Tight bounds for parallel randomized load balancing

Lenzen, Christoph; Wattenhofer, Roger

doi:10.1007/s00446-014-0225-4

Cited by 16 publications

(4 citation statements)

References 42 publications

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“…Upper bounds for various problems in the congested clique are given in [30,32,8,29,28]. Of particular relevance to our work is [8], which gives an upper bound ofÕ(n (d−2)/d ) on the round complexity of detecting a fixed d-vertex subgraph in CLIQUE-UCAST n,O(log n) ; we give upper and lower bounds on subgraph detection in the broadcast version of the congested clique in Section 3.…”

Section: Introductionmentioning

confidence: 99%

On the power of the congested clique model

Drucker

Kühn

Oshman

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

179

211

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We study the computation power of the congested clique, a model of distributed computation where n players communicate with each other over a complete network in order to compute some function of their inputs. The number of bits that can be sent on any edge in a round is bounded by a parameter b. We consider two versions of the model: in the first, the players communicate by unicast, allowing them to send a different message on each of their links in one round; in the second, the players communicate by broadcast, sending one message to all their neighbors.It is known that the unicast version of the model is quite powerful; to date, no lower bounds for this model are known. In this paper we provide a partial explanation by showing that the unicast congested clique can simulate powerful classes of bounded-depth circuits, implying that even slightly super-constant lower bounds for the congested clique would give new lower bounds in circuit complexity. Moreover, under a widely-believed conjecture on matrix multiplication, the triangle detection problem, studied in [8], can be solved in O(n ) time for any > 0.The broadcast version of the congested clique is the wellknown multi-party shared-blackboard model of communication complexity (with number-in-hand input). This version is more amenable to lower bounds, and in this paper we show that the subgraph detection problem studied in [8] requires polynomially many rounds for several classes of subgraphs. We also give upper bounds for the subgraph detection problem, and relate the hardness of triangle detection in the broadcast congested clique to the communication complexity of set disjointness in the 3-party number-on-forehead model.

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Section: Introductionmentioning

confidence: 99%

On the power of the congested clique model

Drucker

Kühn

Oshman

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

179

211

View full text Add to dashboard Cite

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“…. , X k are negatively associated, we follow the same line of arguments as Lenzen and Wattenhofer [12], who showed the same statement for a balls-into-bins process in which the balls are put uniformly at random into the bins. The proof is based on the following statements proven in [4].…”

Section: B Some Probability Theorymentioning

confidence: 61%

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

Englert¹,

Röglin²,

Vöcking³

2023

Preprint

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2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on "real world" Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every p ∈ N, a family of L p instances on which 2-Opt can take an exponential number of steps.Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0, 1] 2 , where it was shown that the expected number of steps is bounded by Õ(n 10 ) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0, 1] d , for an arbitrary d ≥ 2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density φ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of Õ(n 4+1/3 • φ 8/3 ). When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to Õ(n 4+1/3−1/d •φ 8/3 ). If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by Õ(n 4−1/d • φ). In addition, we prove an upper bound of O( d √ φ) on the expected approximation factor with respect to all L p metrics.Let us remark that our probabilistic analysis covers as special cases the uniform input model with φ = 1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with φ ∼ 1/σ d .

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“…At the end of the process, the maximum load denotes the maximum number of balls in a bin. It is well known that such a process results in a maximum load of Âðlog m= log log mÞ when m ¼ n [31], [32]. It was also shown that for n !…”

Section: One Of Two Arbitrary Sets Has No Unique Objectsmentioning

confidence: 98%

Set Reconciliation via Counting Bloom Filters

Guo

2013

IEEE Trans. Knowl. Data Eng.

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In this paper, we study the set reconciliation problem, in which each member of a node pair has a set of objects and seeks to deliver its unique objects to the other member. How could each node compute the set difference, however, is challenging in the set reconciliation problem. To address such an issue, we propose a lightweight but efficient method that only requires the pair of nodes to represent objects using a counting Bloom filter (CBF) of size OðdÞ and exchange with each other, where d denotes the total size of the set differences. A receiving node then subtracts the received CBF from its local one via minus operation proposed in this paper. The resultant CBF can approximately represent the union of the set differences and thus the set difference to each node can be identified after querying the resultant CBF. In this paper, we propose a novel estimator through which each node can accurately estimate not only the value of d but also the size of the set difference to each node. Such an estimation result can be used to optimize the parameter setting of the CBF to achieve less false positives and false negatives. Comprehensive analysis and evaluation demonstrates that our method is more efficient than prior BF-based methods in terms of achieving the same accuracy with less communication cost. Moreover, our reconciliating method needs no prior context logs and it is very useful in networking and distributed applications.

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Tight bounds for parallel randomized load balancing

Cited by 16 publications

References 42 publications

On the power of the congested clique model

On the power of the congested clique model

Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP

Set Reconciliation via Counting Bloom Filters

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