Abstract. We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. In an efficient preprocessing phase our algorithm creates a linear-size structure that facilitates single-source shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverse-Ackermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the all-pairs and single-source shortest paths problems. We solve the all-pairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the single-source problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchy-based approach invented by Thorup. Key words. single-source shortest paths, all-pairs shortest paths, undirected graphs, Dijkstra’s

Abstract—We consider the problem of placing sensors in a network to detect and identify the source of any contamination. We consider two variants of this problem: (1) sensor-constrained: we are allowed a fixed number of sensors and want to minimize contamination detection time; and (2) time-constrained]: we must detect contamination within a given time limit and want to minimize the number of sensors required. Our main results are as follows. First, we give a necessary and sufficient condition for source identification. Second, we show that the sensor and time constrained versions of the problem are polynomially equivalent. Finally, we show that the sensor-constrained version of the problem is polynomially equivalent to the asymmetric k-center problem and that the time-constrained version of the problem is

We present an all-pairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverse-Ackermann function. Our algorithm eliminates the sorting bottleneck inherent in approaches based on Dijkstra's algorithm, and for graphs with O(n) edges our algorithm is within a tiny O(log (n; n)) factor of optimal. Our algorithm can be implemented to run in polynomial time (granted, a large polynomial). We leave open the problem of providing an efficient implementation.

We consider the problem of preprocessing an edge-weighted directed graph G to answer queries that ask for the shortest distance from any given node x to any other node y avoiding an arbitrary failed node or link. We describe an oracle (i.e, a simple data structure) for such queries that can be stored in O(n 2 log n) space, and which allows queries to be answered in O(1) time, where n is the number of nodes in G. We also show that if we are willing to use Θ(n 2.5) space, we can reduce the preprocessing time by a factor of √ n while maintaining the constant query time. We can also keep track of the shortest path avoiding any failed node or link by maintaining for each node the outgoing edge that should be used to get on such a path.

In this paper, we report on our own experience in studying a fundamental problem on graphs: all pairs shortest paths. In particular, we discuss the interplay between theory and practice in engineering a simple variant of Dijkstra's shortest path algorithm. In this context, we show that studying heuristics that are e#cient in practice can yield interesting clues to the combinatorial properties of the problem, and eventually lead to new theoretically e#cient algorithms.

We consider the problem of placing sensors in a network to detect and identify the source of any contamination. We consider two variants of this problem: 1) sensor-constrained: we are allowed a fixed number of sensors and want to minimize con-tamination detection time; and 2) time-constrained: we must detect contamination within a given time limit and want to minimimize the number of sensors required. Our main results are as follows. First, we give a necessary and sufficient condition Preprint submitted to Elsevier Science 15 December 2004 for source identification. Second, we show that the sensor and time constrained ver-sions of the problem are polynomially equivalent. Finally, we show that the sensor-constrained version of the problem is polynomially equivalent to the asymmetric k-center problem and that the time-constrained version of the problem is polyno-mially equivalent to the dominating set problem. 1