Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180--200, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the cost of migrating tasks and maintaining load information, making these other policies suboptimal for such systems. Chapter 4 presents the CPU management policies that are tuned for large shared memory systems, and the experiments show that they perform close to optimal. Some researchers [1, 5, 44] have studied randomized algorithms for load balancing. These algorithms randomly select a small number of processors and choose the least loaded of these processors to run a task. Because these algorithms select processors randomly, they do not require any global load information, which can be ....

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. In Proceedings of the 26th ACM Symposium on the Theory of Computing, pages 593-- 602, 1994.

....sequentially and choose d units uniformly at random. The job is then allocated at a candidate unit with minimum load breaking ties arbitrarily. It is well known that the maximum load for d = 1 is about ln n= ln ln n (see e.g. Gon81] or [RS98] It came as quite a surprise when it was shown in [ABKU94] that for d = 2 the maximum load is exponentially smaller, namely about ln ln n= ln d. This phenomenon is often referred to as the power of two choices [Mit96b] There is a rather broad literature on this phenomenon. An early application of the power of two choices can be found in PRAM ....

....nal destination. However, the amount of communication should be kept small. Here we focus on the following simple and natural strategy: each job chooses d 2 candidate units randomly and checks their current load. Then the job is allocated at a unit with lowest load. This model was introduced in [ABKU94] slight variants and improvements can e.g. be found in [ABKU94] ACMR95] CFM 98] V oc99] BCSV00] Mit96a] Mit97] CS97] consider some related models. 3 A Generalized Approach Using Witness Trees In this section we introduce the technique that we will later use for the analysis of ....

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Yossi Azar, Andrej Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing (STOC-94), pages 593-602, 1994.

....classical hashing methods in [47, Section 6.4] These methods also seem to prevail in practice. The most prominent ones, and the basis for our experiments in Section 4.4, are Chained Hashing (with separate chaining) Linear Probing, and Double Hashing. A more recent scheme called Two Way Chaining [3] will also be investigated. We detail our implementation in Section 4.4. Theoretical Work. Early theoretical analysis of hashing schemes was typically done under the assumption that hash function values were uniformly random and independent. Precise analyses of the average and expected worst ....

.... (x) x Two table accesses are in fact (worst case) optimal among all data structures using linear space, except for special cases, see [62] Remark: The idea of storing keys in one out of two places given by hash functions previously appeared in [44] in the context of PRAM simulation, and in [3] for Two Way Chaining, mentioned in Section 4.1.1. It is shown in [62] that if r 2 (1 #) n for some constant # 0 (i.e. the tables are to be a bit less than half full) and h 1 , h 2 are picked uniformly at random from an (O(1) O(log n) universal family, the probability that there is no ....

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Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180--200 (electronic), 1999.

....nite and in nite or dynamic allocation processes. In the latter case, the system is regarded for an in nite time interval, new balls arrive at the system and other balls are deleted during that time. In the area of parallel or sequential in nite task allocation several work has been done, see [ABKU94] Mit96a] Mit96b] CS97] Mit97] Czu98] and [ABS98] In this paper, we focus work which is done in the area of nite and parallel task allocation. Karp, Luby, and Meyer auf der Heide [KLM92] were the rst to present a process which uses several possible locations in order to lower the ....

....allocation process using O(log log n) communication rounds to guarantee a maximum load of O(log log n) w.h.p. In that case, the memory accesses of the PRAM represent the balls in the game, and the locations xed by the randomly chosen hash functions determine the i.u.r. chosen bins. Azar et al. ABKU94] examine a similar protocol in a sequential setting. For each ball they choose d bins i.u.r. and put the ball into the bin with the minimum load at time of placement. They show that, after placing m balls, the maximum load is log log(n) log(d) 1 o(1) m=n) w.h.p. Furthermore, they ....

[Article contains additional citation context not shown here]

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations (extended abstract). In Proceedings of the 26th Symposium on Theory of Computing (STOC '94), pages 593-602, 1994.

....bin and where D = L Gamma N=M. 1.2 Related work Let us compare our results for the random arc allocation to the well known results for balls into bins processes. These processes are among of the most intensively studied stochastic processes in the context of algorithm analysis (e.g. [14, 22, 16, 12, 2, 23, 15, 19, 4]) The simplest balls into bins process assumes that N balls are placed i.u.r. into M bins [14, 22, 16] Balls into bins are the special case of chains into bins where all chains consist of a single ball, i.e. n = N. We get M ln M r logM if N =W(M logM) 6) The Bounds (5) and ....

Y. Azar, A. Z. Broder, A. R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180--200, February 2000.

....Maximization by Tufte [17] In particular, one should reduce non data ink and redundant data ink from the diagram. The ratio of data ink to total ink used 0.5 2.5 16 64 256 1024 2 24 max Load m n m Fig. 6. m Balls are place into n bins using balanced random allocation [2, 3]. The difference between maximal and average load is plotted for different values of m and n. The experiments have been repeated at least sufficiently often to reduce the standard error oe= repetitions [11] below one percent. In order to minimize artifacts of the random number generator, we ....

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180--200, February 2000.

....on algorithms, Knuth s selection of algorithms is in agreement with current practice for implementation of general purpose dictionaries. In particular, the excellent cache usage of Linear Probing makes it a prime choice on modern architectures. A more recent scheme called Two Way Chaining [2] will also be investigated. All schemes are briefly described in Section 5. 2.1. Analysis of early hashing schemes Early theoretical analysis of hashing schemes was done under the assumption that hash function values are uniformly random and independent. Precise analyses of the average and ....

