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Nicolas Rivera

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Michael Hopkins

Université de Genève

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Université de Bordeaux

American University of Beirut

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Sharif University of Technology

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Papers by Nicolas Rivera

The 2^k Neighborhoods for Grid Path Planning

Journal of Artificial Intelligence Research, 2020

Grid path planning is an important problem in AI. Its understanding has been key for the developm... more Grid path planning is an important problem in AI. Its understanding has been key for the development of autonomous navigation systems. An interesting and rather surprising fact about the vast literature on this problem is that only a few neighborhoods have been used when evaluating these algorithms. Indeed, only the 4- and 8-neighborhoods are usually considered, and rarely the 16-neighborhood. This paper describes three contributions that enable the construction of effective grid path planners for extended 2k-neighborhoods; that is, neighborhoods that admit 2k neighbors per state, where k is a parameter. First, we provide a simple recursive definition of the 2k-neighborhood in terms of the 2k-1-neighborhood. Second, we derive distance functions, for any k ≥ 2, which allow us to propose admissible heuristics that are perfect for obstacle-free grids, which generalize the well-known Manhattan and Octile distances. Third, we define the notion of canonical path for the 2k-neighborhood; t...

Best-of-Three Voting on Dense Graphs

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

Given a graph G of n vertices, where each vertex is initially attached an opinion of either red o... more Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Bestof-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d = n α , where α = Ω (log log n) −1. We prove that if initially each vertex is red with probability greater than 1/2 + δ, and blue otherwise, where δ ≥ (log d) −C for some C > 0, then with high probability this dynamic reaches a final state where all vertices are red within O (log log n) + O log δ −1 steps .

Improved Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures, 2017

We present improved bounds on the cover time of the coalescingbranching random walk process COBRA... more We present improved bounds on the cover time of the coalescingbranching random walk process COBRA. The COBRA process, introduced in [Dutta et al., SPAA 2013], can be viewed as spreading a single item of information throughout an undirected graph in synchronised rounds. In each round, each vertex which has received the information in the previous round (possibly simultaneously from more than one neighbour and possibly not for the first time), 'pushes' the information to b randomly selected neighbours. The COBRA process is typically studied for integer branching rates b ≥ 2 (with the case b = 1 corresponding to a random walk). The aim of the process is to propagate the information quickly, but with a limited number of transmissions per vertex per round. The cover time of COBRA is defined as the expected number of rounds until each vertex has received the information at least once. Our main results are a bound of O(m + (d max) 2 log n) = O(n 2 log n) on the COBRA cover time for an arbitrary connected graph with n vertices, m edges and the maximum vertex degree d max , and a bound of O((r 2 +r /(1−λ)) log n) for r-regular connected graphs with the second eigenvalue λ. Our bounds improve the O(n 11/4 log n) and O((r 4 /ϕ 2) log 2 n) bounds shown in [Mitzenmacher et al., SPAA 2016], where ϕ is the conductance of the graph, and complement the O((1/(1 − λ)) 3 log n) bound shown in [Cooper et al., PODC 2016]. We obtain our bounds by analysing the process called Biased Infection with Persistent Source (BIPS), which was introduced in [Cooper et al., PODC 2016] as a dual process for COBRA .

Multi-Agent Flag Coordination Games

Adaptive Agents and Multi-Agents Systems, May 8, 2017

Many multi-agent coordination problems can be understood as autonomous local choices between a fi... more Many multi-agent coordination problems can be understood as autonomous local choices between a finite set of options, with each local choice undertaken simultaneously without explicit coordination between decision-makers, and with a shared goal of achieving a desired global state or states. Examples of such problems include classic consensus problems between nodes in a distributed computer network and the adoption of competing technology standards. We model such problems as a multi-round game between agents having flags of different colours to represent the finite choice options, and all agents seeking to achieve global patterns of colours through a succession of local colour-selection choices. We generalise and formalise the problem, proving results for the probabilities of achievement of common desired global states when these games are undertaken on bipartite graphs, extending known results for non-bipartite graphs. We also calculate probabilities for the game entering infinite cycles of non-convergence. In addition, we present a game-theoretic approach to the problem that has a mixed-strategy Nash equilibrium where two players can simultaneously flip the colour of one of the opponent's nodes in the bipartite graph before or during a flag-coordination game.

