Counting Sort - Learning Module

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When Counting Sort Is Appropriate

Choosing the Right Tool

Understanding an algorithm's mechanics is only half the battle. The mark of an experienced engineer is knowing when to apply each tool. Counting sort is a powerful but specialized algorithm—a scalpel rather than a Swiss Army knife.

This page synthesizes everything we've learned about counting sort into actionable decision criteria. By the end, you'll have a clear mental framework for identifying counting sort opportunities and avoiding its pitfalls.

The Selection Mindset

Every sorting problem is a decision point: use the built-in sort, implement a comparison-based algorithm, or recognize a special case where counting/radix sort shines. Counting sort is rarely the default choice—but when appropriate, it's dramatically better than alternatives.

Ideal Use Cases for Counting Sort

Counting sort excels in specific scenarios where its constraints align with the problem's characteristics. These are the situations where you should immediately consider counting sort.

Perfect Fit Scenarios

•Sorting characters (ASCII/Unicode subsets): With k = 128 for ASCII or k = 256 for extended ASCII, counting sort is unbeatable. Text processing, anagram detection, and character frequency analysis all benefit.
•Sorting grades, scores, or ratings: Educational scores (0-100), star ratings (1-5), or percentage values have tiny, fixed ranges. Sorting millions of exam results is O(n) with k ≈ 100.
•Age-based sorting: Human ages span roughly 0-120, a minuscule range. Demographics analysis, actuarial calculations, and age-cohort grouping are ideal applications.
•Sorting pixels by intensity: 8-bit grayscale (0-255) or color channels. Image histogram computation and intensity-based processing leverage counting sort naturally.
•Sorting by enumerated categories: Product types, status codes (HTTP 100-599), priority levels (1-10), days of week (0-6)—any small, fixed enumeration.
•Radix sort subroutine: Counting sort on single digits (0-9) or bytes (0-255) is the building block of radix sort, enabling linear-time sorting of larger integers.
•Histogram and frequency analysis: Even when not 'sorting' per se, counting sort's counting phase builds histograms in O(n) time.

Common Thread

Notice the pattern: all ideal cases have small, known, integer-like ranges. The moment you see 'values are between X and Y' where Y - X is small, counting sort should come to mind.

When NOT to Use Counting Sort

Equally important is knowing when counting sort is a poor choice. Using it inappropriately wastes time, space, or both—and may not even work.

Avoid Counting Sort When

•Values are floating-point: 3.14159 and 2.71828 can't be array indices. You'd need to convert to integers (multiply and round), which adds complexity and loses precision.
•Values are strings: Even short strings have astronomical ranges. A 4-character ASCII string has 128^4 ≈ 270 million possible values. Use radix sort on character positions instead.
•Range is very large relative to count: Sorting 100 elements with values up to 10^9 means k/n = 10,000,000. The count array alone uses ~4GB. Unacceptable.
•Values are 64-bit integers with full range: k = 2^64 is physically impossible to allocate. Use radix sort, which handles 8 bytes as 8 passes of k = 256.
•Values are negative without preprocessing: Counting sort assumes non-negative indices. Negative values require offset normalization first.
•Sorting custom objects by complex keys: If comparison involves parsing, dereferencing, or non-trivial logic, counting sort can't help. You need a comparison function.
•Memory is severely constrained: Counting sort uses O(n + k) auxiliary space. If k is large or memory is tight, in-place sorts like heap sort are preferable.
•Data is already partially sorted: Insertion sort achieves O(n) for nearly-sorted data. Counting sort does the same work regardless of initial order.

The Dangerous Middle Ground

The trickiest cases are where k is 'medium'—say, k = 100,000 for n = 10,000 elements. Here, O(n + k) = O(110,000) vs O(n log n) ≈ O(130,000). The theoretical difference is small, but cache effects may favor the comparison sort. Profile before committing.

The Counting Sort Decision Framework

Use this structured approach to decide whether counting sort is appropriate for a given sorting problem.

