# The Impact of Locality in the Broadcast Congested Clique Model

## Abstract

*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 $R \cdot b = \Omega(n^{1/\lfloor{k/2}\rfloor})$.

### Keywords

### MSC codes

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*Analyzing graph structure via linear measurements*, in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, 2012, pp. 459--467.

*Graph sketches: Sparsification, spanners, and subgraphs*, in Proceedings of the 31st Symposium on Principles of Database Systems, 2012, pp. 5--14.

*Local and global properties in networks of processors*, in Proceedings of the 12th ACM Symposium on Theory of Computing, 1980, pp. 82--93.

*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.

*A trade-off between information and communication in broadcast protocols*, J. ACM, 37 (1990), pp. 238--256.

*Allowing each node to communicate only once in a distributed system: Shared whiteboard models*, Distrib. Comput., 28 (2015), pp. 189--200.

*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.

*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.

*Tree-width and circumference of graphs*, J. Graph Theory, 43 (2003), pp. 24--25.

*Cycles of even length in graphs*, J. Combin. Theory Ser. B, 16 (1974), pp. 97--105.

*Graph Classes: A Survey*, Discrete Math. Appl. 3, SIAM, Philadelphia, 1999.

*Algebraic methods in the congested clique*, in Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, 2015, pp. 143--152.

*NC algorithms for recognizing chordal graphs and*, IEEE Trans. Comput., 37 (1988), pp. 1178--1183.

*k*trees*A faster algorithm to recognize even-hole-free graphs*, J. Combin. Theory Ser. B, 113 (2015), pp. 141--161.

*Mémoire sur les nombres premiers*, J. Math. Pures Appl., 17 (1852), pp. 366--390.

*On the power of the congested clique model*, in Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing, 2014, pp. 367--376.

*Extremal problems in graph theory*, in Theory of Graphs and its Applications, Proc. Sympos. Smolenice, 1964.

*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.

*Vertex and hyperedge connectivity in dynamic graph streams*, in Proceedings of the 34th ACM Symposium on Principles of Database Systems, 2015, pp. 241--247.

*Near-constant-time distributed algorithms on a congested clique*, in Proceedings of the 28th International Symposium on Distributed Computing, 2014, pp. 514--530.

*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.

*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.

*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.

*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.

*Communication Complexity*, Cambridge University Press, New York, 1997.

*A new series of dense graphs of high girth*, Bull. Amer. Math. Soc., 32 (1995), pp. 73--79.

*Locality in distributed graph algorithms*, SIAM J. Comput., 21 (1992), pp. 193--201.

*Minimum-weight spanning tree construction in O (log log n) communication rounds*, SIAM J. Comput., 35 (2005), pp. 120--131.

*What can be computed locally?*, SIAM J. Comput., 24 (1995), pp. 1259--1277.

*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.

*The extremal function for complete minors*, J. Combin. Theory Ser. B, 81 (2001), pp. 318--338.

*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

#### Copyright

#### History

**Submitted**: 17 December 2018

**Accepted**: 12 December 2019

**Published online**: 12 March 2020

#### Keywords

#### MSC codes

### Authors

#### Funding Information

#### Funding Information

#### Funding Information

#### Funding Information

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

There are no citations for this item