arXiv subjects
Publications and source records attributed to Asish Mukhopadhyay.
In this paper, we report on a study of the road networks of the provinces of Canada as complex networks. A number of statistical features have been analyzed and compared with two random models. In addition, we have also studied the resilience of these road networks under random failures and targeted attacks.
In this note, we consider the problem of generating $k$-factorable graphic sequences with connected (resp. no connected) $k$-factors.
In this paper, we show that unswitchable graphs are a proper subclass of split graphs, and exploit this fact to propose efficient algorithms for their recognition and generation.
We propose a scheme for generating a weakly chordal graph from a randomly generated input graph, G = (V, E). We reduce G to a chordal graph H by adding fill-edges, using the minimum vertex degree heuristic. Since H is necessarily a weakly chordal graph, we use an algorithm for deleting edges from a weakly chordal graph that preserves the weak chordality property of H. The edges that are candidates for deletion are the fill-edges that were inserted into G. In order to delete a maximal number of fill-edges, we maintain these in a queue. A fill-edge is removed from the front of the queue, which we then try to delete from H. If this violates the weak chordality property of H, we reinsert this edge at the back of the queue. This loop continues till no more fill-edges can be removed from H. Operationally, we implement this by defining a deletion round as one in which the edge at the back of the queue is at the front.We stop when the size of the queue does not change over two successive deletion rounds and output H.
There is an extensive literature on dynamic algorithms for a large number of graph theoretic problems, particularly for all varieties of shortest path problems. Germane to this paper are a number fully dynamic algorithms that are known for chordal graphs. However, to the best of our knowledge no study has been done for the problem of dynamic algorithms for strongly chordal graphs. To address this gap, in this paper, we propose a semi-dynamic algorithm for edge-deletions and a semi-dynamic algorithm for edge-insertions in a strongly chordal graph, $G = (V, E)$, on $n$ vertices and $m$ edges. The query complexity of an edge-deletion is $O(d_u^2d_v^2 (n + m))$, where $d_u$ and $d_v$ are the degrees of the vertices $u$ and $v$ of the candidate edge $\{u, v\}$, while the query-complexity of an edge-insertion is $O(n^2)$.
Strongly chordal graphs are a subclass of chordal graphs. Farber also established a number of different characterisations for this class of graphs. These include an intersection graph characterisation that is analogous to a similar characterisation for chordal graphs. Seker et al. exploited this characterisation of chordal graphs to obtain an algorithm for generating them. In this paper, we propose an algorithm to show that strongly chordal graphs can also be generated using their intersection graph characterisation.
We propose a scheme for generating a weakly chordal graph on n vertices with m edges. In this method, we first construct a tree and then generate an orthogonal layout (which is a weakly chordal graph on the n vertices) based on this tree. In the next and final step, we insert additional edges to give us a weakly chordal graph on m edges. Our algorithm ensures that the graph remains weakly chordal after each edge is inserted. The time complexity of an insertion query is O(n^3) time and an insertion takes constant time. On the other hand, a generation algorithm based on finding a 2-pair takes O(nm) time using the algorithm of Arikati and Rangan [1].
Strongly chordal graphs are a subclass of chordal graphs. The interest in this subclass stems from the fact that many problems which are NP-complete for chordal graphs are solvable in polynomial time for this subclass. However, we are not aware of any algorithm that can generate instances of this class, often necessary for testing purposes. In this paper, we address this issue. Our algorithm first generates chordal graphs, using an available algorithm and then adds enough edges to make it strongly chordal, unless it is already so. The edge additions are based on a totally balanced matrix characterizations of strongly chordal graphs.
Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless sensor network (WSN) is an effective way to save energy and reduce the impact of broadcasting storms. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. This could be modeled as a k-connected, m-fold dominating set ((k,m)-CDS). Given a virtual undirected network G=(V,E), a subset C\subset V is a (k,m)-CDS of G if (i) G[C], the subgraph of G induced by C is k-connected, and (ii) each node in V\C has at least m neighbors in C. We present a two-phase greedy algorithm for computing a (2,2)-CDS that achieves an asymptotic approximation factor of $(3+\ln(Δ+2))$, where $Δ$ is the maximum degree of G. This result improves on the previous best known performance factor of $(4+\lnΔ+2\ln(2+\lnΔ))$ for this problem.
