Counting Sort tutorial

Overview

What is Counting Sort?

Counting Sort is a distribution sort technique. Counts each integer value, then emits values in numeric order. Its central idea is summarized by Time: Θ(n + k) · Auxiliary space: Θ(k).

This guide connects theory to practice. You will trace the input, intermediate process, and output; decode the notation; run a dependency-light implementation; then repeat the workflow with Python standard library and evaluate the result.

Preparation

Prerequisites and tools

You do not need an advanced software stack. Start with a recent Python environment and the fundamentals below, then install only the packages used by the practical example.

• Comfort with Python lists, loops, functions, comparisons, and index manipulation.
• An understanding of input size n, time complexity, auxiliary space, and algorithm stability.
• The educational function from this page when running the benchmark tutorial.

Process

How Counting Sort works

1. 1

   Input

   An unordered array of comparable values.

2. 2

   Prepare and configure

   Check shapes, value ranges, ordering assumptions, missing values, and the parameters that control the algorithm’s behavior.

3. 3

   Algorithm process

   Build a frequency table and reconstruct the array from those counts.

4. 4

   Output

   The same values arranged from smallest to largest.

5. 5

   Validate

   Measure the key range k as well as the number of values n.

Core concept

Formula and intuition

The notation captures the main operation or complexity statement behind Counting Sort. Read it together with the symbol key above and the step-by-step process in this guide.

Applications

When to use Counting Sort

Dense integer keys with a known, reasonably small range.

Implementation

Counting Sort from scratch in Python

This dependency-light example emphasizes the algorithm’s mechanics so each important step remains visible.

def counting_sort(values):
    if not values:
        return []
    low, high = min(values), max(values)
    counts = [0] * (high - low + 1)
    for value in values:
        counts[value - low] += 1
    return [value for value, count in enumerate(counts, low)
            for _ in range(count)]

Practical tutorial

Build Counting Sort with Python standard library

Use the educational implementation above to study this algorithm, then compare it with Python’s production-grade stable sorted() baseline.

1. Prepare the data.An unordered array of comparable values. Validate its shape, type, range, and ordering before training or execution.
2. Configure the algorithm.Begin with explicit, conservative parameters and a fixed random seed whenever the library supports one.
3. Fit or execute.Build a frequency table and reconstruct the array from those counts.
4. Inspect the result.The same values arranged from smallest to largest. Then apply the evaluation checks in the next section.

from random import Random
from timeit import timeit

rng = Random(7)
values = [rng.randrange(10_000) for _ in range(250)]
expected = sorted(values)
result = counting_sort(values)
assert result == expected
assert values != expected  # the educational function leaves its input unchanged

elapsed = timeit(lambda: counting_sort(values), number=100)
baseline = timeit(lambda: sorted(values), number=100)
print("educational implementation:", elapsed)
print("Python sorted baseline:", baseline)

API details and version-specific options: official Python standard library reference →

Evaluation

How to evaluate the result

A successful run is not enough. Evaluate the output against the intended use, compare it with a simple baseline, and preserve a genuinely unseen test case whenever the task involves learned parameters.

• Measure the key range k as well as the number of values n.
• Validate empty, one-item, duplicate-heavy, ordered, reversed, and random inputs against sorted().

Common mistakes

Pitfalls and how to avoid them

These failure modes are common in tutorials and production systems. Treat them as review questions, not just after-the-fact debugging advice.

• A sparse or enormous key range wastes memory.
• Microbenchmarks require repeated runs and representative input distributions.
• Prefer sorted() or list.sort() in production unless this specific algorithm is a deliberate requirement.

Project checklist

Before using Counting Sort in a project

• ✓State the supported value types, ordering rule, stability requirement, and mutation behavior.
• ✓Test empty, singleton, duplicate-heavy, sorted, reversed, and randomized inputs.
• ✓Measure comparisons, writes, auxiliary memory, and elapsed time on relevant distributions.
• ✓Validate every result against Python’s trusted sorted() baseline.
• ✓Prefer the standard library in production unless a specialized algorithm is required.

Further reading

Official documentation and next steps

This guide is an educational introduction, not a substitute for domain validation. For consequential applications, review the source documentation, test against representative data, and involve a subject-matter expert.

Continue learning

Related sorting algorithms guides

• Bubble SortRepeatedly swaps adjacent values that are in the wrong order.
• Selection SortSelects the smallest remaining value and places it into its final position.
• Insertion SortBuilds a sorted prefix by inserting each new value into its correct place.