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 give a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a p-pass Shellsort for any incremental sequence is \Omega\Gamma pn 1+1=p ) for every p. The proof is an example of the use of Kolmogorov complexity (the incompressibility method) in the analysis of algorithms. 1 Introduction The question of a nontrivial general lower bound (or upper bound) on the average complexity of Shellsort (due to D.L. Shell [14]) has been open for about four decades [5, 13]. We present such a lower bound for p-pass Shellsort for every p. Shellsort sorts a list of n elements in p passes using a sequence of increments h 1 ; : : : ; h p . In the kth pass the main list is divided in h k separate sublists of length dn=h k e, where the ith sublist consists of the elements at positions j, where j mod h k = i \Gamma 1, of the main list (i = 1; : : : ; h k ). Every sublist is sorted using a straightforward insertion sort. The effi...

We prove an \Omega\Gamma/1 2 n= lg lg n) lower bound for the depth of n-input sorting networks based on the shuffle permutation. The best previously known lower bound was the trivial \Omega\Gammaiv n) bound, while the best upper bound is given by Batcher's \Theta(lg 2 n)-depth bitonic sorting network. The proof technique employed in the lower bound argument may be of independent interest. 1 Introduction A variety of different classes of sorting networks has been described in the literature. Of particular interest here are the so-called AKS network [1] discovered by Ajtai, Koml'os and Szemer'edi, and the sorting networks proposed by Batcher [2]. The AKS network is the only known sorting network with O(lg n) depth. However, the topology of the network is highly irregular, and the multiplicative constant hidden by the O-notation is impractically large [1, 11]. On the other hand, the networks proposed by Batcher have a relatively simple interconnection structure and a small constant...

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a p-pass Shellsort for any incremental sequence is \Omega\Gamma pn 1+ 1 p ) for all p log n. Using similar arguments, we analyze the average-case complexity of several other sorting algorithms. 1 Introduction The question of a nontrivial general lower bound (or upper bound) on the average complexity of Shellsort (due to D.L. Shell [15]) has been open for about four decades [7, 14]. We present such a lower bound for p-pass Shellsort for every p. Shellsort sorts a list of n elements in p passes using a sequence of increments h 1 ; : : : ; h p . In the kth pass the main list is divided in h k separate sublists of length n=h k , where the ith sublist consists of the elements at positions j, where j mod h k , of the main list (i = 1; : : : ; h k ). Every sublist is sorted using a straightforward insertion sort. The efficiency of the method is gove...

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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Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard average-case analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, Dobosiewicz-sort, Shakersort, and sorting with stacks and queues in sequential or parallel mode. Especially in the case of Shellsort the uses of Kolmogorov complexity surprisingly easily resolved problems that had stayed open for a long time despite strenuous attacks.

We study sorting algorithms based on randomized roundrobin comparisons. Specifically, we study Spin-the-bottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a temperature parameter. Both algorithms are simple, randomized, data-oblivious sorting algorithms, which are useful in privacy-preserving computations, but, as we show, Annealing sort is much more efficient. We show that there is an input permutation that causes Spin-the-bottle sort to require Ω(n 2 log n) expected time in order to succeed, and that in O(n 2 log n) time this algorithm succeeds with high probability for any input. We also show there is an specification of Annealing sort that runs in O(n log n) time and succeeds with very high probability. 1