Phylogenetic trees are leaf-labelled trees, where the leaves correspond to extant species (taxa), and the internal vertices represent ancestral species. The evolutionary history of a set of species can be explained by more than one phylogenetic tree, giving rise to the problem of comparing phylogenetic trees for similarity. Various distance metrics, like the subtree prune-and-regraft (SPR), tree bisection reconnection (TBR) and nearest neighbour interchange (NNI) have been proposed to capture this similarity. The distance between two phylogenetic trees can also be measured by the size of a Maximum Agreement Forest (MAF) on these trees, as it has been shown that the rooted subtree prune-and-regraft distance is 1 less than the size of a MAF. Since computing a MAF of minimum size is an NP-hard problem, approximation algorithms are of interest. Recently, it has been shown that the MAF on k(>=2) trees can be approximated to within a factor of 8. In this paper, we improve this ratio to 3. For certain species, however, the evolutionary history is not completely tree-like. Due to reticulate evolution two gene trees, though related, appear different, making a phylogenetic network a more appropriate representation of reticulate evolution. A phylogenetic network contains hybrid nodes for the species evolved from two parents. The number of such nodes is its hybridization number. It has been shown that this number is 1 less than the size of a Maximum Acyclic Agreement Forest (MAAF). We show that the MAAF for k(>= 2) phylogenetic trees can be approximated to within a factor of 3.
In this paper, we discuss the algorithm engineering aspects of an O(n^2)-time algorithm [6] for computing a minimum-area convex polygon that intersects a set of n isothetic line segments.
In this paper we report on an algorithm for aligning multiple protein structures. The algorithm has been tested on a variety of inputs and it performs well in comparison to well-known algorithms for this problem.
The point placement problem is to determine the positions of a set of $n$ distinct points, P = {p1, p2, p3, ..., pn}, on a line uniquely, up to translation and reflection, from the fewest possible distance queries between pairs of points. Each distance query corresponds to an edge in a graph, called point placement graph ppg, whose vertex set is P. The uniqueness requirement of the placement translates to line rigidity of the ppg. In this paper we show how to construct in 2 rounds a line rigid point placement graph of size 9n/7 + O(1). This improves the existing best result of 4n/3 + O(1). We also improve the lower bound on 2-round algorithms from 17n/16 to 9n/8.
We present algorithms for length-constrained maximum sum segment and maximum density segment problems, in particular, and the problem of finding length-constrained heaviest segments, in general, for a sequence of real numbers. Given a sequence of n real numbers and two real parameters L and U (L <= U), the maximum sum segment problem is to find a consecutive subsequence, called a segment, of length at least L and at most U such that the sum of the numbers in the subsequence is maximum. The maximum density segment problem is to find a segment of length at least L and at most U such that the density of the numbers in the subsequence is the maximum. For the first problem with non-uniform width there is an algorithm with time and space complexities in O(n). We present an algorithm with time complexity in O(n) and space complexity in O(U). For the second problem with non-uniform width there is a combinatorial solution with time complexity in O(n) and space complexity in O(U). We present a simple geometric algorithm with the same time and space complexities. We extend our algorithms to respectively solve the length-constrained k maximum sum segments problem in O(n+k) time and O(max{U, k}) space, and the length-constrained $k$ maximum density segments problem in O(n min{k, U-L}) time and O(U+k) space. We present extensions of our algorithms to find all the length-constrained segments having user specified sum and density in O(n+m) and O(nlog (U-L)+m) times respectively, where m is the number of output. Previously, there was no known algorithm with non-trivial result for these problems. We indicate the extensions of our algorithms to higher dimensions. All the algorithms can be extended in a straight forward way to solve the problems with non-uniform width and non-uniform weight.
Given a finite set of non-collinear points in the plane, there exists a line that passes through exactly two points. Such a line is called an ordinary line. An efficient algorithm for computing such a line was proposed by Mukhopadhyay et al. In this note we extend this result in two directions. We first show how to use this algorithm to compute an ordinary conic, that is, a conic passing through exactly five points, assuming that all the points do not lie on the same conic. Both our proofs of existence and the consequent algorithms are simpler than previous ones. We next show how to compute an ordinary hyperplane in three and higher dimensions.