Showing papers by "Rolf Fagerberg published in 2007"

Optimal resilient dynamic dictionaries

Abstract: We investigate the problem of computing in the presence of faults that may arbitrarily (i.e., adversarially) corrupt memory locations. In the faulty memory model, any memory cell can get corrupted at any time, and corrupted cells cannot be distinguished from uncorrupted ones. An upper bound δ on the number of corruptions and O(1) reliable memory cells are provided. In this model, we focus on the design of resilient dictionaries, i.e., dictionaries which are able to operate correctly (at least) on the set of uncorrupted keys.We first present a simple resilient dynamic search tree, based on random sampling, with O(log n+δ) expected amortized cost per operation, and O(n) space complexity. We then propose an optimal deterministic static dictionary supporting searches in Θ(log n+δ) time in the worst case, and we show how to use it in a dynamic setting in order to support updates in O(log n + δ) amortized time. Our dynamic dictionary also supports range queries in O(log n+δ+t) worst case time, where t is the size of the output. Finally, we show that every resilient search tree (with some reasonable properties) must take Ω(log n + δ) worst-case time per search.

Optimal sparse matrix dense vector multiplication in the I/O-model

Abstract: We analyze the problem of sparse-matrix dense-vector multiplication (SpMV) in the I/O-model. The task of SpMV is to compute y := Ax, where A is a sparse N x N matrix and x and y are vectors. Here, sparsity is expressed by the parameter k that states that A has a total of at most kN nonzeros, i.e., an average number of k nonzeros per column. The extreme choices for parameter k are well studied special cases, namely for k=1 permuting and for k=N dense matrix-vector multiplication.We study the worst-case complexity of this computational task, i.e., what is the best possible upper bound on the number of I/Os depending on k and N only. We determine this complexity up to a constant factor for large ranges of the parameters. By our arguments, we find that most matrices with kN nonzeros require this number of I/Os, even if the program may depend on the structure of the matrix. The model of computation for the lower bound is a combination of the I/O-models of Aggarwal and Vitter, and of Hong and Kung.We study two variants of the problem, depending on the memory layout of A.If A is stored in column major layout, SpMV has I/O complexity Θ(min{kNB(1+logM/BNmax{M,k}), kN}) for k ≤ N1-e and any constant 1> e > 0. If the algorithm can choose the memory layout, the I/O complexity of SpMV is Θ(min{kNB(1+logM/BNkM), kN]) for k ≤ 3√N.In the cache oblivious setting with tall cache assumption M ≥ B1+e, the I/O complexity is Ο(kNB(1+logM/BNk)) for A in column major layout.

Computing the quartet distance between evolutionary trees of bounded degree

Abstract: We present an algorithm for calculating the quartet distance between two evolutionary trees of bounded degree on a common set of n species. The previous best algorithm has running time O(d2n2) when considering trees, where no node is of more than degree d. The algorithm developed herein has running time O(d9n logn)) which makes it the first algorithm for computing the quartet distance between non-binary trees which has a sub-quadratic worst case running time.

Improved Approximate String Matching and Regular Expression Matching on Ziv-Lempel Compressed Texts

Abstract: We study the approximate string matching and regular expression matching problem for the case when the text to be searched is compressed with the Ziv-Lempel adaptive dictionary compression schemes. We present a time-space trade-off that leads to algorithms improving the previously known complexities for both problems. In particular, we significantly improve the space bounds. In practical applications the space is likely to be a bottleneck and therefore this is of crucial importance.

Optimal Resilient Dynamic Dictionaries

Abstract: . In the resilient memory model any memory cell can get cor- rupted at any time, and corrupted cells cannot be distinguished from uncorrupted cells. An upper bound, , on the number of corruptions and O(1) reliable memory cells are provided. In this model, a data structure is denoted resilient if it gives the correct output on the set of uncor- rupted elements. We propose two optimal resilient static dictionaries, a randomized one and a deterministic one. The randomized dictionary supports searches in O(log n + ) expected time using O(log ) random bits in the worst case, under the assumption that corruptions are not performed by an adaptive adversary. The deterministic static dictionary supports searches in O(log n + ) time in the worst case. We also in- troduce a deterministic dynamic resilient dictionary supporting searches in O(log n + ) time in the worst case, which is optimal, and updates in O(log n + ) amortized time. Our dynamic dictionary supports range queries in O(log n + + k) worst case time, where k is the size of the output.

Computing the All-Pairs Quartet Distance on a Set of Evolutionary Trees

Abstract: We present two algorithms for calculating the quartet distance between all pairs of trees in a set of binary evolutionary trees on a common set of species. The algorithms exploit common substructure among the trees to speed up the pairwise distance calculations thus performing significantly better on large sets of trees compared to performing distinct pairwise distance calculations, as we illustrate experimentally, where we see a speedup factor of around 130 in the best case.