Sublinear Algorithms in T-Interval Dynamic Networks
We consider standard T - interval dynamic networks , under the synchronous timing model and the broadcast CONGEST model. In a T - interval dynamic network , the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the to...
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| Published in: | Algorithmica Vol. 86; no. 9; pp. 2959 - 2996 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
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Springer US
01.09.2024
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | We consider standard
T
-
interval dynamic networks
, under the synchronous timing model and the broadcast CONGEST model. In a
T
-
interval dynamic network
, the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the topology may change arbitrarily from round to round, as determined by some
adversary
and subject to the following constraint: For every
T
consecutive rounds, the topologies in those rounds must contain a common connected spanning subgraph. Let
H
r
to be the maximum (in terms of number of edges) such subgraph for round
r
through
r
+
T
-
1
. We define the
backbone diameter
d
of a
T
-interval dynamic network to be the maximum diameter of all such
H
r
’s, for
r
≥
1
. We use
n
to denote the number of nodes in the network. Within such a context, we consider a range of fundamental distributed computing problems including
Count
/
Max
/
Median
/
Sum
/
LeaderElect
/
Consensus
/
ConfirmedFlood
. Existing algorithms for these problems all have time complexity of
Ω
(
n
)
rounds, even for
T
=
∞
and even when
d
is as small as
O
(1). This paper presents a novel approach/framework, based on the idea of
massively parallel aggregation
. Following this approach, we develop a novel deterministic
Count
algorithm with
O
(
d
3
log
2
n
)
complexity, for
T
-interval dynamic networks with
T
≥
c
·
d
2
log
2
n
. Here
c
is a (sufficiently large) constant independent of
d
,
n
, and
T
. To our knowledge, our algorithm is the very first such algorithm whose complexity does not contain a
Θ
(
n
)
term. This paper further develops novel algorithms for solving
Max
/
Median
/
Sum
/
LeaderElect
/
Consensus
/
ConfirmedFlood
, while incurring
O
(
d
3
polylog
(
n
)
)
complexity. Again, for all these problems, our algorithms are the first ones whose time complexity does not contain a
Θ
(
n
)
term. |
|---|---|
| AbstractList | We consider standard
T
-
interval dynamic networks
, under the synchronous timing model and the broadcast CONGEST model. In a
T
-
interval dynamic network
, the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the topology may change arbitrarily from round to round, as determined by some
adversary
and subject to the following constraint: For every
T
consecutive rounds, the topologies in those rounds must contain a common connected spanning subgraph. Let
H
r
to be the maximum (in terms of number of edges) such subgraph for round
r
through
r
+
T
-
1
. We define the
backbone diameter
d
of a
T
-interval dynamic network to be the maximum diameter of all such
H
r
’s, for
r
≥
1
. We use
n
to denote the number of nodes in the network. Within such a context, we consider a range of fundamental distributed computing problems including
Count
/
Max
/
Median
/
Sum
/
LeaderElect
/
Consensus
/
ConfirmedFlood
. Existing algorithms for these problems all have time complexity of
Ω
(
n
)
rounds, even for
T
=
∞
and even when
d
is as small as
O
(1). This paper presents a novel approach/framework, based on the idea of
massively parallel aggregation
. Following this approach, we develop a novel deterministic
Count
algorithm with
O
(
d
3
log
2
n
)
complexity, for
T
-interval dynamic networks with
T
≥
c
·
d
2
log
2
n
. Here
c
is a (sufficiently large) constant independent of
d
,
n
, and
T
. To our knowledge, our algorithm is the very first such algorithm whose complexity does not contain a
Θ
(
n
)
term. This paper further develops novel algorithms for solving
Max
/
Median
/
Sum
/
LeaderElect
/
Consensus
/
ConfirmedFlood
, while incurring
O
(
d
3
polylog
(
n
)
)
complexity. Again, for all these problems, our algorithms are the first ones whose time complexity does not contain a
