Surender Baswana

Bio: Surender Baswana is an academic researcher from Indian Institute of Technology Kanpur. The author has contributed to research in topics: Time complexity & Directed graph. The author has an hindex of 23, co-authored 69 publications receiving 1759 citations. Previous affiliations of Surender Baswana include Indian Institutes of Technology & Max Planck Society.

A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs

Abstract: Let G = (V,E) be an undirected weighted graph on |V | = n vertices and |E| = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V,ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well-motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43-year old girth lower bound conjecture made independently by Erdos, Bollobas, and Bondy & Simonovits. Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information, which involves growing either breadth first search trees up to t(t)-levels or full shortest path trees on a large fraction of vertices. The truly local approach of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 Preliminary version of this work appeared in the 30th International Colloquium on Automata, Languages and Programming, pages 384–396, 2003.

Additive spanners and (α, β)-spanners

Abstract: 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(n1+1/k) and an (additive) (1,2)-spanner of size O(n3/2). 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(n4/3). 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(kn1+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.

Approximate distance oracles for unweighted graphs in expected O(n2) time

Abstract: Let G = (V, E) be an undirected graph on n vertices, and let δ(u, v) denote the distance in G between two vertices u and v. Thorup and Zwick showed that for any positive integer t, the graph G can be preprocessed to build a data structure that can efficiently report t-approximate distance between any pair of vertices. That is, for any u, v ∈ V, the distance reported is at least δ(u, v) and at most tδ(u, v). The remarkable feature of this data structure is that, for t≥3, it occupies subquadratic space, that is, it does not store all-pairs distances explicitly, and still it can answer any t-approximate distance query in constant time. They named the data structure “approximate distance oracle” because of this feature. Furthermore, the trade-off between the stretch t and the size of the data structure is essentially optimal.In this article, we show that we can actually construct approximate distance oracles in expected O(n2) time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first expected linear-time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph. A (2, 1) spanner of an undirected unweighted graph G = (V, E) is a subgraph (V, E), E ⊆ E, such that for any two vertices u and v in the graph, their distance in the subgraph is at most 2δ(u, v) p 1.

Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths

Abstract: Let G = (V,E) be a weighted undirected graph with |V | = n and |E| = m. An estimate \hat \delta \left( {u,v} \right) of the distance \delta \left( {u,v} \right) in G between u, v \in V is said to be of stretch t iff \delta \left( {u,v} \right) \leqslant \hat \delta \left( {u,v} \right) \leqslant t ? \delta \left( {u,v} \right). The most efficient algorithms known for computing small stretch distances in G are the approximate distance oracles of [16] and the three algorithms in [9] to compute all-pairs stretch t distances for t = 2, 7/3, and 3. We present faster algorithms for these problems. For any integer k \geqslant 1, Thorup and Zwick in [16] gave an O(kmn^{1/k}) algorithm to construct a data structure of size O(kn^{1+1/k}) which, given a query (u, v) \in V ? V , returns in O(k) time, a 2k - 1 stretch estimate of \delta \left( {u,v} \right). But for small values of k, the time to construct the oracle is rather high. Here we present an O(n^2 log n) algorithm to construct such a data structure of size O(kn^{1+1/k}) for all integers k \geqslant 2. Our query answering time is O(k) for k \ge 2 and \Theta (log n) for k = 2. We use a new generic scheme for all-pairs approximate shortest paths for these results. This scheme also enables us to design faster algorithms for allpairs t-stretch distances for t = 2 and 7/3, and compute all-pairs almost stretch 2 distances in O(n^2 log n) time.

A simple linear time algorithm for computing a (2k - 1)-spanner of o(n 1+1/k ) size in weighted graphs

Abstract: Let G(V, E) be an undirected weighted graph with |V| = n, and |E| = m A t-spanner of the graph G(V, E) is a sub-graph G(V, ES) such that the distance between any pair of vertices in the spanner is at most t times the distance between the two in the given graph A 1963 girth conjecture of Erdos implies that Ω(n1+1/k) edges are required in the worst case for any (2k - 1)-spanner, which has been proved for k = 1, 2, 3, 5 There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn1/k) expected running time In this paper, we present an extremely simple linear time randomized algorithm that constructs (2k - 1)-spanner of size matching the conjectured lower bound Our algorithm requires local information for computing a spanner, and thus can be adapted suitably to obtain efficient distributed and parallel algorithms

Flows in Networks

Abstract: In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Approximate distance oracles

Abstract: Let G = (V,E) be an undirected weighted graph with vVv = n and vEv = m. Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O(kmn1/k) expected time, constructing a data structure of size O(kn1p1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k−1, that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdos, implies that Ω(n1p1/k) space is needed in the worst case for any real stretch strictly smaller than 2kp1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O(n1p1/k) space had a query time of Ω(n1/k).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.

Approximate distance oracles

Abstract: Let G=(V,E) be an undirected weighted graph with |V|=n and |E|=m. Let k\ge 1 be an integer. We show that G=(V,E) can be preprocessed in O(kmn^{1/k}) expected time, constructing a data structure of size O(kn^{1+1/k}), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k-1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k-1. We show that a 1963 girth conjecture of Erd{\H{o}}s, implies that ω(n^{1+1/k}) space is needed in the worst case for any real stretch strictly smaller than 2k+1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name oracle. Previously, data structures that used only O(n^{1+1/k}) space had a query time of ω(n^{1/k}) and a slightly larger, non-optimal, stretch. Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.}

Graph stream algorithms: a survey

Abstract: Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was two-fold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the state-of-the-art results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas.

The DBLP Computer Science bibliography: Evolution, research issues, perspectives

Abstract: Publications are essential for scientific communication. Access to publications is provided by conventional libraries, digital libraries operated by learned societies or commercial publishers, and a huge number of web sites maintained by the scientists themselves or their institutions. Comprehensive meta-indices for this increasing number of information sources are missing for most areas of science. The DBLP Computer Science Bibliography of the University of Trier has grown from a very specialized small collection of bibliographic information to a major part of the infrastructure used by thousands of computer scientists. This short paper first reports the history of DBLP and sketches the very simple software behind the service. The most time-consuming task for the maintainers of DBLP may be viewed as a special instance of the authority control problem; how to normalize different spellings of person names. The third section of the paper discusses some details of this problem which might be an interesting research issue for the information retrieval community.