Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems-computations on matrices; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems-geometrical problems and computations; sort-ing and searching; G. 1.6 [Mathematics of Computing]: Optimization-linear programming

We present new BSP algorithms for deterministic sorting and randomized median finding. We sort n general keys by using a partitioning scheme that achieves the requirements of efficiency (one-optimality) and insensitivity against data skew (the accuracy of the splitting keys depends solely on the step distance, which can be adapted to meet the worstcase requirements of our application). Although we employ sampling in order to realize efficiency, we can give a precise worst-case estimation of the maximum imbalance which might occur. We also investigate optimal randomized BSP algorithms for the problem of finding the median of n elements that require, with high-probability, 3n=(2p) + o(n=p) number of comparisons, for a wide range of values of n and p. Experimental results for the two algorithms are also presented.

We propose a fast and memory efficient algorithm for lexicographically sorting the suffixes of a string, a problem that has important applications in data compression as well as string matching. Our

Improving a long standing result of Schonhage, Paterson and Pippenger we show that the median of a set containing n elements can be found using at most 2:95n comparisons.

This doctoral dissertation presents a range of results concerning efficient algorithms and data structures for string processing, including several schemes contributing to sequential data compression. It comprises both theoretic results and practical implementations. We study the suffix tree data structure, presenting an efficient representation and several generalizations. This includes augmenting the suffix tree to fully support sliding window indexing (including a practical implementation) in linear time. Furthermore, we consider a variant that indexes naturally word-partitioned data, and present a linear-time construction algorithm for a tree that represents only suffixes starting at word boundaries, requiring space linear in the number of words. By applying our sliding window indexing techniques, we achieve an efficient implementation for dictionary-based compression based on the LZ-77 algorithm. Furthermore, considering predictive source

We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas-convex hulls, intersections, searching, proximity, and combinatorial optimizations-are discussed. Seven algorithmic techniques incremental construction, plane-sweep, locus, divide-andconquer, geometric transformation, prune-and-search, and dynamization-are each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.

We study the exact number of symbol comparisons that are required to solve the string matching problem and present a family of efficient algorithms. Unlike previous string matching algorithms, the algorithms in this family do not "forget" results of comparisons, what makes their analysis much simpler. In particular, we give a linear-time algorithm that finds all occurrences of a pattern of length m in a text of length n in n+d 4 log m+2 m (n \Gamma m)e comparisons. The pattern preprocessing takes linear time and makes at most 2m comparisons. This algorithm establishes that, in general, searching for a long pattern is easier than searching for a short one. We also show that any algorithm in the family of the algorithms presented must make at least n + blog mcb n\Gammam m c symbol comparisons, for m = 2 k \Gamma 1 and any integer k 1.

. There has been recent progress in the selection problem, and in median-finding in particular, after a lull of ten years. This paper reviews some ancient and modern results on this problem, and suggests possibilities for future research. 1 Introduction The selection problem, determining the k th largest out of a set of n elements, is a junior partner of the more fundamental sorting problem, but it has still been studied extensively over several decades. Our focus will be on performing selection using a minimal number of comparisons in the worst case. Let V k (n) be the worst-case minimum number of pairwise comparisons required to find the k th largest out of n distinct elements. Of particular interest is finding the median, the dn=2e th largest say. We denote V dn=2e (n) by M (n). The worst-case comparison complexity of sorting is n log 2 n+O(n), and even the coefficient of the linear term has been fairly closely estimated. However for V k (n) we do not yet have an asymptotic...

In this paper we present a new randomized selection algorithm on the Bulk-Synchronous Parallel (BSP) model of computation along with an application of this algorithm to dynamic data structures, namely Parallel Priority Queues (PPQs). We show that our algorithms improve previous results upon both the communication requirements and the amount of parallel slack required to achieve optimal performance. We also establish that optimality to within small multiplicative constant factors can be achieved for a wide range of parallel machines. While these algorithms are fairly simple themselves, descriptions of their performance in terms of the BSP parameters is somewhat involved. The main reward of quantifying these complications is that it allows transportable software to be written for parallel machines that fit the model. We also present experimental results for the selection algorithm that reinforce our claims.

We analyze the worst-case number of comparisons Tn of Hoare’s selection algorithm find when the input is a random permutation, and worst case is measured with respect to the rank k. We give a new short proof that Tn/n tends to a limit distribution, and provide new bounds for the limiting distribution.