Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability ln n= ln ln n(1 + o(1)) balls in it. Suppose instead, that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only ln ln n= ln 2+O(1) balls -- exponentially less than before. Furthermore, we show that a similar gap exists in the infinite process, where at each step one ball, chosen uniformly at random, is deleted, and one ball is added in the manner above. We discuss consequences of this and related theorems for dynamic resource allocation, hashing, and on-line load balancing. 1 Introduction Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. Properties of this random allocation process have been extensively studied in ...

We consider cuckoo hashing as proposed by Pagh and Rodler in 2001. We show that the expected construction time of the hash table is O(n) as long as the two open addressing tables are each of size at least (1 #)n,where#>0andn is the number of data points. Slightly improved bounds are obtained for various probabilities and constraints. The analysis rests on simple properties of branching processes.

In a standard multiple choice hashing scheme, each item is stored in one of d ≥ 2 hash table buckets. The availability of choice in where items are stored improves space utilization. These schemes are often very amenable to a hardware implementation, such as in a router. Recently, researchers have discovered powerful variants where items already in the hash table may be moved during the insertion of a new item. Unfortunately, these schemes occasionally require a large number of items to be moved during an insertion operation, making them inappropriate for a hardware implementation. We show that it is possible to significantly increase the space utilization of a multiple choice hashing scheme by allowing at most one item to be moved during an insertion. Furthermore, our schemes can be effectively analyzed, optimized, and compared using numerical methods based on fluid limit arguments, without resorting to much slower simulations.

In a multiple-choice hashing scheme, each item is stored in one of d> = 2 possible hash tablebuckets. The availability of these multiple choices allows for a substantial reduction in the maximum load of the buckets. However, a lookup may now require examining each of the d locations. Forapplications where this cost is undesirable, Song et al. propose keeping a summary that allows one to determine which of the d locations is appropriate for each item, where the summary may allowfalse positives for items not in hash table. We propose alternative, simple constructions of such summaries that use less space for both the summary and the underlying hash table. Moreover, ourconstructions are easily analyzable and tunable.

Abstract — Hash tables are a fundamental data structure in many network applications, including route lookups, packet classification and monitoring. Often a part of the data path, they need to operate at wire-speed. However, several associative memory accesses are needed to resolve collisions, making them slower than required. This motivates us to consider minimal perfect hashing schemes, which reduce the number of memory accesses to just 1 and are also space-efficient. Existing perfect hashing algorithms are not tailored for network applications because they take too long to construct and are hard to implement in hardware. This paper introduces a hardware-friendly scheme for minimal perfect hashing, with space requirement approaching 3.7 times the information theoretic lower bound. Our construction is several orders faster than existing perfect hashing schemes. Instead of using the traditional mapping-partitioning-searching methodology, our scheme employs a Bloom filter, which is known for its simplicity and speed. We extend our scheme to the dynamic setting, thus handling insertions and deletions. I.

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

In a multiple-choice hashing scheme, each item is stored in one of P possible hash table buckets. The availability of these multiple choices allows for a substantial reduction in the maximum load of the buckets. However, a lookup may now require examining each of the locations. For applications where this cost is undesirable, Song et al. propose keeping a summary that allows one to determine which of the locations is appropriate for each item, where the summary may allow false positives for items not in hash table. We propose alternative, simple constructions of such summaries that use less space for both the summary and the underlying hash table. Moreover, our constructions are easily analyzable and tunable.

Abstract—This paper is about designing optimal highthroughput hashing schemes that minimize the total number of memory accesses needed to build and access an hash table. Recent schemes often promote the use of multiple-choice hashing. However, such a choice also implies a significant increase in the number of memory accesses to the hash table, which translates into higher power consumption and lower throughput. In this paper, we propose to only use choice when needed. Given some target hash table overflow rate, we provide a lower bound on the total number of needed memory accesses. Then, we design and analyze schemes that provably achieve this lower bound over a large range of target overflow values. Further, for the multilevel hash table scheme, we prove that the optimum occurs when its subtable sizes decrease in a geometric way, thus formally confirming a heuristic rule-of-thumb. A. Background I.

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.

Provision of QoS-related router functions such as traffic regulation, policy routing, and usage-based accounting requires that a flow table stores state information for active flows. The design of such a flow table is not trivial for a high-speed Internet router (e.g., 100+ Gbps) with a large number of active flows (e.g., tens of millions) and a high packet arrival rate (e.g., tens of millions of packets per second). Targeting two different models (centralized and distributed) of router design, we propose a software-based design to be implemented on individual line cards, which is suitable for the distributed model, and a hardware-based design to be implemented in the main forwarding engine of a router, which is suitable for the centralized model. The software-based design, adapted from hash table data structure, employs a practical and effective technique to solve the garbage collection problem caused by the expired flows. The hardware-based design, adapted from the architecture of an N-way set-associative cache, employs a dynamic set-associative scheme to reduce the overflow ratio that traditional set-associative scheme incurs, by a high percentage, and a pipelined design to achieve a throughput of 100+ Gbps. The performance evaluation results from both trace-driven simulation and statistical analysis demonstrate that both designs are cost-effective for their targeted router models.