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...

We study the problem of finding a shortest path between two vertices in a directed graph. This is an important problem with many applications, including that of computing driving directions. We allow preprocessing the graph using a linear amount of extra space to store auxiliary information, and using this information to answer shortest path queries quickly. Our approach uses A ∗ search in combination with a new graph-theoretic lower-bounding technique based on landmarks and the triangle inequality. We also develop new bidirectional variants of A ∗ search and investigate several variants of the new algorithms to find those that are most efficient in practice. Our algorithms compute optimal shortest paths and work on any directed graph. We give experimental results showing that the most efficient of our new algorithms outperforms previous algorithms, in particular A ∗ search with Euclidean bounds, by a wide margin on road networks. We also experiment with several synthetic graph families.

We consider in this paper the shortest-path problem in networks in which the delay (or weight) of the edges changes with time according to arbitrary functions. We present algorithms for finding the shortest-path and minimum-delay under various waiting constraints and investigate the properties of the derived path. We show that if departure time from the source node is unrestricted then a shortest path can be found that is simple and achieves a delay as short as the most unrestricted path. In the case of restricted transit, it is shown that there exist cases where the minimum delay is finite but the path that achieves it is infinite. ____________________________________ This research was supported by the Foundation for Research in Electronics, Computers and Communications, Administered by the Israel Academy of Science and Humanities I. INTRODUCTION Shortest paths algorithms have been the subject of extensive research for many years resulting in a large number of algorithms for vario...

We present an algorithm for accurately controlling delays during the placement of large standard cell integrated circuits. Previous approaches to timing driven placement could not handle circuits containing 20,000 or more cells and yielded placement qualities which were well short of the state of the art. Our timing optimization algorithm has been added to the placement algorithm which has yielded the best results ever reported on the full set of MCNC benchmark circuits, including a circuit containing more than 100,000 cells. A novel pin-pair algorithm controls the delay without the need for user path specification. The timing algorithm is generally applicable to hierarchical, itera-tive placement methods. Using this algorithm, we present results for the only MCNC standard cell benchmark circuits (fract, struct, and avq.small) for which timing information is available. We decreased the delay of the longest path of circuit fract by 36 % at an area cost of only 2.5%. For circuit struct, the delay of the longest path was decreased by 50 % at an area cost of 6%. Finally, for the large (21,000 cell) circuit avq.small, the longest path delay was decreased by 28 % at an area cost of 6%.

We study the point-to-point shortest path problem in a setting where preprocessing is allowed. We improve the reach-based approach of Gutman [16] in several ways. In particular, we introduce a bidirectional version of the algorithm that uses implicit lower bounds and we add shortcut arcs which reduce vertex reaches. Our modifications greatly reduce both preprocessing and query times. The resulting algorithm is as fast as the best previous method, due to Sanders and Schultes [27]. However, our algorithm is simpler and combines in a natural way with A∗ search, which yields significantly better query times.

Shortest Path Problems are among the most studied network flow optimization problems, with interesting applications in various fields. One such field is transportation, where shortest path problems of different kinds need to be solved. Due to the nature of the application, transportation scientists need very flexible and efficient shortest path procedures, both from the running time point of view, and also for the memory requirements. Since no "best" algorithm currently exists for every kind of transportation problem, research in this field has recently moved to the design and implementation of "ad hoc" shortest path procedures, which are able to capture the peculiarities of the problems under consideration. The aim of this work is to present in a unifying framework both the main algorithmic approaches that have been proposed in the past years for solving the shortest path problems arising most frequently in the transportation field, and also some important implementation techniques ...

: This paper solves what appears to be a 30 years old problem dealing with the discovery of most efficient algorithms possible to compute all-to-one shortest paths in discrete dynamic networks. This problem lies at the heart of efficient solution approaches to dynamic network models that arise in dynamic transportation systems, such as Intelligent Transportation Systems (ITS), applications. While the main objective of this paper is the study of the allto -one dynamic shortest paths problem, one-to-all fastest paths problems are studied as well. Early results are revisited and new properties are established. We establish the exact complexity of these problems and develop optimal, in the run time sense, solution algorithms. A new and simple solution algorithm is proposed for all-to-one all departure time intervals shortest path problems. It is proved, theoretically, that the new solution algorithm has an optimal run time complexity that equals the complexity of the problem. Computer impl...

We investigate the minimum weight path problem in networks whose link weights and link delays are both functions of time. We demonstrate that in general there exist cases in which no finite path is optimal leading us to define an infinite path (naturally, containing loops) in such a way that the minimum weight problem always has a solution. We also characterize the structure of an infinite optimal path. In many practical cases, finite optimal paths do exist. We formulate a criterion that guarantees the existence of a finite optimal path and develop an algorithm to find such a path. Some special cases, e.g., optimal loopless paths, are also discussed.

Abstract not found

: 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...