We consider the well-known vertex coloring problem: given a graph G, find a coloring of the vertices so that no two neighbors in G have the same color. It is trivial to see that every graph of maximum degree ∆ can be colored with ∆+1 colors, and distributed algorithms that find a (∆+1)-coloring in a logarithmic number of communication rounds, with high probability, are known since more than a decade. This is in general the best possible if only a constant number of bits can be sent along every edge in each round. In fact, we show that for the n-node cycle the bit complexity of the coloring problem is Ω(log n). More precisely, if only one bit can be sent along each edge in a round, then every distributed coloring algorithm (i.e., algorithms in which every node has the same initial state and initially only knows its own edges) needs at least Ω(log n) rounds, with high probability, to color the cycle, for any finite number of colors. But what if the edges have orientations, i.e., the endpoints of an edge agree on its orientation (while bits may still flow in both directions)? Does this allow one to provide faster coloring algorithms?Interestingly, for the cycle in which all edges have the same orientation, we show that a simple randomized algorithm can achieve a 3-coloring with only O( √ log n) rounds of bit transmissions, with high probability (w.h.p.). Large Scale Information Systems (DELIS).sult is tight because we also show that the bit complexity of coloring an oriented cycle is Ω( √ log n), with high probability, no matter how many colors are allowed. The 3-coloring algorithm can be easily extended to provide a (∆ + 1)-coloring for all graphs of maximum degree ∆ in O( √ log n) rounds of bit transmissions, w.h.p., if ∆ is a constant, the edges are oriented, and the graph does not contain an oriented cycle of length less than √ log n. Using more complex algorithms, we show how to obtain an O(∆)-coloring for arbitrary oriented graphs of maximum degree ∆ using essentially O(log ∆ + √ log n) rounds of bit transmissions, w.h.p., provided that the graph does not contain an oriented cycle of length less than √ log n.
Abstract-Designing an overlay network for publish/subscribe communication in a system where nodes may subscribe to many different topics of interest is of fundamental importance. For scalability and efficiency, it is important to keep the degree of the nodes in the publish/subscribe system low. It is only natural then to formalize the following problem: Given a collection of nodes and their topic subscriptions, connect the nodes into a graph that has least possible maximum degree in such a way that for each topic , the graph induced by the nodes interested in is connected. We present the first polynomial-time logarithmic approximation algorithm for this problem and prove an almost tight lower bound on the approximation ratio. Our experimental results show that our algorithm drastically improves the maximum degree of publish/subscribe overlay systems. We also propose a variation of the problem by enforcing that each topic-connected overlay network be of constant diameter while keeping the average degree low. We present three heuristics for this problem that guarantee that each topicconnected overlay network will be of diameter 2 and that aim at keeping the overall average node degree low. Our experimental results validate our algorithms, showing that our algorithms are able to achieve very low diameter without increasing the average degree by much.
We consider the problem of designing a distributed algorithm that, given an arbitrary connected graph G of nodes with unique labels, converts G into a sorted list of nodes. This algorithm should be as simple as possible and, for scalability, should guarantee a polylogarithmic runtime as well as at most a polylogarithmic increase in the degree of each node during its execution. Furthermore, it should be selfstabilizing, that is, it should be able to eventually construct a sorted list from any state in which the graph is connected. It turns out that satisfying all of these demands at the same time is not easy. Our basic approach towards this goal is the so-called linearization technique: each node v repeatedly does the following with its neighbors:
An important problem for wireless ad hoc networks has been to design overlay networks that allow time-and energy-efficient routing. Many local-control strategies for maintaining such overlay networks have already been suggested, but most of them are based on an oversimplified wireless communication model.In this paper, we suggest a model that is much more general than previous models. It allows the path loss of transmissions to significantly deviate from the idealistic unit disk model and does not even require the path loss to form a metric. Also, our model is apparently the first proposed for algorithm design that does not only model transmission and interference issues but also aims at providing a realistic model for physical carrier sensing. Physical carrier sensing is needed so that our protocols do not require any prior information (not even an estimate on the number of nodes) about the wireless network to run efficiently.Based on this model, we propose a local-control protocol for establishing a constant density spanner among a set of mobile stations (or nodes) that are distributed in an arbitrary way in a 2-dimensional Euclidean space. More precisely, we establish a backbone structure by efficiently electing cluster leaders and gateway nodes so that there is only a constant number of cluster leaders and gateway nodes within the transmission range of any node and the backbone structure satisfies the properties of a topological spanner.Our protocol has the advantage that it is locally self-stabilizing, i.e., it can recover from any initial configuration, even if adversarial nodes participate in it, as long as the honest nodes sufficiently far away from adversarial nodes can in principle form a single connected component. Furthermore, we only need constant size messages and a constant amount of storage at the nodes, irrespective of the distribution of the nodes. Hence, our protocols would even work in extreme situations such as very simple wireless devices (like sensors) in a hostile environment.
Abstract-Designing an overlay network for publish/subscribe communication in a system where nodes may subscribe to many different topics of interest is of fundamental importance. For scalability and efficiency, it is important to keep the degree of the nodes in the publish/subscribe system low. It is only natural then to formalize the following problem: Given a collection of nodes and their topic subscriptions, connect the nodes into a graph that has least possible maximum degree in such a way that for each topic , the graph induced by the nodes interested in is connected. We present the first polynomial-time logarithmic approximation algorithm for this problem and prove an almost tight lower bound on the approximation ratio. Our experimental results show that our algorithm drastically improves the maximum degree of publish/subscribe overlay systems. We also propose a variation of the problem by enforcing that each topic-connected overlay network be of constant diameter while keeping the average degree low. We present three heuristics for this problem that guarantee that each topicconnected overlay network will be of diameter 2 and that aim at keeping the overall average node degree low. Our experimental results validate our algorithms, showing that our algorithms are able to achieve very low diameter without increasing the average degree by much.
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.