Approximating Nearest Neighbor Distances

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we consider the most natural metric with this property, which we call the nearest neighbor metric. Given a point set P and a path $\gamma$, our metric charges each point of $\gamma$ with its distance to P. The total charge along $\gamma$ determines its nearest neighbor length, which is formally defined as the integral of the distance to the input points along the curve. We describe a $(3+\varepsilon)$-approximation algorithm and a $(1+\varepsilon)$-approximation algorithm to compute the nearest neighbor metric. Both approximation algorithms work in near-linear time. The former uses shortest paths on a sparse graph using only the input points. The latter uses a sparse sample of the ambient space, to find good approximate geodesic paths.Comment: corrected author nam

Minimum Eccentricity Shortest Path Problem: an Approximation Algorithm and Relation with the k-Laminarity Problem

The Minimum Eccentricity Shortest Path (MESP) Problem consists in determining a shortest path (a path whose length is the distance between its extremities) of minimum eccentricity in a graph. It was introduced by Dragan and Leitert [9] who described a linear-time algorithm which is an 8-approximation of the problem. In this paper, we study deeper the double-BFS procedure used in that algorithm and extend it to obtain a linear-time 3-approximation algorithm. We moreover study the link between the MESP problem and the notion of laminarity, introduced by Völkel et al [12], corresponding to its restriction to a diameter (i.e. a shortest path of maximum length), and show tight bounds between MESP and laminarity parameters

Approximation Algorithms for Min-Distance Problems in DAGs

Graph parameters such as the diameter, radius, and vertex eccentricities are not defined in a useful way in Directed Acyclic Graphs (DAGs) using the standard measure of distance, since for any two nodes, there is no path between them in one of the two directions. So it is natural to consider the distance between two nodes as the length of the shortest path in the direction in which this path exists, motivating the definition of the min-distance. The min-distance between two nodes u and v is the minimum of the shortest path distances from u to v and from v to u. As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in O?(mn) time. So it is natural to resort to approximation algorithms in O?(mn^{1-?}) time for some positive ?. Abboud, Vassilevska W., and Wang [SODA 2016] first studied min-distance problems achieving constant factor approximation algorithms on DAGs, and Dalirrooyfard et al [ICALP 2019] gave the first constant factor approximation algorithms on general graphs for min-diameter, min-radius and min-eccentricities. Abboud et al obtained a 3-approximation algorithm for min-radius on DAGs which works in O?(m?n) time, and showed that any (2-?)-approximation requires n^{2-o(1)} time for any ? > 0, under the Hitting Set Conjecture. We close the gap, obtaining a 2-approximation algorithm which runs in O?(m?n) time. As the lower bound of Abboud et al only works for sparse DAGs, we further show that our algorithm is conditionally tight for dense DAGs using a reduction from Boolean matrix multiplication. Moreover, Abboud et al obtained a linear time 2-approximation algorithm for min-diameter along with a lower bound stating that any (3/2-?)-approximation algorithm for sparse DAGs requires n^{2-o(1)} time under SETH. We close this gap for dense DAGs by obtaining a 3/2-approximation algorithm which works in O(n^{2.350}) time and showing that the approximation factor is unlikely to be improved within O(n^{? - o(1)}) time under the high dimensional Orthogonal Vectors Conjecture, where ? is the matrix multiplication exponent

Shortest Paths in Geometric Intersection Graphs

This thesis studies shortest paths in geometric intersection graphs, which can model, among others, ad-hoc communication and transportation networks. First, we consider two classical problems in the field of algorithms, namely Single-Source Shortest Paths (SSSP) and All-Pairs Shortest Paths (APSP). In SSSP we want to compute the shortest paths from one vertex of a graph to all other vertices, while in APSP we aim to find the shortest path between every pair of vertices. Although there is a vast literature for these problems in many graph classes, the case of geometric intersection graphs has been only partially addressed. In unweighted unit-disk graphs, we show that we can solve SSSP in linear time, after presorting the disk centers with respect to their coordinates. Furthermore, we give the first (slightly) subquadratic-time APSP algorithm by using our new SSSP result, bit tricks, and a shifted-grid-based decomposition technique. In unweighted, undirected geometric intersection graphs, we present a simple and general technique that reduces APSP to static, offline intersection detection. Consequently, we give fast APSP algorithms for intersection graphs of arbitrary disks, axis-aligned line segments, arbitrary line segments, d-dimensional axis-aligned boxes, and d-dimensional axis-aligned unit hypercubes. We also provide a near-linear-time SSSP algorithm for intersection graphs of axis-aligned line segments by a reduction to dynamic orthogonal point location. Then, we study two problems that have received considerable attention lately. The first is that of computing the diameter of a graph, i.e., the longest shortest-path distance between any two vertices. In the second, we want to preprocess a graph into a data structure, called distance oracle, such that the shortest path (or its length) between any two query vertices can be found quickly. Since these problems are often too costly to solve exactly, we study their approximate versions. Following a long line of research, we employ Voronoi diagrams to compute a (1+epsilon)-approximation of the diameter of an undirected, non-negatively-weighted planar graph in time near linear in the input size and polynomial in 1/epsilon. The previously best solution had exponential dependency on the latter. Using similar techniques, we can also construct the first (1+epsilon)-approximate distance oracles with similar preprocessing time and space and only O(log(1/\epsilon)) query time. In weighted unit-disk graphs, we present the first near-linear-time (1+epsilon)-approximation algorithm for the diameter and for other related problems, such as the radius and the bichromatic closest pair. To do so, we combine techniques from computational geometry and planar graphs, namely well-separated pair decompositions and shortest-path separators. We also show how to extend our approach to obtain O(1)-query-time (1+epsilon)-approximate distance oracles with near linear preprocessing time and space. Then, we apply these oracles, along with additional ideas, to build a data structure for the (1+epsilon)-approximate All-Pairs Bounded-Leg Shortest Paths (apBLSP) problem in truly subcubic time

