Randomized Data Structures for the Dynamic Closest-Pair Problem

We describe a new randomized data structure, the {\em sparse partition}, for solving the dynamic closest-pair problem. Using this data structure the closest pair of a set of $n$ points in $k$-dimensional space, for any fixed $k$, can be found in constant time. If the points are chosen from a finite universe, and if the floor function is available at unit-cost, then the data structure supports insertions into and deletions from the set in expected $O(\log n)$ time and requires expected $O(n)$ space. Here, it is assumed that the updates are chosen by an adversary who does not know the random choices made by the data structure. The data structure can be modified to run in $O(\log^2 n)$ expected time per update in the algebraic decision tree model of computation. Even this version is more efficient than the currently best known deterministic algorithms for solving the problem for $k>1$

Incremental and Decremental Maintenance of Planar Width

We present an algorithm for maintaining the width of a planar point set dynamically, as points are inserted or deleted. Our algorithm takes time O(kn^epsilon) per update, where k is the amount of change the update causes in the convex hull, n is the number of points in the set, and epsilon is any arbitrarily small constant. For incremental or decremental update sequences, the amortized time per update is O(n^epsilon).Comment: 7 pages; 2 figures. A preliminary version of this paper was presented at the 10th ACM/SIAM Symp. Discrete Algorithms (SODA '99); this is the journal version, and will appear in J. Algorithm

Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs

We develop data structures for dynamic closest pair problems with arbitrary distance functions, that do not necessarily come from any geometric structure on the objects. Based on a technique previously used by the author for Euclidean closest pairs, we show how to insert and delete objects from an n-object set, maintaining the closest pair, in O(n log^2 n) time per update and O(n) space. With quadratic space, we can instead use a quadtree-like structure to achieve an optimal time bound, O(n) per update. We apply these data structures to hierarchical clustering, greedy matching, and TSP heuristics, and discuss other potential applications in machine learning, Groebner bases, and local improvement algorithms for partition and placement problems. Experiments show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp. 619-628. For source code and experimental results, see http://www.ics.uci.edu/~eppstein/projects/pairs

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before

Algorithms for Stable Matching and Clustering in a Grid

We study a discrete version of a geometric stable marriage problem originally proposed in a continuous setting by Hoffman, Holroyd, and Peres, in which points in the plane are stably matched to cluster centers, as prioritized by their distances, so that each cluster center is apportioned a set of points of equal area. We show that, for a discretization of the problem to an $n\times n$ grid of pixels with $k$ centers, the problem can be solved in time $O(n^2 \log^5 n)$, and we experiment with two slower but more practical algorithms and a hybrid method that switches from one of these algorithms to the other to gain greater efficiency than either algorithm alone. We also show how to combine geometric stable matchings with a $k$-means clustering algorithm, so as to provide a geometric political-districting algorithm that views distance in economic terms, and we experiment with weighted versions of stable $k$-means in order to improve the connectivity of the resulting clusters.Comment: 23 pages, 12 figures. To appear (without the appendices) at the 18th International Workshop on Combinatorial Image Analysis, June 19-21, 2017, Plovdiv, Bulgari

The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30

Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms

Consider the following 2-respecting min-cut problem. Given a weighted graph $G$ and its spanning tree $T$, find the minimum cut among the cuts that contain at most two edges in $T$. This problem is an important subroutine in Karger's celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present a new approach for this problem which can be easily implemented in many settings, leading to the following randomized min-cut algorithms for weighted graphs. * An $O(m\frac{\log^2 n}{\log\log n} + n\log^6 n)$-time sequential algorithm: This improves Karger's $O(m \log^3 n)$ and $O(m\frac{(\log^2 n)\log (n^2/m)}{\log\log n} + n\log^6 n)$ bounds when the input graph is not extremely sparse or dense. Improvements over Karger's bounds were previously known only under a rather strong assumption that the input graph is simple [Henzinger et al. SODA'17; Ghaffari et al. SODA'20]. For unweighted graphs with parallel edges, our bound can be improved to $O(m\frac{\log^{1.5} n}{\log\log n} + n\log^6 n)$. * An algorithm requiring $\tilde O(n)$ cut queries to compute the min-cut of a weighted graph: This answers an open problem by Rubinstein et al. ITCS'18, who obtained a similar bound for simple graphs. * A streaming algorithm that requires $\tilde O(n)$ space and $O(\log n)$ passes to compute the min-cut: The only previous non-trivial exact min-cut algorithm in this setting is the 2-pass $\tilde O(n)$-space algorithm on simple graphs [Rubinstein et al., ITCS'18] (observed by Assadi et al. STOC'19). In contrast to Karger's 2-respecting min-cut algorithm which deploys sophisticated dynamic programming techniques, our approach exploits some cute structural properties so that it only needs to compute the values of $\tilde O(n)$ cuts corresponding to removing $\tilde O(n)$ pairs of tree edges, an operation that can be done quickly in many settings.Comment: Updates on this version: (1) Minor corrections in Section 5.1, 5.2; (2) Reference to newer results by GMW SOSA21 (arXiv:2008.02060v2), DEMN STOC21 (arXiv:2004.09129v2) and LMN 21 (arXiv:2102.06565v1