We survey routing problems on fixed-connection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, k-relation routing, routing to random destinations, dynamic routing, isotonic routing, fault tolerant routing, and related sorting results. We also provide a list of unsolved problems and numerous references.

. This paper provides an analysis of a natural d-round tournamentover n = 2 d players, and demonstrates that the tournament possesses a surprisingly strong ranking property. The ranking property of this tournament is used to design efficient sorting algorithms for a variety of different models of parallel computation: (i) a comparator network of depth c \Delta lg n, c 7:44, that sorts the vast majority of the n! possible input permutations, (ii) an O(lg n)-depth hypercubic comparator network that sorts the vast majority of permutations, (iii) a hypercubic sorting network with nearly logarithmic depth, (iv) an O(lgn)-time randomized sorting algorithm for any hypercubic machine (other such algorithms have been previously discovered, but this algorithm has a significantly smaller failure probability than any previously known algorithm), and (v) a randomized algorithm for sorting n O(m)-bit records on an (n lg n)-node omega machine in O(m + lg n) bit steps. Key words. parallel sort...

A natural class of "hypercubic" sorting networks is defined. The regular structure of these sorting networks allows for elegant and efficient implementations on any of the so-called hypercubic networks (e.g., the hypercube, shuffle-exchange, butterfly, and cube-connected cycles). This class of sorting networks contains Batcher's O(lg 2 n)-depth bitonic sort, but not the O(lg n)-depth sorting network of Ajtai, Koml'os, and Szemer'edi. In fact, no o(lg 2 n)- depth compare-interchange sort was previously known for any of the hypercubic networks. In this paper, we prove the existence of a family of 2 O( p lg lg n) lg n-depth hypercubic sorting networks. Note that this depth is o(lg 1+ffl n) for any constant ffl ? 0. 1 Introduction A comparator network is an n-input, n-output acyclic circuit made up of wires and 2-input, 2output comparator gates. The input wires of the network are numbered from 0 to n \Gamma 1, as are the output wires. The inputs to the network may be tho...

Hypercubic sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, n-input hypercubic sorting networks with depth 2 O( p lg lg n) lg n have been discovered. These networks are the only known sorting networks of depth o(lg 2 n) that are not based on expanders, and their existence raises the question of whether a depth of O(lg n) can be achieved by any hypercubic sorting network. In this paper, we resolve this question by establishing an\Omega \Gamma lg n lg lg n lg lg lg n \Delta lower bound on the depth of any n-input hypercubic sorting network. Our lower bound can be extended to certain restricted classes of non-oblivious sorting algorithms on hypercubic machines. 1 Introduction A variety of different classes of sorting networks have been described in the literature. Of particular interest here are the so-called AKS network [1] discovered by Ajtai, Koml'os, and Szemer...

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