Approximation Algorithms for Broadcasting in Simple Graphs with Intersecting Cycles

Broadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes by placing a series of calls along the communication lines of the network. Every time the informed nodes aid the originator in distributing the message. Finding the minimum broadcast time of any vertex in an arbitrary graph is NP-Complete. The problem remains NP-Complete even for planar graphs of degree 3 and for a graph whose vertex set can be partitioned into a clique and an independent set. The best theoretical upper bound gives logarithmic approximation. It has been shown that the broadcasting problem is NP-Hard to approximate within a factor of 3-ɛ. The polynomial time solvability is shown only for tree-like graphs; trees, unicyclic graphs, tree of cycles, necklace graphs and some graphs where the underlying graph is a clique; such as fully connected trees and tree of cliques. In this thesis we study the broadcast problem in different classes of graphs where cycles intersect in at least one vertex. First we consider broadcasting in a simple graph where several cycles have common paths and two intersecting vertices, called a k-path graph. We present a constant approximation algorithm to find the broadcast time of an arbitrary k-path graph. We also study the broadcast problem in a simple cactus graph called k-cycle graph where several cycles of arbitrary lengths are connected by a central vertex on one end. We design a constant approximation algorithm to find the broadcast time of an arbitrary k-cycle graph. Next we study the broadcast problem in a hypercube of trees for which we present a 2-approximation algorithm for any originator. We provide a linear algorithm to find the broadcast time in hypercube of trees with one tree. We extend the result for any arbitrary graph whose nodes contain trees and design a linear time constant approximation algorithm where the broadcast scheme in the arbitrary graph is already known. In Chapter 6 we study broadcasting in Harary graph for which we present an additive approximation which gives 2-approximation in the worst case to find the broadcast time in an arbitrary Harary graph. Next for even values of n, we introduce a new graph, called modified-Harary graph and present a 1-additive approximation algorithm to find the broadcast time. We also show that a modified-Harary graph is a broadcast graph when k is logarithmic of n. Finally we consider a diameter broadcast problem where we obtain a lower bound on the broadcast time of the graph which has at least (d+k-1 choose d) + 1 vertices that are at a distance d from the originator, where k >= 1

Approximating the Diameter of Planar Graphs in Near Linear Time

We present a $(1+\epsilon)$-approximation algorithm running in $O(f(\epsilon)\cdot n \log^4 n)$ time for finding the diameter of an undirected planar graph with non-negative edge lengths

Space and Time Efficient Parallel Graph Decomposition, Clustering, and Diameter Approximation

We develop a novel parallel decomposition strategy for unweighted, undirected graphs, based on growing disjoint connected clusters from batches of centers progressively selected from yet uncovered nodes. With respect to similar previous decompositions, our strategy exercises a tighter control on both the number of clusters and their maximum radius. We present two important applications of our parallel graph decomposition: (1) $k$-center clustering approximation; and (2) diameter approximation. In both cases, we obtain algorithms which feature a polylogarithmic approximation factor and are amenable to a distributed implementation that is geared for massive (long-diameter) graphs. The total space needed for the computation is linear in the problem size, and the parallel depth is substantially sublinear in the diameter for graphs with low doubling dimension. To the best of our knowledge, ours are the first parallel approximations for these problems which achieve sub-diameter parallel time, for a relevant class of graphs, using only linear space. Besides the theoretical guarantees, our algorithms allow for a very simple implementation on clustered architectures: we report on extensive experiments which demonstrate their effectiveness and efficiency on large graphs as compared to alternative known approaches.Comment: 14 page

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author