The study of hashing is closely related to the analysis of balls and bins. Azar et. al. [1] showed that instead of using a single hash function if we randomly hash a ball into two bins and place it in the smaller of the two, then this dramatically lowers the maximum load on bins. This leads to the concept of two-way hashing where the largest bucket contains O(log log n) balls with high probability. The hash look up will now search in both the buckets an item hashes to. Since an item may be placed in one of two buckets, we could potentially move an item after it has been initially placed to reduce maximum load. Using this fact, we present a simple, practical hashing scheme that maintains a maximum load of 2, with high probability, while achieving high memory utilization. In fact, with n buckets, even if the space for two items are pre-allocated per bucket, as may be desirable in hardware implementations, more than n items can be stored giving a high memory utilization. Assuming truly random hash functions, we prove the following properties for our hashing scheme. • Each lookup takes two random memory accesses, and reads at most two items per access. • Each insert takes O(log n) time and up to log log n+ O(1) moves, with high probability, and constant time in expectation. • Maintains 83.75 % memory utilization, without requiring dynamic allocation during inserts. We also analyze the trade-off between the number of moves performed during inserts and the maximum load on a bucket. By performing at most h moves, we can maintain a maximum load of O(hlogl((~og~og:n/h)). So, even by performing one move, we achieve a better bound than by performing no moves at all. 1

Cuckoo hashing holds great potential as a high-performance hashing scheme for real applications. Up to this point, the greatest drawback of cuckoo hashing appears to be that there is a polynomially small but practically significant probability that a failure occurs during the insertion of an item, requiring an expensive rehashing of all items in the table. In this paper, we show that this failure probability can be dramatically reduced by the addition of a very small constant-sized stash. We demonstrate both analytically and through simulations that stashes of size equivalent to only three or four items yield tremendous improvements, enhancing cuckoo hashing’s practical viability in both hardware and software. Our analysis naturally extends previous analyses of multiple cuckoo hashing variants, and the approach may prove useful in further related schemes.

Abstract Hashing is an extremely useful technique for a variety of high-speed packet-processing applications in routers. In this chapter, we survey much of the recent work in this area, paying particular attention to the interaction between theoretical and applied research. We assume very little background in either the theory or applications of hashing, reviewing the fundamentals as necessary. 1

We study the the following question in Random Graphs. We are given two disjoint sets L, R with |L | = n = αm and |R | = m. We construct a random graph G by allowing each x ∈ L to choose d random neighbours in R. The question discussed is as to the size µ(G) of the largest matching in G. When considered in the context of Cuckoo Hashing, one key question is as to when is µ(G) = n whp? We answer this question exactly when d is at least three. We also establish a precise threshold for when Phase 1 of the Karp-Sipser Greedy matching algorithm suffices to compute a maximum matching whp.

Abstract. We present an algorithm for hashing ⌊ αn ⌋ elements into a table with n separate chains that requires O(1) deterministic worst-case insert time, and O(1) expected worst-case search time for constant α. We exploit the connection between two-way chaining and random graph theory in our techniques.

In this paper, we provide a polylogarithmic bound that holds with high probability on the insertion time for cuckoo hashing under the random-walk insertion method. Cuckoo hashing provides a useful methodology for building practical, high-performance hash tables. The essential idea of cuckoo hashing is to combine the power of schemes that allow multiple hash locations for an item with the power to dynamically change the location of an item among its possible locations. Previous work on the case where the number of choices is larger than two has required a breadth-first search analysis, which is both inefficient in practice and currently has only a polynomial high probability upper bound on the insertion time. Here we significantly advance the state of the art by proving a polylogarithmic bound on the more efficient randomwalk method, where items repeatedly kick out random blocking items until a free location for an item is found. 1

Locality Sensitive Hash (LSH) is widely used in peerto-peer (P2P) systems. Although it can support range or similarity queries, it breaks the load balance mechanism of traditional Distributed Hash Table (DHT) based system by replacing consistent hash with LSH. To solve the imbalance problem, current systems either weaken the locality preserve ability from similarity preserved to order preserved or adopt load aware peer join mechanism. The first method does not support similarity query as it loses the similarity information and the second method is greatly affected by the dynamic nature of P2P networks. In this paper, we propose a novel system, cuckoo ring, which can preserve similarity information while load balanced. It does not guide the newly joining peer to the hot areas but move the items in the hot areas to cold areas so that the short life time peers are distributed uniformly across the network instead of being guided to the hot areas. Compared to traditional DHT systems, cuckoo ring only maintains a little more information about the global light load peers and the moved indexed items.

This paper presents an FPGA-friendly approach to tracking elephant flows in network traffic. Our approach, Single Step Segmented Least Recently Used (S 3-LRU) policy, is a network traffic-friendly replacement policy for maintaining flow states in a Naïve Hash Table (NHT). We demonstrate that our S 3-LRU approach preserves elephant flows: conservatively promoting potential elephants and evicting lowrate flows in LRU manner. Our approach keeps flow-state of any elephant since start-of-day and provides a significant improvement over filtering approaches proposed in previous work. Our FPGA-based implementation of the S 3-LRU in combination with an NHT suites well the parallel access to block memories while capitalising on the retuning of parameters through dynamic-reprogramming. 1.

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