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┌─────────────────────────────────────────────────────────────┐
│           COUNTING SORT DECISION FLOWCHART                   │
└─────────────────────────────────────────────────────────────┘
                           │
                           ▼
              ┌────────────────────────┐
              │ Are values integers    │
              │ (or easily convertible │
              │ to integers)?          │
              └────────────┬───────────┘
                    │
         ┌──────No──┤──────Yes──────┐
         ▼                          ▼
  ┌──────────────┐      ┌────────────────────────┐
  │ Use          │      │ Is the range k known   │
  │ comparison   │      │ and bounded?           │
  │ sort         │      └────────────┬───────────┘
  └──────────────┘           │
                   ┌────No───┤───────Yes────┐
                   ▼                        ▼
           ┌──────────────┐    ┌────────────────────────┐
           │ Use          │    │ Is k ≤ n?              │
           │ comparison   │    │ (or at least k = O(n)) │
           │ sort         │    └────────────┬───────────┘
           └──────────────┘         │
                          ┌───No────┤───────Yes─────┐
                          ▼                         ▼
                 ┌────────────────┐     ┌────────────────────┐
                 │ Is k ≤ n log n?│     │ Is space O(n + k)  │
                 └───────┬────────┘     │ acceptable?        │
                   │                    └──────────┬─────────┘
        ┌────No────┤───Yes──┐                 │
        ▼                   ▼          ┌──No──┤──Yes──┐
┌──────────────┐   ┌──────────────┐    ▼             ▼
│ Use          │   │ Either works │ ┌────────┐  ┌───────────┐
│ comparison   │   │ Profile to   │ │ Use    │  │ USE       │
│ sort         │   │ decide       │ │ in-    │  │ COUNTING  │
└──────────────┘   └──────────────┘ │ place  │  │ SORT! ✓   │
                                    │ sort   │  └───────────┘
                                    └────────┘

Quick Checklist Version

•Integer (or integer-like) values? If no → comparison sort.
•Known, bounded range k? If no → comparison sort or radix sort.
•k = O(n)? If yes → counting sort is optimal. If k slightly exceeds n, profile.
•k >> n? If k = O(n²) or worse → comparison sort is better.
•Space O(n + k) acceptable? If no → consider in-place alternatives.
•Stability required? Counting sort provides it; use stable version.

Real-World Applications

Counting sort appears throughout industry in contexts you might not expect. Understanding these applications reinforces when the algorithm is truly valuable.

Industry Applications of Counting Sort
Domain	Application	Why Counting Sort?
Computer Graphics	Histogram equalization	Pixel intensities (0-255) have tiny range; process millions of pixels in O(n)
Databases	Sorting by enumerated columns	Status (Active/Inactive), Priority (Low/Med/High) have fixed, small domains
Networking	Packet prioritization by QoS level	QoS levels are typically 0-7 (3 bits); sort millions of packets for scheduling
Bioinformatics	DNA sequence sorting by nucleotide	Only 4 values (A, C, G, T); k = 4 makes counting sort essentially O(n)
Finance	Transaction sorting by day of month	Days 1-31 give k = 31; batch processing millions of transactions
Gaming	Leaderboard by score tiers	Score brackets (Bronze/Silver/Gold/etc) have small fixed count
Log Analysis	Sorting events by severity level	Syslog levels 0-7 (Emergency to Debug); k = 8
Education	Grade distribution analysis	0-100 percentage scores; build histograms for millions of grades

Hidden Counting Sort Opportunities

Often, counting sort applies not to raw data but to derived values. Sorting transactions by 'month' (k = 12). Sorting users by 'age bracket' (k = 10). Sorting files by 'size category' (k = 5). Transform your data into bounded categories to unlock counting sort.

Counting Sort as a Building Block

Perhaps counting sort's most important role is as a subroutine in more sophisticated algorithms. Its stability and linear-time complexity make it the foundation of radix sort and bucket sort.

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/**
 * Radix Sort: Sort n integers by processing one digit at a time.
 * 
 * Key insight: Use stable counting sort on each digit position.
 * Process from least-significant to most-significant digit.
 * Stability ensures earlier digit orderings are preserved.
 * 
 * For 32-bit integers with base-256 (byte-at-a-time):
 *   - 4 passes of counting sort
 *   - Each pass: O(n + 256) = O(n)
 *   - Total: O(4n) = O(n)
 * 
 * This achieves linear-time sorting of integers that couldn't
 * be sorted directly with counting sort (k = 2^32 is too large).
 */
function radixSort32(arr: number[]): number[] {
    const n = arr.length;
    if (n <= 1) return arr.slice();
    
    let current = arr.slice();
    let output = new Array(n);
    
    // Process each of 4 bytes (32 bits / 8 bits per byte)
    for (let byte = 0; byte < 4; byte++) {
        // Stable counting sort on this byte
        const count = new Array(256).fill(0);
        
        // Extract byte and count
        for (const num of current) {
            const digit = (num >>> (byte * 8)) & 0xFF;
            count[digit]++;
        }
        
        // Cumulative sum
        for (let i = 1; i < 256; i++) {
            count[i] += count[i - 1];
        }
        
        // Place elements (right-to-left for stability)
        for (let i = n - 1; i >= 0; i--) {
            const digit = (current[i] >>> (byte * 8)) & 0xFF;
            count[digit]--;
            output[count[digit]] = current[i];
        }
        
        [current, output] = [output, current];
    }
    
    return current;
}
 
// This sorts 32-bit integers in O(n) time!
// Works for any n, regardless of integer values.

