Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 200
5 Key Transformations (Hashing)
5.1 Introduction
The principal question discussed in Chap. 4 at length is the following: Given a set of items characterized
by a key (upon which an ordering relation is defined), how is the set to be organized so that retrieval of an
item with a given key involves as little effort as possible? Clearly, in a computer store each item is ultimately
accessed by specifying a storage address. Hence, the stated problem is essentially one of finding an
appropriate mapping H of keys (K) into addresses (A):
H: K → A
In Chap. 4 this mapping was implemented in the form of various list and tree search algorithms based on
different underlying data organizations. Here we present yet another approach that is basically simple and
very efficient in many cases. The fact that it also has some disadvantages is discussed subsequently.
The data organization used in this technique is the array structure. H is therefore a mapping transforming
keys into array indices, which is the reason for the term key transformation that is generally used for this
technique. It should be noted that we shall not need to rely on any dynamic allocation procedures; the array
is one of the fundamental, static structures. The method of key transformations is often used in problem
areas where tree structures are comparable competitors.
The fundamental difficulty in using a key transformation is that the set of possible key values is much
larger than the set of available store addresses (array indices). Take for example names consisting of up to
16 letters as keys identifying individuals in a set of a thousand persons. Hence, there are 26 16 possible
keys which are to be mapped onto 10 3 possible indices. The function H is therefore obviously a many-toone
function. Given a key k, the first step in a retrieval (search) operation is to compute its associated index
h = H(k), and the second - evidently necessary - step is to verify whether or not the item with the key k is
indeed identified by h in the array (table) T, i.e., to check whether T[H(k)].key = k. We are immediately
confronted with two questions:
1. What kind of function H should be used?
2. How do we cope with the situation that H does not yield the location of the desired item?
The answer to the second question is that some method must be used to yield an alternative location, say
index h', and, if this is still not the location of the wanted item, yet a third index h", and so on. The case in
which a key other than the desired one is at the identified location is called a collision; the task of
generating alternative indices is termed collision handling. In the following we shall discuss the choice of a
transformation function and methods of collision handling.
5.2 Choice of a Hash Function
A prerequisite of a good transformation function is that it distributes the keys as evenly as possible over
the range of index values. Apart from satisfying this requirement, the distribution is not bound to any
pattern, and it is actually desirable that it give the impression of being entirely random. This property has
given this method the somewhat unscientific name hashing, i.e., chopping the argument up, or making a
mess. H is called the hash function. Clearly, it should be efficiently computable, i.e., be composed of very
few basic arithmetic operations.
Assume that a transfer function ORD(k) is available and denotes the ordinal number of the key k in the
set of all possible keys. Assume, furthermore, that the array indices i range over the intergers 0 .. N-1,
where N is the size of the array. Then an obvious choice is