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem.Inventing admissible heuristics
  • Admissible heuristics can also be derived from the solution cost of a subproblem of a given problem.
  • This cost is a lower bound on the cost of the real problem.
  • Pattern databases store the exact solution to for every possible subproblem instance.
  • The complete heuristic is constructed using the patterns in the DB
  • Inventing admissible heuristics
  • Another way to find an admissible heuristic is through learning from experience:
  • Experience = solving lots of 8-puzzles
  • An inductive learning algorithm can be used to predict costs for other states that arise during search.
  • Tree search algorithmfunction TREE-SEARCH(problem,fringe) return a solution or failurefringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe)loop doif EMPTY?(fringe) then return failurenode REMOVE-FIRST(fringe)if GOAL-TEST[problem] applied to STATE[node] succeedsthen return SOLUTION(node)fringe INSERT-ALL(EXPAND(node, problem), fringe)Tree search algorithm (2)function EXPAND(node,problem) return a set of nodessuccessors the empty setfor each <action, result> in SUCCESSOR-FN[problem](STATE[node]) dos a new NODE STATE[s]  result PARENT-NODE[s]  node ACTION[s]  action PATH-COST[s]  PATH-COST[node]+ STEP-COST(node, action,s) DEPTH[s]  DEPTH[node]+1 add s to successorsreturnsuccessors
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