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TIGHT ANALYSES OF TWO LOCAL LOAD BALANCING ALGORITHMS
 SIAM J. COMPUT.
, 1999
"... This paper presents an analysis of the following load balancing algorithm. At each step, each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d + 1 fewer tokens, where d is the maximum degree of any node in the network. We ..."
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Cited by 51 (5 self)
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This paper presents an analysis of the following load balancing algorithm. At each step, each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d + 1 fewer tokens, where d is the maximum degree of any node in the network. We show that within O(∆/α) steps, the algorithm reduces the maximum difference in tokens between any two nodes to at most O((d 2 log n)/α), where ∆ is the global imbalance in tokens (i.e., the maximum difference between the number of tokens at any node initially and the average number of tokens), n is the number of nodes in the network, and α is the edge expansion of the network. The time bound is tight in the sense that for any graph with edge expansion α, and for any value ∆, there exists an initial distribution of tokens with imbalance ∆ for which the time to reduce the imbalance to even ∆/2 is at least Ω(∆/α). The bound on the final imbalance is tight in the sense that there exists a class of networks that can be locally balanced everywhere (i.e., the maximum difference in tokens between any two neighbors is at most 2d), while the global imbalance remains Ω((d 2 log n)/α). Furthermore, we show that upon reaching a state with a global imbalance of O((d 2 log n)/α), the time for this algorithm to locally balance the network can be as large as Ω(n 1/2). We extend our analysis to a variant of this algorithm for dynamic and asynchronous
Efficient Schemes for Nearest Neighbor Load Balancing
, 1998
"... We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diffusion type. Within this framework we develop a new optimal polynomial scheme (OPS) which we show to terminate within a finite number m of steps, where m only depends on the graph ..."
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Cited by 46 (13 self)
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We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diffusion type. Within this framework we develop a new optimal polynomial scheme (OPS) which we show to terminate within a finite number m of steps, where m only depends on the graph and not on the initial load distribution. We show that all existing diffusion load balancing algorithms, including OPS, determine a flow of load on the edges of the graph which is uniquely defined, independent of the method and minimal in the l 2 norm. This result can be extended to edge weighted graphs. The l 2 minimality is achieved only if a diffusion algorithm is used as preprocessing and the real movement of load is performed in a second step. Thus, it is advisable to split the balancing process into the two steps of first determining a balancing flow and afterwards moving the load. We introduce the problem of scheduling a flow and present some first results on its complexity and the ...
Gossiping in Minimal Time
 SIAM J. on Computing
, 1992
"... The gossip problem involves communicating a unique item from each node in a graph to every other node. We study the minimum time required to do this under the weakest model of parallel communication which allows each node to participate in just one communication at a time as either sender or receive ..."
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Cited by 46 (2 self)
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The gossip problem involves communicating a unique item from each node in a graph to every other node. We study the minimum time required to do this under the weakest model of parallel communication which allows each node to participate in just one communication at a time as either sender or receiver. We study a number of topologies including the complete graph, grids, hypercubes and rings. Definitive new optimal time algorithms are derived for complete graphs, rings, regular grids and toroidal grids that significantly extend existing results. In particular, we settle an open problem about minimum time gossiping in complete graphs. Specifically, for a graph with N nodes, at least log ae N communication steps, where the logarithm is in the base of the golden ratio ae, are required by any algorithm under the weakest model of communication. This bound, which is approximately 1:44 log 2 N , can be realized for some networks and so the result is optimal. KEYWORDS: Gossiping, broadcasting. ...
A Dynamic Distributed Load Balancing Algorithm with Provable Good Performance
 In Proceedings of the 5th Annual ACM Symposium on Parallel Algorithms and Architectures
, 1993
"... The overall efficiency of parallel algorithms is most decisively effected by the strategy applied for the mapping of workload. Strategies for balancing dynamically generated workload on a processor network which are also useful for practical applications have intensively been investigated by simulat ..."
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Cited by 44 (5 self)
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The overall efficiency of parallel algorithms is most decisively effected by the strategy applied for the mapping of workload. Strategies for balancing dynamically generated workload on a processor network which are also useful for practical applications have intensively been investigated by simulations and by direct applications. This paper presents the complete theoretical analysis of a dynamically distributed load balancing strategy. The algorithm is adaptive by nature and is therefore useful for a broad range of applications. A similar algorithmic principle has already been implemented for a number of applications in the areas of combinatorial optimization, parallel programming languages and graphical animation. The algorithm performed convincingly for all these applications. In our analysis we will prove that the expected number of packets on each processor varies only by a constant factor compared with that on any other processor, independent of the generation and consumption of ...
Counting Networks and MultiProcessor Coordination (Extended Abstract)
 In Proceedings of the 23rd Annual Symposium on Theory of Computing
, 1991
"... ) James Aspnes Maurice Herlihy y Nir Shavit z Digital Equipment Corporation Cambridge Research Lab CRL 90/11 September 18, 1991 Abstract Many fundamental multiprocessor coordination problems can be expressed as counting problems: processes must cooperate to assign successive values from a g ..."
