Searcharxiv⌕ Search

arXiv subjects

Ching-Lueh Chang

Publications and source records attributed to Ching-Lueh Chang.

11 recordsLinked to original sources
arXiv2022

Deterministic metric $1$-median selection with very few queries

Given an $n$-point metric space $(M,d)$, {\sc metric $1$-median} asks for a point $p\in M$ minimizing $\sum_{x\in M}\,d(p,x)$. We show that for each computable function $f\colon \mathbb{Z}^+\to\mathbb{Z}^+$ satisfying $f(n)=ω(1)$, {\sc metric $1$-median} has a deterministic, $o(n)$-query, $o(f(n)\cdot\log n)$-approximation and nonadaptive algorithm. Previously, no deterministic $o(n)$-query $o(n)$-approximation algorithms are known for {\sc metric $1$-median}. On the negative side, we prove each deterministic $O(n)$-query algorithm for {\sc metric $1$-median} to be not $(δ\log n)$-approximate for a sufficiently small constant $δ>0$. We also refute the existence of deterministic $o(n)$-query $O(\log n)$-approximation algorithms.

cs.DS↗
arXiv2019

On ultrametric $1$-median selection

Consider the problem of finding a point in an ultrametric space with the minimum average distance to all points. We give this problem a Monte Carlo $O((\log^2(1/ε))/ε^3)$-time $(1+ε)$-approximation algorithm for all $ε>0$.

cs.DS↗
arXiv2018

Metric 1-median selection with fewer queries

Let $h\colon\mathbb{Z}^+\to\mathbb{Z}^+\setminus\{1\}$ be any function such that $h(n)$ and $\lceil n^{1/h(n)}\rceil$ are computable from $n$ in $O(h(n)\cdot n^{1+1/h(n)})$ time. We show that given any $n$-point metric space $(M,d)$, the problem of finding $\mathop{\mathrm{argmin}}_{i\in M}\,\sum_{j\in M}\,d(i,j)$ (breaking ties arbitrarily) has a deterministic, $O(h(n)\cdot n^{1+1/h(n)})$-time, $O(n^{1+1/h(n)})$-query, $(2\,h(n))$-approximation and nonadaptive algorithm. Our proofs modify those of Chang~\cite{Cha15, Cha15CMCT} with the following improvements: (1) We improve Chang's~\cite{Cha15} query complexity of $O(h(n)\cdot n^{1+1/h(n)})$ to $O(n^{1+1/h(n)})$, everything else being equal. (2) Chang's~\cite{Cha15CMCT} unpublished work establishes our result only when $n$ is a perfect $(h(n))$th power.

cs.DS↗
arXiv2017

Metric random matchings with applications

Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected value and the variance of $\sum_{i=1}^{\lfloor n/2\rfloor}\,d({\boldsymbolπ}(2i-1),{\boldsymbolπ}(2i))$ for a uniformly random permutation ${\boldsymbolπ}$ of $\{1,2,\ldots,n\}$, leading to the following results: (I) Consider the problem of finding a point in $\{1,2,\ldots,n\}$ with the minimum sum of distances to all points. We show that this problem has a randomized algorithm that (1) always outputs a $(2+ε)$-approximate solution in expected $O(n/ε^2)$ time and that (2) inherits Indyk's~\cite{Ind99, Ind00} algorithm to output a $(1+ε)$-approximate solution in $O(n/ε^2)$ time with probability $Ω(1)$, where $ε\in(0,1)$. (II) The average distance in $(\{1,2,\ldots,n\},d)$ can be approximated in $O(n/ε)$ time to within a multiplicative factor in $[\,1/2-ε,1\,]$ with probability $1/2+Ω(1)$, where $ε>0$. (III) Assume $d$ to be a graph metric. Then the average distance in $(\{1,2,\ldots,n\},d)$ can be approximated in $O(n)$ time to within a multiplicative factor in $[\,1-ε,1+ε\,]$ with probability $1/2+Ω(1)$, where $ε=ω(1/n^{1/4})$.

