Binary search

Binary search

Class	Search algorithm
Data structure	Array
Worst-case performance	O(log n)
Best-case performance	O(1)
Average performance	O(log n)
Worst-case space complexity	O(1)
Optimal	Yes

This article is about searching a finite sorted array. For searching continuous function values, see bisection method.

In computer science, a binary search or half-interval search algorithm finds the position of a specified value (the input "key") within a sorted array. In each step, 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 found so its index, or position, is returned. Otherwise, if the sought key is less than the middle element's key, then the algorithm repeats its action on the sub-array to the left of the middle element or, if the input key is greater, on the sub-array to the right. If the remaining array to be searched is reduced to zero, then the key cannot be found in the array and a special "Not found" indication is returned.

A binary search halves the number of items to check with each iteration, so locating an item (or determining its absence) takes logarithmic time. A binary search is a dichotomic divide and conquer search algorithm.

Overview

Searching a sorted collection is a common task. A dictionary is a sorted list of word definitions. Given a word, one can find its definition. A telephone book is a sorted list of people's names, addresses, and telephone numbers. Knowing someone's name allows one to quickly find their telephone number and address.

If the list to be searched contains more than a few items (a dozen, say) a binary search will require far fewer comparisons than a linear search, but it imposes the requirement that the list be sorted. Similarly, a hash search can be faster than a binary search but imposes still greater requirements. If the contents of the array are modified between searches, maintaining these requirements may take more time than the searches! And if it is known that some items will be searched for much more often than others, and it can be arranged that these items are at the start of the list, then a linear search may be the best.

Examples

Number guessing game

This rather simple game begins something like "I'm thinking of an integer between forty and sixty inclusive, and to your guesses I'll respond 'High', 'Low', or 'Yes!' as might be the case." Supposing that N is the number of possible values (here, twenty-one as "inclusive" was stated), then at most $\lceil \log _{2}N\rceil$ questions are required to determine the number, since each question halves the search space. Note that one less question (iteration) is required than for the general algorithm, since the number is already constrained to be within a particular range.

Even if the number to guess can be arbitrarily large, in which case there is no upper bound N, The number can be found in at most $2\lceil \log _{2}k\rceil$ steps (where k is the (unknown) selected number) by first finding an upper bound by repeated doubling. For example, if the number were 11, the following sequence of guesses could be used to find it: 1, 2, 4, 8, 16, 12, 10, 11

One could also extend the method to include negative numbers; for example the following guesses could be used to find −13: 0, −1, −2, −4, −8, −16, −12, −14, −13.

Word lists

People typically use a mixture of the binary search and interpolative search algorithms when searching a telephone book, after the initial guess we exploit the fact that the entries are sorted and can rapidly find the required entry. For example when searching for Smith, if Rogers and Thomas have been found, one can flip to a page about halfway between the previous guesses. If this shows Samson, it can be concluded that Smith is somewhere between the Samson and Thomas pages so these can be divided.

Applications to complexity theory

Even if we do not know a fixed range the number k falls in, we can still determine its value by asking $2\lceil \log _{2}k\rceil$ simple yes/no questions of the form "Is k greater than x?" for some number x. As a simple consequence of this, if you can answer the question "Is this integer property k greater than a given value?" in some amount of time then you can find the value of that property in the same amount of time with an added factor of $\log _{2}k$. This is called a reduction, and it is because of this kind of reduction that most complexity theorists concentrate on decision problems, algorithms that produce a simple yes/no answer.

For example, suppose we could answer "Does this n x n matrix have determinant larger than k?" in O(n²) time. Then, by using binary search, we could find the (ceiling of the) determinant itself in O(n²log d) time, where d is the determinant; notice that d is not the size of the input, but the size of the output.

Algorithm

Recursive

A straightforward implementation of binary search is recursive. The initial call uses the indices of the entire array to be searched. The procedure then calculates an index midway between the two indices, determines which of the two subarrays to search, and then does a recursive call to search that subarray. Each of the calls is tail recursive, so a compiler need not make a new stack frame for each call. The variables imin and imax are the lowest and highest inclusive indices that are searched.

int binary_search(int A[], int key, int imin, int imax)
{
  // test if array is empty
  if (imax < imin):
    // set is empty, so return value showing not found
    return KEY_NOT_FOUND;
  else
    {
      // calculate midpoint to cut set in half
      int imid = (imin + imax) / 2;
      
      // three-way comparison
      if (A[imid] > key):
        // key is in lower subset
        return binary_search(A, key, imin, imid-1);
      else if (A[imid] < key):
        // key is in upper subset
        return binary_search(A, key, imid+1, imax);
      else:
        // key has been found
        return imid;
    }
}

It is invoked with initial imin and imax values of 0 and N-1 for a zero based array.

