Ilya P. Razenshteyn

Graph Partitioning with Natural Cuts

We present a novel approach to graph partitioning based on the notion of \emph{natural cuts}. Our algorithm, called PUNCH, has two phases. The first phase performs a series of minimum-cut computations to identify and contract dense regions of the graph. This reduces the graph size, but preserves its general structure. The second phase uses a combination of greedy and local search heuristics to assemble the final partition. The algorithm performs especially well on road networks, which have an abundance of natural cuts (such as bridges, mountain passes, and ferries). In a few minutes, it obtains the best known partitions for continental-sized networks, significantly improving on previous results.

Optimal Data-Dependent Hashing for Approximate Near Neighbors

We show an optimal data-dependent hashing scheme for the approximate near neighbor problem. For an n-point dataset in a d-dimensional space our data structure achieves query time O(d ⋅ nρ+o(1)) and space O(n1+ρ+o(1) + d ⋅ n), where ρ=1/(2c2-1) for the Euclidean space and approximation c>1. For the Hamming space, we obtain an exponent of ρ=1/(2c-1). Our result completes the direction set forth in (Andoni, Indyk, Nguyen, Razenshteyn 2014) who gave a proof-of-concept that data-dependent hashing can outperform classic Locality Sensitive Hashing (LSH). In contrast to (Andoni, Indyk, Nguyen, Razenshteyn 2014), the new bound is not only optimal, but in fact improves over the best (optimal) LSH data structures (Indyk, Motwani 1998) (Andoni, Indyk 2006) for all approximation factors c>1. From the technical perspective, we proceed by decomposing an arbitrary dataset into several subsets that are, in a certain sense, pseudo-random.

Optimal hashing-based time-space trade-offs for approximate near neighbors

[See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the

On model-based RIP-1 matrices

c

On the Complexity of Inner Product Similarity Join

-Approximate Near Neighbor Search problem. For the

Weighted low rank approximations with provable guarantees

d

An exact combinatorial algorithm for minimum graph bisection

-dimensional Euclidean space and

Separating Hierarchical and General Hub Labelings

n

Restricted Isometry Property for General p-Norms

-point datasets, we develop a data structure with space

Triangle-free 2-matchings revisited

n^{1 + \rho_u + o(1)} + O(dn)