D. Dor and U. Zwick [1999], SOKOBAN and other motion planning problems, Comput. Geom. 13, 215-228.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... L k 2D NP hard [Wil88] unit k 2D NP hard [DO92] PushPush3D 1 3D NP hard [OT99] PushPush 1 2D NP hard [DDO00] k 2D open [DO92] 2D NP hard [OT99] Push # k 2D NP hard [Hof00] Sokoban 12 2 2D PSPACE compl. [DZ99] 15 Puzzle 2D NP hard [RW90] Rush Hour 1 2,3 1 2D PSPACE compl. FB02] 2D PSPACE compl. HS01] Table 2: Time complexity of some sliding block puzzles. Our implementation handles di#erent possible goal positions by imposing a move limit and trying all possible ....

Dorit Dor and Uri Zwick. SOKOBAN and other motion planning problems. CGTA: Computational Geometry: Theory and Applications, 13(4):215--228, oct 1999.

....form the given molecule Obviously, this problem can be formalized as a state space search problem, which recently was undertaken by H u ner et al. 9] There di erent heuristic search methods were presented. Atomix falls into the category of sliding block puzzles as, e.g. PushPush [3] Sokoban [2, 4], or 15Puzzle [12] where time and space complexity was, and still is, subject of intense research. Though seemingly trivial, most variations are at least NP hard, and contained in PSPACE; some are even PSPACE complete we refer the reader to, e.g. Balc azar et al. 1] for further details on ....

....an obstacle 8. The Dimension of the puzzle: Is it 2D or 3D 1. 2. 3. 4. 5. 6. 7. 8. 9. Game Robot Pull Blocks Fixed # Path Slide Dim. Complexity PushPush3D unit 1 3D NP hard [11] PushPush unit 1 2D NP hard [3] Push unit k 2D NP hard [7] Sokoban 1 2 2 2D PSPACE compl. [4] Sokoban unit 1 2D PSPACE compl. 2] 15 Puzzle unit 1 2D NP hard [12] RushHour 1 f2; 3g 1 2D PSPACE compl. 5] Atomix unit 1 2D PSPACE compl. Bricks variable 1 2D PSPACE compl. 8] 1. 2. 3. 4. 5. 6. 7. 8. 9. Game Robot Pull Blocks Fixed # Path Slide Dim. Complexity PushPush3D ....

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D. Dor and U. Zwick. Sokoban and other motion planning problems. Computational Geometry: Theory and Applications, 13(4):215-228, 1999.

....Blocks is NP Complete for Noncrossing Solution Paths Erik D. Demaine Michael Ho mann y Abstract We prove NP hardness of a wide class of pushing block puzzles like the classic Sokoban, generalizing several previous results [4, 5, 7, 8, 13, 15]. The puzzles consist of unit square blocks on an integer lattice; all blocks are movable. The robot may move horizontally and vertically in order to reach a speci ed goal position. In the Push k puzzle, the robot can push up to k blocks in front of it as long as there is at least one free square ....

....As robots become more powerful at manipulation, an understanding of such models becomes increasingly important. Currentday applications include automated warehouse control and warehouse navigation; see, e.g. 9] A representative abstraction of such applications is the popular Sokoban puzzle [3, 8], which is known to be PSPACEcomplete [3] In this paper we study several variations of simpler puzzles, and show all of these models are NP hard using a reductions from 3 coloring of planar graphs [10] Some variations are additionally known to be NP complete, others PSPACE complete, while the ....

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Dor, D., and Zwick, U. Sokoban and other motion planning problems. Computational Geometry: Theory and Applications 13, 4 (1999), 215-228.

....Blocks is NP Complete for Noncrossing Solution Paths Erik D. Demaine Michael Ho mann y Abstract We prove NP hardness of a wide class of pushing block puzzles like the classic Sokoban, generalizing several previous results [3, 4, 6, 7, 12, 14]. The puzzles consist of unit square blocks on an integer lattice; all blocks are movable. The robot may move horizontally and vertically in order to reach a speci ed goal position. In the Push k puzzle, the robot can push up to k blocks in front of it as long as there is at least one free square ....