Algorithms Built on Counting Sort

•Radix Sort (LSD): Least-significant-digit-first radix sort uses stable counting sort at each digit position. Essential for sorting large integers.
•Radix Sort (MSD): Most-significant-digit-first variant, useful for strings and variable-length data, also uses counting sort for partitioning.
•Bucket Sort: Distributes elements into buckets, then sorts each bucket. Counting sort handles the bucket assignment when keys are integers.
•Suffix Array Construction: Some linear-time suffix array algorithms use counting sort for radix sorting of suffix ranks.
•BWT (Burrows-Wheeler Transform): The transform step benefits from counting sort on character frequencies in text compression.

The Power of Composition

Radix sort is a brilliant example of algorithm composition: it can't sort 32-bit integers directly (k = 2^32), so it decomposes each integer into 4 bytes (k = 256 each) and applies counting sort 4 times. The result: O(n) sorting for any integer type.

Practical Performance Considerations

Beyond asymptotic complexity, several practical factors influence counting sort's real-world performance. Experienced engineers consider these when making algorithmic choices.

Performance Factors

•Cache locality: Small k means the count array fits in L1/L2 cache. Counting increments are fast. Large k causes cache misses on every count update—a hidden performance killer.
•Memory allocation overhead: Creating the count and output arrays takes time. For very small n (say, n < 100), this overhead may exceed the time saved by linear sorting. Simple insertion sort might be faster.
•Parallelization difficulty: Counting sort's phases have dependencies. Counting can be parallelized with atomic increments, but the placement phase is inherently sequential. Parallel merge sort may scale better.
•Branch prediction: Counting sort has few conditionals, benefiting modern CPUs. Comparison sorts have comparison-dependent branches that can cause misprediction penalties.
•Streaming suitability: Counting sort requires two passes (count, then place). If data arrives streaming and can't be buffered, comparison sorts that work in single passes may be preferable.

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PROFILING DECISION GUIDE
═══════════════════════════════════════════════════════════
 
Always profile when:
─────────────────────
• k is in the "gray zone" (n < k < n log n)
• You're on memory-constrained systems
• Data size is very small (n < 1000)
• The sort is in a hot path executed millions of times
 
Trust theoretical analysis when:
────────────────────────────────
• k is clearly tiny (k < 1000) relative to n
• n is large (> 100,000)
• Memory is plentiful
• This is not a performance-critical path
 
Profiling Example:
──────────────────
n = 100,000 elements, k = 50,000
 
Theoretical:
  Counting sort: O(n + k) = 150,000 ops
  Quicksort: O(n log n) ≈ 1,700,000 ops
  Counting sort should win by ~11x
 
Actual (measured):
  Counting sort: 15ms
  Quicksort: 8ms
  Quicksort wins!
 
Why? Count array (200KB) exceeds L2 cache (256KB).
Every count[value]++ causes a cache miss.
Quicksort's in-place partitioning has better locality.
 
Lesson: O(n + k) < O(n log n) doesn't guarantee faster wall-clock time.

Language and Library Considerations

Different programming languages and their standard libraries have different characteristics that affect counting sort decisions.

Language-Specific Factors
Language	Built-in Sort	Counting Sort Considerations
JavaScript/TS	Timsort (O(n log n))	Arrays are optimized; counting sort must beat a very efficient default. Use typed arrays for count array.
Python	Timsort	Python loops are slow. For large n, use NumPy's vectorized operations for counting sort, or stick with sorted().
Java	Dual-pivot Quicksort (primitives)	Arrays.sort is highly optimized. Counting sort wins only for very favorable k. Consider parallelStream for comparison sorts.
C++	Introsort (std::sort)	Manual memory management allows tight control. Counting sort with stack-allocated arrays for small k is very fast.
Rust	Timsort (stable), Pattern-defeating Quicksort (unstable)	Zero-cost abstractions make counting sort competitive. Iterators compile to efficient loops.
Go	Pdqsort	Generic sorting is well-optimized. Counting sort shines for specific, known small ranges like bytes.

Implementation Tip

In TypeScript/JavaScript, use Int32Array instead of Array for the count array when k is large. It's more memory-efficient and enables faster integer operations. For small k, regular arrays are fine.

Counting Sort in Interview Problems

Counting sort knowledge frequently provides optimal solutions to interview problems that appear more complex at first glance.

