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## Disjoint pattern database heuristics (2002)

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### Other Repositories/Bibliography

Venue: | Artificial Intelligence |

Citations: | 136 - 35 self |

### Citations

13831 |
Computers and Intractability; A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1990
(Show Context)
Citation Context ...yperedges that have no vertices in common, so that the sum of the weights of the edges and hyperedges is maximized. Unfortunately, the corresponding three-dimensional matching problem is NP-complete (=-=Garey & Johnson, 1979-=-), as is higher-order matching. For the tile puzzles, however, if we only include tiles whose pairwise or triple distances exceed the sum of their Manhattan distances, the mutual-cost graph becomes ve... |

1935 |
Reducibility among combinatorial problems
- Karp
- 1972
(Show Context)
Citation Context ...t least one of the vertices in the subset. A minimum or optimal vertex cover is one with the fewest number of vertices possible. Vertex cover was one of the earliest problems shown to be NP-complete (=-=Karp, 1972-=-). There are many approaches to solve this problem. A comprehensive survey is beyond the scope of this paper. We refer the reader to a survey that was presented by Balasubramanian et al. (Balasubraman... |

1660 |
Combinatorial Optimization: Algorithms and Complexity
- Papadimitriou, Steiglitz
- 1982
(Show Context)
Citation Context ...hosen edges are incident to the same vertex, such that the sum of the labels of the chosen edges is maximized. This is called the maximum weighted matching problem, and can be solved in O(n 3 ) time (=-=Papadimitriou & Steiglitz, 1982-=-), where n is the number of vertices, or tiles in this case. 3.3 Triple and Higher-Order Distances This idea can generalized to larger groups of tiles as follows. Let k be the size of the group. A k −... |

1494 |
A formal basis for the heuristic determination of minimum cost paths
- Hart, Raphael
- 1968
(Show Context)
Citation Context ...own heuristics for the problem. 1. Introduction and Overview Heuristic search is a general problem-solving method in artificial intelligence. The most important heuristic search algorithms include A*(=-=Hart, Nilsson, & Raphael, 1968-=-), iterativedeepening-A* (IDA*)(Korf, 1985a), and depth-first branch-and-bound (DFBnB). All of these algorithms make use of a heuristic evaluation function h(n), which takes a state n and efficiently ... |

512 | Depth-first iterative-deepening: An optimal admissible tree search
- Korf
- 1985
(Show Context)
Citation Context ...euristic search is a general problem-solving method in artificial intelligence. The most important heuristic search algorithms include A*(Hart, Nilsson, & Raphael, 1968), iterativedeepening-A* (IDA*)(=-=Korf, 1985-=-a), and depth-first branch-and-bound (DFBnB). All of these algorithms make use of a heuristic evaluation function h(n), which takes a state n and efficiently computes an estimate of the cost of an opt... |

185 | Vertex cover: Further observations and further improvements
- Chen, Kanj, et al.
(Show Context)
Citation Context ... one of the vertices to include, and exclude the other. Many attempts have been made recently to reduce the size of the search tree by pruning irrelevant nodes or by combining several nodes together (=-=Chen, Kanj, & Jia, 2001-=-; Balasubramanian et al., 1998; Downey, Fellows, & Stege, 1999; Niedermeier & Rossmanith, 1999). To date, the best of these methods is the work done by (Chen et al., 2001). However, despite the compre... |

168 |
Macro-operators: A weak method for learning
- Korf
- 1987
(Show Context)
Citation Context ...euristic search is a general problem-solving method in artificial intelligence. The most important heuristic search algorithms include A*(Hart, Nilsson, & Raphael, 1968), iterativedeepening-A* (IDA*)(=-=Korf, 1985-=-a), and depth-first branch-and-bound (DFBnB). All of these algorithms make use of a heuristic evaluation function h(n), which takes a state n and efficiently computes an estimate of the cost of an opt... |

165 |
Pattern databases
- Culberson, Schaeffer
- 1998
(Show Context)
Citation Context ...Computer Science, Bar-Ilan University Ramat-Gan, Israel, 52900 Abstract We explore a method for computing admissible heuristic evaluation functions for search problems. It utilizes pattern databases (=-=Culberson & Schaeffer, 1998-=-), which are precomputed tables of the exact cost of solving various subproblems of an existing problem. Unlike standard pattern database heuristics, however, we partition our problems into disjoint s... |

