On the area requirements of Euclidean minimum spanning trees

In their seminal paper on Euclidean minimum spanning trees [Discrete & Computational Geometry, 1992], Monma and Suri proved that any tree of maximum degree 5 admits a planar embedding as a Euclidean minimum spanning tree. Their algorithm constructs embeddings with exponential area; however, the authors conjectured that cn × cn area is sometimes required to embed an n-vertex tree of maximum degree 5 as a Euclidean minimum spanning tree, for some constant c > 1. In this paper, we prove the first exponential lower bound on the area requirements for embedding trees as Euclidean minimum spanning trees. © 2011 Springer-Verlag

On Maximizing the Throughput of Multiprocessor Tasks

We consider the problem of scheduling n independent multiprocessor tasks with due dates and unit processing times, where the objective is to compute a schedule maximizing the throughput. We derive the complexity results and present several approximation algorithms. For the parallel variant of the problem, we introduce the rst- t increasing algorithm and the latest- t increasing algorithm, and prove that their worst-case ratios are 2 and 2 1=m, respectively (m 2 is the number of processors). Then we propose a revised algorithm with a worst-case ratio bounded by 3=2 1=(2m) (m is odd) and 3=2 1=(2m 2) (m is even). For the dedicated variant, we present a simple greedy algorithm. We show that its worst-case ratio is bounded by m + 1. We straighten this result by showing that the problem cannot be approximated within a factor of m for any " > 0, unless NP = ZPP

On the topologies of local minimum spanning trees

This paper is devoted to study the combinatorial properties of Local Minimum Spanning Trees (LMSTs), a geometric structure that is attracting increasing research interest in the wireless sensor networks community. Namely, we study which topologies are allowed for a sensor network that uses, for supporting connectivity, a local minimum spanning tree approach. First, we refine the current definition of LMST realizability, focusing on the role of the power of transmission (i.e., of the radius of the covered area). Second, we show simple planar, connected, and triangle-free graphs with maximum degree 3 that cannot be represented as an LMST. Third, we present several families of graphs that can be represented as LMSTs. Then, we show a relationship between planar graphs and their representability as LMSTs based on homeomorphism. Finally, we show that the general problem of determining whether a graph is LMST representable is NP-hard