Analysing local algorithms in location-aware quasi-unit-disk graphs
Discrete Applied Mathematics
ACM Computing Surveys (CSUR)
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This paper studies the power of nonrestricted preprocessing on a communication graph G, in a synchronous, reliable system. In our scenario, arbitrary preprocessing can be performed on G, after which a sequence of labeling problems has 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 for minimizing 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 $\min\{\chi(G), O((\log n)^{2})\}$. On the other hand, we show the existence of a graph for which even unbounded preprocessing does not allow fast solution of a simple labeling problem. Specifically, it is shown that a processor needs to know its $\Omega(\log n / \log \log n)$-neighborhood in order to pick a label.Finally, we derive some results for the resource allocation problem. In particular, we show that $\Omega(\log n / \log \log n)$ communication rounds are needed if resources are to be fully utilized. In this context, we define the compact coloring problem, for which the orientation preprocessing provides fast distributed labeling algorithm. This algorithm suggests efficient solution for the resource allocation problem.