T-H. Hubert Chan

Multi-Dimensional Range Query over Encrypted Data

We design an encryption scheme called multi-dimensional range query over encrypted data (MRQED), to address the privacy concerns related to the sharing of network audit logs and various other applications. Our scheme allows a network gateway to encrypt summaries of network flows before submitting them to an untrusted repository. When network intrusions are suspected, an authority can release a key to an auditor, allowing the auditor to decrypt flows whose attributes (e.g., source and destination addresses, port numbers, etc.) fall within specific ranges. However, the privacy of all irrelevant flows are still preserved. We formally define the security for MRQED and prove the security of our construction under the decision bilinear Diffie-Hellman and decision linear assumptions in certain bilinear groups. We study the practical performance of our construction in the context of network audit logs. Apart from network audit logs, our scheme also has interesting applications for financial audit logs, medical privacy, untrusted remote storage, etc. In particular, we show that MRQED implies a solution to its dual problem, which enables investors to trade stocks through a broker in a privacy-preserving manner.

Metric embeddings with relaxed guarantees

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler (2004), we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into /spl lscr//sub 1/ which exhibit gracefully degrading distortion: these is a single embedding into /spl lscr//sub 1/ that achieves distortion at most O(log 1//spl epsi/) on all but at most an /spl epsi/ fraction of distances, simultaneously for all /spl epsi/ > 0. We extend this with distortion O(log 1//spl epsi/)/sup 1/p/ to maps into general /spl lscr//sub p/, p /spl ges/ 1 for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight, and give a general technique to obtain lower bounds for /spl epsi/-slack embeddings from lower bounds for low-distortion embeddings.

Small hop-diameter sparse spanners for doubling metrics

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse (1+ε)-spanners with small hop diameter for metrics of low doubling dimension.In this paper, we show that given any metric with constant doubling dimension k and any 0<ε<1, one can find (1+ε)-spanner for the metric with nearly linear number of edges (i.e., only O(nlog *n+nε−O(k)) edges) and constant hop diameter; we can also obtain a (1+ε)-spanner with linear number of edges (i.e., only nε−O(k) edges) that achieves a hop diameter that grows like the functional inverse of Ackermann’s function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.

An SDP primal-dual algorithm for approximating the Lovász-theta function

The Lovasz ϑ-function [Lov79] on a graph G = (V,E) can be defined as the maximum of the sum of the entries of a positive semidefinite matrix X, whose trace Tr(X) equals 1, and X ij = 0 whenever {i, j} ∈ E. This function appears as a subroutine for many algorithms for graph problems such as maximum independent set and maximum clique. We apply Arora and Kales primal-dual method for SDP to design an approximate algorithm for the ϑ-function with an additive error of δ ≫ 0, which runs in time O(α2n2/δ2 log n · M e ), where α = ϑ(G) and M e = O(n3) is the time for a matrix exponentiation operation. Moreover, our techniques generalize to the weighted Lovasz ϑ-function, and both the maximum independent set weight and the maximum clique weight for vertex weighted perfect graphs can be approximated within a factor of (1+∊) in time O(∊−2n5 log n).

Ranking on Arbitrary Graphs: Rematch via Continuous Linear Programming

Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are es...