The Linear Voting Model

We study voting models on graphs. In the beginning, the vertices of a given graph have some initi... more We study voting models on graphs. In the beginning, the vertices of a given graph have some initial opinion. Over time, the opinions on the vertices change by interactions between graph neighbours. Under suitable conditions the system evolves to a state in which all vertices have the same opinion. In this work, we consider a new model of voting, called the Linear Voting Model. This model can be seen as a generalization of several models of voting, including among others, pull voting and push voting. One advantage of our model is that, even though it is very general, it has a rich structure making the analysis tractable. In particular we are able to solve the basic question about voting, the probability that certain opinion wins the poll, and furthermore, given appropriate conditions, we are able to bound the expected time until some opinion wins.

Fast Plurality Consensus in Regular Expanders

Pull voting is a classic method to reach consensus among $n$ vertices with differing opinions in ... more Pull voting is a classic method to reach consensus among $n$ vertices with differing opinions in a distributed network: each vertex at each step takes on the opinion of a random neighbour. This method, however, suffers from two drawbacks. Even if there are only two opposing opinions, the time taken for a single opinion to emerge can be slow and the final opinion is not necessarily the initially held majority. We refer to a protocol where 2 neighbours are contacted at each step as a 2-sample voting protocol. In the two-sample protocol a vertex updates its opinion only if both sampled opinions are the same. Not much was known about the performance of two-sample voting on general expanders in the case of three or more opinions. In this paper we show that the following performance can be achieved on a $d$-regular expander using two-sample voting. We suppose there are $k \ge 3$ opinions, and that the initial size of the largest and second largest opinions is $A_1, A_2$ respectively. We p...

New Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

ACM Transactions on Parallel Computing, 2019

If citing, it is advised that you check and use the publisher's definitive version for pagination... more If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

Discordant Voting Processes on Finite Graphs

SIAM Journal on Discrete Mathematics, 2018

We consider an asynchronous voting process on graphs which we call discordant voting, and which c... more We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T , the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET , the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = n 2 /4 independent of the underlying graph. For the complete graph K n , the push protocol has ET = Θ(n log n), whereas the pull protocol has ET = Θ(2 n). For the cycle C n all three protocols have ET = Θ(n 2). For the star graph however, the pull protocol has ET = O(n 2), whereas the push protocol is slower with ET = Θ(n 2 log n). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders.

Dispersion processes

Random Structures & Algorithms, 2018

If citing, it is advised that you check and use the publisher's definitive version for pagination... more If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

The Coalescing-Branching Random Walk on Expanders and the Dual Epidemic Process

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, 2016

If citing, it is advised that you check and use the publisher's definitive version for pagination... more If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

Coalescing Walks on Rotor-Router Systems

Lecture Notes in Computer Science, 2015

We consider the rotor-router mechanism for distributing particles in an undirected graph. If the ... more We consider the rotor-router mechanism for distributing particles in an undirected graph. If the last particle passing through a vertex v took an edge v,u, then the next time a particle is at v, it will leave v along the next edge v,w according to a fixed cyclic order of edges adjacent to v. The system works in synchronized steps and when two or more particles meet at the same vertex, they coalesce into one particle. A k-particle configuration of such a system is stable, if it does not lead to any coalescing. For 2≤k≤n, we give the full characterization of stable k-particle configurations for cycles. We also show sufficient conditions for regular graphs with n vertices to admit n-particle stable configurations.

