Jakub Pachocki

Jakub Pachocki
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Jakub Pachocki
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Computer Science - Data Structures and Algorithms (15)
 
Computer Science - Computational Complexity (2)
 
Computer Science - Discrete Mathematics (2)
 
Mathematics - Numerical Analysis (1)
 
Computer Science - Numerical Analysis (1)
 
Mathematics - Combinatorics (1)
 
Mathematics - Optimization and Control (1)

Publications Authored By Jakub Pachocki

In the communication problem $\mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x, y \in\{0,1\}^n$ with the promise that $x\neq y$. The last player to receive a message must output an index $i$ such that $x_i\neq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $\Theta(\min\{n,\log(1/\delta)\log^2(\frac n{\log(1/\delta)})\})$ for failure probability $\delta$. Read More

In the communication problem $\mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x$ and $y$ in $\{0,1\}^n$ with the promise that $x\neq y$. The last player to receive a message must output an index $i$ such that $x_i\neq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $\Theta(\min\{n, \log(1/\delta)\log^2(\frac{n}{\log(1/\delta)})\})$ bits for failure probability $\delta$. Read More

A spectral sparsifier of a graph $G$ is a sparser graph $H$ that approximately preserves the quadratic form of $G$, i.e. for all vectors $x$, $x^T L_G x \approx x^T L_H x$, where $L_G$ and $L_H$ denote the respective graph Laplacians. Read More

We develop new methods based on graph motifs for graph clustering, allowing more efficient detection of communities within networks. We focus on triangles within graphs, but our techniques extend to other clique motifs as well. Our intuition, which has been suggested but not formalized similarly in previous works, is that triangles are a better signature of community than edges. Read More

In this paper we provide faster algorithms for solving the geometric median problem: given $n$ points in $\mathbb{R}^{d}$ compute a point that minimizes the sum of Euclidean distances to the points. This is one of the oldest non-trivial problems in computational geometry yet despite an abundance of research the previous fastest algorithms for computing a $(1+\epsilon)$-approximate geometric median were $O(d\cdot n^{4/3}\epsilon^{-8/3})$ by Chin et. al, $\tilde{O}(d\exp{\epsilon^{-4}\log\epsilon^{-1}})$ by Badoiu et. Read More

We show that schemes for sparsifying matrices based on iteratively resampling rows yield guarantees matching classic 'offline' sparsifiers (see e.g. Spielman and Srivastava [STOC 2008]). Read More

Finding a small spectral approximation for a tall $n \times d$ matrix $A$ is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of $A$. Row sampling improves interpretability, saves space when $A$ is sparse, and preserves row structure, which is especially important, for example, when $A$ represents a graph. Read More

We introduce the notion of balance for directed graphs: a weighted directed graph is $\alpha$-balanced if for every cut $S \subseteq V$, the total weight of edges going from $S$ to $V\setminus S$ is within factor $\alpha$ of the total weight of edges going from $V\setminus S$ to $S$. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with $\alpha = 1$) and residual graphs of $(1+\epsilon)$-approximate undirected maximum flows (with $\alpha=O(1/\epsilon)$). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. Read More

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of $|V(H)|^{o(|V(H)|)}$-time algorithm deciding if graph $G$ is a subgraph of $H$. Read More

Subgraph Isomorphism is a very basic graph problem, where given two graphs $G$ and $H$ one is to check whether $G$ is a subgraph of $H$. Despite its simple definition, the Subgraph Isomorphism problem turns out to be very broad, as it generalizes problems such as Clique, $r$-Coloring, Hamiltonicity, Set Packing and Bandwidth. However, for all of the mentioned problems $2^{\mathcal{O}(n)}$ time algorithms exist, so a natural and frequently asked question in the past was whether there exists a $2^{\mathcal{O}(n)}$ time algorithm for Subgraph Isomorphism. Read More

In the Manhattan Sequence Consensus problem (MSC problem) we are given $k$ integer sequences, each of length $l$, and we are to find an integer sequence $x$ of length $l$ (called a consensus sequence), such that the maximum Manhattan distance of $x$ from each of the input sequences is minimized. For binary sequences Manhattan distance coincides with Hamming distance, hence in this case the string consensus problem (also called string center problem or closest string problem) is a special case of MSC. Our main result is a practically efficient $O(l)$-time algorithm solving MSC for $k\le 5$ sequences. Read More

We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. When applied to graph Laplacians, this leads to ultra-sparsifiers that in expectation behave as the nearly-optimal ones given by [Kolla-Makarychev-Saberi-Teng STOC`10]. Combining this with the recursive preconditioning framework by [Spielman-Teng STOC`04] and improved embedding algorithms, this leads to algorithms that solve symmetric diagonally dominant linear systems and electrical flow problems in expected time close to $m\log^{1/2}n$ . Read More

We give a generalized definition of stretch that simplifies the efficient construction of low-stretch embeddings suitable for graph algorithms. The generalization, based on discounting highly stretched edges by taking their $p$-th power for some $0 < p < 1$, is directly related to performances of existing algorithms. This discounting of high-stretch edges allows us to treat many classes of edges with coarser granularity. Read More

We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. Read More