Binary search algorithm

Procedure

Given an array A of n elements with values or records A₀, A₁, ..., A_n−1, sorted such that A₀ ≤ A₁ ≤ ... ≤ A_n−1, and target value T, the following subroutine uses binary search to find the index of T in A.^[7]

1. Set L to 0 and R to n − 1.
2. If L > R, the search terminates as unsuccessful.
3. Set m (the position of the middle element) to the floor of (L + R) / 2, which is the greatest integer less than or equal to (L + R) / 2.
4. If A_m < T, set L to m + 1 and go to step 2.
5. If A_m > T, set R to m − 1 and go to step 2.
6. Now A_m = T, the search is done; return m.

This iterative procedure keeps track of the search boundaries with the two variables L and R. The procedure may be expressed in pseudocode as follows, where the variable names and types remain the same as above, floor is the floor function, and unsuccessful refers to a specific variable that conveys the failure of the search.^[7]

function binary_search(A, n, T):
    L := 0
    R := n − 1
    while L <= R:
        m := floor((L + R) / 2)
        if A[m] < T:
            L := m + 1
        else if A[m] > T:
            R := m - 1
        else:
            return m
    return unsuccessful

Duplicate elements

The procedure may return any index whose element is equal to the target value, even if there are duplicate elements in the array. For example, if the array to be searched was [1, 2, 3, 4, 4, 5, 6, 7] and the target was 4, then it would be correct for the algorithm to either return the 4th (index 3) or 5th (index 4) element. The regular procedure would return the 4th element (index 3). However, it is sometimes necessary to find the leftmost element or the rightmost element if the target value is duplicated in the array. In the above example, the 4th element is the leftmost element of the value 4, while the 5th element is the rightmost element of the value 4. The alternative procedure above will always return the index of the rightmost element if an element is duplicated in the array.^[9]

function binary_search_leftmost(A, n, T):
    L := 0
    R := n
    while L < R:
        m := floor((L + R) / 2)
        if A[m] < T:
            L := m + 1
        else:
            R := m
    return L

function binary_search_rightmost(A, n, T):
    L := 0
    R := n
    while L < R:
        m := floor((L + R) / 2)
        if A[m] > T:
            R := m
        else:
            L := m + 1
    return L - 1

Approximate matches

Binary search can be adapted to compute approximate matches. In the example above, the rank, predecessor, successor, and nearest neighbor are shown for the target value 5, which is not in the array.

The above procedure only performs exact matches, finding the position of a target value. However, it is trivial to extend binary search to perform approximate matches because binary search operates on sorted arrays. For example, binary search can be used to compute, for a given value, its rank (the number of smaller elements), predecessor (next-smallest element), successor (next-largest element), and nearest neighbor. Range queries seeking the number of elements between two values can be performed with two rank queries.^[11]

• Rank queries can be performed with the procedure for finding the leftmost element. The number of elements less than the target value is returned by the procedure.^[11]
• Predecessor queries can be performed with rank queries. If the rank of the target value is r, its predecessor is r − 1.^[12]
• For successor queries, the procedure for finding the rightmost element can be used. If the result of running the procedure for the target value is r, then the successor of the target value is r + 1.^[12]
• The nearest neighbor of the target value is either its predecessor or successor, whichever is closer.
• Range queries are also straightforward.^[12] Once the ranks of the two values are known, the number of elements greater than or equal to the first value and less than the second is the difference of the two ranks. This count can be adjusted up or down by one according to whether the endpoints of the range should be considered to be part of the range and whether the array contains keys matching those endpoints.^[13]

Performance of alternative procedure

Each iteration of the binary search procedure defined above makes one or two comparisons, checking if the middle element is equal to the target in each iteration. Assuming that each element is equally likely to be searched, each iteration makes 1.5 comparisons on average. A variation of the algorithm checks whether the middle element is equal to the target at the end of the search. On average, this eliminates half a comparison from each iteration. This slightly cuts the time taken per iteration on most computers. However, it guarantees that the search takes the maximum number of iterations, on average adding one iteration to the search. Because the comparison loop is performed only times in the worst case, the slight increase in efficiency per iteration does not compensate for the extra iteration for all but enormous .^{[lower-alpha 3][17][18]}

