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24
Randomized PursuitEvasion in a Polygonal Environment
, 2004
"... This paper contains two main results: First, we revisit the wellknown visibility based pursuitevasion problem and show that, in contrast to deterministic strategies, a single pursuer can locate an unpredictable evader in any simplyconnected polygonal environment using a randomized strategy. The ..."
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Cited by 88 (12 self)
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This paper contains two main results: First, we revisit the wellknown visibility based pursuitevasion problem and show that, in contrast to deterministic strategies, a single pursuer can locate an unpredictable evader in any simplyconnected polygonal environment using a randomized strategy. The evader can be arbitrarily faster than the pursuer and it may know the position of the pursuer at all times but it does not have prior knowledge of the random decisions made by the pursuer. Second, using the randomized algorithm together with the solution to a problem called the "lion and man problem" [2] as subroutines, we present a strategy for two pursuers (one of which is at least as fast as the evader) to quickly capture an evader in a simplyconnected polygonal environment. We show how this strategy can be extended to obtain a strategy for (i) a polygonal room with a door, (ii) two pursuers who have only lineofsight communication, and (iii) a single pursuer (at the expense of increased capture time).
Playing games with algorithms: Algorithmic combinatorial game theory
 IN: PROC. 26TH SYMP. ON MATH FOUND. IN COMP. SCI., LECT. NOTES IN COMP. SCI., SPRINGERVERLAG
, 2001
"... Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, ..."
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Cited by 61 (11 self)
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Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, which analyzes ideal play in perfectinformation games. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomialtime algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.
Locating and Capturing an Evader in a Polygonal Environment
, 2003
"... This paper contains two main results: First, we revisit the wellknown visibility based pursuitevasion problem and show that, in contrast to deterministic strategies, a single pursuer can locate an unpredictable evader in any simplyconnected polygonal environment using a randomized strategy. The e ..."
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Cited by 23 (4 self)
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This paper contains two main results: First, we revisit the wellknown visibility based pursuitevasion problem and show that, in contrast to deterministic strategies, a single pursuer can locate an unpredictable evader in any simplyconnected polygonal environment using a randomized strategy. The evader can be arbitrarily faster than the pursuer and it may know the position of the pursuer at all times but it does not have prior knowledge of the random decisions made by the pursuer. Second, using the randomized algorithm together with the solution of a known lion and man problem [1] as subroutines, we present a strategy for two pursuers (one of which is at least as fast as the evader) to quickly capture an evader in a simplyconnected polygonal environment. We show how this strategy can be extended to obtain a strategy for (i) capturing the evader in a polygonal room with a door, (ii) two pursuers who have only lineofsight communication, and (iii) a single pursuer (at the expense of increased capture time).
On Fraenkel's Nheap Wythoff’s conjecture
"... The Nheap Wythoff’s game is a twoplayer impartial game with N piles of tokens of sizes A1,..., AN, A1 ≤ · · · ≤ AN. Players take turns removing any number of tokens from a single pile, or removing (a1,..., aN) from all piles — ai tokens from the ith pile, providing that 0 ≤ ai ≤ Ai, �N i=1 ..."
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Cited by 11 (1 self)
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The Nheap Wythoff’s game is a twoplayer impartial game with N piles of tokens of sizes A1,..., AN, A1 ≤ · · · ≤ AN. Players take turns removing any number of tokens from a single pile, or removing (a1,..., aN) from all piles — ai tokens from the ith pile, providing that 0 ≤ ai ≤ Ai, �N i=1 ai> 0 and a1 ⊕ · · · ⊕ aN = 0, where ⊕ is the nim addition. The first player that cannot make a move loses. Denote all the Ppositions (i.e., losing positions) by (A 1,..., A N−2, A N−1 n
OneDimensional Peg Solitaire, and Duotaire
, 2002
"... We solve the problem of onedimensional Peg Solitaire. In particular, we show that the set of configurations that can be reduced to a single peg forms a regular language, and that a lineartime algorithm exists for reducing any configuration to the minimum number of pegs. We then look ..."
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Cited by 8 (0 self)
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We solve the problem of onedimensional Peg Solitaire. In particular, we show that the set of configurations that can be reduced to a single peg forms a regular language, and that a lineartime algorithm exists for reducing any configuration to the minimum number of pegs. We then look
Bearingonly Pursuit
"... We study a variant of a wellknown pursuit evasion game, the lion and man game. In this game a lion (the pursuer) tries to capture a man (the evader). The players move in turns. At each time step, they can move a unit distance. We focus on a version which takes place in an unbounded arena: the posit ..."
