Sergio Cabello

Secret Sharing Schemes with Detection of Cheaters for a General Access Structure

In a secret sharing scheme, some participants can lie about the value of their shares when reconstructing the secret in order to obtain some illicit benefit. We present in this paper two methods to modify any linear secret sharing scheme in order to obtain schemes that are unconditionally secure against that kind of attack. The schemes obtained by the first method are robust, that is, cheaters are detected with high probability even if they know the value of the secret. The second method provides secure schemes, in which cheaters that do not know the secret are detected with high probability. When applied to ideal linear secret sharing schemes, our methods provide robust and secure schemes whose relation between the probability of cheating and the information rate is almost optimal. Besides, those methods make it possible to construct robust and secure schemes for any access structure.

Testing Homotopy for Paths in the Plane

Abstract In this paper we present an efficient algorithm to test if two given paths are homotopic; that is, whether they wind around obstacles in the plane in the same way. For paths specified by n line segments with obstacles described by n points, several standard ways achieve quadratic running time. For simple paths, our algorithm runs in O(n log n) time, which we show is tight. For self-intersecting paths the problem is related to Hopcroft’s problem; our algorithm runs in O(n3/2log n) time.

Finding shortest non-separating and non-contractible cycles for topologically embedded graphs

We present an algorithm for finding shortest surface non-separating cycles in graphs with given edge-lengths that are embedded on surfaces. The time complexity is O(g3/2V3/2log V+g5/2V1/2), where V is the number of vertices in the graph and g is the genus of the surface. If g=o(V1/3−e), this represents a considerable improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in O(g

Many distances in planar graphs

^{O({\it g})}

Finding Shortest Non-Separating and Non-Contractible Cycles for Topologically Embedded Graphs

V3/2) time, improving previous results for fixed genus. This result can be applied for computing the (non-separating) face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in O(V5/4log V) time.

Facility location problems in the plane based on reverse nearest neighbor queries

We show how to compute in O(n4/3log 1/3n+n2/3k2/3log n) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/log n)5/6≤k≤n2/log 6n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs.

ON THE B-CHROMATIC NUMBER OF REGULAR GRAPHS

We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in

Schematization of networks

O(g^{3/2}V^{3/2}\log V+g^{5/2}V^{1/2})

Schematization of road networks

time, where V is the number of vertices in the graph and g is the genus of the surface. If

Multiple-Source Shortest Paths in Embedded Graphs

g=o(V^{1/3})