Min/Max-Poly Weighting Schemes and the NL versus UL Problem
TL;DR: This article proposes an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme and generalizes the techniques of Immerman, Szelepcsényi, and Reinhardt and Allender to give the desired UL algorithm.
Abstract: For a graph G(V, E) (vVv = n) and a vertex s ∈ V, a weighting scheme (W : E m Z+) is called a min-unique (resp max-unique) weighting scheme if, for any vertex v of the graph G, there is a unique path of minimum (resp maximum) weight from s to v, where weight of a path is the sum of the weights assigned to the edges Instead, if the number of paths of minimum (resp maximum) weight is bounded by nc for some constant c, then the weighting scheme is called a min-poly (resp max-poly) weighting schemeIn this article, we propose an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme This improves the result in Reinhardt and Allender [2000], in which a UL algorithm was given for the case when the weighting scheme is min-uniqueOur main technique involves triple inductive counting and generalizes the techniques of Immerman [1988], Szelepcsenyi [1988], and Reinhardt and Allender [2000], combined with a hashing technique due to Fredman et al [1984] (also used in Garvin et al [2014]) We combine this with a complementary unambiguous verification method to give the desired UL algorithmAt the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes To achieve this, we first reduce reachability in layered DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-unique and max-poly properties of the graph Using our techniques, we generalize the double inductive counting method in Limaye et al [2009], in which the UL algorithm was given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting schemeAn important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for layered DAGs
References
More filters
Book•
[...]
14 Aug 2008
TL;DR: This book provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal, and is suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science.
Abstract: Graph theory is a flourishing discipline containing a body of beautiful and powerful theorems of wide applicability. Its explosive growth in recent years is mainly due to its role as an essential structure underpinning modern applied mathematics computer science, combinatorial optimization, and operations research in particular but also to its increasing application in the more applied sciences. The versatility of graphs makes them indispensable tools in the design and analysis of communication networks, for instance. The primary aim of this book is to present a coherent introduction to the subject, suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science. It provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal. Commonly used proof techniques are described and illustrated, and a wealth of exercises - of varying levels of difficulty - are provided to help the reader master the techniques and reinforce their grasp of the material. A second objective is to serve as an introduction to research in graph theory. To this end, sections on more advanced topics are included, and a number of interesting and challenging open problems are highlighted and discussed in some detail. Despite this more advanced material, the book has been organized in such a way that an introductory course on graph theory can be based on the first few sections of selected chapters. Visit the graph theory book blog at: http://blogs.springer.com/bondyandmurty/.
3,296 citations
20 Apr 2009
TL;DR: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory and can be used as a reference for self-study for anyone interested in complexity.
Abstract: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.
2,965 citations
TL;DR: A data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time and is easy to implement.
Abstract: A data structure for representing a set of n items from a umverse of m items, which uses space n + o(n) and accommodates membership queries m constant time is described. Both the data structure and the query algorithm are easy to ~mplement.
943 citations
TL;DR: It immediately follows that the context-sensitive languages are closed under complementation, thus settling a question raised by Kuroda.
Abstract: In this paper we show that nondeterministic space $s(n)$ is closed under complementation for $s(n)$ greater than or equal to $\log n$. It immediately follows that the context-sensitive languages are closed under complementation, thus settling a question raised by Kuroda [Inform. and Control, 7 (1964), pp. 207–233].
732 citations
"Min/Max-Poly Weighting Schemes and ..." refers background in this paper
...It is known (Immerman [1988]; Szelepcsényi [1988]) that NL = co-NL....
[...]