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We show that there exists a graph G with Opnq nodes, where any forest of n nodes is a node-induced subgraph of G. Furthermore, for constant arboricity k, the result implies the existence of a graph with Opn k q nodes that contains all n-node graphs as node-induced subgraphs, matching a Ωpn k q lower bound. The lower bound and previously best upper bounds… (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 " min hpSqs " 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 Pˇatraşcu and Thorup [SODA'13] makes c " Op1q lookups in tables of size u 1{c.… (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)

Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this… (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 log n + 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… (More)

The power of two choices is a classic paradigm used for assigning m balls to n bins. When placing a ball we pick two bins according to some hash functions h 0 and h 1 , and place the ball in the least full bin. It was shown by Azar et al. [STOC'94] that for m " Opnq with perfectly random hash functions this scheme yields a maximum load of lg lg n ` Op1q… (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)

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 Ωpkq of time per element over k independent samples when estimating the number… (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)