Norbert Zeh

An external memory data structure for shortest path queries

We present results related to satisfying shortest path queries on a planar graph stored in external memory. Let N denote the number of vertices in the graph and sort(N) denote the number of input/output (I/O) operations required to sort an array of length N: (1) We describe a blocking for rooted trees to support bottom-up traversals of these trees in O(K/B) I/Os, where K is the length of the traversed path. The space required to store the tree is O(N/B) blocks, where N is the number of vertices of the tree and B is the block size. (2) We give an algorithm for computing a 2/3-separator of size O(√N) for a given embedded planar graph. Our algorithm takes O(sort(N)) I/Os, provided that a breadth-first spanning tree is given. (3) We give an algorithm for triangulating embedded planar graphs in O(sort(N)) I/Os. We use these results to construct a data structure for answering shortest path queries on planar graphs. The data structure uses O(N3/2/B) blocks of external memory and allows for a shortest path query to be answered in O((√N + K)/DB) I/Os, where K is the number of vertices on the reported path and D is the number of parallel disks.

A unifying view on approximation and FPT of agreement forests

We provide a unifying view on the structure of maximum (acyclic) agreement forests of rooted and unrooted phylogenies. This enables us to obtain linear- or O(n log n)-time 3-approximation and improved fixed-parameter algorithms for the subtree prune and regraft distance between two rooted phylogenies, the tree bisection and reconnection distance between two unrooted phylogenies, and the hybridization number of two rooted phylogenies.

FIXED-PARAMETER ALGORITHMS FOR MAXIMUM AGREEMENT FORESTS ∗

We present new and improved fixed-parameter algorithms for computing maximum agreement forests of pairs of rooted binary phylogenetic trees. The size of such a forest for two trees corresponds to their subtree prune-and-regraft distance and, if the agreement forest is acyclic, to their hybridization number. These distance measures are essential tools for understanding reticulate evolution. Our algorithm for computing maximum acyclic agreement forests is the first depth-bounded search algorithm for this problem. Our algorithms substantially outperform the best previous algorithms for these problems.

Supertrees Based on the Subtree Prune-and-Regraft Distance.

Supertree methods reconcile a set of phylogenetic trees into a single structure that is often interpreted as a branching history of species. A key challenge is combining conflicting evolutionary histories that are due to artifacts of phylogenetic reconstruction and phenomena such as lateral gene transfer (LGT). Many supertree approaches use optimality criteria that do not reflect underlying processes, have known biases, and may be unduly influenced by LGT. We present the first method to construct supertrees by using the subtree prune-and-regraft (SPR) distance as an optimality criterion. Although calculating the rooted SPR distance between a pair of trees is NP-hard, our new maximum agreement forest-based methods can reconcile trees with hundreds of taxa and > 50 transfers in fractions of a second, which enables repeated calculations during the course of an iterative search. Our approach can accommodate trees in which uncertain relationships have been collapsed to multifurcating nodes. Using a series of benchmark datasets simulated under plausible rates of LGT, we show that SPR supertrees are more similar to correct species histories than supertrees based on parsimony or Robinson–Foulds distance criteria. We successfully constructed an SPR supertree from a phylogenomic dataset of 40,631 gene trees that covered 244 genomes representing several major bacterial phyla. Our SPR-based approach also allowed direct inference of highways of gene transfer between bacterial classes and genera. A Small number of these highways connect genera in different phyla and can highlight specific genes implicated in long-distance LGT. [Lateral gene transfer; matrix representation with parsimony; phylogenomics; prokaryotic phylogeny; Robinson–Foulds; subtree prune-and-regraft; supertrees.]

Approximating geometric bottleneck shortest paths

In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: Given two points p and q of S and a real number L, compute (or approximate) a shortest path in the subgraph of the complete graph on S consisting of all edges whose length is less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on minimum spanning trees, spanners, the Delaunay triangulation, and planar separators.

Fast FPT algorithms for computing rooted agreement forests: theory and experiments

We improve on earlier FPT algorithms for computing a rooted maximum agreement forest (MAF) or a maximum acyclic agreement forest (MAAF) of a pair of phylogenetic trees. Their sizes give the subtree-prune-and-regraft (SPR) distance and the hybridization number of the trees, respectively. We introduce new branching rules that reduce the running time of the algorithms from O(3n) and O(3n log n) to O(2.42n) and O(2.42n log n), respectively. In practice, the speed up may be much more than predicted by the worst-case analysis. We confirm this intuition experimentally by computing MAFs for simulated trees and trees inferred from protein sequence data. We show that our algorithm is orders of magnitude faster and can handle much larger trees and SPR distances than the best previous methods, treeSAT and sprdist.

Cache-Oblivious Data Structures and Algorithms for Undirected Breadth-First Search and Shortest Paths

We present improved cache-oblivious data structures and algorithms for breadth-first search and the single-source shortest path problem on undirected graphs with non-negative edge weights. Our results removes the performance gap between the currently best cache-aware algorithms for these problems and their cache-oblivious counterparts. Our shortest-path algorithm relies on a new data structure, called bucket heap, which is the first cache-oblivious priority queue to efficiently support a weak DecreaseKey operation.

External Memory Algorithms for Outerplanar Graphs

We present external memory algorithms for outerplanarity testing, embedding outerplanar graphs, breadth-first search (BFS) and depth-first search (DFS) in outerplanar graphs, and finding a 2/3-separator of size 2 for a given outerplanar graph. Our algorithms take O(sort(N)) I/Os and can easily be improved to take O(perm(N)) I/Os, as all these problems have linear time solutions in internal memory. For BFS, DFS, and outerplanar embedding we show matching lower bounds.

Parallel Computation of Skyline Queries

Skyline queries have received considerable attention in the database community. The goal is to retrieve all records in a database that have the property that no other record is better according to all of a given set of criteria. While this problem has been well studied in the computational geometry literature, the solution of this problem in the database context requires techniques designed particularly to handle large amounts of data. In this paper, we show that parallel computing is an effective method to speed up the answering of skyline queries on large data sets. We also propose to preprocess the set of data points to quickly answer subsequent skyline queries on any subset of the dimensions.

I/O-efficient topological sorting of planar DAGs

We present algorithms that solve a number of fundamental problems on planar directed graphs (planar digraphs) in O((N)) I/Os, where (N) is the number of I/Os needed to sort N elements. The problems we consider are breadth-first search, the single-source shortest path problem, computing a directed ear decomposition of a strongly connected planar digraph, computing an open directed ear decomposition of a strongly connected biconnected planar digraph, and topologically sorting a planar directed acyclic graph.