This thesis examines two largely unrelated problems in applied algorithmics, motivated by the search for efficient geometric algorithms. In the first part of the thesis, we consider the problem of finding efficient parallel algorithms for heterogeneous parallel computers, i.e., parallel computers in which different processors have different computational potential. To this end, we define a formal computational model for heterogeneous systems and develop algorithms for commonly used communication operations. The result is that many existing parallel algorithms which use these communication operations can be adapted to our model with little or no modifications. In the second part of the thesis we consider the problem of geometric models which allow for varying levels of detail. To this end, we extend the progressive mesh representation introduced by Hoppe. The main technical contribution of this part is an efficient scheme for refining only selected regions of a progressive mesh. Using ...

We present a coarse grained parallel algorithm for computing a maximum matching in a convex bipartite graph G = #A; B; E#.Forp processors with N=p memory per processor, N = jAj + jBj, N=p # p, the algorithm requires p ; log p# local computation, where n = jAj, m = jBj and T sequ #n; m# is the sequential time complexity for the problem. For the BSP model, this implies O#log p# supersteps log p# local computation.

Consider the selection problem of determining the kth smallest element of a set of n elements. Under the CGM (coarse-grained multicomputer) model with p processors and O(n=p) local memory, we present a deterministic parallel algorithm for the selection problem that requires O(log p) communication rounds. Besides requiring a low number of communication rounds, the algorithm also attempts to minimize the total amount of data transmitted in each round (only O(p) except in the last round). In addition to showing theoretical complexities, we present very promising experimental results obtained on a parallel machine that show almost linear speedup, indicating the efficiency and scalability of the proposed algorithm.

In this paper, we present parallel algorithms for the coarse grained multicomputer (CGM) and bulk synchronous parallel computer (BSP) for solving two well known graph problems: (1) determining whether a graph G is bipartite, and (2) determining whether a bipartite graph G is convex. Our algorithms require O(...

This paper presents a randomized parallel algorithm for the Maximal Independent Set problem. Our algorithm uses a BSP-like computer with p processors and requires that n+m p = \Omega\Gamma p) for a graph with n vertices and m edges. Under this scalability assumption, and after a preprocessing phase, it computes a maximal independent set after O(log p) communication rounds, with high probability, each round requiring linear computation time O( n+m p ). The preprocessing phase is deterministic and important in order to ensure that degree computations can be implemented efficiently. For this, we give an optimal parallel BSP/CGM algorithm to the p-quantiles search problem, which runs in O( m log p p ) time and a constant number of communication rounds, and could be of interest in its own right, as shown in the text. This paper has been submitted for presentation to ESA'98. Keywords: Parallel Algorithms, Maximal Independent Set, Randomized Algorithms, Graph Algorithms, ...

External memory (EM) algorithms are designed for computational problems in which the size of the internal memory of the computer is only a small fraction of the problem size. The Parallel Disk Model (PDM) of Vitter and Shriver is widely used to discriminate between external memory algorithms on the basis of input/output (I/O) complexity. Parallel algorithms are designed to efficiently utilize the computing power of multiple processing units, interconnected by a communication mechanism. A popular model for developing and analyzing parallel algorithms is the Bulk Synchronous Parallel (BSP) model due to Valiant. In this work we develop simulation techniques, both randomized and deterministic, which produce efficient EM algorithms from efficient algorithms developed under BSPlike parallel computing models. Our techniques can accommodate one or multiple processors on the EM target machine, each with one or more disks, and they also adapt to the disk blocking factor of the target machine. ...

We consider the problem of finding efficient parallel algorithms for heterogeneous parallel computers, i.e., parallel computers in which different processors have different computational potential. To this end, we define a formal computational model for heterogeneous systems and develop algorithms for commonly used communication operations. The result is that many existing parallel algorithms which use these communication operations can be adapted to our model with little or no modifications. Experimental results are give which show that our algorithms are of considerable practical relevance.

Given an array of n real numbers A = (a1 , a2 , ..., an ), define MIN(i, j) = min{a i , ..., a j }. The range minima problem consists of preprocessing array A such that queries MIN(i, j), for any 1 i j n, can be answered in constant time. Range minima is a basic problem that appears in many other important problems such as lowest common ancestor, Euler tour, pattern matching with scaling, etc. In this work we present a parallel algorithm under the CGM model (Coarse Grained Multicomputer), that solves the range minima problem in O( n p ) time and constant number of communication rounds.

This paper reports on solving instances with more than 10,000 data items in a few hours. Keane, Page, Naughton, Travers and McInerney report on a fully cross platform coarse-grained distributed application for building large phylogenetic trees. Their new system overcomes many of the limitations imposed by the current set of parallel phylogenetic programs and it is now publicly available. The next two papers deal with gene and protein sequence comparison. Alves, Caceres and Song present a new coarse-grained parallel algorithm for the all-substrings longest common subsequence operation which is used, e.g. to find approximate tandem repeats and the alignment of one sequence with several others that have a common subsequence. Driga, Lu, Schaeffer, Szafron, Charter and Parsons discuss parallel and sequential sequence alignment via a method called FastLSA. Experiments indicate that their method scales well and can be parameterized to take advantage of cache memory and main memory sizes

This work was financially supported by the University of Bergen through a