....As robots become more powerful at manipulation, an understanding of such models becomes increasingly important. Current day applications include automated warehouse control and warehouse navigation; see, e.g. 8] A representative abstraction of such applications is the popular Sokoban puzzle [2, 7], which is known to be PSPACE complete [2] In this paper we study several variations of simpler puzzles, and show all of these models are NP hard using a reductions from 3 coloring of planar graphs [9] Some variations are additionally known to be NP complete, others PSPACE complete, while the ....

[Article contains additional citation context not shown here]

Dor, D., and Zwick, U. SOKOBAN and other motion planning problems. Computational Geometry: Theory and Applications 13, 4 (1999), 215-228.

....as measured in the size of the search tree needed to solve a problem instance. This paper presents a study on solving challenging single agent search problems for the domain of Sokoban. Sokoban is a one player game and is of general interest as an instance of a robot motion planning problem [ Dor and Zwick, 1995 ] Sokoban is analogous to the problem of having a robot in a warehouse move specified goods from their current location to their final destination, subject to the topology of the warehouse and any obstacles in the way. Sokoban has been shown to be NP hard [ Culberson, 1997; Dor and Zwick, 1995 ....

....[ Dor and Zwick, 1995 ] Sokoban is analogous to the problem of having a robot in a warehouse move specified goods from their current location to their final destination, subject to the topology of the warehouse and any obstacles in the way. Sokoban has been shown to be NP hard [ Culberson, 1997; Dor and Zwick, 1995 ] Previously we reported on our attempts to solve Sokoban problems using the standard single agent search techniques available in the literature [ Junghanns and Schaeffer, 1998c ] When these proved inadequate, solving only 10 of a 90 problem test suite, new algorithms had to be developed to ....

D. Dor and U. Zwick. SOKOBAN and other motion planning problems, 1995. At: http://www.math.tau.ac.il/~ddorit.

....to be searched, or heuristic lower bound functions use speci c domain knowledge. This paper presents a study on solving challenging single agent search problems for the domain of Sokoban. Sokoban is a one player puzzle and is of general interest as an instance of robot motion planning problems [4]. Sokoban is analogous to the problem of having a robot in a warehouse move speci ed goods from their current location to their nal destination, subject to the topology of the warehouse and any obstacles in the way. Sokoban has been shown to be NP hard and PSPACE complete [2, 4] Previously, we ....

....planning problems [4] Sokoban is analogous to the problem of having a robot in a warehouse move speci ed goods from their current location to their nal destination, subject to the topology of the warehouse and any obstacles in the way. Sokoban has been shown to be NP hard and PSPACE complete [2, 4]. Previously, we reported on our attempts to solve Sokoban problems using the standard single agent search techniques available in the literature [10] When these proved inadequate, solving only 10 problems of a 90 problem test suite, new algorithms had to be developed to improve search eciency ....

D. Dor and U. Zwick. SOKOBAN and other motion planning problems, 1995. http://www.math.tau.ac.il/~ddorit. 27

....deadlock is critical to prevent futile searching. For sliding tile puzzles, there are algorithms for generating non optimal solutions. In Sokoban, because of the presence of deadlock, often it is very difficult to find any solution. 2. 2 Related Work Sokoban has been shown to be PSPACE complete [1, 3]. Dor and Zwick show that the game is an instance of a motion planning problem, and compare it to other motion planning problems in the literature [3] For example, Sokoban is similar to Wilfong s work with movable obstacles, where the man is allowed to hold on to the obstacle and move with it, as ....

....of the presence of deadlock, often it is very difficult to find any solution. 2. 2 Related Work Sokoban has been shown to be PSPACE complete [1, 3] Dor and Zwick show that the game is an instance of a motion planning problem, and compare it to other motion planning problems in the literature [3]. For example, Sokoban is similar to Wilfong s work with movable obstacles, where the man is allowed to hold on to the obstacle and move with it, as if they were one object [16] Sokoban can be compared to the problem of having a robot in a warehouse move a number of specified goods from their ....

