Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time algorithm APASP 2 for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP 2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k ? 2, we describe an g) time algorithm APASP k for computing all distances in G with an additive one-sided error of at most k.

We investigate the all-pairs shortest paths problem in weighted graphs. We present an algorithm---the Hidden Paths Algorithm---that finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m* is likely to be small in practice, since m* = O(n log n) with high probability for many probability distributions on edge weights. We also prove an Ω(mn) lower bound on the running time of any path-comparison based algorithm for the all-pairs shortest paths problem. Path-comparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are effective in this more general setting.

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves...

We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.

We show that the All Pairs Shortest Paths (APSP) problem for undirected graphs with integer edge weights taken from the range f1; 2; : : : ; Mg can be solved using only a logarithmic number of distance products of matrices with elements in the range f1; 2; : : : ; Mg. As a result, we get an algorithm for the APSP problem in such graphs that runs in ~ O(Mn ! ) time, where n is the number of vertices in the input graph, M is the largest edge weight in the graph, and ! ! 2:376 is the exponent of matrix multiplication. This improves, and also simplifies, an ~ O(M (!+1)=2 n ! ) time algorithm of Galil and Margalit. 1. Introduction The All Pairs Shortest Paths (APSP) problem is one of the most fundamental algorithmic graph problems. The APSP problem for directed or undirected graphs with real weights can be solved using classical methods, in O(mn + n 2 log n) time (Dijkstra [4], Johnson [10], Fredman and Tarjan [7]), or in O(n 3 ((log log n)= log n) 1=2 ) time (Fredman [6], ...

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in ~ O(n 2+ ) time, where satisfies the equation !(1; ; 1) = 1 + 2 and !(1; ; 1) is the exponent of the multiplication of an n \Theta n matrix by an n \Theta n matrix. The currently best available bounds on !(1; ; 1), obtained by Coppersmith and Winograd, and by Huang and Pan, imply that ! 0:575. The running time of our algorithm is therefore O(n 2:575 ). Our algorithm improves on the ~ O(n (3+!)=2 ) time algorithm, where ! = !(1; 1; 1) ! 2:376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n 2:688 ). The second

In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general real-weighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in real-vertex-weighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs. 1

Let G = (V; E) be a weighted undirected graph. A path between u; v 2 V is said to be of stretch t if its length is at most t times the distance between u and v in the graph. We consider the problem of finding small-stretch paths between all pairs of vertices in the graph G. It is easy to see that finding paths of stretch less than 2 between all pairs of vertices in an undirected graph with n vertices is at least as hard as the Boolean multiplication of two n \Theta n matrices. We describe three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges. The first algorithm, STRETCH 2 , runs in ~ O(n 3=2 m 1=2 ) time and finds stretch 2 paths. The second algorithm, STRETCH 7=3 , runs in ~ O(n 7=3 ) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH 3 , runs in ~ O(n 2 ) and finds stretch 3 paths. Our algorithms are simpler, more efficient and more accurate than the previously best algorithms ...

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 parallel algorithms for computing all pair shortest paths in directed graphs. Our algorithm has time complexity O( f (n)/p + I (n) log n) on the PRAM using p processors, where I (n) is log n on the EREW PRAM, log log n on the CCRW PRAM, f (n) is o(n 3 ). On the randomized CRCW PRAM we are able to achieve time complexity O(n 3 /p + log n) using p processors. Key Words. Analysis of algorithms, Design of algorithms, Parallel algorithms, Graph algorithms, Shortest path. 1. Introduction. A number of known algorithms compute the all pair shortest paths in graphs and digraphs with n vertices by using O(n 3 ) operations [D], [Fl], [J]. All these algorithms, however, use at least n-1 recursive steps in the worst case and thus require at least the order of n time in their parallel implementation, even if the number of available processors is not bounded. O(n) time and n 2 processor bounds can indeed be achieved, for instance, in the straightforward parallelization of th...