The Impact of Locality in the Broadcast Congested Clique Model

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 $b=\mathcal{O}(d \cdot \log n)$. 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 ${{\sc BClique}}[r]$, we study the problem of detecting, in $G$, an induced cycle of length at most $k$ (${\sc Cycle}_{\leq k}$) and the problem of detecting an induced cycle of length at least $k+1$ (${\sc Cycle}_{>k}$). We give upper and lower bounds. We show that if each node is allowed to see up to distance $r={\lfloor k/2 \rfloor + 1}$, then a polylogarithmic bandwidth is sufficient for solving ${\sc Cycle}_{>k}$ with only two rounds. Nevertheless, if nodes were allowed to see up to distance $r=\lfloor k/3 \rfloor$, then any one-round algorithm that solves ${\sc Cycle}_{>k}$ needs the bandwidth $b$ to be at least $\Omega(n/\log n)$. We also show the existence of a one-round, deterministic ${{\sc BClique}}$ algorithm that solves ${\sc Cycle}_{\leq k}$ with bandwitdh $b=\mathcal{O}(n^{1/\lfloor{k/2}\rfloor} \cdot \log n)$. On the negative side, we prove that, if $\epsilon \leq 1/3$ and $0 < r \leq k/4$, then any $\epsilon$-error, $R$-round, $b$-bandwidth algorithm in the ${{\sc BClique}}[r]$ model that solves problem ${\sc Cycle}_{\leq k}$ satisfies $R \cdot b = \Omega(n^{1/\lfloor{k/2}\rfloor})$.

