scandum / binary_search

https://github.com/scandum/binary_search

C
A collection of improved binary search algorithms.



The most commonly used binary search variant was first published by Hermann Bottenbruch in 1962 and hasn't notably changed since. Below I'll describe several novel variants with improved performance. The most notable variant, the quaternary binary search, consistently executes up to 25% faster on 32 bit integers.

A source code implementation in C is available licensed under GPL 3 (a less restrictive license may be negotiated) and is bundled with a bench marking routine. A graph with performance results is included at the bottom of this page. Keep in mind performance will vary depending on hardware and compiler optimizations.

I'll briefly describe each variant below.

Deferred Detection of Equality

By skipping the detection of equality until the binary search has finished (which does not allow for early termination) each loop contains 1 key check, 1 integer check and 2 integer assignments. This is pretty much the standard algorithm that has been used since 1962.

Tailed Binary Search

This binary search variant has the same number of key checks as the standard binary search, except that the calculations are slightly less complicated, giving it a slight performance gain without drawbacks.

Boundless Binary Search

A boundless binary search is faster than the standard binary search since the loop contains 1 key check 1 integer check and (on average) 1.5 integer assignments. The performance gain will vary depending on various factors, but should be between 5% and 40% when comparing 32 bit integers. It performs more key checks.

Boundless Quaternary Binary Search

The boundless quaternary binary search has more key checks than the boundless binary search but runs faster on large arrays. In most of my tests it gives a performance gain between 15% and 25% when comparing 32 bit integers.

Inbound Binary search

Due to boundary issues the mid - 1 optimization used in the standard search cannot easily be used in the boundless binary search. The inbound binary search solves this problem and allows fewer key checks than the boundless search. It might have some utility when key checks are expensive.

Inbound Quaternary Binary search

Like the boundless quaternary binary search, but with fewer key checks. Overall performance is similar.

Boundless Interpolated Binary Search

When you have an even distribution you can make an educated guess as to the location of the index. Due to the expense of the initial check and error correction, the interpolated binary search is unlikely to outperform other binary searches on arrays with less than 100 elements. When the distribution is uneven performance will drop, but not significantly.

Benchmark graph

The graph below shows the execution speed on arrays with 10, 100, 1000, 10000, 100000, and 1000000 elements on an Intel i3 quad-core processor. The Y axis lists the execution time in microseconds.

binary search graph

Benchmark tables

The following benchmark was on WSL gcc version 7.4.0 (Ubuntu 7.4.0-1ubuntu1~18.04.1). The source code was compiled using gcc -O3 binary-search.c

Even distribution with 10 32 bit integers

Name Items Hits Misses Checks Time
standard_binary_search 10 158357 9841643 42991533 0.000284
standard_binary_search 10 158357 9841643 42991533 0.000283
tailed_binary_search 10 158357 9841643 42991533 0.000276
boundless_binary_search 10 158357 9841643 58195179 0.000249
inbound_binary_search 10 158357 9841643 50000000 0.000282
boundless_quaternary_search 10 158357 9841643 61762418 0.000265
inbound_quaternary_search 10 158357 9841643 55400810 0.000271
boundless_interpolated_search 10 158357 9841643 66699093 0.000271

Even distribution with 100 32 bit integers

Name Items Hits Misses Checks Time
standard_binary_search 100 188541 9811459 77168228 0.000492
standard_binary_search 100 188541 9811459 77168228 0.000491
tailed_binary_search 100 188541 9811459 77168228 0.000451
boundless_binary_search 100 188541 9811459 99635413 0.000330
inbound_binary_search 100 188541 9811459 80000000 0.000429
boundless_quaternary_search 100 188541 9811459 109846685 0.000383
inbound_quaternary_search 100 188541 9811459 95228258 0.000384
boundless_interpolated_search 100 188541 9811459 82681266 0.000310

Even distribution with 1000 32 bit integers

Name Items Hits Misses Checks Time
standard_binary_search 1000 194868 9805132 109697960 0.000729
standard_binary_search 1000 194868 9805132 109697960 0.000728
tailed_binary_search 1000 194868 9805132 109697960 0.000638
boundless_binary_search 1000 194868 9805132 136000769 0.000438
inbound_binary_search 1000 194868 9805132 110000000 0.000592
boundless_quaternary_search 1000 194868 9805132 149498206 0.000498
inbound_quaternary_search 1000 194868 9805132 143213946 0.000477
boundless_interpolated_search 1000 194868 9805132 101245246 0.000348

Even distribution with 10000 32 bit integers

Name Items Hits Misses Checks Time
standard_binary_search 10000 196724 9803276 143596482 0.000950
standard_binary_search 10000 196724 9803276 143596482 0.000950
tailed_binary_search 10000 196724 9803276 143596482 0.000873
boundless_binary_search 10000 196724 9803276 168770687 0.000480
inbound_binary_search 10000 196724 9803276 150000000 0.000793
boundless_quaternary_search 10000 196724 9803276 186344773 0.000628
inbound_quaternary_search 10000 196724 9803276 175777209 0.000607
boundless_interpolated_search 10000 196724 9803276 116021998 0.000414

Even distribution with 100000 32 bit integers

Name Items Hits Misses Checks Time
standard_binary_search 100000 197875 9802125 176901757 0.001243
standard_binary_search 100000 197875 9802125 176901757 0.001243
tailed_binary_search 100000 197875 9802125 176901757 0.001139
boundless_binary_search 100000 197875 9802125 200406343 0.000710
inbound_binary_search 100000 197875 9802125 180000000 0.001044
boundless_quaternary_search 100000 197875 9802125 223608878 0.000862
inbound_quaternary_search 100000 197875 9802125 208351360 0.000854
boundless_interpolated_search 100000 197875 9802125 146245804 0.000515

Even distribution with 1000000 32 bit integers

Name Items Hits Misses Checks Time
standard_binary_search 1000000 197423 9802577 209516052 0.001728
standard_binary_search 1000000 197423 9802577 209516052 0.001728
tailed_binary_search 1000000 197423 9802577 209516052 0.001609
boundless_binary_search 1000000 197423 9802577 235273893 0.001343
inbound_binary_search 1000000 197423 9802577 210000000 0.001509
boundless_quaternary_search 1000000 197423 9802577 261110371 0.001314
inbound_quaternary_search 1000000 197423 9802577 254735155 0.001276
boundless_interpolated_search 1000000 197423 9802577 158754364 0.000707
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