Iterative

The binary search algorithm can also be expressed iteratively with two index limits that progressively narrow the search range.

int binary_search(int A[], int key, int imin, int imax)
{
  // continue searching while [imin,imax] is not empty
  while (imax >= imin)
    {
      /* calculate the midpoint for roughly equal partition */
      int imid = (imin + imax) / 2;

      // determine which subarray to search
      if      (A[imid] <  key)
        // change min index to search upper subarray
        imin = imid + 1;
      else if (A[imid] > key )
        // change max index to search lower subarray
        imax = imid - 1;
      else
        // key found at index imid
        return imid;
  }
  // key not found
  return KEY_NOT_FOUND;
}

Deferred detection of equality

The above iterative and recursive versions take three paths based on the key comparison: one path for less than, one path for greater than, and one path for equality. (There are two conditional branches.) The path for equality is taken only when the record is finally matched, so it is rarely taken. That branch path can be moved outside the search loop in the deferred test for equality version of the algorithm. The following algorithm uses only one conditional branch per iteration.

// inclusive indices
//   0 <= imin when using truncate toward zero divide
//     imid = (imin+imax)/2;
//   imin unrestricted when using truncate toward minus infinity divide
//     imid = (imin+imax)<<1; or
//     imid = (int)floor((imin+imax)/2.0);
int binary_search(int A[], int key, int imin, int imax)
{
  // continually narrow search until just one element remains
  while (imin < imax)
    {
      int imid = (imin + imax) / 2;

      // code must guarantee the interval is reduced at each iteration
      assert(imid < imax);
      // note: 0 <= imin < imax implies imid will always be less than imax

      // reduce the search
      if (A[imid] < key)
        imin = imid + 1;
      else
        imax = imid;
    }
  // At exit of while:
  //   if A[] is empty, then imax < imin
  //   otherwise imax == imin

  // deferred test for equality
  if ((imax == imin) && (A[imin] == key))
    return imin;
  else
    return KEY_NOT_FOUND;
}

The deferred detection approach foregoes the possibility of early termination on discovery of a match, so the search will take about log₂(N) iterations. On average, a successful early termination search will not save many iterations. For large arrays that are a power of 2, the savings is about two iterations. Half the time, a match is found with one iteration left to go; one quarter the time with two iterations left, one eighth with three iterations, and so forth. The infinite series sum is 2.

The deferred detection algorithm has the advantage that if the keys are not unique, it returns the smallest index (the starting index) of the region where elements have the search key. The early termination version would return the first match it found, and that match might be anywhere in region of equal keys.

Performance

With each test that fails to find a match at the probed position, the search is continued with one or other of the two sub-intervals, each at most half the size. More precisely, if the number of items, N, is odd then both sub-intervals will contain (N - 1)/2 elements, while if N is even then the two sub-intervals contain N/2 - 1 and N/2 elements.

If the original number of items is N then after the first iteration there will be at most N/2 items remaining, then at most N/4 items, at most N/8 items, and so on. In the worst case, when the value is not in the list, the algorithm must continue iterating until the span has been made empty; this will have taken at most ⌊log₂(N) + 1⌋ iterations, where the ⌊ ⌋ notation denotes the floor function that rounds its argument down to an integer. This worst case analysis is tight: for any N there exists a query that takes exactly ⌊log₂(N) + 1⌋ iterations. When compared to linear search, whose worst-case behaviour is N iterations, we see that binary search is substantially faster as N grows large. For example, to search a list of one million items takes as many as one million iterations with linear search, but never more than twenty iterations with binary search. However, a binary search can only be performed if the list is in sorted order.

Average performance

$\log _{2}(N)-1$ is the expected number of probes in an average successful search, and the worst case is $\log _{2}(N)$, just one more probe. If the list is empty, no probes at all are made. Thus binary search is a logarithmic algorithm and executes in O($\log N$) time. In most cases it is considerably faster than a linear search. It can be implemented using iteration, or recursion. In some languages it is more elegantly expressed recursively; however, in some C-based languages tail recursion is not eliminated and the recursive version requires more stack space.

