computer science

Multiplicative binary search

Multiplicative binary search is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Multiplicative binary search rather than just read about it. In short: 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. Multiplicative binary search was first described by Thomas Standish in 1980.

Key takeaways

Multiplicative binary search belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Multiplicative binary search to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Multiplicative binary search from memory before moving on to harder problems.

Reference excerpt

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. 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. On modern hardware, the cache-friendly nature of multiplicative binary search makes it suitable for out-of-core search on block-oriented storage as an alternative to B-trees and B+ trees. For optimal performance, the branching factor of a B-tree or B+-tree must match the block size of the file system that it is stored on. The permutation used by multiplicative binary search places the optimal number of keys in the first (root) block, regardless of block size. Multiplicative binary search is used by some optimizing compilers to implement switch statements.

Algorithm Multiplicative binary search operates on a permuted sorted array. Keys are stored in the array in a 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 a multiplicative binary search to find the index of T in A.

Set i to 0 if i ≥ n, the search terminates unsuccessfully. if Ai = T, the search is done; return i. if Ai > T, set i to 2×i + 1 and go to step 2. if Ai < T, set i to 2×i + 2 and go to step 2.

See also Binary search tree – Rooted binary tree data structure Methods for storing binary trees – Limited form of tree data structure Ahnentafel – Genealogical numbering system for listing a person's direct ancestors

Citations

Illustrations

Worked examples

Example 1 — a first encounter with Multiplicative binary search

Start with the simplest possible case. Write down what Multiplicative binary search claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Multiplicative binary search before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Multiplicative binary search ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Multiplicative binary search

In research: Multiplicative binary search appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Multiplicative binary search in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Multiplicative binary search is common in secondary-school and first-year university syllabi. It links to neighbouring topics Search algorithms, so understanding it makes those chapters shorter.
In everyday life: Look for Multiplicative binary search outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Multiplicative binary search in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Multiplicative binary search means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Multiplicative binary search out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Multiplicative binary search in simple terms?

Why does Multiplicative binary search matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Multiplicative binary search?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Multiplicative binary search.