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We show that there exists a graph G with O(n) 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 O(nk) nodes that contains all n-node graphs as node-induced subgraphs, matching a Ω(n<sup>k</sup>) lower bound. The lower bound and previously best… (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)

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)

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)

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)

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 Ω(k) of time per element over k independent samples when estimating the… (More)

We consider ancestry labeling schemes: Given a rooted tree T , assign a binary string (label) to each node, such that given the labels of any two nodes, one can determine whether the first is an ancestor of the second in T. Recently, Fraigniaud and Korman [STOC'10] showed that such labels can be assigned using log n + O(log log n) bits per label, solving a… (More)