Fast Consensus for Voting on General Expander Graphs

Lecture Notes in Computer Science, 2015

The 2^k Neighborhoods for Grid Path Planning

Journal of Artificial Intelligence Research, 2020

Grid path planning is an important problem in AI. Its understanding has been key for the developm... more Grid path planning is an important problem in AI. Its understanding has been key for the development of autonomous navigation systems. An interesting and rather surprising fact about the vast literature on this problem is that only a few neighborhoods have been used when evaluating these algorithms. Indeed, only the 4- and 8-neighborhoods are usually considered, and rarely the 16-neighborhood. This paper describes three contributions that enable the construction of effective grid path planners for extended 2k-neighborhoods; that is, neighborhoods that admit 2k neighbors per state, where k is a parameter. First, we provide a simple recursive definition of the 2k-neighborhood in terms of the 2k-1-neighborhood. Second, we derive distance functions, for any k ≥ 2, which allow us to propose admissible heuristics that are perfect for obstacle-free grids, which generalize the well-known Manhattan and Octile distances. Third, we define the notion of canonical path for the 2k-neighborhood; t...

Best-of-Three Voting on Dense Graphs

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

Given a graph G of n vertices, where each vertex is initially attached an opinion of either red o... more Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Bestof-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d = n α , where α = Ω (log log n) −1. We prove that if initially each vertex is red with probability greater than 1/2 + δ, and blue otherwise, where δ ≥ (log d) −C for some C > 0, then with high probability this dynamic reaches a final state where all vertices are red within O (log log n) + O log δ −1 steps .

Improved Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures, 2017

We present improved bounds on the cover time of the coalescingbranching random walk process COBRA... more We present improved bounds on the cover time of the coalescingbranching random walk process COBRA. The COBRA process, introduced in [Dutta et al., SPAA 2013], can be viewed as spreading a single item of information throughout an undirected graph in synchronised rounds. In each round, each vertex which has received the information in the previous round (possibly simultaneously from more than one neighbour and possibly not for the first time), 'pushes' the information to b randomly selected neighbours. The COBRA process is typically studied for integer branching rates b ≥ 2 (with the case b = 1 corresponding to a random walk). The aim of the process is to propagate the information quickly, but with a limited number of transmissions per vertex per round. The cover time of COBRA is defined as the expected number of rounds until each vertex has received the information at least once. Our main results are a bound of O(m + (d max) 2 log n) = O(n 2 log n) on the COBRA cover time for an arbitrary connected graph with n vertices, m edges and the maximum vertex degree d max , and a bound of O((r 2 +r /(1−λ)) log n) for r-regular connected graphs with the second eigenvalue λ. Our bounds improve the O(n 11/4 log n) and O((r 4 /ϕ 2) log 2 n) bounds shown in [Mitzenmacher et al., SPAA 2016], where ϕ is the conductance of the graph, and complement the O((1/(1 − λ)) 3 log n) bound shown in [Cooper et al., PODC 2016]. We obtain our bounds by analysing the process called Biased Infection with Persistent Source (BIPS), which was introduced in [Cooper et al., PODC 2016] as a dual process for COBRA .

Multi-Agent Flag Coordination Games

Adaptive Agents and Multi-Agents Systems, May 8, 2017

Many multi-agent coordination problems can be understood as autonomous local choices between a fi... more Many multi-agent coordination problems can be understood as autonomous local choices between a finite set of options, with each local choice undertaken simultaneously without explicit coordination between decision-makers, and with a shared goal of achieving a desired global state or states. Examples of such problems include classic consensus problems between nodes in a distributed computer network and the adoption of competing technology standards. We model such problems as a multi-round game between agents having flags of different colours to represent the finite choice options, and all agents seeking to achieve global patterns of colours through a succession of local colour-selection choices. We generalise and formalise the problem, proving results for the probabilities of achievement of common desired global states when these games are undertaken on bipartite graphs, extending known results for non-bipartite graphs. We also calculate probabilities for the game entering infinite cycles of non-convergence. In addition, we present a game-theoretic approach to the problem that has a mixed-strategy Nash equilibrium where two players can simultaneously flip the colour of one of the opponent's nodes in the bipartite graph before or during a flag-coordination game.