Θ
(
n
)
term. We consider standard T-interval dynamic networks, under the synchronous timing model and the broadcast CONGEST model. In a T-interval dynamic network, the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the topology may change arbitrarily from round to round, as determined by some adversary and subject to the following constraint: For every T consecutive rounds, the topologies in those rounds must contain a common connected spanning subgraph. Let Hr to be the maximum (in terms of number of edges) such subgraph for round r through r+T-1. We define the backbone diameterd of a T-interval dynamic network to be the maximum diameter of all such Hr’s, for r≥1. We use n to denote the number of nodes in the network. Within such a context, we consider a range of fundamental distributed computing problems including Count/Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood. Existing algorithms for these problems all have time complexity of Ω(n) rounds, even for T=∞ and even when d is as small as O(1). This paper presents a novel approach/framework, based on the idea of massively parallel aggregation. Following this approach, we develop a novel deterministic Count algorithm with O(d3log2n) complexity, for T-interval dynamic networks with T≥c·d2log2n. Here c is a (sufficiently large) constant independent of d, n, and T. To our knowledge, our algorithm is the very first such algorithm whose complexity does not contain a Θ(n) term. This paper further develops novel algorithms for solving Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood, while incurring O(d3polylog(n)) complexity. Again, for all these problems, our algorithms are the first ones whose time complexity does not contain a Θ(n) term. |
| Author | Yu, Haifeng Jahja, Irvan |
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| Copyright | The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| Copyright_xml | – notice: The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Keywords | interval dynamic networks Distributed algorithms Sublinear algorithms |
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| References | Michail, O., Chatzigiannakis, I., Spirakis, P.: Naming and counting in anonymous unknown dynamic networks. In: Proceedings of International Symposium on Stabilization, Safety, and Security of Distributed Systems (2013) ChenBYuHZhaoYGibbonsPBThe cost of fault tolerance in multi-party communication complexityJ. ACM20146131911964321529910.1145/2597633 YuHZhaoYJahjaIThe cost of unknown diameter in dynamic networksJ. ACM20186553113134385654310.1145/3209665 Kuhn, F., Oshman, R.: The complexity of data aggregation in directed networks. In: DISC (2011) Haeupler, B., Karger, D.: Faster information dissemination in dynamic networks via network coding. In: PODC (2011) Luna, G., Bonomi, S., Chatzigiannakis, I., Baldoni, R.: Counting in anonymous dynamic networks: an experimental perspective. In: ALGOSENSORS (2013) PelegDDistributed Computing: A Locality-Sensitive Approach1987Philadelphia, PennsylvaniaSociety for Industrial and Applied Mathematics Hou, R., Jahja, I., Sun, Y., Wu, J., Yu, H.: Achieving sublinear complexity under constant T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document} in T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document}-interval dynamic networks. In: SPAA (2022) SchmidUWeissBKeidarIImpossibility results and lower bounds for consensus under link failuresSIAM J. Comput.200938519121951247628010.1137/S009753970443999X Brandes, P., Meyer auf der Heide, F.: Distributed computing in fault-prone dynamic networks. In: International Workshop on Theoretical Aspects of Dynamic Distributed Systems (2012) Das Sarma, A., Molla, A., Pandurangan, G.: Fast distributed computation in dynamic networks via random walks. In: DISC (2012) KuhnFOshmanRDynamic networks: models and algorithmsSIGACT News2011421829610.1145/1959045.1959064 Kuhn, F., Moses, Y., Oshman, R.: Coordinated consensus in dynamic networks. In: PODC (2011) Augustine, J., Pandurangan, G., Robinson, P.: Fast byzantine agreement in dynamic networks. In: PODC (2013) Coulouma, E., Godard, E.: A characterization of dynamic networks where consensus is solvable. In: SIROCCO (2013) Kempe, D., Dobra, A., Gehrke, J.: Gossip-based computation of aggregate information. In: FOCS (2003) AlmeidaPBaqueroCFarach-ColtonMJesusPMosteiroMAFault-tolerant aggregation: flow-updating meets mass-distributionDistributed Comput.2017304281291368059810.1007/s00446-016-0288-5 Abshoff, S., Benter, M., Cord-Landwehr, A., Malatyali, M., Meyer auf der Heide, F.: Token dissemination in geometric dynamic networks. In: ALGOSENSORS (2013) Ahmadi, M., Kuhn, F.: Multi-message broadcast in dynamic radio networks. In: ALGOSENSORS (2017) Luna, G., Baldoni, R., Bonomi, S., Chatzigiannakis, I.: Conscious and unconscious counting on anonymous dynamic networks. In: International Conference on Distributed Computing and Networking (2014) Terpstra, W.W., Leng, C., Buchmann, A.P.: Brief announcement: practical summation via gossip. In: PODC (2007) Charron-Bost, B., Fugger, M., Nowak, T.: Approximate consensus in highly dynamic networks: the role of averaging algorithms. In: ICALP (2015) Luna, G., Baldoni, R., Bonomi, S., Chatzigiannakis, I.: Counting in anonymous dynamic networks under worst-case adversary. In: IEEE International Conference on Distributed Computing Systems (2014) Jahja, I., Yu, H.: Sublinear algorithms in T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document}-interval dynamic networks. In: SPAA (2020) ChakrabortyMMilaniAMosteiroMA faster exact-counting protocol for anonymous dynamic networksAlgorithmica2018801130233049384092110.1007/s00453-017-0367-4 Kuhn, F., Lynch, N., Oshman, R.: Distributed computation in dynamic networks. In: STOC (2010) LynchNDistributed Algorithms1996San FranciscoMorgan Kaufmann Ingram, R., Shields, P., Walter, J.: An asynchronous leader election algorithm for dynamic networks. In: IPDPS (2009) Chlebus, B., Kowalski, D., Olkowski, J., Olkowski, J.: Disconnected agreement in networks prone to link failures. In: International Symposium on Stabilizing, Safety, and Security of Distributed Systems (2023) DolevSSelf-Stabilization2000CambridgeMIT Press10.7551/mitpress/6156.001.0001 Augustine, J., Pandurangan, G., Robinson, P.: Fast byzantine leader election in dynamic networks. In: DISC (2015) Kowalski, D., Mosteiro, M.: Polynomial counting in anonymous dynamic networks with applications to anonymous dynamic algebraic computations. In: ICALP (2018) N Lynch (1250_CR9) 1996 1250_CR1 1250_CR21 1250_CR20 S Dolev (1250_CR26) 2000 1250_CR23 1250_CR25 1250_CR24 1250_CR27 H Yu (1250_CR10) 2018; 65 B Chen (1250_CR17) 2014; 61 1250_CR28 F Kuhn (1250_CR29) 2011; 42 P Almeida (1250_CR30) 2017; 30 D Peleg (1250_CR8) 1987 1250_CR4 1250_CR32 1250_CR5 1250_CR31 1250_CR2 1250_CR12 1250_CR3 1250_CR14 1250_CR13 1250_CR6 M Chakraborty (1250_CR11) 2018; 80 1250_CR16 1250_CR7 1250_CR15 1250_CR18 1250_CR19 U Schmid (1250_CR22) 2009; 38 |
| References_xml | – reference: PelegDDistributed Computing: A Locality-Sensitive Approach1987Philadelphia, PennsylvaniaSociety for Industrial and Applied Mathematics – reference: Charron-Bost, B., Fugger, M., Nowak, T.: Approximate consensus in highly dynamic networks: the role of averaging algorithms. In: ICALP (2015) – reference: Kuhn, F., Oshman, R.: The complexity of data aggregation in directed networks. In: DISC (2011) – reference: Ahmadi, M., Kuhn, F.: Multi-message broadcast in dynamic radio networks. In: ALGOSENSORS (2017) – reference: Luna, G., Baldoni, R., Bonomi, S., Chatzigiannakis, I.: Counting in anonymous dynamic networks under worst-case adversary. In: IEEE International Conference on Distributed Computing Systems (2014) – reference: Augustine, J., Pandurangan, G., Robinson, P.: Fast byzantine leader election in dynamic networks. In: DISC (2015) – reference: Hou, R., Jahja, I., Sun, Y., Wu, J., Yu, H.: Achieving sublinear complexity under constant T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document} in T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document}-interval dynamic networks. In: SPAA (2022) – reference: YuHZhaoYJahjaIThe cost of unknown diameter in dynamic