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 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

We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.

In this paper we introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive closure. In particular, we devise a deterministic algorithm for general directed graphs that achieves O(n²) amortized time for updates, while preserving unit worstcase cost for queries. In case of deletions only, our algorithm performs updates faster in O(n) amortized time. Our

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 A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m non-zero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multiplication algorithm, on the other hand, may need to perform #(mn) operations to accomplish the same task. For , the new algorithm performs an almost optimal number of only n operations. For m the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n ) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast rectangular matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices.

This paper presents an efficient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form "Is there a directed path from i to j?" can be answered in O(1) time. The algorithm is randomized; it is correct when answering yes, but has O(1/n^c) probability of error when answering no, for any constant c. In acyclic graphs, worst case update time is O(n^2). In general graphs, update time is O(n^(2+alpha)), where alpha = min {.26, maximum size of a strongly connected component}. The space complexity of the algorithm is O(n^2).

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.

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.