Faster Algorithms for the Geometric Transportation Problem

Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric: * For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost. * For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point. * An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2

Diverse near neighbor problem

Motivated by the recent research on diversity-aware search, we investigate the k-diverse near neighbor reporting problem. The problem is defined as follows: given a query point q, report the maximum diversity set S of k points in the ball of radius r around q. The diversity of a set S is measured by the minimum distance between any pair of points in $S$ (the higher, the better). We present two approximation algorithms for the case where the points live in a d-dimensional Hamming space. Our algorithms guarantee query times that are sub-linear in n and only polynomial in the diversity parameter k, as well as the dimension d. For low values of k, our algorithms achieve sub-linear query times even if the number of points within distance r from a query $q$ is linear in $n$. To the best of our knowledge, these are the first known algorithms of this type that offer provable guarantees.Charles Stark Draper LaboratoryNational Science Foundation (U.S.) (Award NSF CCF-1012042)David & Lucile Packard Foundatio

I/O-Efficient Algorithms for Contour Line Extraction and Planar Graph Blocking

For a polyhedral terrain C, the contour at z-coordinate h, denoted Ch, is defined to be the intersection of the plane z = h with C. In this paper, we study the contour-line extraction problem, where we want to preprocess C into a data structure so that given a query z-coordinate h, we can report Ch quickly. This is a central problem that arises in geographic information systems (GIS), where terrains are often stored as Triangular Irregular Networks (TINS). We present an I/O-optimal algorithm for this problem which stores a terrain C with N vertices using O(N/B) blocks, where B is the size of a disk block, so that for any query h, the contour ch can be computed using o(log, N + I&l/B) I/O operations, where l&l denotes the size of Ch. We also present en improved algorithm for a more general problem of blocking bounded-degree planar graphs such as TINS (i.e., storing them on disk so that any graph traversal algorithm can traverse the graph in an I/O-efficient manner), and apply it to two problms that arise in GIS

A divide-and-conquer algorithm for min-cost perfect matching in the plane

Given a set V of 2n points in the plane, the min-cost perfect matching problem is to pair up the points (into n pairs) so that the sum of the Euclidean distances between the paired points is minimized. We present an O(n 3/2 log 5 n)time algorithm for computing a min-cost perfect matching in the plane, which is an improvement over the previous best algorithm of Vaidya [21] by nearly a factor of n. Vaidya’s algorithm is an implementation of the algorithm of Edmonds [8], which runs in n phases, and computes a matching with i edges at the end of the i-th phase. Vaidya shows that geometry can be exploited to implement a single phase in roughly O(n 3/2) time, thus obtaining an O(n 5/2 log 4 n)time algorithm. We improve upon this in two major ways. First, we develop a variant of Edmonds ’ algorithm that uses geometric divide-and-conquer, so that in the conquer step we need only O ( √ n) phases. Second, we show that a single phase can be implemented in O(n log 5 n) time. 1

A new NC-algorithm for finding a perfect matching in bipartite planar and small genus graphs (Extended Abstract)

It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a P-time algorithm for the case of graphs of small genus.) However, it is not known if the corresponding search problem, that of finding one perfect matching in a planar graph, is in NC. This situation is intriguing as it seems to contradict our intuition that search should be easier than counting. For the case of planar bipartite graphs, Miller and Naor [22] showed that a perfect matching can indeed be found using an NC algorithm. We present a very different NC-algorithm for this problem. Unlike the Miller..

High-Dimensional Shape Fitting in Linear Time

The radius of a k-dimensional at F with respect to P , denoted by RD(F ; P ), is de ned to be max p2P dist(F ; p), where dist(F ; p) denotes the Euclidean distance between p and its projection onto F . The k-at radius of P , which we denote by R k (P ), is the minimum, over all k-dimensional ats F , of RD(F ; P ). We consider the problem of computing R k (P ) for a given set of points P

Projective clustering in high dimensions using core-sets

In this paper, we show that there exists a small core-set for the problem of computing the “smallest ” radius k-flat for a given point-set in IR d. The size of the core-set is dimension independent. Such small core-sets yield immediate efficient algorithms for finding the (1+ε)-approximate smallest radius k-flat for the points in dn O(k6 /ε 5 log(1/ε)) time. Furthermore, we can use it to (1 + ε)-approximate the the smallest radius k-flat for a prespecified fraction of the given points, in the same running time. Our algorithm can also be used for computing the min-max such coverage of the point-set by j flats, each one of them of dimension k. No previous efficient approximation algorithms were known for those problems in high-dimensions, when k> 1 or j> 1.

Approximating Shortest Paths on a Nonconvex Polyhedron

We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + ")ae, where ae is the length of the shortest path between s and t on @P , and " ? 0 is an arbitararily small positive constant. The algorithm runs in O(n 5=3 log 5=3 n) time. We also present a slightly faster algorithm that runs in O(n 8=5 log 8=5 n) time and returns a path whose length is at most 15(1 + ")ae. Work on this paper has been supported by National Science Foundation Grant CCR-93--01259, by an Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, by matching funds from Xerox Corporation, and by a grant from the U.S.--Israeli Binational Science Foundation. y Department of Computer Science, Box 90129, Duke University, [email protected] z Department of Computer Science, Box 90129, Duke University, pa..