951,426 research outputs found
Polylogarithmic Approximation for Generalized Minimum Manhattan Networks
Given a set of terminals, which are points in -dimensional Euclidean
space, the minimum Manhattan network problem (MMN) asks for a minimum-length
rectilinear network that connects each pair of terminals by a Manhattan path,
that is, a path consisting of axis-parallel segments whose total length equals
the pair's Manhattan distance. Even for , the problem is NP-hard, but
constant-factor approximations are known. For , the problem is
APX-hard; it is known to admit, for any \eps > 0, an
O(n^\eps)-approximation.
In the generalized minimum Manhattan network problem (GMMN), we are given a
set of terminal pairs, and the goal is to find a minimum-length
rectilinear network such that each pair in is connected by a Manhattan
path. GMMN is a generalization of both MMN and the well-known rectilinear
Steiner arborescence problem (RSA). So far, only special cases of GMMN have
been considered.
We present an -approximation algorithm for GMMN (and, hence,
MMN) in dimensions and an -approximation algorithm for 2D.
We show that an existing -approximation algorithm for RSA in 2D
generalizes easily to dimensions.Comment: 14 pages, 5 figures; added appendix and figure
Path Gain Algebraic Formulation for the Scalar Linear Network Coding Problem
In the algebraic view, the solution to a network coding problem is seen as a
variety specified by a system of polynomial equations typically derived by
using edge-to-edge gains as variables. The output from each sink is equated to
its demand to obtain polynomial equations. In this work, we propose a method to
derive the polynomial equations using source-to-sink path gains as the
variables. In the path gain formulation, we show that linear and quadratic
equations suffice; therefore, network coding becomes equivalent to a system of
polynomial equations of maximum degree 2. We present algorithms for generating
the equations in the path gains and for converting path gain solutions to
edge-to-edge gain solutions. Because of the low degree, simplification is
readily possible for the system of equations obtained using path gains. Using
small-sized network coding problems, we show that the path gain approach
results in simpler equations and determines solvability of the problem in
certain cases. On a larger network (with 87 nodes and 161 edges), we show how
the path gain approach continues to provide deterministic solutions to some
network coding problems.Comment: 12 pages, 6 figures. Accepted for publication in IEEE Transactions on
Information Theory (May 2010
Reachability and Shortest Paths in the Broadcast CONGEST Model
In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter D of the underlying network is constant.
We show that for the case where D = 1 there is, quite surprisingly, a very simple algorithm that solves the reachability problem in 1(!) round. In contrast, for networks with D = 2, we show that any distributed algorithm (possibly randomized) for this problem requires Omega(sqrt{n/ log{n}}) rounds. Our results therefore completely resolve (up to a small polylog factor) the complexity of the single-source reachability problem for a wide range of diameters.
Furthermore, we show that when D = 1, it is even possible to get an almost 3 - approximation for the all-pairs shortest path problem (for directed unweighted graphs) in just 2 rounds. We also prove a stronger lower bound of Omega(sqrt{n}) for the single-source shortest path problem for unweighted directed graphs that holds even when the diameter of the underlying network is 2. As far as we know this is the first lower bound that achieves Omega(sqrt{n}) for this problem
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