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.

Abstract—Counting Bloom Filters (CBFs) are widely used in networking device algorithms. They implement fast set representations to support membership queries with limited error, and support element deletions unlike Bloom Filters. However, they also consume significant amounts of memory. In this paper we introduce a new general method based on variable increments to improve the efficiency of CBFs and their variants. Unlike CBFs, at each packet arrival, the hashed counters increase by a hashed variable increment instead of a unit increment. Then, to query a packet, the exact value of a counter is considered and not just its positiveness. We present two simple schemes based on this method. We demonstrate that this method can always achieve a lower false positive rate and a lower overflow probability bound than CBF in large systems. We also show how it can be easily implemented in hardware, with limited added complexity and memory overhead. We also explain how this method can extend many variants of CBF that have been published in the literature. Last, using simulations, we show how it can improve the false positive rate of CBFs by up to an order of magnitude given the same amount of memory.

Abstract — We show that when memory is bounded, i.e. buckets are finite, dynamic hash tables that allow insertions and deletions behave significantly worse than their static counterparts that only allow insertions. This behavior differs from previous results in which, when memory is unbounded, the two models behave similarly. We show the decrease in performance in dynamic hash tables using several hash-table schemes. We also provide tight upper and lower bounds on the achievable overflow fractions in these schemes. Finally, we propose an architecture with contentaddressable memory (CAM), which mitigates this decrease in performance. A. Background I.

Bloom Filters should particularly suit network devices, because of their low theoretical memory-access rates. However, in practice, since memory is often divided into blocks and Bloom Filters hash elements into several arbitrary memory blocks, Bloom Filters actually need high memory-access rates. On the other hand, hashing all Bloom Filter elements into a single memory block to solve this problem also yields high false positive rates. In this paper, we propose to implement load-balancing schemes for the choice of the memory block, along with an optional overflow list, resulting in improved false positive rates while keeping a high memory-access efficiency. To study this problem, we define, analyze and solve a fundamental access-constrained balancing problem, where incoming elements need to be optimally balanced across resources while satisfying average and instantaneous constraints on the number of memory accesses associated with checking the current load of the resources. We then build on this problem to suggest a new access-efficient Bloom Filter scheme, called the Balanced Bloom Filter. Finally, we show that this scheme can reduce the false positive rate by up to two orders of magnitude, with a worst-case cost of up to 3 memory accesses for each element and an overflow list size of 0.5 % of the elements.

Abstract. The purpose of this brief note is to describe recent work in the area of cuckoo hashing, including a clear description of several open problems, with the hope of spurring further research. 1

Abstract—Hash-based data structures, which use randomization in order to represent efficiently a list of elements, are one of the most-used data structures in networking applications, where both time and fast memory are scarce resources. This paper investigates the realistic scenario in which elements are not only added to the data structure but also deleted. We show that when the memory is bounded, dynamic hash-tables with deletions behave significantly worse than their static counterparts. This is contrast with previous results that show that when the memory is not bounded the two models behave practically the same. We provide tight upper and lower bounds on the achievable overflow fraction of the scheme under various models and system parameters. Then, we propose two architectures using CAMs and TCAMs that allow us to mitigate this decrease in performance. Our analytical results are confirmed using simulations with reallife traces and real hash-functions. A. Background I.

Abstract—This paper defines and analyzes a fundamental energy-constrained balancing problem, in which elements need to be balanced across resources in order to minimize the increasing convex cost function associated with the load at each resource. However, the balancing operation needs to satisfy average and instantaneous constraints on the energy associated with checking the current load of the many resources. In the paper, we first show tight lower and upper bounds on the solution of the problem depending on the specific system parameters. Then, we explain how these solutions can be applied to construct hash tables with optimal variance of the bin size, as well as energy-efficient Bloom filters.