Fat Polygonal Partitions with Applications to Visualization and Embeddings

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in $\mathcal{T}$ is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in $\mathbb{R}^d$. We use these partitions with slack for embedding ultrametrics into $d$-dimensional Euclidean space: we give a $\mathop{\rm polylog}(\Delta)$-approximation algorithm for embedding $n$-point ultrametrics into $\mathbb{R}^d$ with minimum distortion, where $\Delta$ denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in $n$. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.Comment: 26 page

Maintaining a large matching and a small vertex cover

We consider the problem of maintaining a large matching and a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and handles a sequence of K updates in K*polylog(n) time, where n is the number of vertices in the graph. Previous data structures require a polynomial amount of computation per update.National Science Foundation (U.S.). (Grant number 0732334)National Science Foundation (U.S.). (Grant number 0728645)Marie Curie International Reintegration Grants (Grant number PIRG03-GA-2008-231077)Israel Science Foundation (Grant number 1147/09)Israel Science Foundation (Grant number 1675/09

Parallel Algorithms for Geometric Graph Problems

We give algorithms for geometric graph problems in the modern parallel models inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes a $(1+\epsilon)$-approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms). In contrast, for general graphs, achieving the same result for MST (or even connectivity) remains a challenging open problem, despite drawing significant attention in recent years. We develop a general algorithmic framework that, besides MST, also applies to Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic framework has implications beyond the MapReduce model. For example it yields a new algorithm for computing EMD cost in the plane in near-linear time, $n^{1+o_\epsilon(1)}$. We note that while recently Sharathkumar and Agarwal developed a near-linear time algorithm for $(1+\epsilon)$-approximating EMD, our algorithm is fundamentally different, and, for example, also solves the transportation (cost) problem, raised as an open question in their work. Furthermore, our algorithm immediately gives a $(1+\epsilon)$-approximation algorithm with $n^{\delta}$ space in the streaming-with-sorting model with $1/\delta^{O(1)}$ passes. As such, it is tempting to conjecture that the parallel models may also constitute a concrete playground in the quest for efficient algorithms for EMD (and other similar problems) in the vanilla streaming model, a well-known open problem

New sublinear methods in the struggle against classical problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 129-134).We study the time and query complexity of approximation algorithms that access only a minuscule fraction of the input, focusing on two classical sources of problems: combinatorial graph optimization and manipulation of strings. The tools we develop find applications outside of the area of sublinear algorithms. For instance, we obtain a more efficient approximation algorithm for edit distance and distributed algorithms for combinatorial problems on graphs that run in a constant number of communication rounds. Combinatorial Graph Optimization Problems: The graph optimization problems considered by us include vertex cover, maximum matching, and dominating set. A graph algorithm is traditionally called a constant-time algorithm if it runs in time that is a function of only the maximum vertex degree, and in particular, does not depend on the number of vertices in the graph. We show a general local computation framework that allows for transforming many classical greedy approximation algorithms into constant-time approximation algorithms for the optimal solution size. By applying the framework, we obtain the first constant-time algorithm that approximates the maximum matching size up to an additive En, where E is an arbitrary positive constant, and n is the number of vertices in the graph. It is known that a purely additive En approximation is not computable in constant time for vertex cover and dominating set. We show that nevertheless, such an approximation is possible for a wide class of graphs, which includes planar graphs (and other minor-free families of graphs) and graphs of subexponential growth (a common property of networks). This result is obtained via locally computing a good partition of the input graph in our local computation framework. The tools and algorithms developed for these problems find several other applications: " Our methods can be used to construct local distributed approximation algorithms for some combinatorial optimization problems. " Our matching algorithm yields the first constant-time testing algorithm for distinguishing bounded-degree graphs that have a perfect matching from those far from having this property. " We give a simple proof that there is a constant-time algorithm distinguishing bounded-degree graphs that are planar (or in general, have a minor-closed property) from those that are far from planarity (or the given minor-closed property, respectively). Our tester is also much more efficient than the original tester of Benjamini, Schramm, and Shapira (STOC 2008). Edit Distance. We study a new asymmetric query model for edit distance. In this model, the input consists of two strings x and y, and an algorithm can access y in an unrestricted manner (without charge), while being charged for querying every symbol of x. We design an algorithm in the asymmetric query model that makes a small number of queries to distinguish the case when the edit distance between x and y is small from the case when it is large. Our result in the asymmetric query model gives rise to a near-linear time algorithm that approximates the edit distance between two strings to within a polylogarithmic factor. For strings of length n and every fixed E > 0, the algorithm computes a (log n)0(/0) approximation in n1i' time. This is an exponential improvement over the previously known near-linear time approximation factor 20( log (Andoni and Onak, STOC 2009; building on Ostrovsky and Rabani, J. ACM 2007). The algorithm of Andoni and Onak was the first to run in O(n 2 -) time, for any fixed constant 6 > 0, and obtain a subpolynomial, n"(o), approximation factor, despite a sequence of papers. We provide a nearly-matching lower bound on the number of queries. Our lower bound is the first to expose hardness of edit distance stemming from the input strings being "repetitive", which means that many of their substrings are approximately identical. Consequently, our lower bound provides the first rigorous separation on the complexity of approximation between edit distance and Ulam distance.by Krzysztof Onak.Ph.D