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// ═══════════════════════════════════════════════════════════
// PATTERN 1: Sort Characters in a String
// ═══════════════════════════════════════════════════════════
// "Sort the characters of a string alphabetically"
// Naive: O(n log n) with built-in sort
// Optimal: O(n) with counting sort (k = 26 or 128)
 
function sortString(s: string): string {
    const count = new Array(128).fill(0);
    for (const char of s) count[char.charCodeAt(0)]++;
    
    let result = '';
    for (let i = 0; i < 128; i++) {
        result += String.fromCharCode(i).repeat(count[i]);
    }
    return result;
}
 
// ═══════════════════════════════════════════════════════════
// PATTERN 2: Check if Two Strings are Anagrams
// ═══════════════════════════════════════════════════════════
// "Determine if two strings are anagrams"
// Key insight: Anagrams have identical character counts
 
function areAnagrams(s1: string, s2: string): boolean {
    if (s1.length !== s2.length) return false;
    
    const count = new Array(26).fill(0);
    for (let i = 0; i < s1.length; i++) {
        count[s1.charCodeAt(i) - 97]++;  // 'a' = 97
        count[s2.charCodeAt(i) - 97]--;
    }
    return count.every(c => c === 0);
}
 
// ═══════════════════════════════════════════════════════════
// PATTERN 3: Find k-th Smallest Element (Small Range)
// ═══════════════════════════════════════════════════════════
// "Find the k-th smallest element in an array of ages (0-120)"
// Naive: O(n log n) sort, then index
// Optimal: O(n) counting, then linear scan
 
function kthSmallestAge(ages: number[], k: number): number {
    const count = new Array(121).fill(0);
    for (const age of ages) count[age]++;
    
    let remaining = k;
    for (let age = 0; age <= 120; age++) {
        remaining -= count[age];
        if (remaining <= 0) return age;
    }
    return -1;  // k exceeds array length
}
 
// ═══════════════════════════════════════════════════════════
// PATTERN 4: Sort Array with Limited Distinct Values
// ═══════════════════════════════════════════════════════════
// "Sort array containing only 0s, 1s, and 2s" (Dutch Flag variation)
// While Dutch Flag is O(n) in-place, counting sort is simpler
 
function sortColors(nums: number[]): void {
    const count = [0, 0, 0];
    for (const n of nums) count[n]++;
    
    let idx = 0;
    for (let val = 0; val < 3; val++) {
        for (let i = 0; i < count[val]; i++) {
            nums[idx++] = val;
        }
    }
}

Interview Signal

Recognizing when counting sort applies—and stating 'this can be done in O(n) using counting sort because the range is bounded'—demonstrates pattern recognition that interviewers value highly. It shows you're not just memorizing solutions but understanding underlying principles.

Comparison with Related Algorithms

Counting sort belongs to a family of non-comparison sorts. Understanding the relationships helps choose the right variant for each situation.

Non-Comparison Sorting Algorithms
Algorithm	Time	Space	Best For
Counting Sort	O(n + k)	O(n + k)	Small integer ranges, k = O(n)
Radix Sort (LSD)	O(d(n + b))	O(n + b)	Fixed-width integers, d digits, base b
Radix Sort (MSD)	O(d(n + b))	O(n + b)	Strings, variable-length keys
Bucket Sort	O(n + k + n²/k)	O(n + k)	Uniformly distributed floats

Decision Guidelines

•Use Counting Sort when values are directly in a small, bounded integer range and k = O(n).
•Use Radix Sort when integers are large (32-bit, 64-bit) but have fixed width. It decomposes large ranges into multiple small-range counting sort passes.
•Use Bucket Sort for floating-point numbers uniformly distributed over a known range. It partitions into buckets, then uses insertion sort within each.
•Fall back to Comparison Sort when values are arbitrary objects, ranges are unknown/unbounded, or simplicity outweighs the linear-time benefit.

Summary: Mastering Algorithm Selection

Knowing when to use counting sort—and when to avoid it—is the capstone of mastering this algorithm. Selection skill comes from understanding constraints deeply and practicing recognition.

Key Takeaways

•Perfect for small integer ranges: Characters, ages, scores, categories—anything where k is constant or O(n).
•Avoid for large or unbounded ranges: Floats, strings, 64-bit integers, arbitrary objects—these need other approaches.
•Use the decision framework: Integer? Bounded? k = O(n)? Space acceptable? If all yes, counting sort wins.
•Foundation for radix sort: When counting sort directly fails (large k), radix sort often works by decomposing into manageable digits.
•Consider practical factors: Cache effects, memory overhead, and parallelization can override theoretical advantages.
•Profile in the gray zone: When k is neither clearly small nor clearly large, measure real performance.

Module Complete

Congratulations! You've completed the Counting Sort module. You now understand non-comparison sorting's foundational algorithm—from the limited range insight through counting mechanics, complexity analysis, and practical application. This knowledge prepares you for radix sort and gives you a powerful tool for specialized sorting scenarios. You've broken the Ω(n log n) barrier and can now recognize when linear-time sorting is achievable!