On Additive Spanners in Weighted Graphs with Local Error

ACM Transactions on Algorithms, 2010

An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k ) and an (additive) (1, 2)-spanner of size O(n 3/2 ). However no other additive spanners are known to exist.

Talg, 2010

An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3/2). However no other additive spanners are known to exist. In this article we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1, 6)-spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that, for any k, a (k, k − 1)-spanner with size O(kn 1+1/k) can be found in linear time, and, further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.

An (α, β)-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u, v:

Theoretical Computer Science, 2005

In this paper, we show that every chordal graph with n vertices and m edges admits an additive 4-spanner with at most 2n − 2 edges and an additive 3-spanner with at most O(n log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory 13 (1989) 99-116]. Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner is constructable in O(m log n) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)-spanner with at most 2n − 2 edges which is constructable in O(n k + m) time.

Lecture Notes in Computer Science, 2004

Spanners are sparse subgraphs that preserve distances up to a given factor in the underlying graph. Recently spanners have found important practical applications in metric space searching and message distribution in networks. These applications use some variant of the socalled greedy algorithm for constructing the spanner -an algorithm that mimics Kruskal's minimum spanning tree algorithm. Greedy spanners have nice theoretical properties, but their practical performance with respect to total weight is unknown. In this paper we give an exact algorithm for constructing minimum-weight spanners in arbitrary graphs. By using the solutions (and lower bounds) from this algorithm, we experimentally evaluate the performance of the greedy algorithm for a set of realistic problem instances.

Lecture Notes in Computer Science, 2013

We introduce and investigate a new notion of resilience in graph spanners. Let S be a spanner of a graph G. Roughly speaking, we say that a spanner S is resilient if all its point-to-point distances are resilient to edge failures. Namely, whenever any edge in G fails, then as a consequence of this failure all distances do not degrade in S substantially more than in G (i.e., the relative distance increases in S are very close to those in the underlying graph G). In this paper we show that sparse resilient spanners exist, and that they can be computed efficiently. arXiv:1303.1559v3 [cs.DS] 23 Apr 2013 O(n 4 3 ) edges [7]

arXiv (Cornell University), 2023

In the pairwise weighted spanner problem, the input consists of a weighted directed graph on n vertices, where each edge is assigned both a cost and a length. Furthermore, we are given k terminal vertex pairs and a distance constraint for each pair. The goal is to find a minimum-cost subgraph in which the distance constraints are satisfied. A more restricted variant of this problem was shown to be O(2 log 1−ε n)-hard to approximate under a standard complexity assumption, by Elkin and Peleg (Theory of Computing Systems, 2007). This general formulation captures many well-studied network connectivity problems, including spanners, distance preservers, and Steiner forests. We study the weighted spanner problem, in which the edges have positive integral lengths of magnitudes that are polynomial in n, while the costs are arbitrary non-negative rational numbers. Our results include the following in the classical offline setting: • AnÕ(n 4/5+ε)-approximation algorithm for the pairwise weighted spanner problem. When the edges have unit costs and lengths, the best previous algorithm gives anÕ(n 3/5+ε)-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (Transactions on Algorithms, 2020). • AnÕ(n 1/2+ε)-approximation algorithm for the weighted spanner problem when the terminal pairs consist of all vertex pairs and the distances must be preserved exactly. When the edges have unit costs and arbitrary positive lengths, the best previous algorithm gives anÕ(n 1/2)-approximation for the all-pair spanner problem, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). We also prove the first results for the weighted spanners in the online setting. In the online setting, the terminal vertex pairs arrive one at a time, in an online fashion, and edges are required to be added irrevocably to the solution in order to satisfy the distance constraints, while approximately minimizing the cost. Our results include the following: • AnÕ(k 1/2+ε)-competitive algorithm for the online pairwise weighted spanner problem. The stateof-the-art results are anÕ(n 4/5)-competitive algorithm when edges have unit costs and arbitrary positive lengths, and a min{Õ(k 1/2+ε),Õ(n 2/3+ε)}-competitive algorithm when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021).

This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most , i.e., the tree-length graphs. For such graphs we construct additive 2 -spanners with O( n+n logn) edges, and additive 4 -spanners with O( n) edges. This provides new upper bounds for chordal graphs for which = 1. We also show a lower bound, and prove that there are graphs of tree-length for which every multiplicative -spanner (and thus every additive ( 1)-spanner) requires (n1+1/ () ) edges. c 2007 Elsevier B.V. All rights reserved.

Theoretical Computer Science, 2007

This paper concerns construction of additive stretched spanners with few edges for n-vertex graphs having a tree-decomposition into bags of diameter at most δ, i.e., the tree-length δ graphs. For such graphs we construct additive 2δ-spanners with O(δn+n log n) edges, and additive 4δ-spanners with O(δn) edges. This provides new upper bounds for chordal graphs for which δ = 1. We also show a lower bound, and prove that there are graphs of tree-length δ for which every multiplicative δ-spanner (and thus every additive (δ − 1)-spanner) requires Ω (n 1+1/Θ(δ) ) edges.

ArXiv, 2021

Given a graph G = (V, E), a subgraph H is an additive +β spanner if distH(u, v) ≤ distG(u, v) + β for all u, v ∈ V . A pairwise spanner is a spanner for which the above inequality is only required to hold for specific pairs P ⊆ V × V given on input; when the pairs have the structure P = S × S for some S ⊆ V , it is called a subsetwise spanner. Additive spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs. In this paper, we consider a multi-level version of the subsetwise additive spanner in weighted graphs motivated by multi-level network design and visualization, where the vertices in S possess varying level, priority, or quality of service (QoS) requirements. The goal is to compute a nested sequence of spanners with the minimum total number of edges. We first generalize the +2 subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance...

Lecture Notes in Computer Science, 2008

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs the tree t-spanner problem (the problem to decide whether G admits a tree t-spanner) is polynomial time solvable for t ≤ 3 and is NP-complete as long as t is part of the input. They also left as an open problem whether the tree t-spanner problem is polynomial time solvable for every fixed t ≥ 4. In this work we resolve this open problem and extend the solution in several directions. We show that for every fixed t, it is possible in polynomial time not only to decide if a planar graph G has a tree t-spanner, but also to decide if G has a t-spanner of bounded treewidth. Moreover, for every fixed values of t and k, the problem, for a given planar graph G to decide if G has a t-spanner of treewidth at most k, is not only polynomial time solvable, but is fixed parameter tractable (with k and t being the parameters). In particular, the running time of our algorithm is linear with respect to the size of G. We extend this result from planar to a much more general class of sparse graphs containing graphs of bounded genus. An apex graph is a graph obtained from a planar graph G by adding a vertex and making it adjacent to some vertices of G. We show that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. Graphs of bounded treewidth are sparse graphs and our technique can be used to settle the complexity of the parameterized version of the sparse tspanner problem, where for given t and m one asks if a given n-vertex graph has a t-spanner with at most n − 1 + m edges. Our results imply that the sparse t-spanner problem is fixed parameter tractable on apexminor-free graphs with t and m being the parameters. Finally we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs. In particular, we prove that for every t ≥ 4, the problem of finding a tree t-spanner is NP-complete on K6-minor-free graphs. Thus our results are tight, in a sense that the restriction of input graph being apex-minor-free cannot be replaced by H-minor-free for some non-apex fixed graph H.

Theoretical Computer Science, 2014

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph G admits a system of µ collective additive tree r-spanners (resp., multiplicative tree t-spanners) if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r (resp., dT (x, y) ≤ t · dG(x, y)). When µ = 1 one gets the notion of additive tree r-spanner (resp., multiplicative tree t-spanner). It is known that if a graph G has a multiplicative tree t-spanner, then G admits a Robertson-Seymour's tree-decomposition with bags of radius at most ⌈t/2⌉ in G. We use this to demonstrate that there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative tree t-spanner, constructs a system of at most log 2 n collective additive tree O(t log n)-spanners of G. That is, with a slight increase in the number of trees and in the stretch, one can "turn" a multiplicative tree spanner into a small set of collective additive tree spanners. We extend this result by showing that if a graph G admits a multiplicative t-spanner with tree-width k − 1, then G admits a Robertson-Seymour's tree-decomposition each bag of which can be covered with at most k disks of G of radius at most ⌈t/2⌉ each. This is used to demonstrate that, for every fixed k, there is a polynomial time algorithm that, given an n-vertex graph G admitting a multiplicative t-spanner with tree-width k − 1, constructs a system of at most k(1 + log 2 n) collective additive tree O(t log n)-spanners of G. a stretch t [17], and an additive tree r-spanner of G is a spanning tree with a surplus r [59]. If we approximate the graph by a tree spanner, we can solve the problem on the tree and the solution interpret on the original graph. The tree t-spanner problem asks, given a graph G and a positive number t, whether G admits a tree t-spanner. Note that the problem of finding a tree t-spanner of G minimizing t is known in literature also as the Minimum Max-Stretch spanning Tree problem (see, e.g., and literature cited therein).

Lecture Notes in Computer Science, 2008

A graph G = (V, E) is said to admit a system of μ collective additive tree r-spanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices u, v of G a spanning tree T ∈ T (G) exists such that the distance in T between u and v is at most r plus their distance in G. In this paper, we examine the problem of finding "small" systems of collective additive tree r-spanners for small values of r on circle graphs and on polygonal graphs. Among other results, we show that every n-vertex circle graph admits a system of at most 2 log 3 2 n collective additive tree 2-spanners and every n-vertex k-polygonal graph admits a system of at most 2 log 3 2 k+7 collective additive tree 2-spanners. Moreover, we show that every n-vertex k-polygonal graph admits an additive (k + 6)-spanner with at most 6n − 6 edges and every n-vertex 3-polygonal graph admits a system of at most 3 collective additive tree 2spanners and an additive tree 6-spanner. All our collective tree spanners as well as all sparse spanners are constructible in polynomial time.

Information and Computation, 1997

A t-spanner of a graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times their distance in G. Spanners arise in the context of approximating the original graph by a sparse subgraph 23]. The MINIMUM t-SPANNER problem seeks to nd a t-spanner with the minimum number of edges for the given graph. In this paper, we completely settle the complexity status of this problem for various values of t, on Chordal graphs, Split graphs, Bipartite graphs and Convex Bipartite graphs. Our results settle an open question raised in 7] and also greatly simplify some of the proofs presented in 7, 8]. We also give a factor two approximation algorithm for the MINIMUM 2-SPANNER problem on interval graphs. Finally, we provide approximation algorithms for the bandwidth minimization problem on Convex Bipartite graphs and Split graphs using the notion of tree spanners.

Journal of Discrete Algorithms, 2011

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs (p, r) where p is a point in the plane and r is a real number. The distance between two points (p i , r i) and (p j , r j) is defined as |p i p j | − r i − r j. We show that in the case where all r i are positive numbers and |p i p j | ≥ r i + r j for all i, j (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a (1 +)-spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has constant spanning ratio. The straight line embedding of the Additively Weighted Delaunay graph may not be a plane graph. We show how to compute a plane embedding that also has a constant spanning ratio.