Showing papers by "Rolf Fagerberg published in 2001"
08 Jul 2001
TL;DR: An algorithm which constructs an evolutionary tree of n species in time O(nd logd n) using at most n⌈d/2⌉(log2ċd/ 2ċ 1 n+O(1)) experiments for d = 2, and improves the previous best upper bound by a factor Θ(log d).
Abstract: We present tight upper and lower bounds for the problem of constructing evolutionary trees in the experiment model We describe an algorithm which constructs an evolutionary tree of n species in time O(nd logd n) using at most n⌈d/2⌉(log2ċd/2ċ 1 n+O(1)) experiments for d > 2, and at most n(log n+O(1)) experiments for d = 2, where d is the degree of the tree This improves the previous best upper bound by a factor Θ(log d) For d = 2 the previously best algorithm with running time O(n log n) had a bound of 4n log n on the number of experiments By an explicit adversary argument, we show an Ω(nd logd n) lower bound, matching our upper bounds and improving the previous best lower bound by a factor Θ(logd n) Central to our algorithm is the construction and maintenance of separator trees of small height, which may be of independent interest
34 citations
19 Dec 2001
TL;DR: An algorithm for computing the quartet distance between two unrooted evolutionary trees of n species in time O(n log2 n) is presented, which is faster than the previous best algorithm.
Abstract: Evolutionary trees describing the relationship for a set of species are central in evolutionary biology, and quantifying differences between evolutionary trees is an important task. One previously proposed measure for this is the quartet distance. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topology is the topological subtree induced by four species. In this paper, we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species in time O(n log2 n). The previous best algorithm runs in time O(n 2).
20 citations
TL;DR: Bender, Demaine and Farach-Colton as discussed by the authors proposed a cache-oblivious search tree, which avoids the use of weight balanced B-trees and can be implemented as a single array of data elements.
Abstract: We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and Farach-Colton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced B-trees, and can be implemented as just a single array of data elements, without the use of pointers. The structure also improves space utilization. For storing n elements, our proposal uses (1+epsilon)n times the element size of memory, and performs searches in worst case O(log_B n) memory transfers, updates in amortized O((log^2 n)/(epsilon B)) memory transfers, and range queries in worst case O(log_B n + k/B) memory transfers, where k is the size of the output. The basic idea of our data structure is to maintain a dynamic binary tree of height log n + O(1) using existing methods, embed this tree in a static binary tree, which in turn is embedded in an array in a cache oblivious fashion, using the van Emde Boas layout of Prokop. We also investigate the practicality of cache obliviousness in the area of search trees, by providing an empirical comparison of different methods for laying out a search tree in memory. The source code of the programs, our scripts and tools, and the data we present here are available online under ftp.brics.dk/RS/01/36/Experiments/.
9 citations
TL;DR: An algorithm which constructs an evolutionary tree of n species in time O(n d logd n) using at most n d/2 experiments for d > 2, and at least n(log n + O(1), where d is the degree of the tree, improves the previous best upper bound by a factor Theta(log d).
Abstract: We present tight upper and lower bounds for the problem of constructing evolutionary trees in the experiment model. We describe an algorithm which constructs an evolutionary tree of n species in time O(n d logd n) using at most n |d/2| (log2|d/2|−1 n + O(1)) experiments for d > 2, and at most n(log n + O(1)) experiments for d = 2, where d is the degree of the tree. This improves the previous best upper bound by a factor Theta(log d). For d = 2 the previously best algorithm with running time O(n log n) had a bound of 4n log n on the number of experiments. By an explicit adversary argument, we show an Omega(nd logd n) lower bound, matching our upper bounds and improving the previous best lower bound by a factor Theta(logd n). Central to our algorithm is the construction and maintenance of separator trees of small height. We present how to maintain separator trees with height log n + O(1) under the insertion of new nodes in amortized time O(log n). Part of our dynamic algorithm is an algorithm for computing a centroid tree in optimal time O(n). Keywords: Evolutionary trees, Experiment model, Separator trees, Centroid tree, Lower bounds
4 citations
08 Aug 2001
TL;DR: This is the first binary search tree with relaxed balance having a height bound better than c ċ log2 n for a fixed constant c, and a standard k-tree with amortized constant rebalancing per update is defined, which is an improvement over the original definition.
Abstract: We introduce the relaxed k-tree, a search tree with relaxed balance and a height bound, when in balance, of (1 + Ɛ) log2 n + 1, for any Ɛ > 0. The rebalancing work is amortized O(1/Ɛ) per update. This is the first binary search tree with relaxed balance having a height bound better than c ċ log2 n for a fixed constant c. In all previous proposals, the constant is at least 1/ log2 Φ > 1.44, where Φ is the golden ratio.
As a consequence, we can also define a standard (non-relaxed) k-tree with amortized constant rebalancing per update, which is an improvement over the original definition.
Search engines based on main-memory databases with strongly fluctuating workloads are possible applications for this line of work.
2 citations