Ittai Abraham, Yair Bartal, Ofer Neiman
We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result is obtained via a new approach of building ``highways'' between portals and a new strong diameter probabilistic decomposition theorem.
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We prove that every weighted graph contains a spanning tree subgraph of average stretch O((log n log log n)^2). Moreover, we show how to construct such a tree in time O(m log^2 n).
We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph $G$ with a linear arrangement of bandwidth $b$ can be embedded into a distribution $\mathcal T$ of spanning trees such that the expected stretch of each edge of $G$ is $O(b^2)$. Our proof implies a linear time algorithm for sampling from $\mathcal T$. Therefore, we have a linear time algorithm that finds a spanning tree of $G$ with average stretch $O(b^2)$ with high probability. We also describe a deterministic linear-time algorithm for comp…
A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the problem of computing a minimum average stretch spanning tree in polygonal 2-trees, a super class of 2-connected outerplanar graphs. For a polygonal 2-tree on $n$ vertices, we present an algorithm to compute a minimum average stretch spanning tree in $O(n \log n)$ time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees.
Spanning Tree Covers for Path-Separable Graphs: Trading Stretch for Size
研究\(p\)路径可分图等的生成树覆盖问题，核心方法是构造不同伸展度和规模的生成树覆盖，主要贡献是给出了不同类型图的生成树覆盖在伸展度与规模上的多种关系及具体结果。
A Note on Preconditioning by Low-Stretch Spanning Trees
本文通过分析预处理线性系统的特征值分布，证明使用低拉伸生成树的拉普拉斯矩阵进行预处理，共轭梯度法可在 $\softO{m^{4/3} \ln (1/ε)}$ 时间内求解拉普拉斯线性系统。
Light Tree Covers, Routing, and Path-Reporting Oracles via Spanning Tree Covers in Doubling Graphs
本文通过构造倍增图的(1+ε)-拉伸生成树覆盖，解决了树覆盖恒定轻度这一长期开放问题，并由此得到轻量树覆盖、紧凑路由方案和路径报告距离预言机等应用。
Low-Stretch Spanning Trees via Smoothed Analysis of Dijkstra's Algorithm
针对现有低拉伸生成树算法复杂但简单Dijkstra启发式效果好的问题，通过平滑分析证明图权重微扰可使任意根最短路径树成为$\tilde{O}(1)$-近似LSST，且扰动可通过低直径分解高效计算，提出LSST求解新方法。
Near Linear-Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs
提出一种近线性工作的并行算法求解对称对角占优线性系统，通过低直径分解生成低拉伸生成树或子图，实现高效并行求解。
The metric Ramsey problem asks for the largest subset $S$ of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor 2007 devised the so called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this paper we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees. We…
Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice. With this paper we address this gap. We provide the first implementation of the combinatorial solver by [Kelner et al., STOC 2013], which is particularly…
A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant em…
A temporal graph is an undirected graph $G=(V,E)$ along with a function that assigns a time-label to each edge in $E$. A path in $G$ with non-decreasing time-labels is called temporal path and the distance from $u$ to $v$ is the minimum length (i.e., the number of edges) of a temporal path from $u$ to $v$. A temporal $α$-spanner of $G$ is a (temporal) subgraph $H$ that preserves the distances between any pair of vertices in $V$, up to a multiplicative stretch factor of $α$. The size of $H$ is the number of its edges. In this work we study the size-stretch trade-offs of temporal spanners. We sh…
Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees
该研究针对三类度约束下的最小代价生成树计算问题，结合经典图参数给出了不同参数化下的紧复杂度界与算法，完善了该问题的细粒度复杂度研究。
We study the problem of how well a tree metric is able to preserve the sum of pairwise distances of an arbitrary metric. This problem is closely related to low-stretch metric embeddings and is interesting by its own flavor from the line of research proposed in the literature. As the structure of a tree imposes great constraints on the pairwise distances, any embedding of a metric into a tree metric is known to have maximum pairwise stretch of $Ω(\log n)$. We show, however, from the perspective of average performance, there exist tree metrics which preserve the sum of pairwise distances of the …
$(α, β)$ Spanners and Hybrid Spanners with Nearly Tight Bounds
本文提出新的生成子图构造，在$O(n^{1+1/k})$边数下将乘法拉伸改进至$O(\log k)$，并改进$k$-混合生成子图至$O(n^{1+1/k}+kn)$边，逼近最优界。