The Linear Voting Model

We study voting models on graphs. In the beginning, the vertices of a given graph have some initi... more We study voting models on graphs. In the beginning, the vertices of a given graph have some initial opinion. Over time, the opinions on the vertices change by interactions between graph neighbours. Under suitable conditions the system evolves to a state in which all vertices have the same opinion. In this work, we consider a new model of voting, called the Linear Voting Model. This model can be seen as a generalization of several models of voting, including among others, pull voting and push voting. One advantage of our model is that, even though it is very general, it has a rich structure making the analysis tractable. In particular we are able to solve the basic question about voting, the probability that certain opinion wins the poll, and furthermore, given appropriate conditions, we are able to bound the expected time until some opinion wins.

Fast Plurality Consensus in Regular Expanders

Pull voting is a classic method to reach consensus among $n$ vertices with differing opinions in ... more Pull voting is a classic method to reach consensus among $n$ vertices with differing opinions in a distributed network: each vertex at each step takes on the opinion of a random neighbour. This method, however, suffers from two drawbacks. Even if there are only two opposing opinions, the time taken for a single opinion to emerge can be slow and the final opinion is not necessarily the initially held majority. We refer to a protocol where 2 neighbours are contacted at each step as a 2-sample voting protocol. In the two-sample protocol a vertex updates its opinion only if both sampled opinions are the same. Not much was known about the performance of two-sample voting on general expanders in the case of three or more opinions. In this paper we show that the following performance can be achieved on a $d$-regular expander using two-sample voting. We suppose there are $k \ge 3$ opinions, and that the initial size of the largest and second largest opinions is $A_1, A_2$ respectively. We p...

New Cover Time Bounds for the Coalescing-Branching Random Walk on Graphs

ACM Transactions on Parallel Computing, 2019

If citing, it is advised that you check and use the publisher's definitive version for pagination... more If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

Discordant Voting Processes on Finite Graphs

SIAM Journal on Discrete Mathematics, 2018

We consider an asynchronous voting process on graphs which we call discordant voting, and which c... more We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T , the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET , the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = n 2 /4 independent of the underlying graph. For the complete graph K n , the push protocol has ET = Θ(n log n), whereas the pull protocol has ET = Θ(2 n). For the cycle C n all three protocols have ET = Θ(n 2). For the star graph however, the pull protocol has ET = O(n 2), whereas the push protocol is slower with ET = Θ(n 2 log n). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders.

Dispersion processes

Random Structures & Algorithms, 2018

If citing, it is advised that you check and use the publisher's definitive version for pagination... more If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

The Coalescing-Branching Random Walk on Expanders and the Dual Epidemic Process

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, 2016

If citing, it is advised that you check and use the publisher's definitive version for pagination... more If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections.

Coalescing Walks on Rotor-Router Systems

Lecture Notes in Computer Science, 2015

We consider the rotor-router mechanism for distributing particles in an undirected graph. If the ... more We consider the rotor-router mechanism for distributing particles in an undirected graph. If the last particle passing through a vertex v took an edge v,u, then the next time a particle is at v, it will leave v along the next edge v,w according to a fixed cyclic order of edges adjacent to v. The system works in synchronized steps and when two or more particles meet at the same vertex, they coalesce into one particle. A k-particle configuration of such a system is stable, if it does not lead to any coalescing. For 2≤k≤n, we give the full characterization of stable k-particle configurations for cycles. We also show sufficient conditions for regular graphs with n vertices to admit n-particle stable configurations.

Fast Consensus for Voting on General Expander Graphs

Lecture Notes in Computer Science, 2015