Hop-Constrained Oblivious Routing
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We prove the existence of an oblivious routing scheme that is poly(log n)-competitive in terms of (congestion + dilation), thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize (congestion + dilation), defined as follows: The dilation is the maximum path hop-length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have (congestion + dilation) within a poly(log n) factor of the best possible value. More precisely, for any integer hop-bound h, this oblivious routing scheme selects paths of length at most h ·poly(log n) and is poly(log n)-competitive in terms of congestion in comparison to the best possible congestion achievable via paths of length at most h hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of Räcke [FOCS 2002, STOC 2008], which are O(log n)-competitive in terms of congestion, but are not competitive in terms of dilation.
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