The well-known greedy triangulation GT #S# of a #nite point set S is obtained by inserting compatible edges in increasing length order, where an edge is compatible if it does not cross previously inserted ones. Exploiting the concept of so-called light edges, we introduce a new way of de#ning GT #S#. The new de#nition does not rely on the length ordering of the edges. It provides a decomposition of GT #S#into levels, and the number of levels allows us to bound the total edge length of GT #S#. In particular, we show jGT #S#j#3#2 k+1 jMWT#S#j, where k is the number of levels and MWT#S# is the minimum-weight triangulation of S. This constitutes the #rst non-trivial upper bound on jGT #S#j for general points sets S. 1 Introduction A triangulation of a given set S of n points in the plane is a maximal set of non-crossing line segments #called edges# whichhave both endpoints in S. Besides the Delaunay triangulation and the minimum-weight triangulation, the greedy triang...

We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(|V |) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed. Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service. Our results can be easily generalized to other wide classes of graphs – for instance we can take any minor-closed family of graphs.

Abstract. We study the competitive ratio of certain online algorithms for a well-studied class of load balancing problems. These algorithms are obtained and analyzed according to a method by Crescenzi et al (2004). We show that an exact analysis of their competitive ratio on certain “uniform ” instances would resolve a fundamental conjecture by Caccetta and Häggkvist (1978). The conjecture is that any digraph on n nodes and minimum outdegree d must contain a directed cycle involving at most ⌈n/d ⌉ nodes. Our results are the first relating this conjecture to the competitive analysis of certain algorithms, thus suggesting a new approach to the conjecture itself. We also prove that, on “uniform ” instances, the analysis by Crescenzi et al (2004) gives only trivial upper bounds, unless we find a counterexample to the conjecture. This is in contrast with other (notable) examples where the same analysis yields optimal (non-trivial) bounds. Key words: Caccetta-Häggkvist conjecture, online load balancing, competitive analysis 1

Let G(n, m) be an undirected random graph with n vertices and m multiedges that may include loops, where each edge is realized by choosing its two vertices independently and uniformly at random with replacement from the set of all n vertices. The random graph G(n, m) is said to be k-orientable, where k ≥ 2 is an integer, if there exists an orientation of the edges such that the maximum out-degree is at most k. Let ck = sup {c: G(n, cn) is k-orientable w.h.p.}. We prove that for k large enough, 1 − 2 k exp −k + 1 + e −k/4) < ck/k < 1 − exp