On-Line Reevaluation of Functions

Given a finite set S and a function f : S^n -> S^m, we consider the problem of making a data structure which maintains a value of x in S^n and allows us to efficiently change an arbitrary coordinate of x and efficiently evaluate f_i(x) for arbitrary i. We both examine the problem for specific choices of f and relate the possibility of an efficient solution to general properties of f: expressibility as a formula, space complexity and time complexity

On the complexity of strongly connected components in directed hypergraphs

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure

Dynamic Shortest Path Algorithms for Hypergraphs

A hypergraph is a set V of vertices and a set of non-empty subsets of V, called hyperedges. Unlike graphs, hypergraphs can capture higher-order interactions in social and communication networks that go beyond a simple union of pairwise relationships. In this paper, we consider the shortest path problem in hypergraphs. We develop two algorithms for finding and maintaining the shortest hyperpaths in a dynamic network with both weight and topological changes. These two algorithms are the first to address the fully dynamic shortest path problem in a general hypergraph. They complement each other by partitioning the application space based on the nature of the change dynamics and the type of the hypergraph

Dynamic maintenance of directed hypergraphs

AbstractIn this paper we are concerned with the on-line maintenance of directed hypergraphs, a generalization of directed graphs previously introduced in the literature. In particular, we show how to maintain efficiently information about hyperpaths while new hyperarcs are inserted. We present a data structure which allows us to check whether there exists a hyperpath between an arbitrarily given pair of nodes in constant time and to return such a hyperpath in a time which is linear in its size. The total time required to maintain the data structure during the insertion of new hyperarcs is O(mn), where m is the total size of the description of the hyperarcs and n is the number of nodes. This generalizes a previous result known for directed graphs and has applications in several areas of computer science, such as rewriting systems, database schemes, logic programming and problem solving. An extension of these results to hyperpaths between sets of nodes is also presented

Dynamic maintenance of directed hypergraphs

AbstractIn this paper we are concerned with the on-line maintenance of directed hypergraphs, a generalization of directed graphs previously introduced in the literature. In particular, we show how to maintain efficiently information about hyperpaths while new hyperarcs are inserted. We present a data structure which allows us to check whether there exists a hyperpath between an arbitrarily given pair of nodes in constant time and to return such a hyperpath in a time which is linear in its size. The total time required to maintain the data structure during the insertion of new hyperarcs is O(mn), where m is the total size of the description of the hyperarcs and n is the number of nodes. This generalizes a previous result known for directed graphs and has applications in several areas of computer science, such as rewriting systems, database schemes, logic programming and problem solving. An extension of these results to hyperpaths between sets of nodes is also presented