Open addressing

The ChainedHashTable data structure uses an array of lists, where the ithi^{th} list stores all elements x such that hash(x) = i. An alternative, called open addressing, is to store the elements directly in an array, t, with each array location in t storing at most one value. This approach is taken by the LinearHashTable described in this section. In some places, this data structure is described as open addressing with linear probing.

Structure of LinearHashTable

The main idea behind a LinearHashTable is that we would, ideally, like to store the element x with hash value i = hash(x) in the table location t[i]. If we cannot do this (because some element is already stored there) then we try to store it at location t[(i + 1) mod t.length]; if that’s not possible, then we try t[(i+ 2) mod t.length], and so on, until we find a place for x.

There are three types of entries stored in t:

  • Data values: Actual values in the USet that we are representing;
  • null values: At array locations where no data has ever been stored; and
  • del values: At array locations where data was once stored but that has since been deleted.

In addition to the counter, n, that keeps track of the number of elements in the LinearHashTable, a counter, q, keeps track of the number of elements of Types 1 and 3. That is, q is equal to n plus the number of del values in t. To make this work efficiently, we need t to be considerably larger than q, so that there are lots of null values in t. The operations on a LinearHashTable therefore maintain the invariant that t.length2q\text{t.length} \ge 2q.

To summarize, a LinearHashTable contains an array, t, that stores data elements, and integers n and q that keep track of the number of data elements and non-null values of t, respectively. Because many hash functions only work for table sizes that are a power of 22, we also keep an integer d and maintain the invariant that t.length=2d.\text{t.length} = 2^{d}.

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