TY - JOUR

T1 - Latency, capacity, and distributed minimum spanning trees

AU - Augustine, John

AU - Gilbert, Seth

AU - Kuhn, Fabian

AU - Robinson, Peter

AU - Sourav, Suman

N1 - Funding Information:
This work was partially supported by the grant from Ministry of Education, Singapore MOE2018-T2-1-160 (Beyond Worst-Case Analysis: A Tale of Distributed Algorithms). Peter Robinson was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 11213620 ], as well as by a grant from the City University of Hong Kong [Project No. 7200639/CS ]. John Augustine was partially supported by DST/SERB MATRICS Grant (file number MTR/2018/001198 ), IIT Madras ERP grant RF/2021/0653/CS/RFER/008500 , and the Centre for Cryptography Cybersecurity and Distributed Trust.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2022/6

Y1 - 2022/6

N2 - We study the cost of distributed MST construction in the setting where each edge has a latency and a capacity, along with the weight. Edge latencies capture the delay on the links of the communication network, while capacity captures their throughput (in this case, the rate at which messages can be sent). Depending on how the edge latencies relate to the edge weights, we provide several tight bounds on the time and messages required to construct an MST. When edge weights exactly correspond with the latencies, we show that, perhaps interestingly, the bottleneck parameter in determining the running time of an algorithm is the total weight W of the MST (rather than the total number of nodes n, as in the standard CONGEST model). That is, we show a tight bound of Θ˜(D+W/c) rounds, where D refers to the latency diameter of the graph, W refers to the total weight of the constructed MST and edges have capacity c. The proposed algorithm sends O˜(m+W) messages, where m, the total number of edges in the network graph under consideration, is a known lower bound on message complexity for MST construction. We also show that Ω(W) is a lower bound for fast MST constructions. When the edge latencies and the corresponding edge weights are unrelated, and either can take arbitrary values, we show that (unlike the sub-linear time algorithms in the standard CONGEST model, on small diameter graphs), the best time complexity that can be achieved is Θ˜(D+n/c). However, if we restrict all edges to have equal latency ℓ and capacity c while having possibly different weights (weights could deviate arbitrarily from ℓ), we give an algorithm that constructs an MST in O˜(D+nℓ/c) time. In each case, we provide nearly matching upper and lower bounds.

AB - We study the cost of distributed MST construction in the setting where each edge has a latency and a capacity, along with the weight. Edge latencies capture the delay on the links of the communication network, while capacity captures their throughput (in this case, the rate at which messages can be sent). Depending on how the edge latencies relate to the edge weights, we provide several tight bounds on the time and messages required to construct an MST. When edge weights exactly correspond with the latencies, we show that, perhaps interestingly, the bottleneck parameter in determining the running time of an algorithm is the total weight W of the MST (rather than the total number of nodes n, as in the standard CONGEST model). That is, we show a tight bound of Θ˜(D+W/c) rounds, where D refers to the latency diameter of the graph, W refers to the total weight of the constructed MST and edges have capacity c. The proposed algorithm sends O˜(m+W) messages, where m, the total number of edges in the network graph under consideration, is a known lower bound on message complexity for MST construction. We also show that Ω(W) is a lower bound for fast MST constructions. When the edge latencies and the corresponding edge weights are unrelated, and either can take arbitrary values, we show that (unlike the sub-linear time algorithms in the standard CONGEST model, on small diameter graphs), the best time complexity that can be achieved is Θ˜(D+n/c). However, if we restrict all edges to have equal latency ℓ and capacity c while having possibly different weights (weights could deviate arbitrarily from ℓ), we give an algorithm that constructs an MST in O˜(D+nℓ/c) time. In each case, we provide nearly matching upper and lower bounds.

KW - Capacity

KW - Distributed minimum spanning tree

KW - Fragment

KW - Latency

KW - MST construction

UR - http://www.scopus.com/inward/record.url?scp=85122097443&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85122097443&partnerID=8YFLogxK

U2 - 10.1016/j.jcss.2021.11.006

DO - 10.1016/j.jcss.2021.11.006

M3 - Article

AN - SCOPUS:85122097443

SN - 0022-0000

VL - 126

SP - 1

EP - 20

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

ER -