networksJ. ACM20186553113134385654310.1145/3209665 – reference: Luna, G., Baldoni, R., Bonomi, S., Chatzigiannakis, I.: Conscious and unconscious counting on anonymous dynamic networks. In: International Conference on Distributed Computing and Networking (2014) – reference: ChenBYuHZhaoYGibbonsPBThe cost of fault tolerance in multi-party communication complexityJ. ACM20146131911964321529910.1145/2597633 – reference: Brandes, P., Meyer auf der Heide, F.: Distributed computing in fault-prone dynamic networks. In: International Workshop on Theoretical Aspects of Dynamic Distributed Systems (2012) – reference: Michail, O., Chatzigiannakis, I., Spirakis, P.: Naming and counting in anonymous unknown dynamic networks. In: Proceedings of International Symposium on Stabilization, Safety, and Security of Distributed Systems (2013) – reference: Chlebus, B., Kowalski, D., Olkowski, J., Olkowski, J.: Disconnected agreement in networks prone to link failures. In: International Symposium on Stabilizing, Safety, and Security of Distributed Systems (2023) – reference: SchmidUWeissBKeidarIImpossibility results and lower bounds for consensus under link failuresSIAM J. Comput.200938519121951247628010.1137/S009753970443999X – reference: Luna, G., Bonomi, S., Chatzigiannakis, I., Baldoni, R.: Counting in anonymous dynamic networks: an experimental perspective. In: ALGOSENSORS (2013) – reference: Haeupler, B., Karger, D.: Faster information dissemination in dynamic networks via network coding. In: PODC (2011) – reference: Ingram, R., Shields, P., Walter, J.: An asynchronous leader election algorithm for dynamic networks. In: IPDPS (2009) – reference: Kuhn, F., Lynch, N., Oshman, R.: Distributed computation in dynamic networks. In: STOC (2010) – reference: Augustine, J., Pandurangan, G., Robinson, P.: Fast byzantine agreement in dynamic networks. In: PODC (2013) – reference: Kuhn, F., Moses, Y., Oshman, R.: Coordinated consensus in dynamic networks. In: PODC (2011) – reference: Kempe, D., Dobra, A., Gehrke, J.: Gossip-based computation of aggregate information. In: FOCS (2003) – reference: Coulouma, E., Godard, E.: A characterization of dynamic networks where consensus is solvable. In: SIROCCO (2013) – reference: Kowalski, D., Mosteiro, M.: Polynomial counting in anonymous dynamic networks with applications to anonymous dynamic algebraic computations. In: ICALP (2018) – reference: AlmeidaPBaqueroCFarach-ColtonMJesusPMosteiroMAFault-tolerant aggregation: flow-updating meets mass-distributionDistributed Comput.2017304281291368059810.1007/s00446-016-0288-5 – reference: KuhnFOshmanRDynamic networks: models and algorithmsSIGACT News2011421829610.1145/1959045.1959064 – reference: DolevSSelf-Stabilization2000CambridgeMIT Press10.7551/mitpress/6156.001.0001 – reference: Abshoff, S., Benter, M., Cord-Landwehr, A., Malatyali, M., Meyer auf der Heide, F.: Token dissemination in geometric dynamic networks. In: ALGOSENSORS (2013) – reference: Das Sarma, A., Molla, A., Pandurangan, G.: Fast distributed computation in dynamic networks via random walks. In: DISC (2012) – reference: LynchNDistributed Algorithms1996San FranciscoMorgan Kaufmann – reference: Jahja, I., Yu, H.: Sublinear algorithms in T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document}-interval dynamic networks. In: SPAA (2020) – reference: ChakrabortyMMilaniAMosteiroMA faster exact-counting protocol for anonymous dynamic networksAlgorithmica2018801130233049384092110.1007/s00453-017-0367-4 – reference: Terpstra, W.W., Leng, C., Buchmann, A.P.: Brief announcement: practical summation via gossip. 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interval dynamic networks
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T
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,... We consider standard T-interval dynamic networks, under the synchronous timing model and the broadcast CONGEST model. In a T-interval dynamic network, the set... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Distributed processing Graph theory Mathematics of Computing Network topologies Nodes Theory of Computation |
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