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Cited by 7 (1 self)
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We study a variant of a wellknown pursuit evasion game, the lion and man game. In this game a lion (the pursuer) tries to capture a man (the evader). The players move in turns. At each time step, they can move a unit distance. We focus on a version which takes place in an unbounded arena: the positive quadrant of the plane. The novelty of our formulation is in the sensor model. In the original formulation, the lion can sense the precise location of the man at all times. In our version, which is inspired by mobile robots equipped with monocular vision systems, the lion can only obtain bearing information about the man’s location. We present a pursuit strategy which guarantees that the distance between the players is reduced to the step size in a bounded number of steps.
Advances in losing
"... ABSTRACT. We survey recent developments in the theory of impartial combinatorial games in misere play, focusing on how Sprague–Grundy theory of normalplay impartial games generalizes to misere play via the indistinguishability quotient construction [P2]. This paper is based on a lecture given on 21 ..."
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Cited by 6 (2 self)
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ABSTRACT. We survey recent developments in the theory of impartial combinatorial games in misere play, focusing on how Sprague–Grundy theory of normalplay impartial games generalizes to misere play via the indistinguishability quotient construction [P2]. This paper is based on a lecture given on 21 June 2005 at the Combinatorial Game Theory Workshop at the Banff International Research Station. It has been extended to include a survey of results on misere games, a list of open problems involving them, and a summary of MisereSolver [AS2005], the excellent Javalanguage program for misere indistinguishability quotient construction recently developed by Aaron Siegel. Many wild misere games that have long appeared intractable may now lie within the grasp of assiduous losers and their faithful computer assistants, particularly those researchers and computers equipped with MisereSolver. 1.
Complete information pursuit evasion in polygonal environments
 In Proc. of 25th Conference on Artificial Intelligence (AAAI
"... Suppose an unpredictable evader is free to move around in a polygonal environment of arbitrary complexity that is under full camera surveillance. How many pursuers, each with the same maximum speed as the evader, are necessary and sufficient to guarantee a successful capture of the evader? The pursu ..."
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Cited by 4 (2 self)
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Suppose an unpredictable evader is free to move around in a polygonal environment of arbitrary complexity that is under full camera surveillance. How many pursuers, each with the same maximum speed as the evader, are necessary and sufficient to guarantee a successful capture of the evader? The pursuers always know the evader’s current position through the camera network, but need to physically reach the evader to capture it. We allow the evader the knowledge of the current positions of all the pursuers as well—this accords with the standard worstcase analysis model, but also models a practical situation where the evader has “hacked ” into the surveillance system. Our main result is to prove that three pursuers are always sufficient and sometimes necessary to capture the evader. The bound is independent of the number of vertices or holes in the polygonal environment.
Pretending in misere combinatorial games
"... Abstract. We survey results old and new in misère (ie, lastplayerlosing) impartial combinatorial game theory (CGT). Using pretending techniques originally described by Dean Allemang [A1], we obtain complete misère analyses of the octal games.123,.351, and.512. We also solve many nontame (ie, wild ..."
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Cited by 3 (0 self)
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Abstract. We survey results old and new in misère (ie, lastplayerlosing) impartial combinatorial game theory (CGT). Using pretending techniques originally described by Dean Allemang [A1], we obtain complete misère analyses of the octal games.123,.351, and.512. We also solve many nontame (ie, wild [WWI], pgs 405–412) subtractionlike quaternary games—these are octal games without heapsplitting moves, or (equivalently) the octal games specifiable using the code digits [GS] {0,1,2,3} only. We correct minor errors in a published solution [A3] for.53, but uncover complications with a mistaken solution for.54 that we haven’t been able to resolve. We close with summaries of current knowledge about Dawson’s Chess (.137), Guiles (.15), and other open problems. In the appendix, we provide alternative, “pretendingcentric”
The game of EndNim
 Electronic Journal of Combinatorics
, 2001
"... In the game of EndNim two players take turns in removing one or more boxes from a string of nonempty stacks. At each move boxes may only be taken from the two stacks which form the ends of the string (unless only one stack remains!). We give a solution for both impartial and partizan versions of t ..."
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Cited by 3 (0 self)
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In the game of EndNim two players take turns in removing one or more boxes from a string of nonempty stacks. At each move boxes may only be taken from the two stacks which form the ends of the string (unless only one stack remains!). We give a solution for both impartial and partizan versions of the game and explain the significance of the mystic hieroglyphs: