Papers by Surender Baswana
INFORMS Journal on Applied Analytics
Talg, 2010
An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a m... more 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.
Proceedings of the Nineteenth Annual Acm Siam Symposium on Discrete Algorithms, 2008
Page 1. Fully Dynamic Algorithm for Graph Spanners with Poly-Logarithmic Update Time ... Our firs... more Page 1. Fully Dynamic Algorithm for Graph Spanners with Poly-Logarithmic Update Time ... Our first algorithm achieves expected O(7 t 4 ) time per update independent of the size of the graph. This algorithm is particularly of interest for maintaining small stretch spanners. ...
Proceedings of the Sixteenth Annual Acm Siam Symposium on Discrete Algorithms, 2005
An (α, β)-spanner of an unweighted graph G is a subgraph H that approximates distances in G in th... more 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: ΔH (u, v) ≤ αΔG(u, v) + β, where ΔG is the distance w.r.t. G. It is well known that there exist (multiplicative) (2k - 1, 0)-spanners of size O(n1+1/k) and that there
Encyclopedia of Algorithms, 2016
Acm Transactions on Algorithms, Oct 1, 2006
Acm Transactions on Algorithms, 2010
Icalp, 2003
Let be an undirected weighted graph with , and . A -spanner of the graph is a sub-graph , , such ... more Let be an undirected weighted graph with , and . A -spanner of the graph is a sub-graph , , such that the distance between any pair of vertices in the sub-graph is at most times the distance between the two in the given graph. A 1963 girth conjecture of Erd˝ os implies that edges are required in the worst case for any -spanner, which has been proved for . There exist polynomial time algorithms that can compute spanners with size matching the conjectured lower bound, and the previously best known algorithm takes expected running time. In this paper, we present an extremely simple linear time randomized algorithm that computes a -spanner of size matching the conjectured lower bound. Note that linear time is a severe limitation since it implies that only a negligible f raction of the distances can be exploited in the construction of the spanner. Nonetheless, all distances must be faithfully approximated. An important feature of our algorithm is its local approach. Unlike all the previous algorithms that require computation of shortest paths, the new algorithm merely explores the edges in the neighborhood of a vertex or a group of vertices. This feature leads to desig ning simple external-memory and parallel algorithms for computing sparse spanners, whose running times are near optimal.
Corr, Nov 6, 2006
Given an undirected graph G = (V, E) on n vertices, m edges, and an integer t ≥ 1, a subgraph (V,... more Given an undirected graph G = (V, E) on n vertices, m edges, and an integer t ≥ 1, a subgraph (V, E S), E S ⊆ E is called a t-spanner if for any pair of vertices u, v ∈ V , the distance between them in the subgraph is at most t times the actual distance. We present streaming algorithms for computing a t-spanner of essentially optimal size-stretch trade offs for any undirected graph. Our first algorithm is for the classical streaming model and works for unweighted graphs only. The algorithm performs a single pass on the stream of edges and requires O(m) time to process the entire stream of edges. This drastically improves the previous best single pass streaming algorithm for computing a t-spanner which requires θ(mn 2 t) time to process the stream and computes spanner with size slightly larger than the optimal. Our second algorithm is for StreamSort model introduced by Aggarwal et al. [2], which is the streaming model augmented with a sorting primitive. The StreamSort model has been shown to be a more powerful and still very realistic model than the streaming model for massive data sets applications. Our algorithm, which works of weighted graphs as well, performs O(t) passes using O(log n) bits of working memory only. Our both the algorithms require elementary data structures.
Stacs, 2010
Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertic... more Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data structure for a given graph G which is capable of answering the following query for any u, v, z ∈ V and t > 1. report t-approximate shortest path between u and v when vertex z fails We present data structures for the single source as well all-pairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size nearly match the size of their best static counterparts.
Lecture Notes in Computer Science, 2015
Let G = (V, E) be an n-vertices m-edges directed graph. Let s ∈ V be any designated source vertex... more Let G = (V, E) be an n-vertices m-edges directed graph. Let s ∈ V be any designated source vertex, and let T be an arbitrary reachability tree rooted at s. We address the problem of finding a set of edges E ⊆ E\T of minimum size such that on a failure of any vertex w ∈ V , the set of vertices reachable from s in T ∪ E \{w} is the same as the set of vertices reachable from s in G\{w}. We obtain the following results: • The optimal set E for any arbitrary reachability tree T has at most n − 1 edges. • There exists an O(m log n)-time algorithm that computes the optimal set E for any given reachability tree T. For the restricted case when the reachability tree T is a Depth-First-Search (DFS) tree it is straightforward to bound the size of the optimal set E by n − 1 using semidominators with respect to DFS trees from the celebrated work of Lengauer and Tarjan [13]. Such a set E can be computed in O(m) time using the algorithm of Buchsbaum et. al [4]. To bound the size of the optimal set in the general case we define semidominators with respect to arbitrary trees. We also present a simple O(m log n) time algorithm for computing such semidominators. As a byproduct, we get an alternative algorithm for computing dominators in O(m log n) time.
Encyclopedia of Algorithms, 2014
Given an undirected graph G = (V, E) on n vertices, m edges, and an integer t ≥ 1, a subgraph (V,... more Given an undirected graph G = (V, E) on n vertices, m edges, and an integer t ≥ 1, a subgraph (V, E S), E S ⊆ E is called a t-spanner if for any pair of vertices u, v ∈ V , the distance between them in the subgraph is at most t times the actual distance. We present streaming algorithms for computing a t-spanner of essentially optimal size-stretch trade offs for any undirected graph. Our first algorithm is for the classical streaming model and works for unweighted graphs only. The algorithm performs a single pass on the stream of edges and requires O(m) time to process the entire stream of edges. This drastically improves the previous best single pass streaming algorithm for computing a t-spanner which requires θ(mn 2 t) time to process the stream and computes spanner with size slightly larger than the optimal. Our second algorithm is for StreamSort model introduced by Aggarwal et al. [2], which is the streaming model augmented with a sorting primitive. The StreamSort model has been shown to be a more powerful and still very realistic model than the streaming model for massive data sets applications. Our algorithm, which works of weighted graphs as well, performs O(t) passes using O(log n) bits of working memory only. Our both the algorithms require elementary data structures.
We present a fully dynamic algorithm for maintaining approximate maximum weight matching in gener... more We present a fully dynamic algorithm for maintaining approximate maximum weight matching in general weighted graphs. The algorithm maintains a matching M whose weight is at least 1 8 M * where M * is the weight of the maximum weight matching. The algorithm achieves an expected amortized O(log n log C) time per edge insertion or deletion, where C is the ratio of the weights of the highest weight edge to the smallest weight edge in the given graph. Using a simple randomized scaling technique, we are able to obtain a matching whith expected approximation ratio 4.9108.
Computing Research Repository, 2010
Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path $\textbf{P}$ between ... more Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path $\textbf{P}$ between any two vertices $u,v\in V$ is said to be $t$-approximate shortest path if its length is at most $t$ times the length of the shortest path between $u$ and $v$. We consider the problem of building a compact data structure for a given graph
Lecture Notes in Computer Science, 2006
ABSTRACT Let G=(V,E) be an undirected weighted graph on |V|=n vertices and |E|=m edges. For the g... more ABSTRACT Let G=(V,E) be an undirected weighted graph on |V|=n vertices and |E|=m edges. For the graph G, A spanner with stretch t∈ℕ is a subgraph (V,E S ), E S ⊆ E, such that the distance between any pair of vertices in this subgraph is at most t times the distance between them in the graph G. We present simple and efficient dynamic algorithms for maintaining spanners with essentially optimal (expected) size versus stretch trade-off for any given unweighted graph. The main result is a decremental algorithm that takes expected O(polylog n)O(\mathop{\mathrm{polylog}} n) time per edge deletion for maintaining a spanner with arbitrary stretch. This algorithm easily leads to a fully dynamic algorithm with sublinear (in n) time per edge insertion or deletion. Quite interestingly, this paper also reports that for stretch at most 6, it is possible to maintain a spanner fully dynamically with expected constant time per update. All these algorithms use simple randomization techniques on the top of an existing static algorithm [6] for computing spanners, and achieve drastic improvement over the previous best deterministic dynamic algorithms for spanners.
Information Processing Letters, 2008
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Papers by Surender Baswana