Paola Vocca

On the Complexity of Computing Minimum Energy Consumption Broadcast Subgraphs

We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a sub-logarithmic factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension.

Drawing with colors

In this paper, we investigate the volume, aspect ratio, angular resolution, edge-separation, and bit-requirement of crossing-free straight-line 3D drawings. We assume the vertex resolution rule, which requires minimum unit distance between any two vertices. Our main result shows that an N-vertex graph colorable with O(1) colors admits a crossing-free straight-line 3D drawing with O(N√N) volume, O(1) aspect ratio,gW(l/NO(1)) angular resolution, Ω (1/NO(1)) edge-separation, and O(log N) bit-requirement, which can be constructed in O(N) time.

Compact Implicit Representation of Graphs

How to represent a graph in memory is a fundamental data structuring problem. In the usual representations, a graph is stored by representing explicitly all vertices and all edges. The names (labels) assigned to vertices are used only to encode the edges and betray nothing about the structure of the graph itself and hence are a “waste” of space. In this context, we present a general framework for labeling any graph so that adjacency between any two given vertices can be tested in constant time. The labeling schema assigns to each vertex x of a general graph a O(δ(x)log3n) bit label, where n is the number of vertices and δ(x) is x’s degree. The adjacency test can be performed in 5 steps and the schema can be computed in polynomial time. This representation strictly contrasts with usual representations, i.e. adjacency matrix and adjacency list representations, which require O(nlog n) bit label per vertex and constant time adjacency test, and O(δ(x)log n) bit label per vertex and O(logδ (x)) steps to test adjacency, respectively. Additionally, the labeling schema is implicit, that is: no pointers are used.

An Efficient Data Structure for Lattice Operations

In this paper, we consider the representation and management of an element set on which a lattice partial order relation is defined. In particular, let n be the element set size. We present an \nradn-space implicit data structure for performing the following set of basic operations: 1. Test the presence of an order relation between two given elements, in constant time. 2. Find a path between two elements whenever one exists, in O(l) steps, where l is the path length. 3. Compute the successors and/or predecessors set of a given element, in O(h) steps, where h is the size of the returned set. 4. Given two elements, find all elements between them, in time O(k log d), where k is the size of the returned set and d is the maximum in-degree or out-degree in the transitive reduction of the order relation. 5. Given two elements, find the least common ancestor and/or the greatest common successor in

Representing graphs implicitly using almost optimal space

O(\sqrt{n})

A data structure for lattice representation

-time. 6. Given k elements, find the least common ancestor and/or the greatest common successor in

Spatial node distribution of manhattan path based random waypoint mobility models with applications

O(\sqrt{n}+k \log n)

Proximity drawings in polynomial area and volume

time. (Unless stated otherwise, all logarithms are to the base 2.) The preprocessing time is O(n2). Focusing on the first operation, representing the building-box for all the others, we derive an overall \nradn-space\,

Topological Routing Schemes

\times

On the approximability of the L(h, k) -labelling problem on bipartite graphs

\,time bound which beats the order n2 bottleneck representing the present complexity for this problem. Moreover, we will show that the complexity bounds for the first three operations are optimal with respect to the worst case. Additionally, a stronger result can be derived. In particular, it is possible to represent a lattice in space