Elias Vicari

A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs

The efficient distributed construction of a maximal independent set (MIS) of a graph is of fundamental importance. We study the problem in the class of Growth-Bounded Graphs, which includes for example the well-known Unit Disk Graphs. In contrast to the fastest (time-optimal) existing approach [11], we assume that no geometric information (e.g., distances in the graphs embedding) is given. Instead, nodes employ randomization for their decisions. Our algorithm computes a MIS in O(log log n • log* n) rounds with very high probability for graphs with bounded growth, where n denotes the number of nodes in the graph. In view of Linials Ω(log* n) lower bound for computing a MIS in ring networks [12], which was extended to randomized algorithms independently by Naor [18] and Linial [13], our solution is close to optimal. In a nutshell, our algorithm shows that for computing a MIS, randomization is a viable alternative to distance information.

Optimal sparse matrix dense vector multiplication in the I/O-model

We analyze the problem of sparse-matrix dense-vector multiplication (SpMV) in the I/O-model. The task of SpMV is to compute y := Ax, where A is a sparse N x N matrix and x and y are vectors. Here, sparsity is expressed by the parameter k that states that A has a total of at most kN nonzeros, i.e., an average number of k nonzeros per column. The extreme choices for parameter k are well studied special cases, namely for k=1 permuting and for k=N dense matrix-vector multiplication. We study the worst-case complexity of this computational task, i.e., what is the best possible upper bound on the number of I/Os depending on k and N only. We determine this complexity up to a constant factor for large ranges of the parameters. By our arguments, we find that most matrices with kN nonzeros require this number of I/Os, even if the program may depend on the structure of the matrix. The model of computation for the lower bound is a combination of the I/O-models of Aggarwal and Vitter, and of Hong and Kung. We study two variants of the problem, depending on the memory layout of A. If A is stored in column major layout, SpMV has I/O complexity Θ(min{kN B(1+logM/BN max{M,k}), kN}) for k ≤ N1-e and any constant 1> e > 0. If the algorithm can choose the memory layout, the I/O complexity of SpMV is Θ(min{kN B(1+logM/BN kM), kN]) for k ≤ 3√N. In the cache oblivious setting with tall cache assumption M ≥ B1+e, the I/O complexity is Ο(kN B(1+logM/B N k)) for A in column major layout.

Simple Robots with Minimal Sensing: From Local Visibility to Global Geometry

We consider problems of geometric exploration and self-deployment for simple robots that can only sense the combinatorial (non-metric) features of their surroundings. Even with such a limited sensing, we show that robots can achieve complex geometric reasoning and perform many non-trivial tasks. Specifically, we show that one robot equipped with a single pebble can decide whether the workspace environment is a simply connected polygon and, if not, it can also count the number of holes in the environment. Highlighting the subtleties of our sensing model, we show that a robot can decide whether the environment is a convex polygon, yet it cannot resolve whether a given vertex is convex. Finally, we show that by using such local and minimal sensing a robot can compute a proper triangulation of a polygon and that the triangulation algorithm can be implemented collaboratively by a group of m such robots, each with Θ(n/m) word memory. As a corollary of the triangulation algorithm, we derive a distributed analog of the well-known Art Gallery Theorem. A group of [n/3] (bounded memory) robots in our minimal sensing model can self-deploy to achieve visibility coverage of an n-vertex art gallery (polygon). This resolves an open question raised recently.

Counting targets with mobile sensors in an unknown environment

We consider the problem of counting the number of indistinguishable targets using a simple binary sensing model. Our setting includes an unknown number of point targets in a (simply- or multiply-connected) polygonal workspace, and a moving point-robot whose sensory input at any location is a binary vector representing the cyclic order of the polygon vertices and targets visible to the robot. In particular, the sensing model provides no coordinates, distance or angle measurements. We investigate this problem under two natural models of environment, friendly and hostile, which differ only in whether the robot can visit the targets or not, and under three different models of motion capability. In the friendly scenario we show that the robots can count the targets, whereas in the hostile scenario no (2 - Ɛ)-approximation is possible, for any Ɛ > 0. Next we consider two, possibly minimally more powerful robots that can count the targets exactly.

Rendezvous of Mobile Agents When Tokens Fail Anytime

We consider the problem of Rendezvous or gathering of multiple autonomous entities (called mobile agents) moving in an unlabelled environment (modelled as a graph). The problem is usually solved using randomization or assuming distinct identities for the agents, such that they can execute different protocols. When the agents are all identical and deterministic, and the environment itself is symmetrical (e.g. a ring) it is difficult to break the symmetry between them unless, for example, the agents are provided with a token to mark the nodes. We consider fault-tolerant protocols for the problem where the tokens used by the agents may disappear unexpectedly. If all tokens fail, then it becomes impossible, in general, to solve the problem. However, we show that with any number of failures (less than a total collapse), we can always solve the problem if the original instance of the problem was solvable. Unlike previous solutions, we can tolerate failures occurring at arbitrary times during the execution of the algorithm. Our solution can be applied to any arbitrary network even when the topology is unknown.

Reconstructing visibility graphs with simple robots

We consider the problem of finding a minimalistic configuration of sensors that enable a simple robot inside an initially unknown polygon P on n vertices to reconstruct the visibility graph of P. The robot can sense features of its environment through its sensors, and it is allowed to move from vertex to vertex. We aim at understanding which sensorial capabilities are sufficient for the reconstruction of the visibility graph of P. We are able to show that the combinatorial visibilities at every vertex do not contain enough information even when combined with the knowledge of the exact interior angle at each vertex. Using sensors that can put distant vertices into a spatial relation on the other hand can in some cases enable our robot to reconstruct the visibility graph of P. We show that this is true for a sensor that can distinguish whether the angle between the lines toward two visible vertices is convex or reflex, as long as the robot is capable of identifying the vertex it last visited. We also show that measuring angles exactly is enough, if the robot has a compass.

A faster distributed approximation scheme for the connected dominating set problem for growth-bounded graphs

We present a distributed algorithm for finding a (1 + Ɛ)- approximation of a Minimum Connected Dominating Set in the class of Growth-Bounded graphs, which includes Unit Disk graphs. In addition, the computed Connected Dominating Set guarantees a constant stretch factor on the length of a shortest path with respect to the original graph and induces a subgraph of constant degree. The nodes do not require any positioning or distance information. The algorithm runs in O(TMIS+1/ƐO(1) ċ log* n)synchronous rounds, where TMIS is the time for computing a Maximal Independent Set (MIS) in the network graph. Using the fastest known deterministic algorithm for computing a MIS, the total running time is O((logΔ+1/ƐO(1)) ċ log* n), where Δ is the maximum degree of the network graph. If one allows randomization, the running time reduces to O((log log n+1/ƐO(1))ċ log* n) rounds.

Optimal Transitions for Targeted Protein Quantification: Best Conditioned Submatrix Selection

Multiple reaction monitoring (MRM) is a mass spectrometric method to quantify a specified set of proteins. In this paper, we identify a problem at the core of MRM peptide quantification accuracy. In mathematical terms, the problem is to find for a given matrix a submatrix with best condition number. We show this problem to be NP-hard, and we propose a greedy heuristic. Our numerical experiments show this heuristic to be orders of magnitude better than currently used methods.