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35
Flocking for MultiAgent Dynamic Systems: Algorithms and Theory
, 2006
"... In this paper, we present a theoretical framework for design and analysis of distributed flocking algorithms. Two cases of flocking in freespace and presence of multiple obstacles are considered. We present three flocking algorithms: two for freeflocking and one for constrained flocking. A compre ..."
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Cited by 436 (2 self)
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In this paper, we present a theoretical framework for design and analysis of distributed flocking algorithms. Two cases of flocking in freespace and presence of multiple obstacles are considered. We present three flocking algorithms: two for freeflocking and one for constrained flocking. A comprehensive analysis of the first two algorithms is provided. We demonstrate the first algorithm embodies all three rules of Reynolds. This is a formal approach to extraction of interaction rules that lead to the emergence of collective behavior. We show that the first algorithm generically leads to regular fragmentation, whereas the second and third algorithms both lead to flocking. A systematic method is provided for construction of cost functions (or collective potentials) for flocking. These collective potentials penalize deviation from a class of latticeshape objects called αlattices. We use a multispecies framework for construction of collective potentials that consist of flockmembers, or αagents, and virtual agents associated with αagents called β and γagents. We show that migration of flocks can be performed using a peertopeer network of agents, i.e. “flocks need no leaders.” A “universal” definition of flocking for particle systems with similarities to Lyapunov stability is given. Several simulation results are provided that demonstrate performing 2D and 3D flocking, split/rejoin maneuver, and squeezing maneuver for hundreds of agents using the proposed algorithms.
Extremal graph theory for metric dimension and diameter
 ELECTRONIC NOTES IN DISCR. MATH
, 2008
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Graph Treewidth and Geometric Thickness Parameters
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restri ..."
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Cited by 20 (9 self)
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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and stararboricity.
Analysis and design tools for distributed motion coordination
 ACC 2005, TO APPEAR
, 2005
"... This paper surveys recentlydeveloped theoretical tools for the analysis and design of coordination algorithms for networks of mobile autonomous agents. First, various motion coordination tasks are encoded into aggregate cost functions from Geometric Optimization. Second, the limited communication ..."
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Cited by 17 (0 self)
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This paper surveys recentlydeveloped theoretical tools for the analysis and design of coordination algorithms for networks of mobile autonomous agents. First, various motion coordination tasks are encoded into aggregate cost functions from Geometric Optimization. Second, the limited communication capabilities of the mobile agents are modeled via the notions of proximity graphs from Computational Geometry and of spatially distributed maps. Finally, we illustrate how to apply these tools to design and analyze scalable cooperative strategies in a variety of motion coordination problems such as deployment, rendezvous, and flocking.
Polynomial treewidth forces a large gridlikeminor
, 2008
"... Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. ..."
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Cited by 12 (2 self)
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Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A gridlikeminor of order ℓ in a graph G is a set of paths in G whose intersection graph is bipartite and contains a Kℓminor. For example, the rows and columns of the ℓ × ℓ grid are a gridlikeminor of order ℓ + 1. We prove that polynomial treewidth forces a large gridlikeminor. In particular, every graph with treewidth at least cℓ 4 √ log ℓ has a gridlikeminor of order ℓ. As an application of this result, we prove that the cartesian product G □ K2 contains a Kℓminor whenever G has treewidth at least cℓ 4 √ log ℓ.
ERROR SCALING LAWS FOR LINEAR OPTIMAL ESTIMATION FROM RELATIVE MEASUREMENTS
, 2009
"... We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the diffe ..."
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Cited by 8 (1 self)
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We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the difference between two variables. We take an arbitrary variable as the reference and consider the optimal (minimum variance) linear unbiased estimate of the remaining variables. We investigate how the error in the optimal linear unbiased estimate of a node variable grows with the distance of the node to the reference node. We establish a classification of graphs, namely, dense or sparse in R d, 1 ≤ d ≤ 3, that determines how the linear unbiased optimal estimation error of a node grows with its distance from the reference node. In particular, if a graph is dense in 1,2, or 3D, then a node variable’s estimation error is upper bounded by a linear, logarithmic, or bounded function of distance from the reference, respectively. Corresponding lower bounds are obtained if the graph is sparse in 1, 2 and 3D. Our results also show that naive measures of graph density, such as node degree, are inadequate predictors of the estimation error. Being true for the optimal linear unbiased estimate, these scaling laws determine algorithmindependent limits on the estimation accuracy achievable in large graphs.
Long paths and cycles in random subgraphs of graphs with large minimum degree
, 2013
"... For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if p ≥ ω/k for a function ω = ω(k) that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and ..."
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Cited by 8 (3 self)
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For a given finite graph G of minimum degree at least k, let Gp be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if p ≥ ω/k for a function ω = ω(k) that tends to infinity as k does, then Gp asymptotically almost surely contains a cycle (and thus a path) of length at least (1 − o(1))k, and (ii) if p ≥ (1 + o(1)) ln k/k, then Gp asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on
CLIQUE MINORS IN CARTESIAN PRODUCTS OF GRAPHS
, 2008
"... A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number ηÔGÕis the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G¥H. Our main result is a rough s ..."
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Cited by 8 (6 self)
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A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number ηÔGÕis the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G¥H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: a planar grid with a vortex of bounded width in the outerface, a cylindrical grid with a vortex of bounded width in each of the two ‘big ’ faces, or a toroidal grid. Motivation for studying the Hadwiger number of a graph includes Hadwiger’s Conjecture, which states that the chromatic number χÔGÕ�ηÔGÕ. It is open whether Hadwiger’s Conjecture holds for every Cartesian product. We prove that if�VÔHÕ�¡1�χÔGÕ�χÔHÕthen Hadwiger’s Conjecture holds for G¥H. On the other hand, we prove that Hadwiger’s Conjecture holds for all Cartesian products if and only if it holds for all G¥K2. We then show that
Complete graph minors and the GRAPH MINOR STRUCTURE THEOREM
 JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2012
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