Papers by Ilya Razenshteyn
Sketching and Embedding are Equivalent for Norms
Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15, 2015
A Linear Time Algorithm for Finding Three Edge-Disjoint Paths in Eulerian Networks
Lecture Notes in Computer Science, 2010
ABSTRACT Consider an undirected graph $G = (VG, EG)$ and a set of six \emph{terminals} $T = \set{... more ABSTRACT Consider an undirected graph $G = (VG, EG)$ and a set of six \emph{terminals} $T = \set{s_1, s_2, s_3, t_1, t_2, t_3} \subseteq VG$. The goal is to find a collection $\calP$ of three edge-disjoint paths $P_1$, $P_2$, and $P_3$, where $P_i$ connects nodes $s_i$ and $t_i$ ($i = 1, 2, 3$). Results obtained by Robertson and Seymour by graph minor techniques imply a polynomial time solvability of this problem. The time bound of their algorithm is $O(m^3)$ (hereinafter we assume $n := \abs{VG}$, $m := \abs{EG}$, $n = O(m)$). In this paper we consider a special, \emph{Eulerian} case of $G$ and $T$. Namely, construct the \emph{demand graph} $H = (VG, \set{s_1t_1, s_2t_2, s_3t_3})$. The edges of $H$ correspond to the desired paths in $\calP$. In the Eulerian case the degrees of all nodes in the (multi-) graph $G + H$ ($ = (VG, EG \cup EH)$) are even. Schrijver showed that, under the assumption of Eulerianess, cut conditions provide a criterion for the existence of $\calP$. This, in particular, implies that checking for existence of $\calP$ can be done in $O(m)$ time. Our result is a combinatorial $O(m)$-time algorithm that constructs $\calP$ (if the latter exists). Comment: SOFSEM 2010
Beyond Locality-Sensitive Hashing
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013
ABSTRACT We present a new data structure for the c-approximate near neighbor problem (ANN) in the... more ABSTRACT We present a new data structure for the c-approximate near neighbor problem (ANN) in the Euclidean space. For n points in R^d, our algorithm achieves O(n^{\rho} + d log n) query time and O(n^{1 + \rho} + d log n) space, where \rho <= 7/(8c^2) + O(1 / c^3) + o(1). This is the first improvement over the result by Andoni and Indyk (FOCS 2006) and the first data structure that bypasses a locality-sensitive hashing lower bound proved by O'Donnell, Wu and Zhou (ICS 2011). By a standard reduction we obtain a data structure for the Hamming space and \ell_1 norm with \rho <= 7/(8c) + O(1/c^{3/2}) + o(1), which is the first improvement over the result of Indyk and Motwani (STOC 1998).
Exact Combinatorial Branch-and-Bound for Graph Bisection
2012 Proceedings of the Fourteenth Workshop on Algorithm Engineering and Experiments (ALENEX), 2012
Separating Hierarchical and General Hub Labelings
Lecture Notes in Computer Science, 2013
Optimal Data-Dependent Hashing for Approximate Near Neighbors
Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15, 2015
Robust Hierarchical k-Center Clustering
Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science - ITCS '15, 2015
2011 IEEE International Parallel & Distributed Processing Symposium, 2011
We present a novel approach to graph partitioning based on the notion of natural cuts. Our algori... more We present a novel approach to graph partitioning based on the notion of 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 significantly, 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.
The restricted isometry property for the general p-norms
ABSTRACT The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enabl... more ABSTRACT The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an $m \times n$ matrix satisfies RIP of order $k$ for the $\ell_p$ norm, if $\|Ax\|_p \approx \|x\|_p$ for every $x$ with at most $k$ non-zero coordinates. For every $1 \leq p < \infty$ we obtain almost tight bounds on the minimum number of rows $m$ necessary for the RIP property to hold. Before, only the cases $p \in \big\{1, 1 + \frac{1}{\log k}, 2\big\}$ were studied. Interestingly, our results show that the case $p = 2$ is a `singularity' in terms of the optimum value of $m$. We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.
Discrete Mathematics, Algorithms and Applications, 2010
A 2-matching in an undirected graph G = (V G, EG) is a function x : EG → {0, 1, 2} such that for ... more A 2-matching in an undirected graph G = (V G, EG) is a function x : EG → {0, 1, 2} such that for each node v ∈ V G the sum of values x(e) on all edges e incident to v does not exceed 2. The size of x is the sum e x(e). If {e ∈ EG | x(e) = 0} contains no triangles then x is called triangle-free. Cornuéjols and Pulleyblank devised a combinatorial O(mn)-algorithm that finds a triangle free 2-matching of maximum size (hereinafter n := |V G|, m := |EG|) and also established a min-max theorem. We claim that this approach is, in fact, superfluous by demonstrating how their results may be obtained directly from the Edmonds-Gallai decomposition. Applying the algorithm of Micali and Vazirani we are able to find a maximum triangle-free 2-matching in O(m √ n)-time. Also we give a short self-contained algorithmic proof of the min-max theorem. Next, we consider the case of regular graphs. It is well-known that every regular graph admits a perfect 2-matching. One can easily strengthen this result and prove that every d-regular graph (for d ≥ 3) contains a perfect triangle-free 2-matching. We give the following algorithms for finding a perfect triangle-free 2-matching in a d-regular graph: an O(n)algorithm for d = 3, an O(m + n 3/2 )-algorithm for d = 2k (k ≥ 2), and an O(n 2 )-algorithm for d = 2k + 1 (k ≥ 2). ⋆
Lecture Notes in Computer Science, 2013
The Restricted Isometry Property (RIP) is a fundamental property of a matrix enabling sparse reco... more The Restricted Isometry Property (RIP) is a fundamental property of a matrix enabling sparse recovery . Informally, an m × n matrix satisfies RIP of order k in the ℓ p norm if Ax p ≈ x p for any vector x that is k-sparse, i.e., that has at most k non-zeros. The minimal number of rows m necessary for the property to hold has been extensively investigated, and tight bounds are known. Motivated by signal processing models, a recent work of Baraniuk et al [BCDH10] has generalized this notion to the case where the support of x must belong to a given model, i.e., a given family of supports. This more general notion is much less understood, especially for norms other than ℓ 2 .
An exact combinatorial algorithm for minimum graph bisection
Mathematical Programming, 2014
Theoretical Computer Science, 2011
C. Calude, A. Nies, L. Staiger, and F. Stephan posed the following question about the relation be... more C. Calude, A. Nies, L. Staiger, and F. Stephan posed the following question about the relation between plain and prefix Kolmogorov complexities (see their paper in DLT 2008 conference proceedings): does the domain of every optimal decompressor contain the domain of some optimal prefix-free decompressor? In this paper we provide a negative answer to this question.
Uploads
Papers by Ilya Razenshteyn