Sorting number

Key takeaways

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

Reference excerpt

In mathematics and computer science, the sorting numbers are a sequence of numbers introduced in 1950 by Hugo Steinhaus for the analysis of comparison sort algorithms. These numbers give the worst-case number of comparisons used by both binary insertion sort and merge sort. However, there are other algorithms that use fewer comparisons.

Formula and examples The n {\displaystyle n} th sorting number is given by the formula

The sequence of numbers given by this formula (starting with n = 1 {\displaystyle n=1} ) is

The same sequence of numbers can also be obtained from the recurrence relation,

A ( n ) = A ( ⌊ n / 2 ⌋ ) + A ( ⌈ n / 2 ⌉ ) + n − 1 {\displaystyle A(n)=A{\bigl (}\lfloor n/2\rfloor {\bigr )}+A{\bigl (}\lceil n/2\rceil {\bigr )}+n-1} . It is an example of a 2-regular sequence. Asymptotically, the value of the n {\displaystyle n} th sorting number fluctuates between approximately n log 2 ⁡ n − n {\displaystyle n\log _{2}n-n} and n log 2 ⁡ n − 0.915 n , {\displaystyle n\log _{2}n-0.915n,} depending on the ratio between n {\displaystyle n} and the nearest power of two.

Application to sorting In 1950, Hugo Steinhaus observed that these numbers count the number of comparisons used by binary insertion sort, and conjectured (incorrectly) that they give the minimum number of comparisons needed to sort n {\displaystyle n} items using any comparison sort. The conjecture was disproved in 1959 by L. R. Ford Jr. and Selmer M. Johnson, who found a different sorting algorithm, the Ford–Johnson merge-insertion sort, using fewer comparisons. The same sequence of sorting numbers also gives the worst-case number of comparisons used by merge sort to sort n {\displaystyle n} items.

Other applications The sorting numbers (shifted by one position) also give the sizes of the shortest possible superpatterns for the layered permutations.

References

Worked examples

Example 1 — a first encounter with Sorting number

Start with the simplest possible case. Write down what Sorting number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Sorting number 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 Sorting number 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 Sorting number

In research: Sorting number appears in mathematics 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 Sorting number 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: Sorting number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Comparison sorts, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Sorting number 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.

How to study Sorting number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sorting number 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 Sorting number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sorting number in simple terms?

In mathematics and computer science, the sorting numbers are a sequence of numbers introduced in 1950 by Hugo Steinhaus for the analysis of comparison sort algorithms. These numbers give the worst-case number of comparisons used by both binary insertion sort and merge sort.

Why does Sorting number matter?

Because it connects several mathematics 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 Sorting number?

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 Sorting number.