Nicolas Hanusse

Connections between theta-graphs, delaunay triangulations, and orthogonal surfaces

Θk-graphs are geometric graphs that appear in the context of graph navigation. The shortest-path metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TD-Delaunay graphs, a.k.a. triangular-distance Delaunay triangulations, introduced by Chew, have been shown to be plane 2-spanners of the 2D Euclidean complete graph, i.e., the distance in the TD-Delaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6-graph defined in the 2D Euclidean space, namely the half-Θ6-graph, composed of the even-cone edges of the Θ6-graph. Our main contribution is to show that these graphs are exactly the TD-Delaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish: - Every Θ6-graph is the union of two spanning TD-Delaunay graphs. In particular, Θ6-graphs are 2-spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6-graphs are t-spanners for some constant t, and Θ7-graphs were only known to be t-spanners for t ≈ 7.562. - Every plane triangulation is TD-Delaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TD-Delaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.

An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation

We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2?n+O(log n), where ? ? 5.007. The current lower bound is 2sn+?(log n) for s ? 4.71. We also show that almost all unlabeled and almost all labeled n-node planar graphs have at least 1.70n edges and at most 2.54n edges.

Towards small world emergence

We investigate the problem of optimizing the routing performance of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1-<i>O</i>(1/<i>n</i>). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node <i>u</i> and any radius <i>r</i> the number of nodes within distance 2<i>r</i> from <i>u</i> is at most a constant times the number of nodes within distance <i>r</i>, our augmentation process proceeds with high probability in <i>O</i>(log <i>n</i> log <i>D</i>) communication rounds, with <i>O</i>(log <i>n</i> log <i>D</i>) messages of size <i>O</i>(log <i>n</i>) bits sent per node and requiring only <i>O</i>(log <i>n</i> log <i>D</i>) bit space in each node, where <i>n</i> is the number of nodes, and <i>D</i> the diameter. In particular, with the only knowledge of original distances, greedy routing computes, between any pair of nodes in the augmented network, a path of length at most <i>O</i>(log² <i>n</i> log² <i>D</i>) with probability 1 - <i>O</i>(1/<i>n</i>), and of expected length <i>O</i>(log <i>n</i> log² <i>D</i>). Hence, we provide a distributed scheme to augment any bounded growth graph into a small world with high probability in polylogarithmic time while requiring polylogarithmic memory. We consider that the existence of such a lightweight process might be a first step towards the definition of a more general construction process that would validate Kleinbergs model as a plausible explanation for the small world phenomenon in large real interaction networks.

Euler Tour Lock-In Problem in the Rotor-Router Model

The rotor-router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. It is known that the route in an undirected graph G = (V, E), where |V| = n and |E| = m, adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In recent work [11] Yanovski et al. proved that independently of the initial configuration of the rotor-router mechanism in G the agent locks-in in time bounded by 2mD, where D is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. The case study is performed in the form of a game between a player P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary A with the counter objective. First, we observe that in certain (easy) cases the lock-in can be achieved in time O(m). On the other hand we show that if adversary A is solely responsible for the assignment of ports and pointers, the lock-in time Ω(mċD) can be enforced in any graph with m edges and diameter D. Furthermore, we show that if A provides its own port numbering after the initial setup of pointers by P, the complexity of the lock-in problem is bounded by O(mċmin{log m, D}). We also propose a class of graphs in which the lock-in requires time Ω(m ċ log m). In the remaining two cases we show that the lock-in requires time Ω(m ċ D) in graphs with the worst-case topology. In addition, however, we present non-trivial classes of graphs with a large diameter in which the lock-in time is O(m).

Could any graph be turned into a small-world?

In the last decade, effective measurements of real interaction networks have revealed specific unexpected properties. Among these, most of these networks present a very small diameter and a high clustering. Furthermore, very short paths can be effciently found between any pair of nodes without global knowledge of the network (i.e., in a decentralized manner) which is known as the small-world phenomenon [1]. Several models have been proposed to explain this phenomenon [2,3]. However, Kleinberg showed in [4] that these models lack the essential navigability property: in spite of a polylogarithmic diameter, decentralized routing requires the visit of a polynomial number of nodes in these models.

Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation

In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive:

A view selection algorithm with performance guarantee

A view selection algorithm takes as input a fact table and computes a set of views to store in order to speed up queries. The performance of view selection algorithm is usually measured by three criteria: (1) the amount of memory to store the selected views, (2) the query response time and (3) the time complexity of this algorithm. The two first measurements deal with the output of the algorithm. No existing solutions give good trade-off between amount of memory and queries cost with a small time complexity. We propose in this paper an algorithm guaranteeing a constant approximation factor of queries response time with respect to the optimal solution. Moreover, the time complexity for a D-dimensional fact table is O (D * 2D) corresponding to the fastest known algorithm. We provide an experimental comparison with two other well known algorithms showing that our approach also gives good performance in terms of memory.

Optimal Randomized Self-stabilizing Mutual Exclusion on Synchronous Rings

We propose several self-stabilizing protocols for unidirectional, anonymous, and uniform synchronous rings of arbitrary size, where processors communicate by exchanging messages. When the size of the ring n is unknown, we better the service time by a factor of n (performing the best possible complexity for the stabilization time and the memory consumption). When the memory size is known, we present a protocol that is optimal in memory (constant and independant of n), stabilization time, and service time (both are in Θ(n)).

Searching with mobile agents in networks with liars

Abstract We present deterministic algorithms to search for an item s contained in a node of a network, without prior knowledge of its exact location. Each node of the network has a database that will answer queries of the form “how do I get to s ?” by responding with the first edge on a shortest path to the node containing s . It may happen that some nodes, called liars , give bad advice. If the number of liars k is bounded, we show different strategies to find the item depending on the topology of the network. In particular we consider the complete graph, ring, torus, hypercube and bounded degree trees.

Compact Routing Tables for Graphs of Bounded Genus (Extended Abstract)

For planar graphs on n nodes we show how to construct in linear time shortest path routing tables that require 8n + o(n) bits per node, and O(log2+∈ n) bit-operations per node to extract the route, with constant ∈ > 0. We generalize the result for every graph of bounded crossing-edge number. We also extend our result to any graph of genus bounded by γ, by building shortest path routing tables of n log (γ + 1)+ O(n) bits per node, and with O(log2+∈ n) bit-operations per node to extract the route. This result is obtained by the use of dominating sets, compact coding of non-crossing partitions, and k-page representation of graphs.