153 | Finding optimal solutions to Rubik’s Cube using pattern databases
- Korf
- 1997
(Show Context)
Citation Context ...of Paper This paper is organized as follows. In section 2 we present previous work on designing more accurate heuristic functions, including pattern databases in general (Culberson & Schaeffer, 1998; =-=Korf, 1997-=-) and statically-partitioned additive pattern databases in particular (Korf & Felner, 2002). In section 3 we present the idea of dynamically-partitioned additive pattern databases, initially in the co... |

85 | Parameterized complexity: A framework for systematically confronting computational intractability
- Downey, Fellows, et al.
- 1999
(Show Context)
Citation Context ... Many attempts have been made recently to reduce the size of the search tree by pruning irrelevant nodes or by combining several nodes together (Chen, Kanj, & Jia, 2001; Balasubramanian et al., 1998; =-=Downey, Fellows, & Stege, 1999-=-; Niedermeier & Rossmanith, 1999). To date, the best of these methods is the work done by (Chen et al., 2001). However, despite the comprehensive and deep work that they have done, they all use a simp... |

63 | Parameterized computational feasibility
- Downey, Fellows
- 1995
(Show Context)
Citation Context ...ranch excludes the vertex from the vertex cover. The complete tree will have depth n, where n is the number of vertices, and 2 n leaf nodes, each corresponding to a different subset of the vertices. (=-=Downey & Fellows, 1995-=-) classify this method as the bounded search tree method. Of course, much of this tree can be pruned in the course of the search. For example: • If a vertex is excluded from the vertex cover, every ad... |

58 | Finding optimal solutions to the twenty-four puzzle
- Korf, Taylor
- 1996
(Show Context)
Citation Context ...suggest further work and present our conclusions in Section 8. The basic idea of dynamically-partitioned pattern databases was developed independently by Gasser (Gasser, 1995) and by Korf and Taylor (=-=Korf & Taylor, 1996-=-) in the context of the sliding-tile puzzles. It was also briefly mentioned in the earlier work(Korf & Felner, 2002) where statically-partitioned databases were introduced. Much of the material on pre... |

55 |
Edelkamp, Time complexity of Iterative-Deepening-A
- Korf, Reid, et al.
- 2001
(Show Context)
Citation Context ...ive Puzzle is beyond the reach of current techniques and machines. However, we can predict the relative performance of IDA* with different heuristics, using the new theory that was introduced lately (=-=Korf, Reid, & Edelkamp, 2001-=-). All that is needed is the brute-force branching factor of the problem space, and a large random sample of solvable states, along with their heuristic values. A rather surprising result of this theo... |

54 |
Heuristics
- Pearl
- 1983
(Show Context)
Citation Context ...es. 2.1 Heuristics as Optimal Solutions to Relaxed Problems In general, admissible heuristic functions represent the cost of exact solutions to simplified or relaxed versions of the original problem (=-=Pearl, 1984-=-). For example, in a sliding-tile puzzle, to move a tile from position x to position y, x and y must be adjacent, and position y must be empty. By ignoring the empty constraint, we get a simplified pr... |

52 | Divide-and-conquer frontier search applied to optimal sequence alignment
- Korf, Zhang
- 2000
(Show Context)
Citation Context ...of available memory, typically exhausting memory in a matter of minutes on current machines. 5.2.3 Frontier A* Algorithm Frontier-A* (FA*) is a modification of A* designed to save memory (Korf, 1999; =-=Korf & Zhang, 2000-=-). Instead of saving both the Open and Closed lists, frontier-A* saves only the Open list, and deletes nodes from memory once they have been expanded. Its name derives from the fact that the Open list... |

45 | Upper bounds for vertex cover further improved
- Niedermeier, Rossmanith
- 1999
(Show Context)
Citation Context ...cently to reduce the size of the search tree by pruning irrelevant nodes or by combining several nodes together (Chen, Kanj, & Jia, 2001; Balasubramanian et al., 1998; Downey, Fellows, & Stege, 1999; =-=Niedermeier & Rossmanith, 1999-=-). To date, the best of these methods is the work done by (Chen et al., 2001). However, despite the comprehensive and deep work that they have done, they all use a simple g(n) cost function for each i... |