cs.DS↗
arXiv2017

A Las Vegas approximation algorithm for metric $1$-median selection

Given an $n$-point metric space, consider the problem of finding a point with the minimum sum of distances to all points. We show that this problem has a randomized algorithm that {\em always} outputs a $(2+ε)$-approximate solution in an expected $O(n/ε^2)$ time for each constant $ε>0$. Inheriting Indyk's algorithm, our algorithm outputs a $(1+ε)$-approximate $1$-median in $O(n/ε^2)$ time with probability $Ω(1)$.

cs.DS↗
arXiv2015

Metric $1$-median selection: Query complexity vs. approximation ratio

Consider the problem of finding a point in a metric space $(\{1,2,\ldots,n\},d)$ with the minimum average distance to other points. We show that this problem has no deterministic $o(n^{1+1/(h-1)})$-query $(2h-Ω(1))$-approximation algorithms for any constant $h\in\mathbb{Z}^+\setminus\{1\}$.

cs.CC↗
arXiv2014

A lower bound for metric 1-median selection

Consider the problem of finding a point in an n-point metric space with the minimum average distance to all points. We show that this problem has no deterministic $o(n^2)$-query $(4-Ω(1))$-approximation algorithms.

cs.DS↗
arXiv2010

On reversible cascades in scale-free and Erdős-Rényi random graphs

Consider the following cascading process on a simple undirected graph $G(V,E)$ with diameter $Δ$. In round zero, a set $S\subseteq V$ of vertices, called the seeds, are active. In round $i+1,$ $i\in\mathbb{N},$ a non-isolated vertex is activated if at least a $ρ\in(\,0,1\,]$ fraction of its neighbors are active in round $i$; it is deactivated otherwise. For $k\in\mathbb{N},$ let $\text{min-seed}^{(k)}(G,ρ)$ be the minimum number of seeds needed to activate all vertices in or before round $k$. This paper derives upper bounds on $\text{min-seed}^{(k)}(G,ρ)$. In particular, if $G$ is connected and there exist constants $C>0$ and $γ>2$ such that the fraction of degree-$k$ vertices in $G$ is at most $C/k^γ$ for all $k\in\mathbb{Z}^+,$ then $\text{min-seed}^{(Δ)}(G,ρ)=O(\lceilρ^{γ-1}\,|\,V\,|\rceil)$. Furthermore, for $n\in\mathbb{Z}^+,$ $p=Ω((\ln{(e/ρ)})/(ρn))$ and with probability $1-\exp{(-n^{Ω(1)})}$ over the Erdős-Rényi random graphs $G(n,p),$ $\text{min-seed}^{(1)}(G(n,p),ρ)=O(ρn)$.

cs.DM↗
arXiv2010

On irreversible dynamic monopolies in general graphs

Consider the following coloring process in a simple directed graph $G(V,E)$ with positive indegrees. Initially, a set $S$ of vertices are white, whereas all the others are black. Thereafter, a black vertex is colored white whenever more than half of its in-neighbors are white. The coloring process ends when no additional vertices can be colored white. If all vertices end up white, we call $S$ an irreversible dynamic monopoly (or dynamo for short) under the strict-majority scenario. An irreversible dynamo under the simple-majority scenario is defined similarly except that a black vertex is colored white when at least half of its in-neighbors are white. We derive upper bounds of $(2/3)\,|\,V\,|$ and $|\,V\,|/2$ on the minimum sizes of irreversible dynamos under the strict and the simple-majority scenarios, respectively. For the special case when $G$ is an undirected connected graph, we prove the existence of an irreversible dynamo with size at most $\lceil |\,V\,|/2 \rceil$ under the strict-majority scenario. Let $ε>0$ be any constant. We also show that, unless $\text{NP}\subseteq \text{TIME}(n^{O(\ln \ln n)}),$ no polynomial-time, $((1/2-ε)\ln |\,V\,|)$-approximation algorithms exist for finding the minimum irreversible dynamo under either the strict or the simple-majority scenario. The inapproximability results hold even for bipartite graphs with diameter at most 8.

cs.DM↗