Hashing

For implementing associative arrays, hash tables, a data structure that maps keys to records using a hash function, are generally faster than binary search on a sorted array of records.^[21] Most hash table implementations require only amortized constant time on average.^{[lower-alpha 4][23]} However, hashing is not useful for approximate matches, such as computing the next-smallest, next-largest, and nearest key, as the only information given on a failed search is that the target is not present in any record.^[24] Binary search is ideal for such matches, performing them in logarithmic time. Binary search also supports approximate matches. Some operations, like finding the smallest and largest element, can be done efficiently on sorted arrays but not on hash tables.^[20]

Trees

Binary search trees are searched using an algorithm similar to binary search.

A binary search tree is a binary tree data structure that works based on the principle of binary search. The records of the tree are arranged in sorted order, and each record in the tree can be searched using an algorithm similar to binary search, taking on average logarithmic time. Insertion and deletion also require on average logarithmic time in binary search trees. This can be faster than the linear time insertion and deletion of sorted arrays, and binary trees retain the ability to perform all the operations possible on a sorted array, including range and approximate queries.^[20][25]

However, binary search is usually more efficient for searching as binary search trees will most likely be imperfectly balanced, resulting in slightly worse performance than binary search. This even applies to balanced binary search trees, binary search trees that balance their own nodes, because they rarely produce optimally-balanced trees. Although unlikely, the tree may be severely imbalanced with few internal nodes with two children, resulting in the average and worst-case search time approaching comparisons.^{[lower-alpha 5]} Binary search trees take more space than sorted arrays.^[27]

Binary search trees lend themselves to fast searching in external memory stored in hard disks, as binary search trees can efficiently be structured in filesystems. The B-tree generalizes this method of tree organization. B-trees are frequently used to organize long-term storage such as databases and filesystems.^[28][29]

Linear search

Linear search is a simple search algorithm that checks every record until it finds the target value. Linear search can be done on a linked list, which allows for faster insertion and deletion than an array. Binary search is faster than linear search for sorted arrays except if the array is short, although the array needs to be sorted beforehand.^{[lower-alpha 6][31]} All sorting algorithms based on comparing elements, such as quicksort and merge sort, require at least comparisons in the worst case.^[32] Unlike linear search, binary search can be used for efficient approximate matching. There are operations such as finding the smallest and largest element that can be done efficiently on a sorted array but not on an unsorted array.^[33]

Set membership algorithms

A related problem to search is set membership. Any algorithm that does lookup, like binary search, can also be used for set membership. There are other algorithms that are more specifically suited for set membership. A bit array is the simplest, useful when the range of keys is limited. It compactly stores a collection of bits, with each bit representing a single key within the range of keys. Bit arrays are very fast, requiring only time.^[34] The Judy1 type of Judy array handles 64-bit keys efficiently.^[35]

For approximate results, Bloom filters, another probabilistic data structure based on hashing, store a set of keys by encoding the keys using a bit array and multiple hash functions. Bloom filters are much more space-efficient than bit arrays in most cases and not much slower: with hash functions, membership queries require only time. However, Bloom filters suffer from false positives.^{[lower-alpha 7][lower-alpha 8][37]}

Other data structures

There exist data structures that may improve on binary search in some cases for both searching and other operations available for sorted arrays. For example, searches, approximate matches, and the operations available to sorted arrays can be performed more efficiently than binary search on specialized data structures such as van Emde Boas trees, fusion trees, tries, and bit arrays. These specialized data structures are usually only faster because they take advantage of the properties of keys with a certain attribute (usually keys that are small integers), and thus will be time or space consuming for keys that lack that attribute.^[20] As long as the keys can be ordered, these operations can always be done at least efficiently on a sorted array regardless of the keys. Some structures, such as Judy arrays, use a combination of approaches to mitigate this while retaining efficiency and the ability to perform approximate matching.^[35]

Uniform binary search

Uniform binary search stores the difference between the current and the two next possible middle elements instead of specific bounds.