D. Dor and U. Zwick. SOKOBAN and other motion planning problems, 1995. At: http://www.math.tau.ac.il/~ddorit.

....PSPACE completeness in an unfinished manuscript [BOS94] More firm are the results on Sokoban, a computer game that restricts the pushing robot to only push one block at a time, and requires the storing of (some or all) blocks into designated storage locations. This game was proved NP hard in [DZ95], and PSPACE complete by Culberson [Cul98] Here we emphasize another theme: finding a nontrivial version of the game that is not intractable. To date only the most uninteresting versions are known to be solvable in polynomial time, for example, where the robot s path must be monotonic [DO92] To ....

.... pushed, or do they slide the maximal amount of their free range If our goal is to find the weakest robot and most unconstrained puzzle conditions that still lead to intractability, it is reasonable to consider robots who can only push (1) and to restrict all blocks to be unit squares (2) as in [DO92, DZ95, Cul98], for permitting robots to pull, and permitting blocks of other shapes, makes it relatively easy to construct intractable puzzles. It also makes sense to explore the goal of simply finding a path (5) as in [Wil91, DO92] rather than the more challenging task of storing the blocks as in Sokoban ....

[Article contains additional citation context not shown here]

D. Dor and U. Zwick. SOKOBAN and other motion planning problems. Comput. Geom. Theory Appl., 13(4):215--228, 1999.

....PSPACE completeness in an unfinished manuscript [BOS94] More firm are the results on Sokoban, a computer game that restricts the pushing robot to only push one block at a time, and requires the storing of (some or all) blocks into designated storage locations. This game was proved NP hard in [DZ95], and PSPACE complete by Culberson [Cul98] Here we emphasize another theme: finding a nontrivial version of the game that is not intractable. To date only the most uninteresting versions are known to be solvable in polynomial time, for example, where the robot s path must be monotonic [DO92] To ....

.... pushed, or do they slide the maximal amount of their free range If our goal is to find the weakest robot and most unconstrained puzzle conditions that still lead to intractability, it is reasonable to consider robots who can only push (1) and to restrict all blocks to be unit squares (2) as in [DO92, DZ95, Cul98], for permitting robots to pull, and permitting blocks of other shapes, makes it relatively easy to construct intractable puzzles. It also makes sense to explore the goal of simply finding a path (5) as in [Wil91, DO92] rather than the more challenging task of storing the blocks as in Sokoban ....

[Article contains additional citation context not shown here]

D. Dor and U. Zwick. SOKOBAN and other motion planning problems. Comput. Geom. Theory Appl., 13(4):215--228, 1999.

.... nd these results to be surprising considering the simplicity of the game and because our proofs do not rely on any form of special purpose hardware such as explicit switching devices or barriers which are needed for similar proofs done on games that are super cially more complex (e.g. Sokoban [8, 7], and Trains [4, 24] Keywords games, PSPACE completeness, reversible logic, motion planning, Boolean logic. Contents 1 Introduction . 1 1.1 Original Game . 2 1.2 Generalized Rush Hour . 2 2 Rush Hour Logic . ....

Dorit Dor and Uri Zwick. SOKOBAN and other motion planning problems, 1995.

....techniques, Sokoban o ers a signi cant challenge to researchers, since many of the core problems of arti cial intelligence need to be addressed to build a program that rivals the best human performance in solving Sokoban problems. 3. 3 Related Work Unbounded Sokoban has been shown to be NP hard [DZ95] and P SPACE complete [Cul97] Dor and Zwick [DZ95] show that Sokoban is an instance of a motion planning problem, and compare the game to other motion planning problems in the literature. For example, Sokoban is similar to Wilfong s work with movable obstacles, where the man is allowed to hold ....

....to researchers, since many of the core problems of arti cial intelligence need to be addressed to build a program that rivals the best human performance in solving Sokoban problems. 3. 3 Related Work Unbounded Sokoban has been shown to be NP hard [DZ95] and P SPACE complete [Cul97] Dor and Zwick [DZ95] show that Sokoban is an instance of a motion planning problem, and compare the game to other motion planning problems in the literature. For example, Sokoban is similar to Wilfong s work with movable obstacles, where the man is allowed to hold on to the obstacle and move with it, as if they were ....

