Krishnaiyan Thulasiraman

Polynomial time approximation algorithms for multi-constrained QoS routing

We study the multi-constrained quality-ofservice (QoS) routing problem where one seeks to find a path from a source to a destination in the presence of <i>K</i> ≥ 2 additive end-to-end QoS constraints. This problem is NP-hard and is commonly modeled using a graph with <i>n</i> vertices and <i>m</i> edges with <i>K</i> additive QoS parameters associated with each edge. For the case of <i>K</i> = 2, the problem has been well studied, with several provably good polynomial time-approximation algorithms reported in the literature, which enforce one constraint while approximating the other. We first focus on an optimization version of the problem where we enforce the first constraint and approximate the other <i>K</i> - 1 constraints. We present an <i>O</i>(<i>mn</i> log log <i>n</i> + <i>mn</i>/ε) time (1 + ε) (<i>K</i> - 1) -approximation algorithm and an <i>O</i>(<i>mn</i> log log <i>n</i> + <i>m</i>(<i>n</i>/ε)^<i>K</i>-1) time (1 + ε) -approximation algorithm, for any ε > 0. When <i>K</i> is reduced to 2, both algorithms produce an (1 + ε) -approximation with a time complexity better than that of the best-known algorithm designed for this special case. We then study the decision version of the problem and present an <i>O</i>(<i>m</i>(<i>n</i>/ε)^<i>K</i>-1) time algorithm which either finds a feasible solution or confirms that there does not exist a source-destination path whose first weight is bounded by the first constraint and whose every other weight is bounded by (1 - ε) times the corresponding constraint. If there exists an <i>H</i>-hop source-destination path whose first weight is bounded by the first constraint and whose every other weight is bounded by (1 - ε) times the corresponding constraint, our algorithm finds a feasible path in <i>O</i>(<i>m</i>(<i>H</i>/ε)^<i>K</i>-1 time. This algorithm improves previous best-known algorithms with <i>O</i>((<i>m</i> + <i>n</i> log <i>n</i>)<i>n</i>/ε) time for <i>K</i> = 2 and <i>O</i>(<i>mn</i>(<i>n</i>/ε)^<i>K</i>-1) time for <i>K</i> ≥ 2.

Quality-of-service and quality-of-protection issues in preplanned recovery schemes using redundant trees

We study quality-of-service (QoS) and quality-of-protection (QoP) issues in redundant tree based preplanned recovery schemes for a single-link failure in two-edge connected graphs and for a single-node failure in two-connected graphs. We present schemes (to be called G-MFBG schemes) that generalize the schemes (to be called MFBG schemes) developed by Me/spl acute/dard et al. (1997) to construct a pair of redundant trees, called red and blue trees, which guarantees fast recovery from any single-link/node failure, as long as the failed node is not the root node. Using the G-MFBG schemes, we study QoS issues relating to red/blue trees. We present effective heuristics for computing a pair of redundant trees with low average delay or small total cost. We develop an optimal algorithm for computing a pair of red/blue trees with maximum bandwidth. Furthermore, a pair of red/blue trees guarantees fast recovery from simultaneous multiple failures if it satisfies certain properties. This leads us to define the concept of QoP of a pair of red/blue trees. We present an effective heuristic to construct a pair of red/blue trees with high QoP. The paper concludes with a discussion of computational results that demonstrate the effectiveness of the different algorithms presented.

Finding a path subject to many additive QoS constraints

A fundamental problem in quality-of-service (QoS) routing is to find a path between a source-destination node pair that satisfies two or more end-to-end QoS constraints. We model this problem using a graph with n vertices and m edges with K additive QoS parameters associated with each edge, for any constant K ≥ 2. This problem is known to be NP-hard. Fully polynomial time approximation schemes (FPTAS) for the case of K = 2 have been reported in the literature. We concentrate on the general case and make the following contributions. 1) We present a very simple O(Km + nlogn) time K -approximation algorithm that can be used in hop-by-hop routing protocols. 2) We present an FPTAS for one optimization version of the QoS routing problem with a time complexity of O(m(n/e)K-1).3) We present an FPTAS for another optimization version of the QoS routing problem with a time complexity of O(n log n + m(H/e)K-1) when there exists an H-hop path satisfying all QoS constraints. When K is reduced to 2, our results compare favorably with existing algorithms. The results of this paper hold for both directed and undirected graphs. For ease of presentation, undirected graph is used.

O(n/sup 2/) algorithms for graph planarization

The authors present two O(n/sup 2/) planarization algorithms, PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on A. Lempel, S. Even, and I. Cederbaums (1967) planarity testing algorithm and its implementation using PQ-trees. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. The algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G/sub p/ of a nonplanar graph G, the MAXIMAL-PLANARIZE algorithm constructs a maximal planar subgraph of G which contains G/sub p/. This latter algorithm can also be used to planarize maximally a biconnected planar graph. >

Conditional Diagnosability of Matching Composition Networks Under the PMC Model

In the work of Lai in 2005, they proposed a new measure for fault diagnosis of systems, namely, conditional diagnosability. It assumes that no fault set can contain all the neighbors of any vertex in the system. In the same paper, they showed that the conditional diagnosability of hypercube Qn is 4(n - 2) + 1 for n ges 5. In this brief, we generalize this result by considering a family of more popular networks, namely, matching composition networks (MCNs), which are a class of networks composed of two components of the same order linked by a perfect matching under PMC (Preparata, Metze and Chien) model. We determine in Theorem 7 the conditional diagnosability for some MCNs, from which we deduce that the hypercube Qn, the crossed cube CQn, the twisted cube TQn, and the MOumlbius cube MQn all have the same conditional diagnosability of 4(n - 2) + 1 for n ges 5. We show that the bijective connection (BC) networks in the work of Fan and He in 2003 and the work of Zhu in 2008 satisfy the conditions of Theorem 7, and thus, our conditional diagnosability result also applies to BC networks. Finally, we show that the MCNs satisfying the conditions of Theorem 7 are more general than the BC networks.