Spanning trees of low average stretch on the non-tree edges, as introduced by Alon et al. [SICOMP 1995], are a natural graph-theoretic object. In recent years, they have found significant applications in solvers for symmetric diagonally dominant (SDD) linear systems. In this work, we provide the first dynamic algorithm for maintaining such trees under edge insertions and deletions to the input graph. Our algorithm has update time $ n^{1/2 + o(1)} $ and the average stretch of the maintained tree is $ n^{o(1)} $, which matches the stretch in the seminal result of Alon et al. Similar to Alon et a…
We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph $G$ and a root node $r$, we seek a single spanning tree $T$ of minimum {\em stretch}, where the stretch of $T$ is defined to be the maximum ratio, over all terminal sets $X$, of the cost of the minimal sub-tree $T_X$ of $T$ that connects $X$ to $r$ to the cost of an optimal Steiner tree connecting $X$ to $r$ in $G$. Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example…
Computing \emph{all best swap edges} (ABSE) of a spanning tree $T$ of a given $n$-vertex and $m$-edge undirected and weighted graph $G$ means to select, for each edge $e$ of $T$, a corresponding non-tree edge $f$, in such a way that the tree obtained by replacing $e$ with $f$ enjoys some optimality criterion (which is naturally defined according to some objective function originally addressed by $T$). Solving efficiently an ABSE problem is by now a classic algorithmic issue, since it conveys a very successful way of coping with a (transient) \emph{edge failure} in tree-based communication netw…
This paper focuses on finding a spanning tree of a graph to maximize the number of its internal vertices. We present an approximation algorithm for this problem which can achieve a performance ratio $\frac{4}{3}$ on undirected simple graphs. This improves upon the best known approximation algorithm with performance ratio $\frac{5}{3}$ before. Our algorithm benefits from a new observation for bounding the number of internal vertices of a spanning tree, which reveals that a spanning tree of an undirected simple graph has less internal vertices than the edges a maximum path-cycle cover of that gr…
We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor $C_{del\approx 2.42$; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor $ρ=1+2π(k\cos{\frac{π{k)^{-1$ and degree bounded by $k$, for any integer parameter $k\geq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean gra…
We prove tight mixing time bounds for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomials. For a matroid of rank $k$ on a ground set of $n$ elements, or more generally distributions associated with log-concave polynomials of homogeneous degree $k$ on $n$ variables, we show that the down-up random walk, started from an arbitrary point in the support, mixes in time $O(k\log k)$. Our bound has no dependence on $n$ or the starting point, unlike the previous analyses [ALOV19,CGM19], and is tight up to const…
Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm with approximation factor $7/12$. They asked whether it is possible to improve these ratios, in particular for claw-free graphs and cubic graphs. We improve the approximation factors for the Ma…
Stochastic Minimum Spanning Trees with a Single Sample
研究边权重未知分布仅凭单样本的随机最小生成树问题，证明基于采样算法在指数分布下性能保证等于最大键大小且紧致，并推广至拟阵。
An out-tree $T$ of a directed graph $D$ is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By $l(D)$ and $l_s(D)$ we denote the maximum number of leaves over all out-trees and out-branchings of $D$, respectively. We give fixed parameter tractable algorithms for deciding whether $l_s(D)\geq k$ and whether $l(D)\geq k$ for a digraph $D$ on $n$ vertices, both with time complexity $2^{O(k\log k)} \cdot n^{O(1)}$. This improves on previous algorithms with complexity $2^{O(k^3\log k)} \cdot n^{O(1)}$ and $2^{O(k\log^2 k)} \cdot n^{O(1)}$…
A classic result in the study of spanners is the existence of light low-stretch spanners for Euclidean spaces. These spanners ahve arbitrary low stretch, and weight only a constant factor greater than that of the minimum spanning tree of the points (with dependence on the stretch and Euclidean dimention). A central open problem in this field asks whether other spaces admit low weight spanners as well - for example metric space with low intrinsic dimension - yet only a handful of results of this type are known. In this paper, we consider snowflake metric spaces of low intrinsic dimension. The α…
On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch
本文研究亚2拉伸近似距离预言机的空间使用与图最大度的关系，提出在最大度受限条件下可实现亚二次空间的(2,1-k)-拉伸ADO，并给出匹配的下界。