44 | Compressing pattern databases
- Felner, Meshulam, et al.
- 2004
(Show Context)
Citation Context ...ation of the blank and stored the minimum among all possible blank locations in order to save memory. In a sense, we have compressed the pattern databases according to the location of the blank (see (=-=Felner, Meshulam, Holte, & Korf, 2004-=-) about compressing pattern databases). 289sFelner, Korf, & Hanan A 5-5-5 partitioning A 6-6-3 partitioning A 7-8 partitioning Figure 3: Different Statically-partitioned Databases for Fifteen Puzzle H... |

42 | Best-first frontier search with delayed duplicate detection
- Korf
(Show Context)
Citation Context ...ed the minimum among all possible blank locations in order to save memory. In a sense, we have compressed the pattern databases according to the location of the blank (see (Felner, Meshulam, Holte, & =-=Korf, 2004-=-) about compressing pattern databases). 289sFelner, Korf, & Hanan A 5-5-5 partitioning A 6-6-3 partitioning A 7-8 partitioning Figure 3: Different Statically-partitioned Databases for Fifteen Puzzle H... |

33 |
Depth- rst iterative-deepening: an optimal admissible tree search. Arti cial Intelligence 27(1):97{109
- Korf
- 1985
(Show Context)
Citation Context ...Heuristic search is a general problem-solving method in arti cial intelligence. The most important heuristic search algorithms include A*(Hart, Nilsson, & Raphael, 1968), iterativedeepening-A* (IDA*)(=-=Korf, 1985-=-a), and depth- rst branch-and-bound (DFBnB). All of these algorithms make use of a heuristic evaluation function h(n), which takes a state n and e ciently computes an estimate of the cost of an optima... |

32 | An improved fixed-parameter algorithm for vertex cover
- Balasubramanian, Fellows, et al.
- 1998
(Show Context)
Citation Context ... (Karp, 1972). There are many approaches to solve this problem. A comprehensive survey is beyond the scope of this paper. We refer the reader to a survey that was presented by Balasubramanian et al. (=-=Balasubramanian, Fellows, & Raman, 1998-=-). 302s6.2 The Search Space Additive Pattern Database Heuristics To avoid confusing the search space tree with the original input graph, we will use the terms “nodes” and “operators” to refer to parts... |

31 |
Criticizing solutions to relaxed models yields powerful admissible heuristics
- Hansson, Mayer, et al.
- 1992
(Show Context)
Citation Context ...nhattan distance heuristic. The second row is for Manhattan distance enhanced by linear con icts. Historically, the linear-con ict heuristic was the rst signi cant improvement over Manhattan distance =-=[5]-=-. It applies to tiles in their goal row or column, but reversed relative to each other. For example, assume the top row of a given state contains the tiles (2 1) in that order, but in the goal state t... |

28 | The Tower of Hanoi - Hinz - 1999 |

28 |
Divide-and-conquer bidirectional search: First results
- Korf
- 1999
(Show Context)
Citation Context ... the amount of available memory, typically exhausting memory in a matter of minutes on current machines. 5.2.3 Frontier A* Algorithm Frontier-A* (FA*) is a modification of A* designed to save memory (=-=Korf, 1999-=-; Korf & Zhang, 2000). Instead of saving both the Open and Closed lists, frontier-A* saves only the Open list, and deletes nodes from memory once they have been expanded. Its name derives from the fac... |

25 |
Solution to advanced problem 3918
- Stewart
- 1941
(Show Context)
Citation Context ...Figure 6, is more interesting. There exists a deterministic algorithm for finding a solution, and a conjecture that it generates an optimal solution, but the conjecture remains unproven (Frame, 1941; =-=Stewart, 1941-=-; Dunkel, 1941). Thus, systematic search is currently the only method guaranteed to find optimal solutions, or to verify the conjecture for problems with a given number of disks. 5.2 Search Algorithms... |

25 |
Notes on the 15 puzzle
- Johnson
(Show Context)
Citation Context ... 22 23 24 Figure 1: The Fifteen and Twenty-Four Puzzles in their Goal States The Fifteen Puzzle was invented by Sam Loyd in the 1870s [13], and appeared in the scienti c literature shortly thereafter =-=[7]-=-. The editor of the journal added the following comment tothe paper: \The `15' puzzle for the 2slast few weeks has been prominently before the American public, and may safely be said to have engaged t... |

