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

Cuckoo hashing is an efficient and practical dynamic dictionary. It provides expected amortized constant update time, worst case constant lookup time, and good memory utilization. Various experiments demonstrated that cuckoo hashing is highly suitable for modern computer architectures and distributed settings, and offers significant improvements compared to other schemes. In this work we construct a practical history-independent dynamic dictionary based on cuckoo hashing. In a history-independent data structure, the memory representation at any point in time yields no information on the specific sequence of insertions and deletions that led to its current content, other than the content itself. Such a property is significant when preventing unintended leakage of information, and was also found useful in several algorithmic settings. Our construction enjoys most of the attractive properties of cuckoo hashing. In particular, no dynamic memory allocation is required, updates are performed in expected amortized constant time, and membership queries are performed in worst case constant time. Moreover, with high probability, the lookup procedure queries only two memory entries which are independent and can be queried in parallel. The approach underlying our construction is to enforce a canonical memory representation on cuckoo hashing. That is, up to the initial randomness, each set of elements has a unique memory representation.

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.

Cuckoo hashing is a highly practical dynamic dictionary: it provides amortized constant insertion time, worst case constant deletion time and lookup time, and good memory utilization. However, with a noticeable probability during the insertion of n elements some insertion requires Ω(log n) time. Whereas such an amortized guarantee may be suitable for some applications, in other applications (such as high-performance routing) this is highly undesirable. Kirsch and Mitzenmacher (Allerton ’07) proposed a de-amortization of cuckoo hashing using queueing techniques that preserve its attractive properties. They demonstrated a significant improvement to the worst case performance of cuckoo hashing via experimental results, but left open the problem of constructing a scheme with provable properties. In this work we present a de-amortization of cuckoo hashing that provably guarantees constant worst case operations. Specifically, for any sequence of polynomially many operations, with overwhelming probability over the randomness of the initialization phase, each operation is performed in constant time. In addition, we present a general approach for proving that the performance guarantees are preserved when using hash functions with limited independence

The performance of a dynamic dictionary is measured mainly by its update time, lookup time, and space consumption. In terms of update time and lookup time there are known constructions that guarantee constant-time operations in the worst case with high probability, and in terms of space consumption there are known constructions that use essentially optimal space. In this paper we settle two fundamental open problems: • We construct the first dynamic dictionary that enjoys the best of both worlds: we present a two-level variant of cuckoo hashing that stores n elements using (1+ϵ)n memory words, and guarantees constant-time operations in the worst case with high probability. Specifically, for any ϵ = Ω((log log n / log n) 1/2) and for any sequence of polynomially many operations, with high probability over the randomness of the initialization phase, all operations are performed in constant time which is independent of ϵ. The construction is based on augmenting cuckoo hashing with a “backyard ” that handles a large fraction of the elements, together with a de-amortized perfect hashing scheme for eliminating the dependency on ϵ.

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

We reinvestigate the oblivious RAM concept introduced by Goldreich and Ostrovsky, which enables a client, that can store locally only a constant amount of data, to store remotely n data items, and access them while hiding the identities of the items which are being accessed. Oblivious RAM is often cited as a powerful tool, which can be used, for example, for search on encrypted data or for preventing cache attacks. However, oblivious RAM it is also commonly considered to be impractical due to its overhead, which is asymptotically efficient but is quite high: each data request is replaced by O(log 4 n) requests, or by O(log 3 n) requests where the constant in the “O ” notation is a few thousands. In addition, O(n log n) external memory is required in order to store the n data items. We redesign the oblivious RAM protocol using modern tools, namely Cuckoo hashing and a new oblivious sorting algorithm. The resulting protocol uses only O(n) external memory, and replaces each data request by only O(log 2 n) requests (with a small constant). This analysis is validated by experiments that we ran. Keywords: Secure two-party computation, oblivious RAM.

With the gaining popularity of remote storage (e.g. in the Cloud), we consider the setting where a small, protected local machine wishes to access data on a large, untrusted remote machine. This setting was introduced in the RAM model in the context of software protection by Goldreich and Ostrovsky. A secure Oblivious RAM simulation allows for a client, with small (e.g., constant size) protected memory, to hide not only the data but also the sequence of locations it accesses (both reads and writes) in the unprotected memory of size n. Our main results are as follows: • We analyze several schemes from the literature, observing a repeated design flaw that leaks information on the memory access pattern. For some of these schemes, the leakage is actually non-negligible, while for others it is negligible. • On the positive side, we present a new secure oblivious RAM scheme, extending a recent scheme by Goodrich and Mitzenmacher. Our scheme uses only O(1) local memory, and its (amortized) overhead is O(log 2 n / log log n), outperforming the previously-best O(log 2 n) overhead (among schemes where the client only uses O(1) additional local 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.