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Extendible hashing is a dynamic data structure which accommodates expansion and contraction of any stored data efficiently. In this article, an algorithm has been developed for managing concurrent operations on extendible hashing by achieving optimal memory utilization by supporting directly expansion and contraction, page split, and merge. The results of this study have been encouraging in the sense that it seems to provide a higher degree of concurrency compared to other algorithms on an extendible hash file.

The coalesced hashing method is one of the faster searching methods known today. This paper is a practical study of coalesced hashing for use by those who intend to implement or further study the algorithm. Techniques are developed for tuning an important parameter that relates the sizes of the address region and the cellar in order to optimize the average running times of different implementations. A value for the parameter is reported that works well in most cases. Detailed graphs explain how the parameter can be tuned further to meet specific needs. The resulting tuned algorithm outperforms several well-known methods including standard coalesced hashing, separate (or direct) chaining, linear probing, and double hashing. A variety of related methods are also analyzed including deletion algorithms, a new and improved insertion strategy called varied-insertion, and applications to external searching on secondary storage devices.

Linear hashing and spiral storage are two dynamic hashing schemes originally designed for external files. This paper shows how to adapt these two methods for hash tables stored in main memory. The necessary data structures and algorithms are described, the expected performance is analyzed mathematically, and actual execution times are obtained and compared with alternative techniques. Linear hashing is found to be both faster and easier to implement than spiral storage. Two alternative techniques are considered: a simple unbalanced binary tree and double hashing with periodic rehashing into a larger table. The retrieval time of linear hashing is similar to double hashing and substantially faster than a binary tree, except for very small trees. The loading times of double hashing (with periodic reorganization), a binary tree, and linear hashing are similar. Overall, linear hashing is a simple and efficient technique for applications where the cardinality of the key set is not known in advance.

Various addressing schemes making use of segment tables are examined. The inadequacies of these schemes when dealing with shared addresses are explained. These inadequacies are traced to the lack of an efficient absolute address for objects in these systems. The direct use of a capability as an address is shown to overcome these difficulties because it provides the needed absolute address. Implementation of capability-based addressing is discussed. It is predicted that the use of tags to identify capabilities will dominate. A hardware address translation scheme which never requires the modification of the representation of capabilities is suggested. The scheme uses a main memory hash table for obtaining a segment's location in main memory given its unique code. The hash table is avoided for recently accessed segments by means of a set of associative registers. A computer using capability-based addressing may be substantially superior to present systems on the basis of protection, simplicity of programming conventions, and efficient implementation.

A refinement of hashing which allows retrieval of an item in a static table with a single probe is considered. Given a set I of identifiers, two methods are presented for building, in a mechanical way, perfect hashing functions, i.e. functions transforming the elements of I into unique addresses. The first method, the “quotient reduction” method, is shown to be complete in the sense that for every set I the smallest table in which the elements of I can be stored and from which they can be retrieved by using a perfect hashing function constructed by this method can be found. However, for nonuniformly distributed sets, this method can give rather sparse tables. The second method, the “remainder reduction” method, is not complete in the above sense, but it seems to give minimal (or almost minimal) tables for every kind of set. The two techniques are applicable directly to small sets. Some methods to extend these results to larger sets are also presented. A rough comparison with ordinary hashing is given which shows that this method can be used conveniently in several practical applications.

A method is presented for building minimal perfect hash functions, i.e., functions which allow single probe retrieval from minimally sized tables of identifier sets. A proof of existence for minimal perfect hash functions of a special type (reciprocal type) is given. Two algorithms for determining hash functions of reciprocal type are presented and their practical limitations are discussed. Further, some application results are given and compared with those of earlier approaches for perfect hashing.

This paper presents pseudochaining as a new collision-resolution method. Pseudochaining is half way between open addressing and chaining. It owes its name to the fact that link fields are present in each cell of the hash table which permits “chaining” of the first overflow items in the table. The efficiency of the method is derived and a tradeoff analysis is given.

A new method for entering and retrieving information in a hash table is described. The method is intended to be efficient if most entries are looked up several times. The expected number of probes to look up an entry, predicted theoretically and verified by Monte Carlo experiments, is considerably less than for other comparable methods if the table is nearly full. An example of a possible Fortran implementation is given.

When the space allocation for a hash-coded table is altered, the table entries must be rescattered over the new space. A technique for accomplishing this rescattering is presented. The technique is independent of both the length of the table and the hashing function used, and can be utilized in conjunction with a linear reallocation of the table being rescattered. Moreover, it can be used to eliminate previously flagged deletions from any hash-coded table, or to change from one hashing method to another. The efficiency of the technique is discussed and theoretical statistics are given.

A general method of file structuring is proposed which uses a hashing function to define tree structure. Two types of such trees are examined, and their relation to trees studied in the past is explained. Results for the probability distributions of path lengths are derived and illustrated.

Introduced is a hash coding method based on fixed-point division rather than multiplication or logical operations. This new method allows the hash table to have almost any length. Also a new method of handling collisions is discussed. Known as quadratic search, this method is faster than random search and free from the “clusters” that build up with a linear search.

A new linear search for hash tables whose increment step is a function of the key being addressed is presented. Comparisons with known methods are given, in terms of efficiency and computation complexity. In particular, the new method applies to tables of size n = 2 r . It allows full table searching, and practically eliminates primary clustering at a very low cost.

Secondary clustering as a cause of hash code inefficiency is discussed, and a new hashing method based on its eliminiation is presented. Comparisons with previous methods are made both analytically and empirically.

This note supplements the paper by V.Y. Lum [2] and elaborates on Table II in McIlroy [3].

The quadratic residue search method for hash tables avoids much of the clustering experienced with a linear search method. The simple quadratic search only accesses half the table. It has been shown that when the length of the table is a prime of the form 4 n + 3, where n is an integer, the whole table may be accessed by two quadratic searches plus a separate access for the original entry point. A search method is presented which is computationally simple, has all the advantages of the quadratic search, and yet accesses all the table in one sweep.

A new method of hash coding is presented and is shown to possess desirable attributes. Specifically, the algorithm is simple, efficient, and exhaustive, while needing little time per probe and using few probes per lookup. Performance data and implementation hints are also given.

It has previously been claimed [1 and 2] that the quadratic hash table search method of Maurer cannot usefully be applied to tables of size 2 n . This is not so; the method can in fact be applied to tables of size p n for any prime p . It is shown below that rather simple conditions on the coefficients suffice to guarantee that all table locations will be examined once and only once. Specifically, if the equation is k + bi 2 + ai mod p n (*) where k is the initial hash address and 0 ≤ i < p n , then, if p divides b but not a , the range of values is all the least positive residues of p n . To prove that all values are covered, we consider some fixed value, say k + bi 2 0 + ai 0 mod p n and ask, what conditions must be true if the congruence equation k + bi 2 + ai ≡ k + bi 0 2 + ai 0 mod p n is to have solutions i , 0 ≤ i < p n , other than i 0 ?