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Register a new userGeodesic diameter of a polygonal domain in O(n^4 log n) time
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Added by
Mikko Koivisto
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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We show that the geodesic diameter of a polygonal domain with n vertices can be computed in O(n^4 log n) time by considering O(n^3) candidate diameter endpoints; the endpoints are a subset of vertices of the overlay of shortest path maps from vertices of the domain.
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For a polygonal domain with $h$ holes and a total of $n$ vertices, we present algorithms that compute the $L_1$ geodesic diameter in $O(n^2+h^4)$ time and the $L_1$ geodesic center in $O((n^4+n^2 h^4)alpha(n))$ time, respectively, where $alpha(cdot)$ denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in $O(n^{7.73})$ or $O(n^7(h+log n))$ time, and compute the geodesic center in $O(n^{11}log n)$ time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on $L_1$ shortest paths in polygonal domains.
Consider a metric space $(P,dist)$ with $N$ points whose doubling dimension is a constant. We present a simple, randomized, and recursive algorithm that computes, in $O(N log N)$ expected time, the closest-pair distance in $P$. To generate recursive calls, we use previous results of Har-Peled and Mendel, and Abam and Har-Peled for computing a sparse annulus that separates the points in a balanced way.
In population protocols, the underlying distributed network consists of $n$ nodes (or agents), denoted by $V$, and a scheduler that continuously selects uniformly random pairs of nodes to interact. When two nodes interact, their states are updated by applying a state transition function that depends only on the states of the two nodes prior to the interaction. The efficiency of a population protocol is measured in terms of both time (which is the number of interactions until the nodes collectively have a valid output) and the number of possible states of nodes used by the protocol. By convention, we consider the parallel time cost, which is the time divided by $n$. In this paper we consider the majority problem, where each node receives as input a color that is either black or white, and the goal is to have all of the nodes output the color that is the majority of the input colors. We design a population protocol that solves the majority problem in $O(log^{3/2}n)$ parallel time, both with high probability and in expectation, while using $O(log n)$ states. Our protocol improves on a recent protocol of Berenbrink et al. that runs in $O(log^{5/3}n)$ parallel time, both with high probability and in expectation, using $O(log n)$ states.
We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in $min {,O(n^omega), O(n^2 + nh log h + chi^2),}$ time, where $omega<2.373$ denotes the matrix multiplication exponent and $chiin Omega(n)cap O(n^2)$ is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in $O(n^2 log n)$ time.
Tree comparison metrics have proven to be an invaluable aide in the reconstruction and analysis of phylogenetic (evolutionary) trees. The path-length distance between trees is a particularly attractive measure as it reflects differences in tree shape as well as differences between branch lengths. The distance equals the sum, over all pairs of taxa, of the squared differences between the lengths of the unique path connecting them in each tree. We describe an $O(n log n)$ time for computing this distance, making extensive use of tree decomposition techniques introduced by Brodal et al. (2004).
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