Multi-level Cooperative Search: A New Paradigm for Combinatorial Optimization and an Application to Graph Partitioning

Cooperative search is a parallelization strategy for search algorithms where parallelism is obtained by concurrently executing several search programs. The solution space is implicitly decomposed according to the search strategy of each program. The programs cooperate by exchanging information on previously explored regions of the solution space. In this paper we propose a new design for cooperative search algorithms which is also a new parallel problem solving paradigm for combinatorial optimization problems. Our new design is based on an innovative approach to decompose the solution space which is inspired from the modeling of cooperative algorithms based on dynamical systems theory. Our design also gives a new purpose to the sharing of information among cooperating tasks based on principles borrowed from scatter search evolutionary algorithms. We have applied this paradigm to the graph partitioning problem. We describe the parallel implementation of this algorithm on a cluster of workstations and compare our results with other well known graph partitioning methods.

Circuits/Cutsets Duality and a Unified Algorithmic Framework for Survivable Logical Topology Design in IP-over-WDM Optical Networks

Given a logical topology G L and a physical topology G, the survivable logical topology design problem in an IP-over- WDM optical network is to map the logical links into lightpaths in G such that G L remains connected after the failure of any edge in G. In view of its fundamental nature and its practical importance, this problem has received considerable attention in the literature. The SMART algorithmic framework based on the circuits in G L is a novel and very significant contribution to this problem. Taking advantage of the dual relationship between circuits and cutsets in a graph, we first present in this paper the primal algorithm CIRCUIT-SMART (similar to SMART) and algorithm CUTSET-SMART that is dual of CIRCUIT-SMART and proofs of correctness of these algorithms. To guarantee survivability we add additional logical links called protection edges, if necessary. This investigation has provided much insight into the structural properties of solutions to this problem and the structure of survivable logical graphs. Specifically, we present a highly simplified version of CUTSET-SMART that always provides a survivable mapping as long as G is 3-edge connected, and a survivable logical topology structure. We also present algorithm INCIDENCE-SMART that uses incidence sets that are special cases of a cut. Two efficient heuristics, one based on maximum matching theory and the other based on both the primal and dual algorithms are also presented. Simulation results comparing the different algorithms in terms of computational time, protection capacity and survivability success rate are also presented.

Linear time construction of redundant trees for recovery schemes enhancing QoP and QoS

Medard, Finn, Barry and Gallager proposed an elegant recovery scheme (known as the MFBG scheme) using redundant trees. Xue, Chen and Thulasiraman extended the MFBG scheme and introduced the concept of quality of protection (QoP) as a metric of multifailure recovery capabilities for single failure recovery schemes. In this paper, we present three linear time algorithms for constructing redundant trees for single link failure recovery in 2-edge connected graphs and for single node failure recovery in 2-connected graphs. Our first algorithm aims at high QoP for single link recovery schemes in 2-edge connected graphs. The previous best algorithm has a running time of O(n/sup 2/(m+n)), where n and m are the number of nodes and links in the network. Our algorithm has a running time of O(m+n), with comparable performance. Our second algorithm aims at high QoS for single link recovery schemes in 2-edge connected graphs. Our algorithm improves the previous best algorithm with O(n/sup 2/(m+n)) time complexity to O(m+n) time complexity with comparable performance. Our third algorithm aims at high QoS for single node recovery schemes in 2-connected graphs. Again, our algorithm improves the previous best algorithm with O(n/sup 2/(m+n)) time complexity to O(m+n) time complexity with comparable performance. Simulation results show that our new algorithms outperform previously known linear time algorithms significantly in terms of QoP or QoS, and outperform other known algorithms in terms of running time, with comparable QoP of QoS performance.

A time-optimal message-efficient distributed algorithm for depth-first-search

Abstract In this paper we study the problem of distributed construction of a depth-first-search tree for an asynchronous communication network. First, we point out that any algorithm requires at least 2n−2 time units and 2m messages in the worst case, where n and m are the number of nodes and the number of edges in the network, respectively. We then provide a modification to a recent algorithm due to Awerbuch (1985), and show that the new algorithm is time-optimal, while requiring less then 4m−(n−1) messages.

Faster algorithms for construction of recovery trees enhancing QoP and QoS

Médard et al. proposed an elegant recovery scheme (known as the MFBG scheme) using red/blue recovery trees for multicast path protection against single link or node failures. Xue et al. extended the MFBG scheme and introduced the concept of quality of protection (QoP) as a metric for multifailure recovery capabilities of single failure recovery schemes. They also presented polynomial time algorithms to construct recovery trees with good QoP and quality of service (QoS). In this paper, we present faster algorithms for constructing recovery trees with good QoP and QoS performance. For QoP enhancement, our <i>O</i>(<i>n</i> + <i>m</i>) time algorithm has comparable performance with the previously best <i>O</i>(<i>n</i>²(<i>n</i> + <i>m</i>)) time algorithm, where and denote the number of nodes and the number of links in the network, respectively. For cost reduction, our <i>O</i>(<i>n</i> + <i>m</i>) time algorithms have comparable performance with the previously best <i>O</i>(<i>n</i>²(<i>n</i> + <i>m</i>)) time algorithms. For bottleneck bandwidth maximization, our <i>O</i>(<i>m</i>log <i>n</i>) time algorithms improve the previously best <i>O</i>(<i>nm</i>) time algorithms. Simulation results show that our algorithms significantly outperform previously known algorithms in terms of running time, with comparable QoP or QoS performance.