We present the first fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with real-valued edge weights. Given a dynamic directed graph G such that each edge can assume at most S di#erent real values, we show how to support updates in O(n amortized time and queries in optimal worst-case time. No previous fully dynamic algorithm was known for this problem. In the special case where edge weights can only be increased, we give a randomized algorithm with one-sided error which supports updates faster in O(S We also show how to obtain query/update trade-o#s for this problem, by introducing two new families of algorithms. Algorithms in the first family achieve an update bound of O(n/k), and improve over the best known update bounds for k in the . Algorithms in the second family achieve an update bound of ), and are competitive with the best known update bounds (first family included) for k in the range (n/S) # Work partially supported by the IST Programme of the EU under contract n. IST-199914. 186 (ALCOM-FT) and by CNR, the Italian National Research Council, under contract n. 01.00690.CT26. Portions of this work have been presented at the 42nd Annual Symp. on Foundations of Computer Science (FOCS 2001) [8] and at the 29th International Colloquium on Automata, Languages, and Programming (ICALP'02) [9].

We present the results of an extensive computational study on dynamic algorithms for all pairs shortest path problems. We describe our implementations of the recent dynamic algorithms of King and of Demetrescu and Italiano, and compare them to the dynamic algorithm of Ramalingam and Reps and to static algorithms on random, real-world and hard instances. Our experimental data suggest that some of the dynamic algorithms and their algorithmic techniques can be really of practical value in many situations. 1

We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the transitive closure of a directed graph, through an arbitrary sequence of edge deletions, in O(mn) total expected time, essentially the time needed for computing the transitive closure of the initial graph. Such a result was previously known only for acyclic graphs.

Dynamic shortest path algorithms update the shortest paths to take into account a change in an edge weight. This paper describes a new technique that allows the reduction of heap sizes used by several dynamic shortest path algorithms. For unit weight change, the updates can be done without heaps. These reductions almost always reduce the computational times for these algorithms. In computational testing, several dynamic shortest path algorithms with and without the heap-reduction technique are compared. Speedups of up to a factor of 1.8 were observed using the heap-reduction technique on random weight changes and of over a factor of five on unit weight changes. We compare as well with Dijkstra 's algorithm, which recomputes the paths from scratch. With respect to Dijkstra's algorithm, speedups of up to five orders of magnitude are observed. 1.

In this paper, we survey fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. In this survey, we will make a special effort to abstract some combinatorial and algebraic properties, and some common data-structural tools that are at the base of those techniques. This will help us try to present some of the newest results in a unifying framework so that they can be better understood and deployed also by non-specialists.

Let G be a directed graph with n vertices, subject to dynamic updates, and such that each edge weight can assume at most S different arbitrary real values throughout the sequence of updates. We present a new algorithm for maintaining all pairs shortest paths in G in O(S n) amortized time per update and in O(1) worst-case time per distance query. This improves over previous bounds. We also show how to obtain query/update trade-offs for this problem, by introducing two new families of algorithms. Algorithms in the first family achieve an update bound of e O(S \Delta k \Delta n and a query bound of e O(n=k), and improve over the best known update bounds for k in the range (n=S) . Algorithms in the second family achieve an update e O e O(n ), and are competitive with the best known update bounds (first family included) for k in the range (n=S) k ! .

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.

We present improved algorithms for maintaining transitive closure and all-pairs shortest paths in a digraph under deletion of edges. For the problem of transitive closure, the previous best known algorithms achieving O(1) query time require O(min(m; n =m)) amortized update time, thus establish an upper bound of O(n ) on update time per edge-deletion where m and n denote the number of edges and vertices respectively in the given graph. We present an algorithm that achieves O(1) query time for answering a query and O(n log log n) update time per edge-deletion, thus improving the upper bound to O(n 3 log n).

A dynamic shortest-path algorithm is called a batch algorithm if it is able to handle graph changes that consist of multiple edge updates at a time. In this paper we focus on fully-dynamic batch algorithms for singlesource shortest paths in directed graphs with positive edge weights. We give an extensive experimental study of the existing algorithms for the single-edge and the batch case, including a broad set of test instances. We further present tuned variants of the already existing SWSF-FP-algorithm being up to 15 times faster than SWSF-FP. A surprising outcome of the paper is the astonishing level of data dependency of the algorithms.

Abstract. In this paper we present the first dynamic data structure for testing- in constant time- the existence of two edge- or quasi-internally vertex-disjoint paths p1 from s to t1 and p2 from s to t2 for any three given vertices s, t1, and t2 of a digraph. By quasi-internally vertex-disjoint we mean that no inner vertex of p1 appears on p2 and vice versa. Moreover, for two vertices s and t, the data structure supports the output of all vertices and all edges whose removal would disconnect s and t in a time linear in the size of the output. The update operations consist of edge insertions and edge deletions, where the implementation of edge deletions will be given only in the full version of this paper. The update time after an edge deletion is competitive with the reconstruction of a static data structure for testing the existence of disjoint paths in constant time, whereas our data structure performs much better in the case of edge insertions. 1