Deciding the Winner of an Arbitrary Finite Poset Game is PSPACE-Complete

A poset game is a two-player game played over a partially ordered set (poset) in which the players alternate choosing an element of the poset, removing it and all elements greater than it. The first player unable to select an element of the poset loses. Polynomial time algorithms exist for certain restricted classes of poset games, such as the game of Nim. However, until recently the complexity of arbitrary finite poset games was only known to exist somewhere between NC^1 and PSPACE. We resolve this discrepancy by showing that deciding the winner of an arbitrary finite poset game is PSPACE-complete. To this end, we give an explicit reduction from Node Kayles, a PSPACE-complete game in which players vie to chose an independent set in a graph

Theory of annihilation games—I

AbstractPlace tokens on distinct vertices of an arbitrary finite digraph with n vertices which may contain cycles or loops. Each of two players alternately selects a token and moves it from its present position u to a neighboring vertex v along a directed edge which may be a loop. If v is occupied, and u ≠ v, both tokens get annihilated and phase out of the game. The player first unable to move is the loser, the other the winner. If there is no last move, the outcome is declared a draw. An O(n6) algorithm for computing the previous-player-winning, next-player-winning and draw positions of the game is given. Furthermore, an algorithm is given for computing a best strategy in O(n6) steps and winning—starting from a next-player-winning position—in O(n5) moves

Geography

AbstractGeneralized Geography is an impartial two-person game played on a digraph G=(V,A). In impartial Arc (Vertex) Geography, a token is initially placed on a special start vertex, and the players alternately move the token along unused arcs (vertices) of G. The player first unable to move loses and his opponent wins. The question of who wins these games IAG and IVG is known to be PSPACE-complete.Both impartial versions with two tokens on special start vertices are proved PSPACE-complete even for DAGs but polynomial for directed trees. The partizan variations, PAG and PVG, with one token per player are PSPACE-complete even for bipartite degree-restricted digraphs. They are NP-hard for DAGs, but polynomial for directed trees

A d-Step Approach for Distinct Squares in Strings

We present an approach to the problem of maximum number of distinct squares in a string which underlines the importance of considering as key variables both the length n and n − d where d is the size of the alphabet. We conjecture that a string of length n and containing d distinct symbols has no more than n − d distinct squares, show the critical role played by strings satisfying n = 2d, and present some properties satisfied by strings of length bounded by a constant times the size of the alphabet

Fast Algorithm for Partial Covers in Words

A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$. In this article we introduce a new notion of $\alpha$-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least $\alpha$ positions in $w$. We develop a data structure of $O(n)$ size (where $n=|w|$) that can be constructed in $O(n\log n)$ time which we apply to compute all shortest $\alpha$-partial covers for a given $\alpha$. We also employ it for an $O(n\log n)$-time algorithm computing a shortest $\alpha$-partial cover for each $\alpha=1,2,\ldots,n$

Conway games, algebraically and coalgebraically

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on non-losing strategies. Hypergames are a fruitful metaphor for non-terminating processes, Conway's sum being similar to shuffling. We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we generalize Conway's results on game determinacy and characterization of strategies. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized Grundy-Sprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. We indicate a number of intriguing directions for future work. We briefly compare hypergames with other notions of games used in computer science.Comment: 30 page

A Minimal Periods Algorithm with Applications

Kosaraju in ``Computation of squares in a string'' briefly described a linear-time algorithm for computing the minimal squares starting at each position in a word. Using the same construction of suffix trees, we generalize his result and describe in detail how to compute in O(k|w|)-time the minimal k-th power, with period of length larger than s, starting at each position in a word w for arbitrary exponent $k\geq2$ and integer $s\geq0$. We provide the complete proof of correctness of the algorithm, which is somehow not completely clear in Kosaraju's original paper. The algorithm can be used as a sub-routine to detect certain types of pseudo-patterns in words, which is our original intention to study the generalization.Comment: 14 page

Predicting the deleterious effects of mutation load in fragmented populations.

Human-induced habitat fragmentation constitutes a major threat to biodiversity. Both genetic and demographic factors combine to drive small and isolated populations into extinction vortices. Nevertheless, the deleterious effects of inbreeding and drift load may depend on population structure, migration patterns, and mating systems and are difficult to predict in the absence of crossing experiments. We performed stochastic individual-based simulations aimed at predicting the effects of deleterious mutations on population fitness (offspring viability and median time to extinction) under a variety of settings (landscape configurations, migration models, and mating systems) on the basis of easy-to-collect demographic and genetic information. Pooling all simulations, a large part (70%) of variance in offspring viability was explained by a combination of genetic structure (F(ST)) and within-deme heterozygosity (H(S)). A similar part of variance in median time to extinction was explained by a combination of local population size (N) and heterozygosity (H(S)). In both cases the predictive power increased above 80% when information on mating systems was available. These results provide robust predictive models to evaluate the viability prospects of fragmented populations

FibLSS: A scalable label storage scheme for dynamic XML updates

Dynamic labeling schemes for XML updates have been the focus of significant research activity in recent years. However the label storage schemes underpinning the dynamic labeling schemes have not received as much attention. Label storage schemes specify how labels are physically encoded and stored on disk. The size of the labels and their logical representation directly influence the computational costs of processing the labels and can limit the functionality provided by the dynamic labeling scheme to an XML update service. This has significant practical implications when merging XML repositories such as clinical studies. In this paper, we provide an overview of the existing label storage schemes. We present a novel label storage scheme based on the Fibonacci sequence that can completely avoid relabeling existing nodes under dynamic insertions. Theoretical analysis and experimental results confirm the scalability and performance of the Fibonacci label storage scheme in comparison to existing approaches