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.

We analyze various stochastic processes for generating permutations almost uniformly at random in distributed and parallel systems. All our protocols are simple, elegant and are based on performing disjoint transpositions executed in parallel. The challenging problem of our concern is to prove that the output configurations in our processes reach almost uniform probability distribution very rapidly, i.e. in a (low) polylogarithmic time. For the analysis of the aforementioned protocols we develop a novel technique, called delayed path coupling, for proving rapid mixing of Markov chains. Our approach is an extension of the path coupling method of Bubley and Dyer. We apply delayed path coupling to three stochastic processes for generating random permutations. For one

A central issue in the design of modern communication networks is that of providing pe:formance guarantees. This issue is particularly important if the networks support real-time traffic such as voice and video. The most critical pe:formance parameter to bound is the delay experienced by a packet as it travels fi'om its source to its destination.

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Abstract. In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain infected forever. Percolation occurs if eventually every vertex is infected. The elements of the set of initially infected vertices, A ⊂ V (G), are normally chosen independently at random, each with probability p, say. This process has been extensively studied on the sequence of torus graphs [n] d, for n = 1, 2,..., where d = d(n) is either fixed or a very slowly growing function of n. For example, Cerf and Manzo [14] showed that the critical probability is o(1) if d(n) � log ∗ n, i.e., if p = p(n) is bounded away from zero then the probability of percolation on [n] d tends to one as n → ∞. In this paper we study the case when the growth of d to ∞ is not excessively slow; in particular, we show that the critical probability is 1/2+o(1) if d � (log log n) 2 log log log n, and give much stronger bounds in the case that G is the hypercube, [2] d. 1.

In this paper, we describe a randomized Shellsort algorithm. This algorithm is a simple, randomized, data-oblivious version of the Shellsort algorithm that always runs in O(n log n) time and succeeds in sorting any given input permutation with very high probability. Taken together, these properties imply applications in the design of new efficient privacypreserving computations based on the secure multi-party computation (SMC) paradigm. In addition, by a trivial conversion of this Monte Carlo algorithm to its Las Vegas equivalent, one gets the first version of Shellsort with a running time that is provably O(n log n) with very high probability. 1

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...

sorting algorithm called Shuffle Ring Sort (SRS) for distributed memory systems with a regular bounded degree interconnection network. Our algorithm is optimal in the sense that it achieves a time complexity of O(N log N ) for sorting at least N keys on N processors.

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We consider three scheduling problems that arise in studies of packet routing, load balancing and disk scheduling. A fundamental problem in the design of packet-switched communication networks is to provide effective methods for resolving contention when many packets wish to cross a link. End-to-end packet delays should be low and queue sizes should be small. For an adversarial connectionless model we provide upper and lower bounds on delay for many simple algorithms. For an adversarial session-oriented model we prove the existence of an asymptotically optimal schedule with per-packet delay guarantees of O(distance + 1/session rate) and constant queue size. We also describe randomized schedules with near-optimal bounds. In the on-line load balancing problem, jobs arrive on-line and must be assigned to one of a set of machines, thereby increasing the load on that machine by a certain weight. Jobs also depart on-line. The goal is to minimize both the maximum load on a machine and the a...