Local labeling and resource allocation using preprocessing

Abstract

This paper studies the power of non-restricted preprocessing on a communication graph G, in a synchronous, reliable system. In our scenario, arbitrary preprocessing can be performed on G, and afterwards a sequence of labeling problems have to be solved on different subgraphs of G. We suggest a preprocessing that produces an orientation of G. The goal is to exploit this preprocessing to minimize the radius of the neighborhood around each vertex from which data has to be collected in order to determine a label. We define a set of labeling problems for which this can be done. The time complexity of labeling a subgraph depends on the topology of the graph G and is always less than min{χ(G), O((log n)²)}. On the other hand, we show the existence of a graph where even unbounded preprocessing does not allow a fast solution of a simple labeling problem. Specifically, it is shown that a processor needs to know its Ω(log n/log log n)-neighborhood in order to pick a legal label. Finally, we derive some results on the resource allocation problem. In particular, we show that Ω(log nlog log n) communication rounds are needed in order to provide short response time, and we give an efficient distributed algorithm that employs the same preprocessing as the labeling algorithm.

This research was supported by grant No. 92-0233 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, Technion V.P.R.—Argentinian Research Fund and the fund for the promotion of research in the Technion.