Data stream processing has recently received increasing attention as a computational paradigm for dealing with massive data sets. Surprisingly, no algorithm with both sublinear space and passes is known for natural graph problems in classical read-only streaming. Motivated by technological factors of modern storage systems, some authors have recently started to investigate the computational power of less restrictive models where writing streams is allowed. In this paper, we show that the use of intermediate temporary streams is powerful enough to provide effective spacepasses tradeoffs for natural graph problems. In particular, for any space restriction of s bits, we show that single-source shortest paths in directed graphs with small positive integer edge weights can be solved in O((n log 3/2 n)/ √ s) passes. The result can be generalized to deal with multiple sources within the same bounds. This is the first known streaming algorithm for shortest paths in directed graphs. For undirected connectivity, we devise an O((n log n)/s) passes algorithm. Both problems require Ω(n/s) passes under the restrictions we consider. We also show that the model where intermediate temporary streams are allowed can be strictly more powerful than classical streaming for some problems, while maintaining all of its hardness for others.
Data stream processing has recently received increasing attention as a computational paradigm for dealing with massive data sets. While major progress has been achieved for several fundamental data sketching and statistics problems, there are many problems that seem to be hard in a streaming setting, including most classical graph problems. Relevant examples are graph connectivity and shortest paths, for which linear lower bounds on the "space x passes" product are known. Some recent papers have shown that several graph problems can be solved with one or few passes, if the working memory is large enough to contain all the vertices of the graph. A natural question is whether we can reduce the space usage at the price of increasing the number of passes. Surprisingly, no algorithm with both sublinear space and passes is known for natural graph problems in classical streaming models. Motivated by technological factors of modern storage systems, some authors have recently started to investigate the computational power of less restrictive streaming models. In a FOCS'04 paper, Aggarwal et al. have shown that the use of intermediate temporary streams, combined with the ability to reorder them at each pass for free, yields enough power to solve in polylogarithmic space and passes a variety of problems, including graph connectivity. They leave however as an open question whether problems such as shortest paths can be solved efficiently in this more powerful model. In this paper, we show that the "streaming with sorting" model by Aggarwal et al. can yield interesting results even without using sorting at all: by just using intermediate temporary streams, we provide the first effective spacepasses tradeoffs for natural graph problems. In particular, for any space restriction of s bits, we show that single-source shortest paths in directed graphs with small positive integer edge weights can be solved in O((n log(3/2) n)/root 7s) passes. This is the first known streaming algorithm for shortest paths in directed graphs. For undirected connectivity, we devise an O((n log n)/s) passes algorithm. Both problems require Omega(n/s) passes under the restrictions we consider. We also show that the model where intermediate temporary streams are allowed can be strictly more powerful than classical streaming for some problems, while maintaining all of its hardness for others
We introduce and investigate a new notion of resilience in graph spanners. Let S be a spanner of a weighted graph G. Roughly speaking, we say that S is resilient if all its pointto-point distances are resilient to edge failures. Namely, whenever any edge in G fails, then as a consequence of this failure all distances do not degrade in S substantially more than in G (i.e., the relative distance increases in S are very close to those in the underlying graph G). In this paper we show that sparse resilient spanners exist, and that they can be computed efficiently.
The vitality of an arc/node of a graph with respect to the maximum flow between two fixed nodes s and t is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper, we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only 2(n − 1) max flow instances and, in st‐planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in O(n) worst‐case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a “contiguous” set of arcs/nodes in time proportional to the size of the set.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.