Minimum Maximum Degree Publish-Subscribe Overlay Network Design

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 which has least possible maximum degree and in such a way that for each topic t, the graph induced by the nodes interested in t 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 a heuristic for this problem which guarantees that each topic-connected overlay network will be of diameter 2 and which aims at keeping the overall average node degree low. Our experimental results validate our algorithm showing that our algorithm is able to achieve very low diameter without increasing the average degree by much. I

Linearization: Locally self-stabilizing sorting in graphs

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: • for its left (i.e., smaller) neighbors u1,..., uk in the order of decreasing labels, v replaces {v, u1},..., {v, uk} by {v, u1}, {u1, u2},..., {uk−1, uk}, and • for its right (i.e., larger) neighbors w1,..., wℓ in the order of increasing labels, v replaces {v, w1},..., {v, wℓ} by {v, w1}, {w1, w2},..., {wℓ−1, wℓ}. As shown in this paper, this technique transforms any connected graph into a sorted list, but there are graphs for which this can take a long time. Hence, we propose several extensions of the linearization technique and experimentally evaluate their performance. Our results indicate that some of these have a polylogarithmic performance, so there is hope that there are distributed algorithms that can achieve all of our goals above.