Abs t rac t . An edge-ranking of an undirected graph G is a labeling of the edges of G with integers such that all paths between two edges with the same label i contain an edge with label j > i. The problem of finding an edge-ranking of G using a minimum number of ranks has applications in scheduling the manufacture of complex multi-part products; it is equivalent to finding the minimum height ...

We present eecient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O(log h + logn) (plus O(k) time for reporting the k edges...

We present e cient algorithms for shortest path and minimum link path queries between two convex polygons inside a simple polygon P which acts as an obstacle to be avoided Let n be the number of vertices of P and h the total number of vertices of the query polygons We show that shortest path queries can be performed optimally in time O logh logn plus O k time for reporting the k edges of the pa...

For a polygon P with n vertices, the vertex guarding problem asks for the minimum subset G of P ’s vertices such that every point in P is seen by at least one point in G. This problem is NP-complete and APX-hard. The first approximation algorithm (Ghosh, 1987) involves decomposing P into O ( n ) cells that are equivalence classes for visibility from the vertices of P . This discretized problem ...

A minimum cycle basis of a weighted undirected graph G is a basis of the cycle space of G such that the total weight of the cycles in this basis is minimized. If G is a planar graph with non-negative edge weights, such a basis can be found in O(n2) time and space, where n is the size of G. We show that this is optimal if an explicit representation of the basis is required. We then present an O(...

In the k-dispersion problem, we need to select k nodes of a given graph so as to maximize the minimum distance between any two chosen nodes. This can be seen as a generalization of the independent set problem, where the goal is to select nodes so that the minimum distance is larger than 1. We design an optimal O(n) time algorithm for the dispersion problem on trees consisting of n nodes, thus i...

In this paper, we introduce an O(nm) time algorithm to determine the minimum length directed cycle in a directed network with n nodes and m arcs and with no negative length directed cycles. This result improves upon the previous best time bound of O(nm + n log log n). Our algorithm first determines the cycle with minimum mean length λ∗ in O(nm) time. Subsequently, it chooses node potentials so ...

We address the problem of Minimum Enclosing Ball (MEB) and its generalization to Minimum Enclosing Convex Polytope (MECP). Given n points in a d dimensional Euclidean space, we give a O(nd/ √ ) algorithm for producing an enclosing ball whose radius is at most away from the optimum. In the case of MECP our algorithm takes O(1/ ) iterations to converge. In both cases we improve the existing resul...

It is well-known that, given a probability distribution over n characters, in the worst case it takes Θ(n logn) bits to store a prefix code with minimum expected codeword length. However, in this paper we first show that, for any with 0 < < 1/2 and 1/ = O(polylog(n)), it takes O(n log log(1/ )) bits to store a prefix code with expected codeword length within an additive of the minimum. We then ...

We show that a minimum edge-colouring of a bipartite graph can be found in O((m) time, where and m denote the maximum degree and the number of edges of G, respectively. It is equivalent to nding a perfect matching in a k-regular bipartite graph in O(km) time. By sharpening the methods, a minimum edge-colouring of a bipartite graph can be found in O((p max (() + log)m) time, where p max (() is t...