21 |
Solution to advanced problem 3918
- Frame
- 1941
(Show Context)
Citation Context ...and shown in Figure 6, is more interesting. There exists a deterministic algorithm for finding a solution, and a conjecture that it generates an optimal solution, but the conjecture remains unproven (=-=Frame, 1941-=-; Stewart, 1941; Dunkel, 1941). Thus, systematic search is currently the only method guaranteed to find optimal solutions, or to verify the conjecture for problems with a given number of disks. 5.2 Se... |

19 | Recent progress in the design and analysis of admissible heuristic functions
- Korf
- 2000
(Show Context)
Citation Context ...ding optimal solutions to the Fifteen Puzzle 2000 times faster than with Manhattan distance, and nding optimal solutions to 50 random Twenty-Four Puzzles. Initial results of this work rst appeared in =-=[11]-=-. 2 Existing Heuristics 2.1 Manhattan Distance Where did the Manhattan distance heuristic come from? In addition to the standard answer to this question, we present an alternative that suggests the di... |

11 | Institute of Technology - Thesis - 1981 |

9 |
Pruning duplicate nodes in depth- rst search
- Taylor, Korf
(Show Context)
Citation Context ...used in [9] was much more complex. Secondly, [9] used a technique, based on nite-state machines (FSMs), to prune duplicate nodes representing the same state arrived at via di erent paths in the graph =-=[16]-=-. FSM pruning reduced the number of nodes generated by IDA* on ve of the problems by a factor that ranged from 2.4 to 3.6. For this work, we did not use FSM pruning, because the technique is complex, ... |

8 |
Editorial note concerning advanced problem 3918
- Dunkel
- 1941
(Show Context)
Citation Context ...re interesting. There exists a deterministic algorithm for finding a solution, and a conjecture that it generates an optimal solution, but the conjecture remains unproven (Frame, 1941; Stewart, 1941; =-=Dunkel, 1941-=-). Thus, systematic search is currently the only method guaranteed to find optimal solutions, or to verify the conjecture for problems with a given number of disks. 5.2 Search Algorithms Once we elimi... |

8 | ZRAM: a library of parallel search algorithms and its use in enumeration and combinatorial optimization
- Marzetta
- 1998
(Show Context)
Citation Context ...ed, the edge between them will not be covered. A single edge is the special case of a clique of size two. The idea of using cliques as a lower bound on the size of the minimum vertex cover is due to (=-=Marzetta, 1998-=-). 6.3.1 Computing an Admissible Heuristic with Additive Pattern Databases In order to compute an admissible heuristic for the size of the minimum vertex cover of a given graph, we partition the verti... |

7 |
Results and open problems on the Tower of Hanoi
- Bode, Hinz
- 1999
(Show Context)
Citation Context ...ted in solving the standard initial state with all disks on a single peg, however, we can do better by capitalizing on the fact that the initial and goal states are completely symmetric in this case (=-=Bode & Hinz, 1999-=-). Consider moving all n disks from the initial to the goal peg. In order to be able to move the largest disk to the goal peg, all the n − 1 smaller disks must be removed from the initial peg, cannot ... |

7 |
An improved algorithm for vertex cover
- Balasubramanian, Fellows, et al.
- 1998
(Show Context)
Citation Context ... (Karp, 1972). There are many approaches to solve this problem. A comprehensive survey is beyond the scope of this paper. We refer the reader to a survey that was presented by Balasubramanian et al. (=-=Balasubramanian, Fellows, & Raman, 1998-=-). 302 Additive Pattern Database Heuristics 6.2 The Search Space To avoid confusing the search space tree with the original input graph, we will use the terms \nodes" and \operators" to refer to parts... |

5 | The Royal Institute of - Thesis - 2000 |

3 |
An improved xed-parameter algorithm for vertex
- Balasubramanian, Fellows, et al.
- 1998
(Show Context)
Citation Context ... (Karp, 1972). There are many approaches to solve this problem. A comprehensive survey is beyond the scope of this paper. We refer the reader to a survey that was presented by Balasubramanian et al. (=-=Balasubramanian, Fellows, & Raman, 1998-=-). 302s6.2 The Search Space Additive Pattern Database Heuristics To avoid confusing the search space tree with the original input graph, we will use the terms \nodes" and \operators" to refer to parts... |

