In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
In computer science, a search algorithm is an algorithm for finding an item with specified properties among a collection of items. The items may be stored individually as records in a database; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots...
for a given key value in an indexed array that has been ordered
Collation
Collation is the assembly of written information into a standard order. One common type of collation is called alphabetization, though collation is not limited to ordering letters of the alphabet...
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
Search space
Search space may refer to one of the following.*In optimization, the domain of the function to be optimized*In search algorithms of computer science, the set of all possible solutions...
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. This method will only work if calculations on the size of differences between key values are sensible.
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
Linear search
In computer science, linear search or sequential search is a method for finding a particular value in a list, that consists of checking every one of its elements, one at a time and in sequence, until the desired one is found....
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
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
(n) comparisons.
In interpolation-sequential search, interpolation is used to find an item near the one being searched for, then linear search
Linear search
In computer science, linear search or sequential search is a method for finding a particular value in a list, that consists of checking every one of its elements, one at a time and in sequence, until the desired one is found....
is used to find the exact item.
Performance
Using big-O notation, the performance of the interpolation algorithm on a data set of size N is O(N); however under the assumption of a uniform distribution of the data on the linear scale used for interpolation, the performance can be shown to be O(log log N).
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.
In computer science, a B-tree is a tree data structure that keeps data sorted and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree is a generalization of a binary search tree in that a node can have more than two children...
s 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
Online algorithm
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from...
. 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.
Some interpolation search implementations may not work as expected when a run of equal key values exists. The simplest implementation of interpolation search won't necessarily select the first (or last) element of such a run. The interpolation calculation also must be written to avoid division by zero.
Book-based searching
The conversion of names in a telephone book to some sort of number clearly will not provide numbers having a uniform distribution (except via immense effort such as sorting the names and calling them name #1, name #2, etc.) and further, it is well-known that some names are much more common than others (Smith, Jones,) Similarly with dictionaries, where there are many more words starting with some letters than others. Some publishers go to the effort of preparing marginal annotations or even cutting into the side of the pages to show markers for each letter so that at a glance a segmented interpolation can be performed.
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...
is a simple implementation. At each stage it computes a probe position then as with the binary search, moves either the upper or lower bound in to define a smaller interval containing the sought value. Unlike the binary search which guarantees a halving of the interval's size with each stage, a misled interpolation may reduce/increase the mid index by only one, thus resulting in a worst-case efficiency of O(n).
public int interpolationSearch(int[] sortedArray, int toFind){
// Returns index of toFind in sortedArray, or -1 if not found
int low = 0;
int high = sortedArray.length - 1;
int mid;
while (sortedArray[low] <= toFind && sortedArray[high] >= toFind) {
mid = low +
((toFind - sortedArray[low]) * (high - low)) /
(sortedArray[high] - sortedArray[low]); //out of range is possible here
if (sortedArray[mid] < toFind)
low = mid + 1;
else if (sortedArray[mid] > toFind)
// Repetition of the comparison code is forced by syntax limitations.
high = mid - 1;
else
return mid;
}
if (sortedArray[low] toFind)
return low;
else
return -1; // Not found
}
Notice that having probed the list at index mid, for reasons of loop control administration, this code sets either high or low to be not mid but an adjacent index, which location is then probed during the next iteration. Since an adjacent entry's value will not be much different the interpolation calculation is not much improved by this one step adjustment, at the cost of an additional reference to distant memory such as disc.
Each iteration of the above code requires between five and six comparisons (the extra is due to the repetitions needed to distinguish the three states of < > and = via binary comparisons in the absence of a three-way comparison
Three-way comparison
In computer science, a three-way comparison takes two values A and B belonging to a type with a total order and determines whether A B in a single operation, in accordance with the mathematical law of trichotomy.- Machine-level computation :...
In computer science, a binary search or half-interval search algorithm finds the position of a specified value within a sorted array. At each stage, the algorithm compares the input key value with the key value of the middle element of the array. If the keys match, then a matching element has been...
can be written with one comparison per iteration and uses only trivial integer arithmetic. It would thereby search an array of a million elements with no more than twenty comparisons (involving accesses to slow memory where the array elements are stored); to beat that the interpolation search as written above would be allowed no more than three iterations.
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