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86
Finding Optimal Solutions to Rubik's Cube Using Pattern Databases
, 1997
"... We have found the first optimal solutions to random instances of Rubik's Cube. The median optimal solution length appears to be 18 moves. The algorithm used is iterativedeepeningA* (IDA*), with a lowerbound heuristic function based on large memorybased lookup tables, or "pattern databas ..."
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Cited by 159 (7 self)
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We have found the first optimal solutions to random instances of Rubik's Cube. The median optimal solution length appears to be 18 moves. The algorithm used is iterativedeepeningA* (IDA*), with a lowerbound heuristic function based on large memorybased lookup tables, or "pattern databases" (Culberson and Schaeffer 1996). These tables store the exact numberofmoves required to solve various subgoals of the problem, in this case subsets of the individual movable cubies. We characterize the effectiveness of an admissible heuristic function by its expected value, and hypothesize that the overall performance of the program obeys a relation in which the product of the time and space used equals the size of the state space. Thus, the speed of the program increases linearly with the amount of memory available. As computer memories become larger and cheaper, we believe that this approach will become increasingly costeffective.
Planning with Pattern Databases
 PROCEEDINGS OF THE 6TH EUROPEAN CONFERENCE ON PLANNING (ECP01)
, 2001
"... Heuristic search planning eectively #12;nds solutions for large planning problems, but since the estimates are either not admissible or too weak, optimal solutions are found in rare cases only. In contrast, heuristic pattern databases are known to signi#12;cantly improve lowerbound estimates for opt ..."
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Cited by 132 (17 self)
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Heuristic search planning eectively #12;nds solutions for large planning problems, but since the estimates are either not admissible or too weak, optimal solutions are found in rare cases only. In contrast, heuristic pattern databases are known to signi#12;cantly improve lowerbound estimates for optimally solving challenging singleagent problems like the 24Puzzle or Rubik's Cube.
This paper studies the eect of pattern databases in the context of deterministic planning. Given afixed state description based on instantiated predicates, we provide a general abstraction scheme to automatically create admissible domainindependent memorybased heuristics for planning problems, where abstractions are found in factorizing the planning space. We evaluate the impact of pattern database heuristics in A* and hill climbing algorithms for a collection of benchmark domains.
New admissible heuristics for domainindependent planning
 In Proc
, 2005
"... Admissible heuristics are critical for effective domainindependent planning when optimal solutions must be guaranteed. Two useful heuristics are the h m heuristics, which generalize the reachability heuristic underlying the planning graph, and pattern database heuristics. These heuristics, however, ..."
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Cited by 60 (11 self)
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Admissible heuristics are critical for effective domainindependent planning when optimal solutions must be guaranteed. Two useful heuristics are the h m heuristics, which generalize the reachability heuristic underlying the planning graph, and pattern database heuristics. These heuristics, however, have serious limitations: reachability heuristics capture only the cost of critical paths in a relaxed problem, ignoring the cost of other relevant paths, while PDB heuristics, additive or not, cannot accommodate too many variables in patterns, and methods for automatically selecting patterns that produce good estimates are not known. We introduce two refinements of these heuristics: First, the additive h m heuristic which yields an admissible sum of h m heuristics using a partitioning of the set of actions. Second, the constrained PDB heuristic which uses constraints from the original problem to strengthen the lower bounds obtained from abstractions. The new heuristics depend on the way the actions or problem variables are partitioned. We advance methods for automatically deriving additive h m and PDB heuristics from STRIPS encodings. Evaluation shows improvement over existing heuristics in several domains, although, not surprisingly, no heuristic dominates all the others over all domains.
Exhibiting Knowledge in Planning Problems to Minimize State Encoding Length
 In ECP
, 1999
"... In this paper we present a generalpurposed algorithm for transforming a planning problem specified in Strips into a concise state description for single state or symbolic exploration. The process of finding a state description consists of four phases. In the first phase we symbolically analyze the ..."
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Cited by 57 (19 self)
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In this paper we present a generalpurposed algorithm for transforming a planning problem specified in Strips into a concise state description for single state or symbolic exploration. The process of finding a state description consists of four phases. In the first phase we symbolically analyze the domain specification to determine constant and oneway predicates, i.e. predicates that remain unchanged by all operators or toggle in only one direction, respectively. In the second phase we symbolically merge predicates which lead to a drastic reduction of state encoding size, while in the third phase we constrain the domains of the predicates to be considered by enumerating the operators of the planning problem. The fourth phase combines the result of the previous phases.
Heuristic Search
, 2011
"... Heuristic search is used to efficiently solve the singlenode shortest path problem in weighted graphs. In practice, however, one is not only interested in finding a short path, but an optimal path, according to a certain cost notion. We propose an algebraic formalism that captures many cost notions ..."
