Superconcentration on a Pair of Butterflies
نویسنده
چکیده
Suppose we concatenate two directed graphs, each isomorphic to a d dimensional butterfly (but not necessarily identical to each other). Select any set of 2k input and 2k output nodes on the resulting graph. Then there exist node disjoint paths from the input nodes to the output nodes. If we take two standard butterflies and permute the order of the layers, then the result holds on sets of any size, not just powers of two. This paper will examine some problems in node-disjoint circuit switching. The motivating problem can be described as follows. Suppose we have a directed graph with N input and N output nodes, both labelled from 1 to N . For each input node v, we choose an output node π(v) to be its destination, for some permutation π. The problem is to find a collection of N node-disjoint paths which each run from v to π(v) for all v. A directed graph that can route all permutations π is called rearrangeable. (For some real-world applications of node-disjoint routing, see, for example, [5].) A classic example of rearrangeability is the Beneš network (see [3]). This network (i.e. directed graph) consists of a “forward” butterfly adjoined to a “reversed” butterfly. A natural question to ask is: if we attach two “forward” butterflies, is this network (the double butterfly) still rearrangeable? This problem has been open for several decades. At least one proof is currently under review [1]. This suggests a more general hypothesis. Suppose that we have two graphs, each isomorphic to a butterfly, but not necessarily identical to each other. If we attach the output nodes of the first to the input nodes of the second, is the resulting graph rearrangeable?
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عنوان ژورنال:
- CoRR
دوره abs/1401.7263 شماره
صفحات -
تاریخ انتشار 2014