.... 2 (x) x end Two table accesses for lookup is in fact optimal among all dictionaries using linear space, except for special cases, see [24] Remark: The idea of storing keys in one out of two places given by hash functions previously appeared in [16] in the context of PRAM simulation, and in [2] for Two Way Chaining. It is shown in [24] that if r # (1 #) n for some constant # 0 (i.e. the tables are to be a bit less than half full) and h 1 , h 2 are picked uniformly at random from an (O(1) O(log n) universal family, the probability that there is no way of arranging the keys of ....

[Article contains additional citation context not shown here]

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM J. Comput., 29(1):180--200 (electronic), 1999.

....algorithms (which perform loadbalancing every few steps, regardless of the system state) or for sender initiated approaches. First, there has been a lot of work on static problems, in which the number of jobs to be serviced is fixed and may even be known in advance. For these results, see [ABKU94, Voc00, CS97, ACMR95, BCSV00] In our paper, we work on dynamic load balancing, in which tasks are generated over time. We will now describe previous work on this problem. In [ABS98] Adler, Berenbrink and Schroder consider a process in which m jobs arrive in each round to n servers and each ....

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations (extended abstract). In Proceedings of the 26th Symposium on Theory of Computing (STOC'94), pages 593--602, 1994.

....nite and in nite or dynamic allocation processes. In the latter case, the system is regarded for an in nite time interval, new balls arrive at the system and other balls are deleted during that time. In the area of parallel or sequential in nite task allocation several work has been done, see [ABKU94] Mit96a] Mit96b] CS97] Mit97] Czu98] and [ABS98] In this paper, we focus work which is done in the area of nite and parallel task allocation. Karp, Luby, and Meyer auf der Heide [KLM92] were the rst to present a process which uses several possible locations in order to lower the ....

....allocation process using O(log log n) communication rounds to guarantee a maximum load of O(log log n) w.h.p. In that case, the memory accesses of the PRAM represent the balls in the game, and the locations xed by the randomly chosen hash functions determine the i.u.r. chosen bins. Azar et al. ABKU94] examine a similar protocol in a sequential setting. For each ball they choose d bins i.u.r. and put the ball into the bin with the minimum load at time of placement. They show that, after placing m balls, the maximum load is log log(n) log(d) 2 (1 o(1) m=n) w.h.p. Furthermore, they ....

[Article contains additional citation context not shown here]

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations (extended abstract). In Proceedings of the 26th Symposium on Theory of Computing (STOC '94), pages 593-602, 1994.

....copies for every data item instead of one leads to an exponential improvement of the maximum load. They used the approach of several copies to construct fast distributed algorithms for the simulation of parallel random access machines (PRAMs) on distributed memory machines (DMMs) Azar et al. ABKU94] study the problem of assigning requests to resources in a sequential setting, i.e. the requests arrive one by one. They show that, if every data item has c copies and m = n, there is a simple strategy that ensures a maximum load of (log log n= log c) with high probability. They also present ....

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations (extended abstract). In Proceedings of the 26th Symposium on Theory of Computing, pages 593--602, 1994.

....copies for every data item instead of one leads to an exponential improvement of the maximum load. They used the approach of several copies to construct fast distributed algorithms for the simulation of parallel random access machines (PRAMs) on distributed memory machines (DMMs) Azar et al. ABKU94] study the problem of assigning requests to resources in a sequential setting, i.e. the requests arrive one by one. They show that, if every data item has c copies and m = n, there is a simple strategy that ensures a maximum load of Theta(log log n= log c) with high probability. They also ....

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations (extended abstract). In Proceedings of the 26th Symposium on Theory of Computing, pages 593--602, 1994.

....showed that using several copies for each data item instead of one can result in an exponential improvement of the maximum load. They used this to construct fast distributed algorithms for the simulation of parallel random access machines (PRAMs) by distributed memory machines (DMMs) Azar et al. ABKU94] study the problem of assigning requests to resources in a sequential setting, i.e. the requests arrive one by one. They show that, if each data item has c copies and m = n, there is a simple strategy that ensures a maximum load of Theta(log log n= log c) with high probability. They also ....

Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations (extended abstract). In Proceedings of the 26th Symposium on Theory of Computing, pages 593--602, 1994.

....and it has been extensively studied in the probability and statistics literature. See e.g. 8, 9] A standard result is that when the process has terminated, the fullest box has, with high probability (that is, 1 Gamma o(1) ln n= ln ln n(1 o(1) balls in it. Azar, Broder, Karlin, and Upfal [4] consider a variant whereby each ball comes with d possible destinations, chosen independently and uniformly at random. The ball is placed in the least full box among the d possible locations at the time of its insertion. Surprisingly when the process terminates the fullest box has only ln ln n= ....

....in an obvious way. So without loss of generality we can assume that G is connected which means G is actually a tree plus one extra edge. Finally it is easy to show that the addition of one edge changes M(n) by at most one, so we can assume that G is a tree T , which is our model. The proofs of [4] can be modified to show that if T is chosen uniformly at random then E(M(n) Theta(log log n) The connection with on line load balancing is as follows: We are given a set of n servers and a sequence of tasks. Each task comes with a list of servers on which it can be executed. The load ....

Yossi Azar, Andrei Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 593--602, May 1994.

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Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180--200, 1999.

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Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180--200, 1999.

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Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. Siam J. Comput., 29(1):180--200, 1994. http://citeseer.nj.nec.com/azar94balanced.html.

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Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM J. Comput., 29(1):180--200, 1999.

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Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations. SIAM Journal on Computing, 29(1):180--200, 1999.

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