Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Simple Randomized $O(n log n)$--Time Closest-Pair Algorithm in Doubling Metrics
96
0
0.0
(
0
)
Added by
Michiel Smid
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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).
It is known for some time that a random graph $G(n,p)$ contains w.h.p. a Hamiltonian cycle if $p$ is larger than the critical value $p_{crit}= (log n + log log n + omega_n)/n$. The determination of a concrete Hamiltonian cycle is even for values much larger than $p_{crit}$ a nontrivial task. In this paper we consider random graphs $G(n,p)$ with $p$ in $tilde{Omega}(1/sqrt{n})$, where $tilde{Omega}$ hides poly-logarithmic factors in $n$. For this range of $p$ we present a distributed algorithm ${cal A}_{HC}$ that finds w.h.p. a Hamiltonian cycle in $O(log n)$ rounds. The algorithm works in the synchronous model and uses messages of size $O(log n)$ and $O(log n)$ memory per node.
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.
Resolving an open question from 2006, we prove the existence of light-weight bounded-degree spanners for unit ball graphs in the metrics of bounded doubling dimension, and we design a simple $mathcal{O}(log^*n)$-round distributed algorithm that given a unit ball graph $G$ with $n$ vertices and a positive constant $epsilon < 1$ finds a $(1+epsilon)$-spanner with constant bounds on its maximum degree and its lightness using only 2-hop neighborhood information. This immediately improves the algorithm of Damian, Pandit, and Pemmaraju which runs in $mathcal{O}(log^*n)$ rounds but has a $mathcal{O}(log Delta)$ bound on its lightness, where $Delta$ is the ratio of the length of the longest edge in $G$ to the length of the shortest edge. We further study the problem in the two dimensional Euclidean plane and we provide a construction with similar properties that has a constant average number of edge intersection per node. This is the first distributed low-intersection topology control algorithm to the best of our knowledge. Our distributed algorithms rely on the maximal independent set algorithm of Schneider and Wattenhofer that runs in $mathcal{O}(log^*n)$ rounds of communication. If a maximal independent set is known beforehand, our algorithms run in constant number of rounds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
