We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O(m + n log n +kn). We describe applications to dynamic programming problems including the knapsack problem, sequence alignment, maximum inscribed polygons, and genealogical relationship discovery. 1 Introduction We consider a long-studied generalization of the shortest path problem, in which not one but several short paths must be produced. The k shortest paths problem is to list the k paths connecting a given source-destination pair in the digraph with minimum total length. Our techniques also apply to the problem of listing all paths shorter than some given threshhold length. In the version of these problems studi...

Recent advances in wireless sensor networks have led to many new routing protocols specifically designed for sensor networks. Almost all of these routing protocols considered energy efficiency as the ultimate objective in order to maximize the whole network lifetime. However, the introduction of video and imaging sensors has posed additional challenges. Transmission of video and imaging data requires both energy and QoS aware routing in order to ensure efficient usage of the sensors and effective access to the gathered measurements. In this paper, we propose an energy-aware QoS routing protocol for sensor networks which can also run efficiently with best-effort traffic. The protocol finds a least-cost, delay-constrained path for real-time data in terms of link cost that captures nodes ’ energy reserve, transmission energy, error rate and other communication parameters. Moreover, the throughput for non-real-time data is maximized by adjusting the service rate for both real-time and non-real-time data at the sensor nodes. Simulation results have demonstrated the effectiveness of our approach for different metrics. 1.

: In this paper, the optimal path problem will be studied from a global point of view and having no restrictions imposed on the network. The concepts of boundness and finiteness will be presented for the general problem and will be studied in two particular cases. Special emphasis will be given to the optimality principle since it allows one to design a class of algorithms -- the labelling algorithms -- which determine an optimal path when the weak optimality principle is satisfied. Its importance will be stressed by two problems which, in its turn, are similar in their description and completly different in what concerns their resolution. Keywords: network, path, optimality principle, labelling algorithms. 1 Introduction In the optimal path problem, a real function is considered which assigns a value to each path that can be defined between a given pair of nodes in a given network; a path with the best value in a subset of paths between that pair of nodes is what has to be determin...

: Ranking shortest paths is a classical network problem consisting of the determination of the K shortest paths connecting a given initial--destination pair of nodes such as the distance of the k th path that is determined is no greater than the distance of any j th one, for some j ? k . In this paper an algorithm for ranking paths is reviewed being its complexity improved in terms of the required memory space. This improvement allows the ranking of really larger problems in reasonably small execution times, which is comproved by the presented computational experiments. In fact, for randomly generated networks with 10000 nodes, the algorithm ranks more than half of a million of paths in a few cents of seconds of CPU execution time. This performance can be very important because of the potential practical applications of the problem, nameley as a subproblem of the constrained shortest path problem and of the multiobjective shortest path problem. Keywords: network, tree, path, pat...

The shortest path problem is a classical network problem that has been extensively studied. The problem of determining not only the shortest path, but also listing the K shortest paths (for a given integer K ? 1) is also a classical one but has not been studied so intensively, despite its obvious practical interest. Two different types of problems are usually considered: the unconstrained and the constrained K shortest paths problem. While in the former no restriction is considered in the definition of a path, in the constrained K shortest paths problem all the paths have to satisfy some condition -- for example, to be loopless. In this paper new algorithms are proposed for the unconstrained problem, which compute a super set of the K shortest paths. It is also shown that ranking loopless paths does not hold in general the Optimality Principle and how the proposed algorithms for the unconstrained problem can be adapted for ranking loopless paths. Keywords: Network, tree, path, path d...

. It is widely accepted that the optimal alignmentbetween a pair of proteins or nucleic acid sequences that minimizes the edit distance may not necessarily re#ect the correct biological alignment. Alignments of proteins based on their structures or of DNA sequences based on evolutionary changes are often di#erent from alignments that minimize edit distance. However, in many cases #e.g. when the sequences are close#, the edit distance alignment is a good approximation to the biological one. Since, for most sequences, the true alignment is unknown, a method that either assesses the signi#cance of the optimal alignment, or that provides few #close" alternatives to the optimal one, is of great importance. A suboptimal alignment is an alignment whose score lies within the neighborhood of the optimal score. Enumeration of suboptimal alignments #Wa83, WaBy# is not very practical since there are many such alignments. Other approaches #Zuk, Vi, ViAr# that use only partial informat...

: In this paper an algorithm for the ranking of loopless paths problem is proposed which is valid for directed and undirected networks. Despite its theoretical computational complexity be an open problem yet, the algorithm appears to perform well in practice as the reported comparative computational experiments allow us to conclude. This conclusion is reinforced with some results obtained with larger networks; more than 500; 000 loopless paths were ranked in 10; 000 nodes and 100; 000 arcs euclidian networks in about 0:35 seconds of CPU execution time when all the arcs are undirected and in about 0:15 seconds for directed euclidian networks, using a server with 128 Mbytes of RAM and a 275 MHz processor running DEC Unix 3.2. Keywords: path, loopless path, path distance, paths ranking, network. 1 Introduction Let (N ,A) denote a given network, where N = fv 1 ; : : : ; v n g is a finite set whose elements are called nodes and A = fa 1 ; : : : ; am g is a proper subset of N \The...

: In this paper the ranking of shortest paths problem is viewed as a generalization of the shortest path problem, since the K shortest paths have to be computed. Finitness and boundness are studied as well as conditions are established in order that an Optimality Principle generalization is satisfied. Under these conditions the K shortest paths can be determined by a natural generalization of forms of the labeling algorithm for the shortest path problem. This generalization is the main subject of the paper. Keywords: network, path, path cost, paths ranking, optimality principle, labeling. 1 Introduction In this paper the ranking of shortest paths is considered as a generalization of the well known shortest path problem, [5, 6, 7, 8, 9, 10, 12], since several paths must be determined. In fact, in the ranking of shortest paths problem, for a given integer K (K 1), the K shortest paths between a given pair of nodes have to be listed by nondecreasing order of their costs. Sometimes K de...

Several strategies have been proposed for routing in a low earth orbit (LEO) satellite system. Some of them are based on the Internet Protocol (IP) and a few on asynchronous transfer mode (ATM) switching. However, issues like memory requirements of the satellites in the LEO network and the overheads involved in transmitting packets over the network have tended to be ignored. This paper focuses on developing an efficient solution to the routing issue in a LEO network. It also discusses the issue of the placement of the Network Operations and Control Centers (NOCCs) required to support the satellite system. Keywords: low earth orbit satellite network, intersatellite links, snapshot, stored program model, network operations & control centers. I. INTRODUCTION Of late, low earth orbit (LEO) and medium earth orbit (MEO) based satellite networks have become the focus of attention as they promise lower delays and better biterror rate performance for Internet and multi-media services than geo...

Abstract: Traffic congestion is becoming a serious problem in more and more modern cities. Encouraging more private-vehicle drivers to use public transportation is one of the most effective and economical ways to reduce the ever increasing congestion problem on the streets (Hartley and Bargiela, 2001). To make public transport services more attractive and competitive, providing travellers with individual travel advice for journeys becomes crucial. However, with the massive and complex network of a modern city, finding one or several suitable route(s) according to user preferences from one place to another is not a simple task. In this paper, the author presents an approach that uses K-shortest paths algorithms to compute a reasonable number of ranked shortest paths, with the ultimate ‘most optimal ’ path being selected by consideration of the preferences. Some experiments have been done based on the public transportation network of Nottingham City. Keywords: User preferences, K-shortest paths algorithms, Multi-objective algorithms 1.