arXiv 1608.044562016-08-16cs.DScs.CG

An Improved Algorithm for Diameter-Optimally Augmenting Paths in a Metric Space

Haitao Wang

英文摘要：

Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ such that the diameter of the resulting graph is minimized. Previously (in ICALP 2015) the problem was solved in $O(n\log^3 n)$ time. In this paper, based on new observations and different algorithmic techniques, we present an $O(n\log n)$ time algorithm.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A Linear-Time Algorithm for Radius-Optimally Augmenting Paths in a Metric Space

Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ to minimize the radius of the resulting graph. Previously, a similar problem for minimizing the diameter of the graph was solved in $O(n\log n)$ time. To the best of our knowledge, the problem of minimizing the radius has not been studied before. In this paper, we present an $O(n)$ time algorithm for the problem, which is optimal.

2019-04-26cs.CGcs.DS

A Linear-Time Algorithm for Discrete Radius Optimally Augmenting Paths in a Metric Space

Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ so that the radius of the resulting graph is minimized, where any center is constrained to be one of the vertices of $P$. Previously, the "continuous" version of the problem where a center may be a point in the interior of an edge of the graph was studied and a linear-time algorithm was known. Our "discrete" version of the problem has not been studied before. We present a linear-time algorithm for the problem.

2020-06-24cs.CGcs.DS

Algorithms for Diameters of Unicycle Graphs and Diameter-Optimally Augmenting Trees

We consider the problem of computing the diameter of a unicycle graph (i.e., a graph with a unique cycle). We present an O(n) time algorithm for the problem, where n is the number of vertices of the graph. This improves the previous best O(n \log n) time solution [Oh and Ahn, ISAAC 2016]. Using this algorithm as a subroutine, we solve the problem of adding a shortcut to a tree so that the diameter of the new graph (which is a unicycle graph) is minimized; our algorithm takes O(n^2 \log n) time and O(n) space. The previous best algorithms solve the problem in O(n^2 \log^3 n) time and O(n) space…

2020-11-19cs.CGcs.DS

Augmenting graphs to minimize the diameter

We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.

2013-09-20cs.DS

Almost optimal algorithms for diameter-optimally augmenting trees

We consider the problem of augmenting an $n$-vertex tree with one shortcut in order to minimize the diameter of the resulting graph. The tree is embedded in an unknown space and we have access to an oracle that, when queried on a pair of vertices $u$ and $v$, reports the weight of the shortcut $(u,v)$ in constant time. Previously, the problem was solved in $O(n^2 \log^3 n)$ time for general weights [Oh and Ahn, ISAAC 2016], in $O(n^2 \log n)$ time for trees embedded in a metric space [Große et al., {\tt arXiv:1607.05547}], and in $O(n \log n)$ time for paths embedded in a metric space [Wang, W…

2018-10-02cs.DS

最大低直径稠密子图问题的分支定界方法

A Branch-and-Bound Approach for Maximum Low-Diameter Dense Subgraph Problems

本文针对最大直径至多为二的 $f(\cdot)$-稠密子图问题，提出基于分解的分支定界算法，结合退化序分解策略和基于排序的上界，在真实图上显著优于MIP求解器与纯分支定界方法。

2025-11-06cs.DS

更多相关论文收起更多论文

Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded Hop-Diameter Spanners

Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional Euclidean spaces -- due to their numerous applications. Euclidean spanners can be viewed as means of compressing the $\binom{n}{2}$ pairwise distances of a $d$-dimensional Euclidean space into $O(n) = O_{ε,d}(n)$ spanner edges, so that the spanner distances preserve the original distances to within a factor of $1+ε$, for any $ε> 0$. Moreover, one can compute such spanners in optimal $O(n \log n)$ time. Once the spanner has been computed, it serves…

2022-06-02cs.CGcs.DS

最优放置绿色桥梁，并在稀疏图中考虑近距离栖息地

Placing Green Bridges Optimally, with Close-Range Habitats in Sparse Graphs

本研究针对野生动物通道放置的网络设计问题，提出在稀疏图中以最小成本实现栖息地直径二连接，并给出不同参数下的算法与NP困难性结果。

2025-10-24cs.CCcs.DS

在树中寻找减小直径的捷径

Finding Diameter-Reducing Shortcuts in Trees

本文研究树中多条捷径的直径优化增广问题，证明 k≥3 时树上下界为 10/9，并为路径设计线性时间 (1+ε)-近似算法，同时提出专用数据结构以支持树的 4-近似和一般图直径计算。

2023-05-27cs.DS

一种用于负权单源最短路径（SSSP）且集成负环检测的自底向上算法

A Bottom-Up Algorithm for Negative-Weight SSSP with Integrated Negative Cycle Finding

提出一种简化的负权SSSP自底向上算法，以图直径为递归参数增强鲁棒性，集成负环检测消除单独步骤，虽无理论效率提升但简化设计，利于实现分析。

2024-11-29cs.DS

Distributed Maximum Matching Verification in CONGEST

We study the maximum cardinality matching problem in a standard distributed setting, where the nodes $V$ of a given $n$-node network graph $G=(V,E)$ communicate over the edges $E$ in synchronous rounds. More specifically, we consider the distributed CONGEST model, where in each round, each node of $G$ can send an $O(\log n)$-bit message to each of its neighbors. We show that for every graph $G$ and a matching $M$ of $G$, there is a randomized CONGEST algorithm to verify $M$ being a maximum matching of $G$ in time $O(|M|)$ and disprove it in time $O(D + \ell)$, where $D$ is the diameter of $G$ …

2020-02-18cs.DCcs.DS

Fully Polynomial-Time Distributed Computation in Low-Treewidth Graphs

We consider global problems, i.e. problems that take at least diameter time, even when the bandwidth is not restricted. We show that all problems considered admit efficient solutions in low-treewidth graphs. By ``efficient'' we mean that the running time has polynomial dependence on the treewidth, a linear dependence on the diameter (which is unavoidable), and only a polylogarithmic dependence on $n$, the number of nodes in the graph. We present the algorithms solving the following problems in the CONGEST model which all attain $\tilde{O(τ^{O(1)}D)}$-round complexity (where $τ$ and $D$ denote …

2022-05-30cs.DCcs.DS

Budgeted Dominating Sets in Uncertain Graphs

We study the {\em Budgeted Dominating Set} (BDS) problem on uncertain graphs, namely, graphs with a probability distribution $p$ associated with the edges, such that an edge $e$ exists in the graph with probability $p(e)$. The input to the problem consists of a vertex-weighted uncertain graph $\G=(V, E, p, ω)$ and an integer {\em budget} (or {\em solution size}) $k$, and the objective is to compute a vertex set $S$ of size $k$ that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of $S$. We refer to the problem as the {\em Probabilistic Budgeted …

2021-07-07cs.DS