Fast Distributed Backup Placement in Sparse and Dense Networks

Barenboim, Leonid; Oren, Gal

doi:10.1137/1.9781611976021.7

Cited by 7 publications

(5 citation statements)

References 19 publications

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“…Their algorithm remained the state-of-the-art for general graphs, as well as several specific graph families. Barenboim and Oren [8] obtained significantly improved algorithms for various graph topologies. Specifically, they showed that an O(1)-approximation to optimal 1-Backup Placement can be computed deterministically in O(1) rounds in wireless networks, and more generally, in any graph with neighborhood independence bounded by a constant.…”

Section: The Backup Placement Problem In Networkmentioning

confidence: 99%

Distributed K-Backup Placement and Applications to Virtual Memory in Real-World Wireless Networks

Oren¹,

Barenboim²

2020

Preprint

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The Backup Placement problem in networks in the CON GE ST distributed setting considers a network graph G = (V, E), in which the goal of each vertex v ∈ V is selecting a neighbor, such that the maximum number of vertices in V that select the same vertex is minimized [10]. Previous backup placement algorithms suffer from obliviousness to main factors of real-world heterogeneous wireless network. Specifically, there is no consideration of the nodes memory and storage capacities, and no reference to a case in which nodes have different energy capacity, and thus can leave (or join) the network at any time. These parameters are strongly correlated in wireless networks, as the load on different parts of the network can differ greatly, thus requiring more communication, energy, memory and storage. In order to fit the real-world attributes of wireless networks, this work addresses a generalized version of the original problem, namely K-Backup Placement, in which each vertex selects K neighbors, for a positive parameter K. Our K-Backup Placement algorithm terminates within just one round. In addition we suggest two complementary algorithms which employ K-Backup-Placement to obtain efficient virtual memory schemes for wireless networks. The first algorithm divides the memory of each node to many small parts. Each vertex is assigned the memories of a large subset of its neighbors. Thus more memory capacity for more vertices is gained, but with much fragmentation. The second algorithm requires greater round-complexity, but produces larger virtual memory for each vertex without any fragmentation.

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Section: The Backup Placement Problem In Networkmentioning

confidence: 99%

Distributed K-Backup Placement and Applications to Virtual Memory in Real-World Wireless Networks

Oren¹,

Barenboim²

2020

Preprint

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“…Above graphs appear naturally in the context of symmetry breaking problems (for instance in the study of wireless networks), and there have been numerous works on MIS and MM in graphs with bounded neighborhood independence and their special cases (see, e.g. [7,8,10,11,20,29,30,37,49] and references therein).…”

Section: Our Contributionsmentioning

confidence: 99%

When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear-Time

Assadi¹,

Solomon²

2020

Preprint

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Maximal independent set (MIS), maximal matching (MM), and (∆ + 1)-coloring in graphs of maximum degree ∆ are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for (∆ + 1)-coloring that runs in O(n √ n) time 1 , which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM.The neighborhood independence number of a graph G, denoted by β(G), is the size of the largest independent set in the neighborhood of any vertex. We identify β(G) as the "right" parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in O(nβ(G)) and O(n log n • β(G)) time respectively on any n-vertex graph G. We complement this positive result by observing that a simple extension of the lower bound of Assadi et al. implies that Ω(nβ(G)) time is also necessary for any algorithm to either problem for all values of β(G) from 1 to Θ(n). We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires Ω(n 2 ) time even for β(G) = 2.Graphs with bounded neighborhood independence, already for constant β = β(G), constitute a rich family of possibly dense graphs, including line graphs, proper interval graphs, unit-disk graphs, claw-free graphs, and graphs of bounded growth. Our results suggest that even though MIS and MM do not admit sublinear-time algorithms in general graphs, one can still solve both problems in sublinear time for a wide range of β(G) ≪ n.

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“…Consequently, the running time becomes just one round, which improves the number of rounds required in the previous solution for such graphs by a constant. More importantly, this improves the approximation ratio as well, which becomes c, rather than 2c + 1 of [5]. Furthermore, this instruction is solely a function of the IDs of a vertex and its neighbors.…”

Section: Introductionmentioning

confidence: 95%

“…In particular, this is the case in wireless networks, certain social networks, claw-free graphs, line graphs, and more generally, any graph with neighborhood-independence bounded by a constant c. Neighborhood independence I(G) of a graph G = (V, E) is the maximum size of an independent set contained in a neighborhood Γ(v), v ∈ V . For graphs with I(G) ≤ c = O(1), a constant-time deterministic distributed algorithm with approximation ratio 2c + 1 = O(1) was devised by Barenboim and Oren [5]. Although not so complicated, this algorithm still consists of several stages, including a computation of a tree cover, and then handling differently the different parts of the trees.…”

Section: Introductionmentioning

confidence: 99%

“…This is a natural relaxation of the perfect matching problem, in which each node is selected just by one neighbor. Previous (approximate) solutions for backup placement are non-trivial, even for simple graph topologies, such as dense graphs [5]. In this paper we devise an algorithm for dense graph topologies, including unit disk graphs, unit ball graphs, line graphs, graphs with bounded diversity, and many more.…”

mentioning

confidence: 99%

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Distributed Backup Placement in One Round and its Applications to Maximum Matching Approximation and Self-Stabilization

Barenboim¹,

Oren²

2020

Symposium on Simplicity in Algorithms

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In the distributed backup-placement problem [8] each node of a network has to select one neighbor, such that the maximum number of nodes that make the same selection is minimized. This is a natural relaxation of the perfect matching problem, in which each node is selected just by one neighbor. Previous (approximate) solutions for backup placement are non-trivial, even for simple graph topologies, such as dense graphs [5]. In this paper we devise an algorithm for dense graph topologies, including unit disk graphs, unit ball graphs, line graphs, graphs with bounded diversity, and many more. Our algorithm requires just one round, and is as simple as the following operation. Consider a circular list of neighborhood IDs, sorted in an ascending order, and select the ID that is next to the selecting vertex ID. Surprisingly, such a simple one-round strategy turns out to be very efficient for backup placement computation in dense networks. Not only that it improves the number of rounds of the solution, but also the approximation ratio is improved by a multiplicative factor of at least 2.Our new algorithm has several interesting implications. In particular, it gives rise to a (2 + ǫ)approximation to maximum matching within O(log * n) rounds in dense networks. The resulting algorithm is very simple as well, in sharp contrast to previous algorithms that compute such a solution within this running time. Moreover, these algorithms are applicable to a narrower graph family than our algorithm. For the same graph family, the best previously-known result has O(log ∆ + log * n) running time [4]. Another interesting implication is the possibility to execute our backup placement algorithm as-is in the self-stabilizing setting. This makes it possible to simplify and improve other algorithms for the self-stabilizing setting, by employing helpful properties of backup placement.

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Fast Distributed Backup Placement in Sparse and Dense Networks

Cited by 7 publications

References 19 publications

Distributed K-Backup Placement and Applications to Virtual Memory in Real-World Wireless Networks

Distributed K-Backup Placement and Applications to Virtual Memory in Real-World Wireless Networks

When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear-Time

Distributed Backup Placement in One Round and its Applications to Maximum Matching Approximation and Self-Stabilization

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