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Author Becker, F.; Montealecre, P.; Rapaport, I.; Todinca, I. doi  openurl
  Title The Impact Of Locality In The Broadcast Congested Clique Model Type
  Year 2020 Publication Siam Journal On Discrete Mathematics Abbreviated Journal SIAM Discret. Math.  
  Volume 34 Issue 1 Pages 682-700  
  Keywords broadcast congested clique; induced cycles; graph degeneracy  
  Abstract The broadcast congested clique model (BCLIQUE) is a message-passing model of distributed computation where n nodes communicate with each other in synchronous rounds. First, in this paper we prove that there is a one-round, deterministic algorithm that reconstructs the input graph G if the graph is d-degenerate, and rejects otherwise, using bandwidth b = O(d . log n). Then, we introduce a new parameter to the model. We study the situation where the nodes, initially, instead of knowing their immediate neighbors, know their neighborhood up to a fixed radius r. In this new framework, denoted BCLIQuE[r], we study the problem of detecting, in G, an induced cycle of length at most k (CYCLE <= k) and the problem of detecting an induced cycle of length at least k +1 (CYCLE>k). We give upper and lower bounds. We show that if each node is allowed to see up to distance r = left perpendicular k/2 right perpendicular + 1, then a polylogarithmic bandwidth is sufficient for solving CYCLE>k with only two rounds. Nevertheless, if nodes were allowed to see up to distance r = left perpendicular k/3 right perpendicular, then any one-round algorithm that solves CYCLE>k needs the bandwidth b to be at least Omega(n/ log n). We also show the existence of a one-round, deterministic BCLIQUE algorithm that solves CYCLE <= k with bandwitdh b = O(n(1/left perpendicular k/2 right perpendicular). log n). On the negative side, we prove that, if epsilon <= 1/3 and 0 < r <= k/4, then any epsilon-error, R-round, b-bandwidth algorithm in the BCLIQUE[r] model that solves problem CYCLE(<= k )satisfies R . b = Omega(n(1/left perpendicular k/2 right perpendicular)).  
  Address [Becker, F.; Todinca, I] Univ Orleans, INSA Ctr Val Loire, LIFO EA 4022, Orleans, France, Email: florent.becker@univ-orleans.fr;  
  Corporate Author Thesis  
  Publisher Siam Publications Place of Publication Editor  
  Language English Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 0895-4801 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000546886700033 Approved  
  Call Number UAI @ eduardo.moreno @ Serial 1182  
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Author Bitar, N.; Goles, E.; Montealegre, P. doi  openurl
  Title COMPUTATIONAL COMPLEXITY OF BIASED DIFFUSION-LIMITED AGGREGATION Type
  Year 2022 Publication Siam Journal On Discrete Mathematics Abbreviated Journal SIAM Discret. Math.  
  Volume 36 Issue 1 Pages 823-866  
  Keywords diffusion-limited aggregation; computational complexity; space complexity; NL-completeness; P-completeness  
  Abstract Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists in a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote by k-DLA the model where the particles move only in k possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid c and a sequence S of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site c when sequence S is realized. The second problem is Realization, where the input is a set of positions of the grid, P. The question is whether there exists a sequence S that realizes P, i.e. all particles of S exactly occupy the positions in P. Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is P-Complete for 2-DLA (thus for 3-DLA). Later, we show that Prediction can be solved much more efficiently for 1-DLA. In fact, we show that in that case the problem is NL-Complete. With respect to Realization, we show that restricted to 2-DLA the problem is in P, while in the 1-DLA case, the problem is in L.  
  Address  
  Corporate Author Thesis  
  Publisher Place of Publication Editor  
  Language Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 0895-4801 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000778502000037 Approved  
  Call Number UAI @ alexi.delcanto @ Serial 1558  
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