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Cited by 46 (24 self)
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Heuristic search is used to efficiently solve the singlenode shortest path problem in weighted graphs. In practice, however, one is not only interested in finding a short path, but an optimal path, according to a certain cost notion. We propose an algebraic formalism that captures many cost notions, like typical Quality of Service attributes. We thus generalize A*, the popular heuristic search algorithm, for solving optimalpath problem. The paper provides an answer to a fundamental question for AI search, namely to which general notion of cost, heuristic search algorithms can be applied. We proof correctness of the algorithms and provide experimental results that validate the feasibility of the approach.
Bidirectional Heuristic Search Reconsidered
 Journal of Artificial Intelligence Research
, 1997
"... The assessment of bidirectional heuristic search has been incorrect since it was first published more than a quarter of a century ago. For quite a long time, this search strategy did not achieve the expected results, and there was a major misunderstanding about the reasons behind it. Although there ..."
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Cited by 41 (2 self)
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The assessment of bidirectional heuristic search has been incorrect since it was first published more than a quarter of a century ago. For quite a long time, this search strategy did not achieve the expected results, and there was a major misunderstanding about the reasons behind it. Although there is still widespread belief that bidirectional heuristic search is afflicted by the problem of search frontiers passing each other, we demonstrate that this conjecture is wrong. Based on this finding, we present both a new generic approach to bidirectional heuristic search and a new approach to dynamically improving heuristic values that is feasible in bidirectional search only. These approaches are put into perspective with both the traditional and more recently proposed approaches in order to facilitate a better overall understanding. Empirical results of experiments with our new approaches show that bidirectional heuristic search can be performed very efficiently and also with limited mem...
permission. Combined Task and Motion Planning for Mobile Manipulation
, 2010
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Complexity analysis of admissible heuristic search
, 1998
"... We analyze the asymptotic time complexity of admissible heuristic search algorithms such as A*, IDA*, and depthfirst branchandbound. Previous analyses relied on an abstract analytical model, and characterize the heuristic function in terms of its accuracy, but do not apply to real problems. In ..."
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Cited by 35 (1 self)
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We analyze the asymptotic time complexity of admissible heuristic search algorithms such as A*, IDA*, and depthfirst branchandbound. Previous analyses relied on an abstract analytical model, and characterize the heuristic function in terms of its accuracy, but do not apply to real problems. In contrast, our analysis allows us to accurately predict the performance of these algorithms on problems such as the slidingtile puzzles and l~ubik’s Cube. The heuristic function is characterized simply by the distribution of heuristic values in the problem space. Contrary to conventional wisdom, our analysis shows that the asymptotic heuristic branching factor is the same as the bruteforce branching factor, and that the effect of a heuristic function is to reduce the effective depth of search, rather than the effective branching factor.
State abstraction in realtime heuristic search
 Journal of Artificial Intelligence Research
, 2006
"... Realtime heuristic search methods are used by situated agents in applications that require the amount of planning per move to be constantbounded regardless of the problem size. Such agents plan only a few actions in a local search space and avoid getting trapped in heuristic local minima by improv ..."
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Cited by 33 (10 self)
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Realtime heuristic search methods are used by situated agents in applications that require the amount of planning per move to be constantbounded regardless of the problem size. Such agents plan only a few actions in a local search space and avoid getting trapped in heuristic local minima by improving their heuristic function over time. We extend a wide class of realtime search algorithms with automatically built graph abstraction. Extensive empirical evaluation in the domain of goaldirected navigation demonstrates that the use of abstraction accelerates learning of the heuristic function while maintaining realtime performance. The resulting algorithm outperforms virtually all tested algorithms simultaneously along negatively correlated performance measures.
On the implementation of mips
 In Proceedings of Workshop on DecisionTheoretic Planning, Artificial Intelligence Planning and Scheduling (AIPS
"... Planning is a central topic of AI and provides solutions to problems given in a problem independent formalism. Recent successes in the exploration of model checking and singleagent search problems have led to a generalization of the symbolic exploration method with binary decision diagrams (BDDs). ..."
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Cited by 30 (4 self)
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Planning is a central topic of AI and provides solutions to problems given in a problem independent formalism. Recent successes in the exploration of model checking and singleagent search problems have led to a generalization of the symbolic exploration method with binary decision diagrams (BDDs). In this paper we present the use, architecture, implementation and performance of our STRIPS planner MIPS abbreviating intelligent model checking and planning system. With BDD refinements, symbolic and single state heuristic search engines we highlight recent improvements that have been added to the system.