Søren Dahlgaard

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In this article, we show that there exists a graph <i>G</i> with <i>O</i>(<i>n</i>) nodes such that any forest of <i>n</i> nodes is an induced subgraph of <i>G</i>. Furthermore, for constant arboricity <i>k</i>, the result implies the existence of a graph with <i>O</i>(<i>n</i><sup><i>k</i></sup>) nodes that contains all <i>n</i>-node graphs of arboricity(More)
The dynamic shortest paths problem on planar graphs asks us to preprocess a planar graph G such that we may support insertions and deletions of edges in G as well as distance queries between any two nodes u, v subject to the constraint that the graph remains planar at all times. This problem has been extensively studied in both the theory and experimental(More)
Finding important nodes in a graph and measuring their importance is a fundamental problem in the analysis of social networks, transportation networks, biological systems, etc. Among the most popular such metrics of importance are graph centrality, betweenness centrality (BC), and reach centrality (RC). These measures are also very related to classic(More)
A distance labeling scheme labels the n nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A D-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least D from each(More)
A random hash function h is ε-minwise if for any set S, |S| “ n, and element x P S, Prrhpxq “ minhpSqs “ p1 ̆ εq{n. Minwise hash functions with low bias ε have widespread applications within similarity estimation. Hashing from a universe rus, the twisted tabulation hashing of Pǎtraşcu and Thorup [SODA’13] makes c “ Op1q lookups in tables of size u1{c.(More)
In this paper we analyze a hash function for k-partitioning a set into bins, obtaining strong concentration bounds for standard algorithms combining statistics from each bin. This generic method was originally introduced by Flajolet and Martin [FOCS'83] in order to save a factor &#x03A9;(k) of time per element over k independent samples when estimating the(More)
For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an Opn5{3q-space distance oracle which answers exact distance queries in Oplog nq time for n-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space (i.e., space Opn2 ́ q(More)
We investigate labeling schemes supporting adjacency, ancestry, sibling, and connectivity queries in forests. In the course of more than 20 years, the existence of logn + O(log log) labeling schemes supporting each of these functions was proven, with the most recent being ancestry [Fraigniaud and Korman, STOC ’10]. Several multi-functional labeling schemes(More)