Multiplicative binary search

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Multiplicative binary search
Multiplicative Binary Search Depiction.svg
Visualization of the multiplicative binary search algorithm where 7 is the target value.
ClassSearch algorithm
Data structureArray
Worst-case performanceO(log n)
Best-case performanceO(1)
Average performanceO(log n)
Worst-case space complexityO(1)

In computer science, multiplicative binary search is a variation of binary search that uses a specific permutation of keys in an array instead of the sorted order used by regular binary search.[1] Multiplicative binary search was first described by Thomas Standish in 1980. This algorithm was originally proposed to simplify the midpoint index calculation on small computers without efficient division or shift operations, but its cache-friendly nature makes it suitable for out-of-memory static search on block-oriented storage as an alternative to B+ trees.

Multiplicative binary search is used by some optimizing compilers to implement switch statements.[2][3]


Multiplicative binary search operates on a permuted sorted array. Keys are stored in the array in level-order sequence of the corresponding balanced binary search tree. This places the first pivot of a binary search as the first element in the array. The second pivots are placed at the next two positions.

Given an array A of n elements with values A0 ... An−1, and target value T, the following subroutine uses multiplicative binary search to find the index of T in A.

  1. Set i to 0
  2. if in, the search terminates unsuccessful.
  3. if Ai = T, the search is done; return i.
  4. if Ai < T, set i to 2×i + 1 and go to step 2.
  5. if Ai > T, set i to 2×i + 2 and go to step 2.

See also[edit]


  1. ^ Standish, Thomas A. (1980). "Chapter 4.2.2: Ordered Table Search". Data Structure Techniques. Addison-Wesley. pp. 136–141. ISBN 978-0201072563.
  2. ^ Sayle, Roger A. (17 June 2008). "A Superoptimizer Analysis of Multiway Branch Code Generation" (PDF). Proceedings of the GCC Developers’ Summit: 103–116. Retrieved 4 March 2017.
  3. ^ Spuler, David A. (January 1994). Compiler Code Generation for Multiway Branch Statements as a Static Search Problem (Technical report). Department of Computer Science, James Cook University, Australia. 94/03.