Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla-cian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,

Let A and B be two sets of n objects in R d , and let M be a (one-to-one) matching between A and B. Let min(M ), max(M ), and \Sigma(M ) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of M respectively. Bottleneck matching---a matching that minimizes max(M )---is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is close to O(n 1:5 ). For instance, if the objects are points in the plane, the running time of the exact algorithm is O(n 1:5 log n). A semi-dynamic data-structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest. Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion...

. The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating k-tuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...

Flashsort [RV83,86] and Samplesort [HC83] are related parallel sorting algorithms proposed in the literature. Both utilize a sophisticated randomized sampling technique to form a splitter set, but Samplesort distributes the splitter set to each processor while Flashsort uses splitter-directed routing. In this

This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to number-theoretic properties of the algorithm; lower bounds on Shellsort and Shellsort-based networks; average-case results; proposed probabilistic sorting networks based on the algorithm; and a list of open problems. 1 Shellsort The basic Shellsort algorithm is among the earliest sorting methods to be discovered (by D. L. Shell in 1959 [36]) and is among the easiest to implement, as exhibited by the following C code for sorting an array a[l],..., a[r]: shellsort(itemType a[], int l, int r) { int i, j, h; itemType v;

Abstract We give an optimal deterministic O(n log n)-time algorithm for slope selection. The algorithm borrows from the optimal solution given in [?], but avoids the complicated machinery of the AKS sorting network and parametric searching. This is achieved by redesigning and refining the O(n log2 n)-time algorithm of [?] with the help of additional approximation tools. 1 Optimal Slope Selection The problem is computing the line defined by two of n given points that has the median slope among all \Gamma n2 \Delta such lines. Equivalently, the problem can be stated as that of selecting the medianabscissa vertex of the arrangement A(L) of a set L of n lines [?]. For generality, we set out to compute the vertex with rank I\Lambda from left to right, for any given 1 ^ I \Lambda ^ \Gamma n2 \Delta.

This paper is accepted for publication in ALGORITHMICA Abstract Let A and B be two sets of n objects in Rd, and let Match be a (one-to-one)matching between A and B. Let min(Match), max(Match), and \Sigma (Match) denote thelength of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match respectively. Bottleneck matching--a matching that minimizesmax( Match)--is suggested as a convenient way for measuring the resemblance between A and B. Several algorithms for computing, as well as approximating, this resemblanceare proposed. The running time of all the algorithms involving planar objects is roughly O(n1.5). For instance, if the objects are points in the plane, the running time of the exactalgorithm is O(n1.5 log n). A semi-dynamic data-structure for answering containmentproblems for a set of congruent disks in the plane is developed. This data structure may be of independent interest.Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are point-setsin the plane, an O(n5 log n) time algorithm for determining whether for some translatedcopy the resemblance gets below a given ae is presented, thus improving the previousresult of Alt, Mehlhorn, Wagener and Welzl by a factor of almost n. This result is usedto compute the smallest such ae in time O(n5 log2 n), and an efficient approximationscheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or thefair matching problem) is to find Match*U, a matching that minimizes max(Match)-min ( Match). A minimum deviation matching Match*D is a matching that minimizes(1 /n)\Sigma (Match)- min(Match). Algorithms for computing Match*U and Match*D inroughly O(n10/3) time are presented. These algorithms are more efficient than theprevious

We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spot-checks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the P-NP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...

We present optimal algorithms for sorting on parallel CREW and EREW versions of the pointer machine model. Intuitively, one can view our methods as being based on a parallel mergesort using linked lists rather than arrays (the usual parallel data structure). We also show how to exploit the "locality" of our approach to solve the set expression evaluation problem, a problem with applications to database querying and logic-programming, in O(log n) time using O(n) processors. Interestingly, this is an asymptotic improvement over what seems possible using previous techniques. Categories and Subject Descriptors: E.1 [Data Structures]: arrays, lists; F.2.2. [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems---sorting and searching General Terms: Algorithms, Theory, Verification Additional Key Words and Phrases: parallel algorithms, PRAM, pointer machine, linking automaton, expression evaluation, mergesort, cascade merging 1 Introduction One of the primar...

. Comparator networks of constant depth can be used for sorting in the following way. The computation consists of a number of iterations, say t, each iteration being a single run through the comparator network. The output of a round j (j ! t) is used as the input for the round j + 1. The output of the round t is the output of the computation. In such a way, it is possible to apply a network with a small number of comparators for sorting long input sequences. However, it is not clear how to make such a computation fast. Odd-Even Transposition Sort gives a periodic sorting network of depth 2, that sorts n numbers in n=2 iterations. The network of depth 8 proposed by Schwiegelshohn [8] sorts n numbers in O( p n log n) iterations. Krammer [5] modified the algorithm and obtained a network of depth 6 sorting in O( p n log n) iterations. For a fixed but arbitrary k 2 N , we present a periodic sorting network of depth O(k) that sorts n input numbers in O(k 2 \Delta n 1=k ) steps. 1 In...