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Cited by 43 (7 self)
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) James Aspnes Maurice Herlihy y Nir Shavit z Digital Equipment Corporation Cambridge Research Lab CRL 90/11 September 18, 1991 Abstract Many fundamental multiprocessor coordination problems can be expressed as counting problems: processes must cooperate to assign successive values from a given range, such as addresses in memory or destinations on an interconnection network. Conventional solutions to these problems perform poorly because of synchronization bottlenecks and high memory contention. Motivated by observations on the behavior of sorting networks, we offer a completely new approach to solving such problems. We introduce a new class of networks called counting networks, i.e., networks that can be used to count. We give a counting network construction of depth log 2 n using n log 2 n "gates," avoiding the sequential bottlenecks inherent to former solutions, and having a provably lower contention factor on its gates. Finally, to show that counting networks are not ...
Approximate Load Balancing on Dynamic and Asynchronous Networks
 In Proceedings of the 25th Annual ACM Symposium on Theory of Computing
, 1993
"... This paper presents a simple local algorithm for load balancing in a distributed network. The algorithm makes no assumption about the structure of the network. It can be executed on a synchronous network with fixed topology, a synchronous network with dynamically changing topology, or an asynchronou ..."
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Cited by 41 (3 self)
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This paper presents a simple local algorithm for load balancing in a distributed network. The algorithm makes no assumption about the structure of the network. It can be executed on a synchronous network with fixed topology, a synchronous network with dynamically changing topology, or an asynchronous network. It works quickly and balances well when the network has an expansion property. In particular, we show that in an nnode networkwith maximumdegree d whose live edges, at every time step, form a ¯expander, the algorithm will balance the load to within an additive O(d log n=¯) term in O(\Delta log(n\Delta)=¯) time, where \Delta is the initial imbalance. The algorithm improves upon previous approaches that yield O(n) time bounds in dynamic and asynchronous networks. 1 Introduction One of the most fundamental problems to solve on a parallel computer or distributed network is to balance the load or work that must be performed among the various processors. This paper analyzes a sim...
Packet Routing In FixedConnection Networks: A Survey
, 1998
"... We survey routing problems on fixedconnection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, krelation routing, routing to random destinations, dynamic routing, isotonic routing ..."
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Cited by 29 (3 self)
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We survey routing problems on fixedconnection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, krelation routing, routing to random destinations, dynamic routing, isotonic routing, fault tolerant routing, and related sorting results. We also provide a list of unsolved problems and numerous references.
A SelfRepairing PeertoPeer System Resilient to Dynamic Adversarial Churn
 In Proc. of the 4th International Workshop on PeertoPeer Systems (IPTPS
, 2005
"... Abstract We present a dynamic distributed hash table where peers may join and leave at any time. Our system tolerates a powerful adversary which has complete visibility of the entire state of the system and can repeatedly add and remove peers. By dynamic analysis we prove that our system provides wo ..."
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Cited by 28 (9 self)
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Abstract We present a dynamic distributed hash table where peers may join and leave at any time. Our system tolerates a powerful adversary which has complete visibility of the entire state of the system and can repeatedly add and remove peers. By dynamic analysis we prove that our system provides worstcase faulttolerance, maintaining desirable properties such as a low peer degree and a low network diameter. 1
Load Balancing, Selection and Sorting on the Hypercube
 In Proceedings of the 1989 ACM Symposium on Parallel Algorithms and Architectures
, 1989
"... This paper presents novel load balancing, selection and sorting algorithms for the hypercube with 1port communication. The main result is an algorithm for sorting n values on p processors, SmoothSort, that runs asymptotically faster (in the worst case) than any previously known algorithm over a ..."
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Cited by 26 (4 self)
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This paper presents novel load balancing, selection and sorting algorithms for the hypercube with 1port communication. The main result is an algorithm for sorting n values on p processors, SmoothSort, that runs asymptotically faster (in the worst case) than any previously known algorithm over a wide range of the ratio n=p. The load balancing and selection algorithms upon which SmoothSort is based are expected to be of independent interest. Although the analysis of our algorithms is limited to obtaining asymptotic bounds, the constant factors being ignored are quite small. 1 Introduction This paper presents novel load balancing, selection and sorting algorithms for the hypercube. The following model of computation is assumed. Each processor has an infinite local memory configured in O(log p) bit words and can perform the usual set of ALU operations in constant time on wordsized operands. Processors communicate with one another by sending packets over the bidirectional channel...
Lower Bounds on the Competitive Ratio for Mobile User Tracking and Distributed Job Scheduling
 Theoretical Computer Science
, 1992
"... 1 We prove a lower bound of Ω(log n / log log n) on the competitive ratio of any (deterministic or randomized) distributed algorithm for solving the mobile user problem introduced by Awerbuch and Peleg [5], on certain networks of n processors. Our lower bound holds for various networks, including th ..."
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Cited by 24 (4 self)
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1 We prove a lower bound of Ω(log n / log log n) on the competitive ratio of any (deterministic or randomized) distributed algorithm for solving the mobile user problem introduced by Awerbuch and Peleg [5], on certain networks of n processors. Our lower bound holds for various networks, including the hypercube, any network with sufficiently large girth, and any highly expanding graph. A similar Ω(log n / log log n) lower bound is proved for the competitive ratio of the maximum job delay of any distributed algorithm for solving the distributed scheduling problem of Awerbuch, Kutten and Peleg [4] on any of these networks. The proofs combine combinatorial techniques with tools from linear algebra and harmonic analysis and apply, in particular, a generalization of the vertex isoperimetric problem on the hypercube, which may be of independent interest. 2 Footnotes for title page