Optimal Ball Recycling

Bender, Michael A.; Christensen, Jake; Conway, Alex; Farach-Colton, Martı́n; Johnson, Rob; Tsai, Meng-Tsung

doi:10.1137/1.9781611975482.155

Cited by 6 publications

(5 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Table 1: The relationship between the average fill, 𝒃, and the maximum fill, 𝒉, in a balls-and-bins system is well understood [4,32].…”

Section: Iceberg Hashingmentioning

confidence: 99%

“…Afterwards, we explain how to synchronize among threads when a level resizes. 4 Formally, we can reduce to this case via standard pruning arguments, as in, e.g., [44].…”

Section: Multi-threadingmentioning

confidence: 99%

“…Assume for simplicity that all polylog(𝑛) of 𝑇 's nodes are distinct balls. 4 We have shown that, for each ball 𝑢, the expected number of ways that we can assign children to 𝑢 is 1/4. Using this, one can argue that the expected number of valid configurations for the full tree 𝑇 with polylog(𝑁 ) parent/child relationships is at most 1/4 polylog(𝑁 ) ≤ 1/poly(𝑁 ).…”

mentioning

confidence: 96%

See 2 more Smart Citations

IcebergHT: High Performance PMEM Hash Tables Through Stability and Low Associativity

Pandey¹,

Bender²,

Conway³

et al. 2022

Preprint

View full text Add to dashboard Cite

Modern hash table designs strive to minimize space while maximizing speed. The most important factor in speed is the number of cache lines accessed during updates and queries. This is especially important on PMEM, which is slower than DRAM and in which writes are more expensive than reads.This paper proposes two stronger design objectives: stability and low-associativity. A stable hash table doesn't move items around, and a hash table has low associativity if there are only a few locations where an item can be stored. Low associativity ensures that queries need to examine only a few memory locations, and stability ensures that insertions write to very few cache lines. Stability also simplifies scaling and crash safety.We present IcebergHT, a fast, crash-safe, concurrent, and space-efficient hash table for PMEM based on the design principles of stability and low associativity. IcebergHTcombines in-memory metadata with a new hashing technique, iceberg hashing, that is (1) space efficient, (2) stable, and (3) supports low associativity. In contrast, existing hash-tables either modify numerous cache lines during insertions (e.g. cuckoo hashing), access numerous cache lines during queries (e.g. linear probing), or waste space (e.g. chaining). Moreover, the combination of ( 1)-(3) yields several emergent benefits: IcebergHT scales better than other hash tables, supports crash-safety, and has excellent performance on PMEM (where writes are particularly expensive).In our benchmarks, IcebergHT inserts are 50% to 3× faster than state-of-the-art PMEM hash tables Dash and CLHT and queries are 20% to 2× faster. IcebergHT space overhead is 17%, whereas Dash and CLHT have space overheads of 2× and 3×, respectively. IcebergHT also exhibits linear scaling and is crash safe. In DRAM, IcebergHT outperforms state-of-the-art hash tables libcuckoo and CLHT by almost 2× on insertions while offering good query throughput and much better space efficiency.

show abstract

“…Table 1: The relationship between the average fill, 𝒃, and the maximum fill, 𝒉, in a balls-and-bins system is well understood [4,32].…”

Section: Iceberg Hashingmentioning

confidence: 99%

“…Afterwards, we explain how to synchronize among threads when a level resizes. 4 Formally, we can reduce to this case via standard pruning arguments, as in, e.g., [44].…”

Section: Multi-threadingmentioning

confidence: 99%

mentioning

confidence: 96%

See 1 more Smart Citation

IcebergHT: High Performance PMEM Hash Tables Through Stability and Low Associativity

Pandey¹,

Bender²,

Conway³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Cup games have also been used to model memory-access heuristics in databases [9]. Here, the emptier is allowed to completely empty a cup at each step, but the water from that cup is then "recycled" among the cups according to some probability distribution.…”