Binary search can interact poorly with the memory hierarchy (i.e. caching), because of its random-access nature. For in-memory searching, if the span to be searched is small, a linear search may have superior performance simply because it exhibits better locality of reference. For external searching, care must be taken or each of the first several probes will lead to a disk seek. A common method is to abandon binary searching for linear searching as soon as the size of the remaining span falls below a small value such as 8 or 16 or even more in recent computers. The exact value depends entirely on the machine running the algorithm.

Notice that for multiple searches with a fixed value for N, then (with the appropriate regard for integer division), the first iteration always selects the middle element at N/2, and the second always selects either N/4 or 3N/4, and so on. Thus if the array's key values are in some sort of slow storage (on a disc file, in virtual memory, not in the cpu's on-chip memory), keeping those three keys in a local array for a special preliminary search will avoid accessing widely separated memory. Escalating to seven or fifteen such values will allow further levels at not much cost in storage. On the other hand, if the searches are frequent and not separated by much other activity, the computer's various storage control features will more or less automatically promote frequently accessed elements into faster storage.

When multiple binary searches are to be performed for the same key in related lists, fractional cascading can be used to speed up successive searches after the first one.

Even though in theory binary search is almost always faster than linear search, in practice even on medium sized arrays (around 100 items or less) it might be infeasible to ever use binary search. On larger arrays, it only makes sense to binary search if the number of searches is large enough, because the initial time to sort the array is comparable to many linear searches

Variations

There are many, and they are easily confused. Also, using a binary search within a sorting method is debatable.

Exclusive or inclusive bounds

The most significant differences are between the "exclusive" and "inclusive" forms of the bounds. In the "exclusive" bound form the span to be searched is (L + 1) to (R − 1), and this may seem clumsy when the span to be searched could be described in the "inclusive" form, as L to R. Although the details differ the two forms are equivalent as can be seen by transforming one version into the other. The inclusive bound form can be attained by replacing all appearances of "L" by "(L − 1)" and "R" by "(R + 1)" then rearranging. Thus, the initialisation of L:=0 becomes (L − 1):=0 or L:=1, and R:=N + 1 becomes (R + 1):=N + 1 or R:=N. So far so good, but note now that the changes to L and R are no longer simply transferring the value of p to L or R as appropriate but now must be (R + 1):=p or R:=p − 1, and (L − 1):=p or L:=p + 1.

Thus, the gain of a simpler initialisation, done once, is lost by a more complex calculation, and which is done for every iteration. If that is not enough, the test for an empty span is more complex also, as compared to the simplicity of checking that the value of p is zero. Nevertheless, the inclusive bound form is found in many publications, such as Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition.

Another common variation uses inclusive bounds for the left bound, but exclusive bounds for the right bound. This is derived from the fact that the bounds in a language with zero-based arrays can be simply initialized to 0 and the size of the array, respectively. This mirrors the way array slices are represented in some programming languages.

Midpoint and width

An entirely different variation involves abandoning the L and R pointers in favour of a current position p and a width w where at each iteration, p is adjusted by + or − w and w is halved. Professor Knuth remarks "It is possible to do this, but only if extreme care is paid to the details" – Section 6.2.1, page 414 of The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition, outlines an algorithm, with the further remark "Simpler approaches are doomed to failure!"

Search domain

There is no particular requirement that the array being searched has the bounds 1 to N. It is possible to search a specified range, elements first to last instead of 1 to N. All that is necessary is that the initialisation of the bounds be L:=first − 1 and R:=last + 1, then all proceeds as before.

The elements of the list are not necessarily all unique. If one searches for a value that occurs multiple times in the list, the index returned will be of the first-encountered equal element, and this will not necessarily be that of the first, last, or middle element of the run of equal-key elements but will depend on the positions of the values. Modifying the list even in seemingly unrelated ways such as adding elements elsewhere in the list may change the result.

If the location of the first and/or last equal element needs to be determined, this can be done efficiently with a variant of the binary search algorithms which perform only one inequality test per iteration.