Learning Spanning Forests Optimally using CUT Queries in Weighted Undirected Graphs
本文针对加权无向图最大生成森林学习问题，提出期望O(n)次CUT查询的最优随机算法，还提出无向无权图下查询复杂度为O(n log n / log log n)的确定性算法，小幅提升了现有最优水平。
A tree $σ$-spanner of a positively real-weighted $n$-vertex and $m$-edge undirected graph $G$ is a spanning tree $T$ of $G$ which approximately preserves (i.e., up to a multiplicative stretch factor $σ$) distances in $G$. Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in $T$, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational …
Finding Almost Tight Witness Trees
本文研究Witness Tree图优化问题，通过更优的见证树选择，改进了Node-Tree Augmentation及特殊图类中Steiner Tree问题的近似算法效果。
With few exceptions (namely, algorithms for maximal matching, $2$-approximate vertex cover, and certain constant-stretch spanners), all known fully dynamic algorithms in general graphs require (amortized) $Ω(\log n)$ update/query time. Showing for the first time that techniques from property testing can lead to constant-time fully dynamic graph algorithms we prove the following results: (1) We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per update that maintains a proper $(Δ+1)$-vertex coloring of a graph with maximum degree at most $Δ$. This improves…
We present approximation algorithms for the following NP-hard optimization problems related to bottleneck spanning trees in metric spaces. 1. The disjoint bottleneck spanning tree problem: Given $n$ pairs of points in a metric space, find two disjoint trees each containing exactly one point from each pair and minimize the largest edge length (over all edges of both trees). It is known that approximating this problem by a factor better than 2 is NP-hard. We present a 4-approximation algorithm for this problem. This improves upon the previous best known approximation ratio of $9$. Our algorithm …
Random-Shift Revisited: Tight Approximations for Tree Embeddings and L1-Oblivious Routings
本文重新分析随机平移分解，证明其在多尺度下具有紧常数权衡，从而得到树嵌入和L1-oblivious路由的紧近似，并支持高效并行与分布式实现。
We show that for every $n$-point metric space $M$ there exists a spanning tree $T$ with unweighted diameter $O(\log n)$ and weight $ω(T) = O(\log n) \cdot ω(MST(M))$. Moreover, there is a designated point $rt$ such that for every point $v$, $dist_T(rt,v) \le (1+ε) \cdot dist_M(rt,v)$, for an arbitrarily small constant $ε> 0$. We extend this result, and provide a tradeoff between unweighted diameter and weight, and prove that this tradeoff is \emph{tight up to constant factors} in the entire range of parameters. These results enable us to settle a long-standing open question in Computational Ge…
The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an $n$-vertex spanning tree $T$ and an additional set $L$ of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links $F$ such that every terminal pair that has arrived so far is $2$-edge-connected in $T \cup F$. This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network desig…
In STOC'95 [ADMSS'95] Arya et al. showed that any set of $n$ points in $\mathbb R^d$ admits a $(1+ε)$-spanner with hop-diameter at most 2 (respectively, 3) and $O(n \log n)$ edges (resp., $O(n \log \log n)$ edges). They also gave a general upper bound tradeoff of hop-diameter at most $k$ and $O(n α_k(n))$ edges, for any $k \ge 2$. The function $α_k$ is the inverse of a certain Ackermann-style function at the $\lfloor k/2 \rfloor$th level of the primitive recursive hierarchy, where $α_0(n) = \lceil n/2 \rceil$, $α_1(n) = \left\lceil \sqrt{n} \right\rceil$, $α_2(n) = \lceil \log{n} \rceil$, $α_3…
Tight Bounds on the Message Complexity of Distributed Tree Verification
研究分布式树验证的消息复杂度，提出通用框架建立紧致下界，揭示生成树与最小生成树验证复杂度的多项式分离，并分析其对网络规模近似质量和BFS树拉伸参数的敏感性。
Better Learning-Augmented Spanning Tree Algorithms via Metric Forest Completion
本文提出了一种改进的学习增强生成树算法，通过度量森林补全框架，在任意度量空间中实现更优的近似最小生成树算法。
In this paper, we study problems of connecting classes of points via noncrossing structures. Given a set of colored terminal points, we want to find a graph for each color that connects all terminals of its color with the restriction that no two graphs cross each other. We consider these problems both on the Euclidean plane and in planar graphs. On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored traveling salesman problem as well as for the red-blue-green separation problem (in which we want to separate terminals of three colors with two noncrossing polygons of minimum …
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