The time complexity of A* with approximate heuristics on multiple-solution search spaces

  • Authors:

  • Hang Dinh;Hieu Dinh;Laurent Michel;Alexander Russell

  • Affiliations:

  • Department of Computer & Information Sciences, Indiana University South Bend, South Bend, IN;MathWorks, Natick, MA;Department of Computer Science & Engineering, University of Connecticut, Storrs, CT;Department of Computer Science & Engineering, University of Connecticut, Storrs, CT

  • Venue:
  • Journal of Artificial Intelligence Research
  • Year:
  • 2012

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Abstract

We study the behavior of the A* search algorithm when coupled with a heuristic h satisfying (1 -- ε1)h* ≤ h ≤ (1 + ε2)h*, where ε1; ε2 ∈ [0; 1) are small constants and h* denotes the optimal cost to a solution. We prove a rigorous, general upper bound on the time complexity of A* search on trees that depends on both the accuracy of the heuristic and the distribution of solutions. Our upper bound is essentially tight in the worst case; in fact, we show nearly matching lower bounds that are attained even by non-adversarially chosen solution sets induced by a simple stochastic model. A consequence of our rigorous results is that the effective branching factor of the search will be reduced as long as ε1 + ε2 A* search on graphs and in this context establish a bound on running time determined by the spectrum of the graph. We then experimentally explore to what extent our rigorous upper bounds predict the behavior of A* in some natural, combinatorially-rich search spaces. We begin by applying A* to solve the knapsack problem with near-accurate admissible heuristics constructed from an efficient approximation algorithm for this problem. We additionally apply our analysis of A* search for the partial Latin square problem, where we can provide quite exact analytic bounds on the number of near-optimal solutions. These results demonstrate a dramatic reduction in effective branching factor of A* when coupled with near-accurate heuristics in search spaces with suitably sparse solution sets.