Graph reconstruction in the congested clique

Montealegre, Pedro; Perez-Salazar, Sebastian; Rapaport, Ivan; Todinca, Ioan

doi:10.1016/j.jcss.2020.04.004

Cited by 6 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 3 we show that the recognition of cographs is in dAM[log n] using shared randomness. Our result consists of adapting an algorithm given in [28,33], originally designed for the Broadcast Congested Clique model. In this regard, we exploit the natural high connectivity of this class, combined with the use of non-determinism in order to route all messages in the network to a leader node, which is delegated to act as a referee.…”

Section: Our Resultsmentioning

confidence: 99%

“…In [28], a public-coin randomized protocol for recognizing cographs is obtained. In [33], randomized protocols recognizing both classes of graphs are given. Interestingly, these protocols not only recognize the classes but reconstruct them, meaning that the referee learns all the edges of the input graph.…”

Section: Related Workmentioning

confidence: 99%

“…In the centralized setting, both cographs and distance-hereditary graphs can be recognized in linear time [10]. They have also been addressed in the context of other computation models such as parallel random access machine [9] or the broadcast congested clique model [28,33] (see the related work section for more details).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

Montealegre¹,

Ramírez-Romero²,

Rapaport³

2020

Preprint

View full text Add to dashboard Cite

We present compact distributed interactive proofs for the recognition of two important graph classes, well-studied in the context of centralized algorithms, namely complement reducible graphs and distance-hereditary graphs. Complement reducible graphs (also called cographs) are defined as the graphs not containing a four-node path P4 as an induced subgraph. Distance-hereditary graphs are a super-class of cographs, defined as the graphs where the distance (shortest paths) between any pair of vertices is the same on every induced connected subgraph.First, we show that there exists a distributed interactive proof for the recognition of cographs with two rounds of interaction. More precisely, we give a dAM protocol with a proof size of O(log n) bits that uses shared randomness and recognizes cographs with high probability. Moreover, our protocol can be adapted to verify any Turing-decidable predicate restricted to cographs in dAM with certificates of size O(log n).Second, we give a three-round, dMAM interactive protocol for the recognition of distancehereditary graphs, still with a proof size of O(log n) bits and also using shared randomness.Finally, we show that any one-round (denoted dM) or two-round, dMA protocol for the recognition of cographs or distance-hereditary graphs requires certificates of size Ω(log n) bits. Moreover, we show that any constant-round dAM protocol using shared randomness requires certificates of size Ω(log log n).

show abstract

Section: Our Resultsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

Montealegre¹,

Ramírez-Romero²,

Rapaport³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The task of topology recognition was also investigated in models other than the radio model: in [13] the authors use the LOCAL model, and in [23] the model used is the congested clique.…”

Section: Related Workmentioning

confidence: 99%

Deterministic Size Discovery and Topology Recognition in Radio Networks with Short Labels

Gańczorz¹,

Jurdziński²,

Lewko³

et al. 2021

Preprint

View full text Add to dashboard Cite

We consider the fundamental problems of size discovery and topology recognition in radio networks modeled by simple undirected connected graphs. Size discovery calls for all nodes to output the number of nodes in the graph, called its size, and in the task of topology recognition each node has to learn the topology of the graph and its position in it. In radio networks, nodes communicate in synchronous rounds and start in the same round. In each round a node can either transmit the same message to all its neighbors, or stay silent and listen. At the receiving end, a node v hears a message from a neighbor w in a given round, if v listens in this round, and if w is its only neighbor that transmits in this round. If more than one neighbor of a node v transmits in a given round, there is a collision at v. We do not assume collision detection: in case of a collision, node v does not hear anything (except the background noise that it also hears when no neighbor transmits). The time of a deterministic algorithm for each of the above problems is the worst-case number of rounds it takes to solve it. Nodes have labels which are (not necessarily different) binary strings. Each node knows its own label and can use it when executing the algorithm. The length of a labeling scheme is the largest length of a label.Our goal is to construct short labeling schemes for size discovery and topology recognition in arbitrary radio networks, and to design efficient deterministic algorithms for each of these tasks, using these short schemes. It turns out that the optimal length of labeling schemes for both problems depends on the maximum degree ∆ of the graph. For size discovery, we construct a labeling scheme of length O(log log ∆) (which is known to be optimal, even if collision detection is available) and we design an algorithm for this problem using this scheme and working in time O(log 2 n), where n is the size of the graph. We also show that time complexity O(log 2 n) is optimal for the problem of size discovery, whenever the labeling scheme is of optimal length O(log log ∆). For topology recognition, we construct a labeling scheme of length O(log ∆), and we design an algorithm for this problem using this scheme and working in time O D∆ + min(∆ 2 , n) , where D is the diameter of the graph. We also show that the length of our labeling scheme is asymptotically optimal, by proving that topology recognition in the class of arbitrary radio networks requires labeling schemes of length Ω(log ∆).

show abstract

Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

Montealegre

Ramírez-Romero

Rapaport

2021

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Graph reconstruction in the congested clique

Cited by 6 publications

References 11 publications

Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

Deterministic Size Discovery and Topology Recognition in Radio Networks with Short Labels

Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary Graphs

Contact Info

Product

Resources

About