Interpolation search

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Interpolation search (sometimes referred to as extrapolation search) is an algorithm for searching for a given key value in an indexed array that has been ordered by the values of the key. It parallels how humans search through a telephone book for a particular name, the key value by which the book's entries are ordered. In each search step it calculates where in the remaining search space the sought item might be, based on the key values at the bounds of the search space and the value of the sought key, usually via a linear interpolation. The key value actually found at this estimated position is then compared to the key value being sought. If it is not equal, then depending on the comparison, the remaining search space is reduced to the part before or after the estimated position. Only if calculations on the size of differences between key values are sensible will this method work.

By comparison, the binary search always chooses the middle of the remaining search space, discarding one half or the other, again depending on the comparison between the key value found at the estimated position and the key value sought. The remaining search space is reduced to the part before or after the estimated position. The linear search uses equality only as it compares elements one-by-one from the start, ignoring any sorting.

On average the interpolation search makes about log(log(n)) comparisons (if the elements are uniformly distributed), where n is the number of elements to be searched. In the worst case (for instance where the numerical values of the keys increase exponentially) it can make up to O(n) comparisons.

In interpolation-sequential search, interpolation is used to find an item near the one being searched for, then linear search is used to find the exact item.



Using big-O notation, the performance of the interpolation algorithm can be shown to be O(log log N).[1][2]

Practical performance of interpolation search depends on whether the reduced number of probes is outweighed by the more complicated calculations needed for each probe. It can be useful for locating a record in a large sorted file on disk, where each probe involves a disk seek and is much slower than the interpolation arithmetic.

Index structures like B-trees also reduce the number of disk accesses, and are more often used to index on-disk data in part because they can index many types of data and can be updated online. Still, interpolation search may be useful when one is forced to search certain sorted but unindexed on-disk datasets.

Adaptation to different datasets

When sort keys for a dataset are uniformly distributed numbers, linear interpolation is straightforward to implement and will find an index very near the sought value.

On the other hand, for a phone book sorted by name, the straightforward approach to interpolation search doesn't apply. The same high-level principles can still apply, though: one can estimate a name's position in the phone book using the relative frequencies of letters in names and use that as a probe location.

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