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...

The bounded shortest multicast algorithm (BSMA) is presented for constructing minimum-cost multicast trees with delay constraints. BSMA can handle asymmetric link characteristics and variable delay bounds on destinations, specified as real values, and minimizes the total cost of a multicast routing tree. Instead of the single-pass tree construction approach used in most previous heuristics, the new algorithm is based on a feasiblesearch optimization strategy that starts with the minimum-delay multicast tree and monotonically decreases the cost by iterative improvement of the delay-bounded multicast tree. BSMA's expected time complexity is analyzed, and simulation results are provided showing that BSMA can achieve near-optimal cost reduction with fast execution.

We prove super-linear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = {e_1, e_2, ..., e_p} between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges e_i. We show that the replacement paths problem requires Ω(m√n) time in the worst case whenever m = O(n√n). This also establishes a similar...

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...

Efficient management of networks requires that the shortest route from one point (node) to another is known; this is termed the shortest path. It is often necessary to be able to determine alternative routes through the network, in case any part of the shortest path is damaged or busy. The k-shortest paths represent an ordered list of the alternative routes available. Four algorithms were selected for more detailed study from over seventy papers written on this subject since the 1950's. These four were implemented in the `C' programming language and, on the basis of the results, an assessment was made of their relative performance. 1 The Background The shortest path through a network is the least cost route from a given node to another given node, and this path will usually be the preferred route between those two nodes. When the shortest path between two nodes is not available for some reason, it is necessary to determine the second shortest path. If this too is not available, a thir...

Let G = (V (G), E(G)) be a weighted directed graph and let P be a shortest path from s to t in G. In the replacement paths problem we are required to compute for every edge e in P, the length of a shortest path from s to t that avoids e. The fastest known algorithm for solving the problem in weighted directed graphs is the trivial one: each edge in P is removed from the graph in its turn and the distance from s to t in the modified graph is computed. The running time of this algorithm is O � mn + n2 log n � , where n = |V (G) | and m = |E(G)|. The replacement paths problem is strongly motivated by two different applications. First, the fastest algorithm to compute the k simple shortest paths from s to t in directed graphs [21, 13] repeatedly computes the replacement paths from s to t. Its running time is O(kn(m + n log n)). Second, the computation of Vickrey pricing of edges in distributed networks can be reduced to the replacement paths problem. An open question raised by Nisan and Ronen [16] asks whether it is possible to compute the Vickrey pricing faster than the trivial algorithm described in the previous paragraph. In this paper we present a near-linear time algorithm for computing replacement paths in

In the navigation system, it is very important not only to find the shortest path but also a detour, in case of a traffic jam for example. This paper surveys algorithms for the shortest path problem and the k shortest path problem at first, extends the latter algorithm for the 2-terminal k shortest paths problem, using AI search techniques such as the bidirectional A 3 algorithm, then defines `detour' precisely, and proposes algorithms for finding a realistic detour based on these algorithms. The efficiency and property of the algorithms are examined through experiments on an actual road network. 1. INTRODUCTION The shortest path problem is very important in various fields. For example, route navigation systems must show the shortest route to the destination as fast as possible. Thus, the shortest path problem is studied very well for a long time. For example, the Dijkstra method is the most famous and traditional algorithm for this problem. To make this algorithm more efficient, m...

: 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...

Measuring distance or some other form of proximity between objects is a standard data mining tool. Connection subgraphs were recently proposed as a way to demonstrate proximity between nodes in networks. We propose a new way of measuring and extracting proximity in networks called “cycle free effective conductance” (CFEC). Importantly, the measured proximity is accompanied with a proximity subgraph, which allows assessing and understanding measured values. Our proximity calculation can handle more than two endpoints, directed edges, is statistically well-behaved, and produces an effectiveness score for the computed subgraphs. We provide an efficient algorithm to measure and extract proximity. Also, we report experimental results and show examples for four large network data sets: a telecommunications calling graph, the IMDB actors graph, an academic co-authorship network, and a movie recommendation system.

: 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...