D. Dor and U. Zwick. SOKOBAN and other motion planning problems, 1995. http://www.math.tau.ac.il/~ddorit.

....program, as measured in size of the search tree needed to solve a problem instance. This paper presents a study on solving challenging single agent search problems for the domain of Sokoban. Sokoban is a one player game and is of general interest as an instance of a robot motion planning problem [ Dor and Zwick, 1995 ] Sokoban is analogous to the problem of having a robot in a warehouse move specified goods from their current location to their final destination, subject to the topology of the warehouse and any obstacles in the way. Sokoban has been shown to be NP hard [ Culberson, 1997; Dor and Zwick, 1995 ....

....[ Dor and Zwick, 1995 ] Sokoban is analogous to the problem of having a robot in a warehouse move specified goods from their current location to their final destination, subject to the topology of the warehouse and any obstacles in the way. Sokoban has been shown to be NP hard [ Culberson, 1997; Dor and Zwick, 1995 ] Previously we reported on our attempts to solve Sokoban problems using the standard single agent search techniques available in the literature [ Junghanns and Schaeffer, 1998c ] When these proved inadequate, solving only 10 of a 90 problem test suite, new algorithms had to be developed to ....

D. Dor and U. Zwick. SOKOBAN and other motion planning problems, 1995. At: http://www.math.tau.ac.il/~ddorit.

....file is maintained showing who has solved which problems and how efficient their solution is (also at http: xsokoban.lcs.mit.edu xsokoban.html) Thus solving a problem is only part of the satisfaction; improving on your solution is equally important. Sokoban has been shown to be PSPACE complete [2, 3]. Dor and Zwick show that the game is an instance of a motion planning problem, and compare the game to other motion planning problems in the literature [3] For example, Sokoban is similar to Wilfong s work with movable obstacles, where the man is allowed to hold on to the obstacle and move with ....

....a problem is only part of the satisfaction; improving on your solution is equally important. Sokoban has been shown to be PSPACE complete [2, 3] Dor and Zwick show that the game is an instance of a motion planning problem, and compare the game to other motion planning problems in the literature [3]. For example, Sokoban is similar to Wilfong s work with movable obstacles, where the man is allowed to hold on to the obstacle and move with it, as if they were one object [12] Sokoban can be compared to the problem of having a robot in a warehouse move a number of specified goods from their ....

D. Dor and U. Zwick. SOKOBAN and other motion planning problems, 1995. At: http://www.math.tau.ac.il/~ddorit.

....accepts, then the pusher will make Theta(n t(n) moves and pushes, where n is the number of symbols on the input tape, and t(n) is the number of transitions made by the TM during its computation. This construction shows that the puzzles are PSPACE complete, solving the open problem stated in [1]. 1 Introduction Sokoban is a puzzle game that can be found at various sites on the Internet [2, 3, 5] and through commercial vendors. If sources are correct, Sokoban is Japanese for warehouse person. We will refer to this person as the pusher . The game consists of the pusher who must push a ....

....emulate a Turing Machine (TM) in linear time using an infinite version of the puzzle in which only a finite number of containers are initially out of storage. Restricting the tape to finite length (linear bounded automata) shows that finite puzzles are PSPACE hard. Since the problem is in PSPACE [1], this means the puzzles are PSPACE complete, solving the open problem posed by Dorit and Zwick[1] We refer the reader to this paper for a summary of related research. Unlike the proof of PSPACE completeness of SOKOBAN presented in [1] our constructions rely critically on the the fact that ....

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Dorit Dor and Uri Zwick. Sokoban and other motion planning problems (extended abstract). Preprint http://www.math.tau.ac.il/ ddorit/, 1995.

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D. Dor and U. Zwick [1999], SOKOBAN and other motion planning problems, Comput. Geom. 13, 215-228.

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