References

1.
K. J. Ahn, S. Guha, and A. McGregor, Analyzing graph structure via linear measurements, in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, 2012, pp. 459--467.
2.
K. J. Ahn, S. Guha, and A. McGregor, Graph sketches: Sparsification, spanners, and subgraphs, in Proceedings of the 31st Symposium on Principles of Database Systems, 2012, pp. 5--14.
3.
D. Angluin, Local and global properties in networks of processors, in Proceedings of the 12th ACM Symposium on Theory of Computing, 1980, pp. 82--93.
4.
H. Arfaoui, P. Fraigniaud, D. Ilcinkas, and F. Mathieu, Distributedly testing cycle-freeness, in Proceedings of the 40th International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci. 8747, 2014, pp. 15--28.
5.
B. Awerbuch, O. Goldreich, R. Vainish, and D. Peleg, A trade-off between information and communication in broadcast protocols, J. ACM, 37 (1990), pp. 238--256.
6.
F. Becker, A. Kosowski, M. Matamala, N. Nisse, I. Rapaport, K. Suchan, and I. Todinca, Allowing each node to communicate only once in a distributed system: Shared whiteboard models, Distrib. Comput., 28 (2015), pp. 189--200.
7.
F. Becker, M. Matamala, N. Nisse, I. Rapaport, K. Suchan, and I. Todinca, Adding a referee to an interconnection network: What can(not) be computed in one round, in Proceedings of the 25th IEEE International Parallel and Distributed Processing Symposium, 2011, pp. 508--514.
8.
F. Becker, P. Montealegre, I. Rapaport, and I. Todinca, The simultaneous number-in-hand communication model for networks: Private coins, public coins and determinism, in Proceedings of the 21st International Colloquium on Structural Information and Communication Complexity, 2014, pp. 83--95.
9.
E. Birmelé, Tree-width and circumference of graphs, J. Graph Theory, 43 (2003), pp. 24--25.
10.
J. A. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combin. Theory Ser. B, 16 (1974), pp. 97--105.
11.
A. Brandstädt, V. B. Le, and J. P. Spinrad, Graph Classes: A Survey, Discrete Math. Appl. 3, SIAM, Philadelphia, 1999.
12.
K. Censor-Hillel, P. Kaski, J. H. Korhonen, C. Lenzen, A. Paz, and J. Suomela, Algebraic methods in the congested clique, in Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, 2015, pp. 143--152.
13.
N. Chandrasekharan and S. S. Iyengar, NC algorithms for recognizing chordal graphs and k trees, IEEE Trans. Comput., 37 (1988), pp. 1178--1183.
14.
H.-C. Chang and H.-I. Lu, A faster algorithm to recognize even-hole-free graphs, J. Combin. Theory Ser. B, 113 (2015), pp. 141--161.
15.
P. L. Chebyshev, Mémoire sur les nombres premiers, J. Math. Pures Appl., 17 (1852), pp. 366--390.
16.
A. Drucker, F. Kuhn, and R. Oshman, On the power of the congested clique model, in Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing, 2014, pp. 367--376.
17.
P. Erdös, Extremal problems in graph theory, in Theory of Graphs and its Applications, Proc. Sympos. Smolenice, 1964.
18.
M. Ghaffari, T. Gouleakis, C. Konrad, S. Mitrović, and R. Rubinfeld, Improved massively parallel computation algorithms for mis, matching, and vertex cover, in Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC '18, 2018, ACM, New York, pp. 129--138.
19.
S. Guha, A. McGregor, and D. Tench, Vertex and hyperedge connectivity in dynamic graph streams, in Proceedings of the 34th ACM Symposium on Principles of Database Systems, 2015, pp. 241--247.
20.
J. W. Hegeman, S. V. Pemmaraju, and V. Sardeshmukh, Near-constant-time distributed algorithms on a congested clique, in Proceedings of the 28th International Symposium on Distributed Computing, 2014, pp. 514--530.
21.
S. Holzer and N. Pinsker, Approximation of distances and shortest paths in the broadcast congest clique, in Proceedings of the 19th International Conference on Principles of Distributed Systems, LIPIcs Leibniz Int. Proc. Inform. 46, 2016, pp. 1--16.
22.
H. Jowhari, M. Saglam, and G. Tardos, Tight bounds for lp samplers, finding duplicates in streams, and related problems, in Proceedings of the 30th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, 2011, pp. 49--58.
23.
T. Jurdziński and K. Nowicki, MST in O(1) rounds of congested clique, in Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2018, pp. 2620--2632, https://doi.org/10.1137/1.9781611975031.167.
24.
J. Kari, M. Matamala, I. Rapaport, and V. Salo, Solving the induced subgraph problem in the randomized multiparty simultaneous messages model, in Proceedings of the 21st International Colloquium on Structural Information and Communication Complexity, 2015, pp. 370--384.
25.
E. Kushilevitz and N. Nisan, Communication Complexity, Cambridge University Press, New York, 1997.
26.
F. Lazebnik, V. A. Ustimenko, and A. J. Woldar, A new series of dense graphs of high girth, Bull. Amer. Math. Soc., 32 (1995), pp. 73--79.
27.
N. Linial, Locality in distributed graph algorithms, SIAM J. Comput., 21 (1992), pp. 193--201.
28.
Z. Lotker, B. Patt-Shamir, E. Pavlov, and D. Peleg, Minimum-weight spanning tree construction in O (log log n) communication rounds, SIAM J. Comput., 35 (2005), pp. 120--131.
29.
M. Naor and L. Stockmeyer, What can be computed locally?, SIAM J. Comput., 24 (1995), pp. 1259--1277.
30.
B. Patt-Shamir and M. Perry, Proof-labeling schemes: Broadcast, unicast and in between, in Proceedings of the 19th International Symposium on Stabilization, Safety, and Security of Distributed Systems, Lecture Notes in Comput. Sci. 10616, 2017, pp. 1--17.
31.
A. Thomason, The extremal function for complete minors, J. Combin. Theory Ser. B, 81 (2001), pp. 318--338.
32.
C. Thomassen, On the presence of disjoint subgraphs of a specified type, J. Graph Theory, 12 (1988), pp. 101--111, https://doi.org/10.1002/jgt.3190120111.

Information & Authors

Information

Published In

SIAM Journal on Discrete Mathematics
Pages: 682 - 700
ISSN (online): 1095-7146

History

Submitted: 17 December 2018
Accepted: 12 December 2019
Published online: 12 March 2020

Authors

Funding Information

Fondo Nacional de Desarrollo Científico y Tecnológico https://doi.org/10.13039/501100002850 : 1170021

Funding Information

Fondo Nacional de Desarrollo Científico y Tecnológico https://doi.org/10.13039/501100002850 : 11190482

Funding Information

Comisión Nacional de Investigación Científica y Tecnológica https://doi.org/10.13039/501100002848

Funding Information

Comisión Nacional de Investigación Científica y Tecnológica https://doi.org/10.13039/501100002848

Metrics & Citations

Citations

Cited By

There are no citations for this item