Noisy search

Several algorithms closely related to or extending binary search exist. For instance, noisy binary search solves the same class of projects as regular binary search, with the added complexity that any given test can return a false value at random. (Usually, the number of such erroneous results are bounded in some way, either in the form of an average error rate, or in the total number of errors allowed per element in the search space.) Optimal algorithms for several classes of noisy binary search problems have been known since the late seventies, and more recently, optimal algorithms for noisy binary search in quantum computers (where several elements can be tested at the same time) have been discovered.

Implementation issues

Although the basic idea of binary search is comparatively straightforward, the details can be surprisingly tricky… — Professor Donald Knuth

When Jon Bentley assigned it as a problem in a course for professional programmers, he found that an astounding ninety percent failed to code a binary search correctly after several hours of working on it, and another study shows that accurate code for it is only found in five out of twenty textbooks. Furthermore, Bentley's own implementation of binary search, published in his 1986 book Programming Pearls, contains an error that remained undetected for over twenty years.

Arithmetic

In a practical implementation, the variables used to represent the indices will be of finite size, hence only capable of representing a finite range of values. For example, 32-bit unsigned integers can only hold values from 0 to 4294967295. Most binary search algorithms use 32-bit signed integers, which can only hold values from -2147483648 to 2147483647. If the binary search algorithm is to operate on large arrays, this has two implications:

• The values first − 1 and last + 1 must both be representable within the finite bounds of the chosen integer type . Therefore, continuing the 32-bit unsigned example, the largest value that last may take is +4294967294, not +4294967295. A problem exists even for the "inclusive" form of the method, as if x > A(4294967295).Key, then on the final iteration the algorithm will attempt to store 4294967296 into L and fail. Equivalent issues apply to the lower limit, where first − 1 could become negative as when the first element of the array is at index zero.
• If the midpoint of the span is calculated as p := (L + R)/2, then the value (L + R) will exceed the number range if last is greater than (for unsigned) 4294967295/2 or (for signed) 2147483647/2 and the search wanders toward the upper end of the search space. This can be avoided by performing the calculation as p := (R - L)/2 + L. When the problem arises for signed integers, a more efficient alternative is by performing the calculation as p := (R + L) >>> 1, where >>> denotes the right logical shift operator. For example, this bug existed in Java SDK at Arrays.binarySearch() from 1.2 to 5.0 and fixed in 6.0.

Language support

Many standard libraries provide a way to do a binary search:

• C provides algorithm function bsearch in its standard library.
• C++'s STL provides algorithm functions binary_search, lower_bound and upper_bound.
• Java offers a set of overloaded binarySearch() static methods in the classes Arrays and Collections in the standard java.util package for performing binary searches on Java arrays and on Lists, respectively. They must be arrays of primitives, or the arrays or Lists must be of a type that implements the Comparable interface, or you must specify a custom Comparator object.
• Microsoft's .NET Framework 2.0 offers static generic versions of the binary search algorithm in its collection base classes. An example would be System.Array's method BinarySearch<T>(T[] array, T value).
• Python provides the bisect module.
• COBOL can perform binary search on internal tables using the SEARCH ALL statement.
• Perl can perform a generic binary search using the CPAN module Search::Binary.
• Go's sort standard library package contains functions Search, SearchInts, SearchFloat64s, and SearchStrings, which implement general binary search, as well as specific implementations for searching slices of integers, floating-point numbers, and strings, respectively.
• For Objective-C, the Cocoa framework provides the NSArray -indexOfObject:inSortedRange:options:usingComparator: method in Mac OS X 10.6+. Apple's Core Foundation C framework also contains a CFArrayBSearchValues() function.

See also

• Interpolation search, similar method with better average complexity
• Index (information technology), very fast 'lookup' using an index to directly select an entry
• Branch table, alternative indexed 'lookup' method for decision making
• Self-balancing binary search tree
• Run-time analysis, illustrating binary search method on machines of differing speeds
• Bisection method, the same idea used to solve equations in the real numbers

References

Other sources

• Kruse, Robert L.: "Data Structures and Program Design in C++", Prentice-Hall, 1999, ISBN 0-13-768995-0, page 280.
• van Gasteren, Netty; Feijen, Wim (1995). "The Binary Search Revisited" (PDF). AvG127/WF214. (investigates the foundations of the binary search, debunking the myth that it applies only to sorted arrays)

External links

• NIST Dictionary of Algorithms and Data Structures: binary search
• Binary search implemented in 12 languages