Uniform binary search stores, instead of the lower and upper bounds, the index of the middle element and the change in the middle element from the current iteration to the next iteration. Each step reduces the change by about half. For example, if the array to be searched is [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], the middle element would be 6. Uniform binary search works on the basis that the difference between the index of middle element of the array and the left and right subarrays is the same. In this case, the middle element of the left subarray ([1, 2, 3, 4, 5]) is 3 and the middle element of the right subarray ([7, 8, 9, 10, 11]) is 9. Uniform binary search would store the value of 3 as both indices differ from 6 by this same amount.^[38] To reduce the search space, the algorithm either adds or subtracts this change from the middle element. The main advantage of uniform binary search is that the procedure can store a table of the differences between indices for each iteration of the procedure. Uniform binary search may be faster on systems where it is inefficient to calculate the midpoint, such as on decimal computers.^[39]

Exponential search

Visualization of exponential searching finding the upper bound for the subsequent binary search

Exponential search extends binary search to unbounded lists. It starts by finding the first element with an index that is both a power of two and greater than the target value. Afterwards, it sets that index as the upper bound, and switches to binary search. A search takes iterations of the exponential search and at most iterations of the binary search, where is the position of the target value. Exponential search works on bounded lists, but becomes an improvement over binary search only if the target value lies near the beginning of the array.^[40]

Interpolation search

Visualization of interpolation search. In this case, no searching is needed because the estimate of the target's location within the array is correct. Other implementations may specify another function for estimating the target's location.

Instead of calculating the midpoint, interpolation search estimates the position of the target value, taking into account the lowest and highest elements in the array as well as length of the array. This is only possible if the array elements are numbers. It works on the basis that the midpoint is not the best guess in many cases. For example, if the target value is close to the highest element in the array, it is likely to be located near the end of the array.^[41] When the distribution of the array elements is uniform or near uniform, it makes comparisons.^[41][42][43]

In practice, interpolation search is slower than binary search for small arrays, as interpolation search requires extra computation. Its time complexity grows more slowly than binary search, but this only compensates for the extra computation for large arrays.^[41]

Fractional cascading

In fractional cascading, each array has pointers to every second element of another array, so only one binary search has to be performed to search all the arrays.

Fractional cascading is a technique that speeds up binary searches for the same element in multiple sorted arrays. Searching each array separately requires time, where is the number of arrays. Fractional cascading reduces this to by storing specific information in each array about each element and its position in the other arrays.^[44][45]

Fractional cascading was originally developed to efficiently solve various computational geometry problems. Fractional cascading has been applied elsewhere, such as in data mining and Internet Protocol routing.^[44]

Noisy binary search

In noisy binary search, there is a certain probability that a comparison is incorrect.

Noisy binary search algorithms solve the case where the algorithm cannot reliably compare elements of the array. For each pair of elements, there is a certain probability that the algorithm makes the wrong comparison. Noisy binary search can find the correct position of the target with a given probability that controls the reliability of the yielded position.^{[lower-alpha 9][lower-alpha 10][49][50]}

Quantum binary search

Classical computers are bounded to the worst case of exactly iterations when performing binary search. Quantum algorithms for binary search are still bounded to a proportion of queries (representing iterations of the classical procedure), but the constant factor is less than one, providing for faster performance on quantum computers. Any exact quantum binary search procedure—that is, a procedure that always yields the correct result—requires at least queries in the worst case, where is the natural logarithm.^[51] There is an exact quantum binary search procedure that runs in queries in the worst case.^[52] In comparison, Grover's algorithm is the optimal quantum algorithm for searching an unordered list of elements, and it requires queries.^[53]

Notes

Citations

Works