Section: Introductionmentioning

confidence: 99%

How Asymmetry Helps Buffer Management: Achieving Optimal Tail Size in Cup Games

Kuszmaul¹

2021

Preprint

View full text Add to dashboard Cite

The cup game on n cups is a multi-step game with two players, a filler and an emptier. At each step, the filler distributes 1 unit of water among the cups, and then the emptier selects a single cup to remove (up to) 1 unit of water from.There are several objective functions that the emptier might wish to minimize. One of the strongest guarantees would be to minimize tail size, which is defined to be the number of cups with fill 2 or greater. A simple lower-bound construction shows that the optimal tail size for deterministic emptying algorithms is Θ(n), however.We present a simple randomized emptying algorithm that achieves tail size Õ(log n) with high probability in n for poly n steps. Moreover, we show that this is tight up to doubly logarithmic factors. We also extend our results to the multi-processor cup game, achieving tail size Õ(log n + p) on p processors with high probability in n. We show that the dependence on p is near optimal for any emptying algorithm that achieves polynomial-bounded backlog.A natural question is whether our results can be extended to give unending guarantees, which apply to arbitrarily long games. We give a lower bound construction showing that no monotone memoryless emptying algorithm can achieve an unending guarantee on either tail size or the related objective function of backlog. On the other hand, we show that even a very small (i.e., 1/ poly n) amount of resource augmentation is sufficient to overcome this barrier.

show abstract

“…This is important, for example, in the study of multiprocessor scheduling with conflicts between tasks [13,14], and in the study of sensor radio networks [11]. Recently, cup emptying games have also found applications to modeling memory-access heuristics in databases [10]. Here, whenever the emptier removes the water from a cup, the water is then redistributed according to some fixed probability distribution.…”

Section: Introductionmentioning

confidence: 99%

Achieving optimal backlog in multi-processor cup games

Bender

Farach-Colton

Kuszmaul

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

Self Cite

View full text Add to dashboard Cite

The single-and multi-processor cup games can be used to model natural problems in areas such as processor scheduling, deamortization, and buffer management.At the beginning of the single-processor cup game, n cups sit in a row, initially empty. In each step of the game, a filler distributes 1 unit of water among the cups, and then an emptier selects a cup and removes 1 + ε units from that cup. The goal of the emptier is to minimize the amount of water in the fullest cup, also known as the backlog. It is known that the greedy algorithm (i.e., empty the fullest cup) achieves backlog O(log n), and that no deterministic algorithm can do better.We show that the performance of the greedy algorithm can be greatly improved with a small amount of randomization: After any step i, and for any k ≥ Ω(log ε −1 ), the emptier achieves backlog at most O(k) with probability at least 1 − O(2 −2 k ). Our algorithm, which we call the smoothed greedy algorithm, can also be interpreted as a one-shot smoothed analysis of the standard greedy algorithm.Whereas bounds for the single-processor cup game have been known for more than fifteen years, proving nontrivial bounds on backlog for the multi-processor extension has remained open. We present a simple analysis of the greedy algorithm for the multi-processor cup game, establishing a backlog of O(ε −1 log n), as long as δ, the game's other speed-augmentation constant, is at least 1/poly(n).Turning to randomized algorithms, we encounter an unexpected phenomenon: When the number of processors p is large, the backlog after each step drops to constant with large probability. Specifically, we show that if δ and ε satisfy reasonable constraints, then there exists an algorithm that bounds the backlog after a given step by three or less with probability at least 1 − O(exp(−Ω(ε 2 p)). We further extend the guarantees of our randomized algorithm to consider larger backlogs.When ε is constant, we prove that our results are asymptotically optimal, in the sense that no algorithms can achieve better bounds, up to constant factors in the backlog and in p. Moreover, we prove robustness results, demonstrating that our randomized algorithms continue to behave well even when placed in bad starting states.The small amount of resource augmentation used by the smoothed greedy algorithm makes it robust to the setting in which cups begin in a bad initial starting state. In particular, we show that if b units of water are maliciously placed into cups at the beginning of the game, then for steps i > b ε , the b units of water have no affect on the guarantees given by the algorithm.The multi-processor cup game. The multi-processor version of the same scheduling question has proven to be much harder. In each step of the multi-processor cup game, the filler distributes some amount of water proportional to p, the number of processors, among the cups (i.e., threads), and the emptier picks p cups and removes a unit of water from each. If the filler is unrestricted in their placement of the water, then they can ensure an...

show abstract

Optimal Ball Recycling

Cited by 6 publications

References 26 publications

IcebergHT: High Performance PMEM Hash Tables Through Stability and Low Associativity

IcebergHT: High Performance PMEM Hash Tables Through Stability and Low Associativity

How Asymmetry Helps Buffer Management: Achieving Optimal Tail Size in Cup Games

Achieving optimal backlog in multi-processor cup games

Contact Info

Product

Resources

About