Plane Gossip: Approximating rumor spread in planar graphs.
Lecture Notes in Computer Science(2018)
摘要
We study the design of schedules for multi-commodity multicast. In this problem, we are given an undirected graph G and a collection of source-destination pairs, and the goal is to schedule a minimum-length sequence of matchings that connects every source with its respective destination. The primary communication constraint of the multi-commodity multicast model is the number of connections that a given node can make, not link bandwidth. Multi-commodity multicast and its special cases, (single-commodity) broadcast and multicast, are all NP-complete. Multi-commodity multicast is closely related to the problem of finding a subgraph of optimal poise, where the poise is defined as the sum of the maximum degree and the maximum distance between any source-destination pair. We show that for any instance of the multicast problem, the minimum poise subgraph can be approximated to within a factor of O(log k) with respect to the value of a natural LP relaxation in a graph with k source-destination pairs. This is the first upper bound on the integrality gap of the natural LP; all previous algorithms yielded approximations with respect to the integer optimum. Using this integrality gap upper bound and shortest-path separators in planar graphs, we obtain our main result: an O(log(3) klog n/log log n)-approximation for multicommodity multicast for planar graphs which improves on the 2 (O) over tilde ((root log n)) - approximation for general graphs. We also study the minimum-time radio gossip problem in planar graphs where a message from each node must be transmitted to all other nodes under a model where nodes can broadcast to all neighbors and only nodes with a single broadcasting neighbor get a non-interfered message. In earlier work Iglesias et al. (FSTTCS 2015), we showed a strong Omega(n(1/2 - epsilon))-hardness of approximation for computing a minimum gossip schedule in general graphs. Using our techniques for the telephone model, we give an O(log(2) n)-approximation for radio gossip in planar graphs breaking this barrier. Moreover, this is the first bound for radio gossip given that doesn't rely on the maximum degree of the graph.
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