3 |
Mathematical Puzzles of Sam Loyd, Selected and Edited by
- Loyd
- 1959
(Show Context)
Citation Context ... 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Figure 1: The Fifteen and Twenty-Four Puzzles in their Goal States The Fifteen Puzzle was invented by Sam Loyd in the 1870s =-=[13]-=-, and appeared in the scienti c literature shortly thereafter [7]. The editor of the journal added the following comment tothe paper: \The `15' puzzle for the 2slast few weeks has been prominently bef... |

2 |
Improving search techniques and using them on different environments
- Felner
- 2001
(Show Context)
Citation Context ...eater than the sum of their Manhattan distances. Covering hyperedges with weights larger than two is a little more complicated but the principle is the same and all possibilities are considered. See (=-=Felner, 2001-=-) for more details. The general problem here can be called “weighted vertex cover”. Given a hypergraph with integer-weighted edges, assign an integer value to each vertex, such that for each hyperedge... |

1 |
Harnessing computational resources for efficient exhastive search
- Gasser
- 1995
(Show Context)
Citation Context ... in general in Section 7, and finally suggest further work and present our conclusions in Section 8. The basic idea of dynamically-partitioned pattern databases was developed independently by Gasser (=-=Gasser, 1995-=-) and by Korf and Taylor (Korf & Taylor, 1996) in the context of the sliding-tile puzzles. It was also briefly mentioned in the earlier work(Korf & Felner, 2002) where statically-partitioned databases... |

1 |
Improving search techniques and using them on di erent environments
- Felner
- 2001
(Show Context)
Citation Context ...eater than the sum of their Manhattan distances. Covering hyperedges with weights larger than two is a little more complicated but the principle is the same and all possibilities are considered. See (=-=Felner, 2001-=-) for more details. The general problem here can be called \weighted vertex cover". Given a hypergraph with integer-weighted edges, assign an integer value to each vertex, such that for each hyperedge... |

1 |
Harnessing computational resources for e cient exhastive search
- Gasser
- 1995
(Show Context)
Citation Context ...od in general in Section 7, and nally suggest further work and present our conclusions in Section 8. The basic idea of dynamically-partitioned pattern databases was developed independently by Gasser (=-=Gasser, 1995-=-) and by Korf and Taylor (Korf & Taylor, 1996) in the context of the sliding-tile puzzles. It was also brie y mentioned in the earlier work(Korf & Felner, 2002) where statically-partitioned databases ... |

1 |
Best- rst frontier search with delayed duplicate detection
- Korf
- 2004
(Show Context)
Citation Context ...ed the minimum among all possible blank locations in order to save memory. In a sense, we have compressed the pattern databases according to the location of the blank (see (Felner, Meshulam, Holte, & =-=Korf, 2004-=-) about compressing pattern databases). 289sFelner, Korf, & Hanan A 5-5-5 partitioning A 6-6-3 partitioning A 7-8 partitioning Figure 3: Di erent Statically-partitioned Databases for Fifteen Puzzle He... |

1 |
Harnessing Computational Resources for E cient Exhaustive Search
- Gasser
- 1995
(Show Context)
Citation Context ...drawback of disjoint database heuristics is that they don't capture interactions between tiles in di erent groups of the partition. This requires a di erent approach, developed independently by Gasser=-=[4]-=- and Korf[9]. Consider a table for a sliding-tile puzzle which contains for each pair of tiles, and each possible pair of positions they could occupy, the number of moves required of both tiles to mov... |

1 |
Harnessing computational resources for ecient exhastive search
- Gasser
- 1995
(Show Context)
Citation Context ...od in general in Section 7, andsnally suggest further work and present our conclusions in Section 8. The basic idea of dynamically-partitioned pattern databases was developed independently by Gasser (=-=Gasser, 1995-=-) and by Korf and Taylor (Korf & Taylor, 1996) in the context of the sliding-tile puzzles. It was also brie y mentioned in the earlier work(Korf & Felner, 2002) where statically-partitioned databases ... |