Algorithms: Divide and Conquer

Algorithms: Divide and Conquer

Explore algorithms: divide and conquer code snippets and tutorials

Algorithms: Divide and Conquer

Merge Based Inversion Count in Java

An inversion is a pair i<j with a[i]>a[j].

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// Merge Based Inversion Count in Java
//
// PURPOSE
// An inversion is a pair i<j with a[i]>a[j]. Counting them during merging avoids testing
// every pair.
//
// METHOD
// Merge sorted runs bottom up. Whenever a right-run value precedes a larger left-run value,
// add the number of remaining left-run values.
//
// COMPLEXITY
// O(n log n) time, O(n) auxiliary space.
//
// CONTRACT
// Mutates input into sorted order. n <= 65536 so the inversion count fits signed 32-bit.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 2.
// Expected a after call: [1, 2, 3].
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class AlgorithmExample {
    static void alg_require(boolean valid) { if (!valid) throw new IllegalArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int inversion_count(int[] a) {
        java.util.Objects.requireNonNull(a, "a");
        int n = a.length;
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((a.length == n));
        alg_require((n <= 65536));
        int[] buffer = new int[alg_size(Math.addExact(n, 1))];
        int width = 1;
        int left = 0;
        int mid = 0;
        int right = 0;
        int i = 0;
        int j = 0;
        int out = 0;
        int count = 0;
        while ((width < n)) {
            left = 0;
            while ((left < n)) {
                mid = Math.min(Math.addExact(left, width), n);
                right = Math.min(Math.addExact(left, Math.multiplyExact(2, width)), n);
                i = left;
                j = mid;
                out = left;
                while ((out < right)) {
                    if (((i < mid) && ((j >= right) || (a[i] <= a[j])))) {
                        buffer[out] = a[i];
                        i = Math.addExact(i, 1);
                    } else {
                        buffer[out] = a[j];
                        j = Math.addExact(j, 1);
                        count = Math.addExact(count, Math.subtractExact(mid, i));
                    }
                    out = Math.addExact(out, 1);
                }
                left = Math.addExact(left, Math.multiplyExact(2, width));
            }
            i = 0;
            while ((i < n)) {
                a[i] = buffer[i];
                i = Math.addExact(i, 1);
            }
            width = Math.multiplyExact(width, 2);
        }
        return count;
    }

    public static void main(String[] args) {
        int[] a = new int[] {3, 1, 2};
        int result = inversion_count(a);
        System.out.println(result);
        // Expected a: [1, 2, 3]
        System.out.println(java.util.Arrays.toString(a));
    }
}
Nov 15, 2026 Java
Algorithms: Divide and Conquer

Merge Based Inversion Count in JavaScript

An inversion is a pair i<j with a[i]>a[j].

javascript 
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"use strict";
// Merge Based Inversion Count in JavaScript
//
// PURPOSE
// An inversion is a pair i<j with a[i]>a[j]. Counting them during merging avoids testing
// every pair.
//
// METHOD
// Merge sorted runs bottom up. Whenever a right-run value precedes a larger left-run value,
// add the number of remaining left-run values.
//
// COMPLEXITY
// O(n log n) time, O(n) auxiliary space.
//
// CONTRACT
// Mutates input into sorted order. n <= 65536 so the inversion count fits signed 32-bit.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// Node.js 18+: Int32Array inputs and workspaces provide compact integer storage. Automatic
// lengths and checked arithmetic reject values outside signed 32-bit. Invalid inputs or
// overflow throw RangeError.
//
// EXAMPLE
// Expected return: 2.
// Expected a after call: [1, 2, 3].
// Run: node example.js
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(valid) { if (!valid) throw new RangeError("invalid algorithm input"); }
function alg_checked(v) { if (!Number.isInteger(v) || v < -2147483648 || v > 2147483647) throw new RangeError("32-bit result overflow"); return v; }
function alg_add(a,b) { return alg_checked(a+b); }
function alg_sub(a,b) { return alg_checked(a-b); }
function alg_mul(a,b) { return alg_checked(a*b); }
function alg_size(n) { alg_require(Number.isInteger(n) && n>=0 && n<=4000000); return n; }

function inversion_count(a) {
    alg_require(a instanceof Int32Array);
    let n = a.length;
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((a.length == n));
    alg_require((n <= 65536));
    let buffer = new Int32Array(alg_size(alg_add(n, 1)));
    let width = 1;
    let left = 0;
    let mid = 0;
    let right = 0;
    let i = 0;
    let j = 0;
    let out = 0;
    let count = 0;
    while ((width < n)) {
        left = 0;
        while ((left < n)) {
            mid = Math.min(alg_add(left, width), n);
            right = Math.min(alg_add(left, alg_mul(2, width)), n);
            i = left;
            j = mid;
            out = left;
            while ((out < right)) {
                if (((i < mid) && ((j >= right) || (a[i] <= a[j])))) {
                    buffer[out] = a[i];
                    i = alg_add(i, 1);
                } else {
                    buffer[out] = a[j];
                    j = alg_add(j, 1);
                    count = alg_add(count, alg_sub(mid, i));
                }
                out = alg_add(out, 1);
            }
            left = alg_add(left, alg_mul(2, width));
        }
        i = 0;
        while ((i < n)) {
            a[i] = buffer[i];
            i = alg_add(i, 1);
        }
        width = alg_mul(width, 2);
    }
    return count;
}

let a = new Int32Array([3, 1, 2]);
let result = inversion_count(a);
console.log(result);
// Expected a: [1, 2, 3]
console.log(Array.from(a));
Nov 15, 2026 javascript
Algorithms: Divide and Conquer

Merge Based Inversion Count in Python

An inversion is a pair i<j with a[i]>a[j].

python 
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from bisect import bisect_left

# Merge Based Inversion Count in Python
#
# PURPOSE
# An inversion is a pair i<j with a[i]>a[j]. Counting them during merging avoids testing
# every pair.
#
# METHOD
# Merge sorted runs bottom up. Whenever a right-run value precedes a larger left-run value,
# add the number of remaining left-run values.
#
# COMPLEXITY
# O(n log n) time, O(n) auxiliary space.
#
# CONTRACT
# Mutates input into sorted order. n <= 65536 so the inversion count fits signed 32-bit.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 2.
# Expected a after call: [1, 2, 3].
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def inversion_count(a: list[int]) -> int:
    alg_require(isinstance(a, list) and len(a) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    alg_require(n <= 65536)
    buffer: list[int] = [0] * alg_size(n + 1)
    width: int = 1
    left: int = 0
    mid: int = 0
    right: int = 0
    i: int = 0
    j: int = 0
    out: int = 0
    count: int = 0
    while width < n:
        left = 0
        while left < n:
            mid = min(left + width, n)
            right = min(left + 2 * width, n)
            i = left
            j = mid
            out = left
            while out < right:
                if i < mid and (j >= right or a[i] <= a[j]):
                    buffer[out] = a[i]
                    i += 1
                else:
                    buffer[out] = a[j]
                    j += 1
                    count += mid - i
                out += 1
            left += 2 * width
        i = 0
        while i < n:
            a[i] = buffer[i]
            i += 1
        width *= 2
    return count

a = [3, 1, 2]
result = inversion_count(a)
print(result)
# Expected a: [1, 2, 3]
print(a)
Nov 15, 2026 python
Algorithms: Divide and Conquer

Merge Based Inversion Count in C

An inversion is a pair i<j with a[i]>a[j].

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#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Merge Based Inversion Count in C
//
// PURPOSE
// An inversion is a pair i<j with a[i]>a[j]. Counting them during merging avoids testing
// every pair.
//
// METHOD
// Merge sorted runs bottom up. Whenever a right-run value precedes a larger left-run value,
// add the number of remaining left-run values.
//
// COMPLEXITY
// O(n log n) time, O(n) auxiliary space.
//
// CONTRACT
// Mutates input into sorted order. n <= 65536 so the inversion count fits signed 32-bit.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 2.
// Expected a after call: [1, 2, 3].
// Run: gcc -std=c99 -O2 example.c -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int inversion_count(int *a, size_t a_length) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *buffer = NULL;
    if (a_length > 1000000 || (a_length && !a)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int n = (int)a_length;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 0) && (n <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)a_length == n))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!((n <= 65536))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    {
        int alg_length = alg_add(n, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        buffer = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*buffer));
        if (!buffer) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int width = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int left = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int mid = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int right = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int i = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int j = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int out = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int count = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (width < n);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        left = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (left < n);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            mid = alg_min(alg_add(left, width, &alg_error), n);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            right = alg_min(alg_add(left, alg_mul(2, width, &alg_error), &alg_error), n);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            i = left;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            j = mid;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            out = left;
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            while (1) {
                int alg_condition = (out < right);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (!alg_condition) break;
                int alg_condition_1 = ((i < mid) && ((j >= right) || (a[i] <= a[j])));
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                if (alg_condition_1) {
                    buffer[out] = a[i];
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    i = alg_add(i, 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                } else {
                    buffer[out] = a[j];
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    j = alg_add(j, 1, &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                    count = alg_add(count, alg_sub(mid, i, &alg_error), &alg_error);
                    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
                }
                out = alg_add(out, 1, &alg_error);
                if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            }
            left = alg_add(left, alg_mul(2, width, &alg_error), &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        i = 0;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (i < n);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            a[i] = buffer[i];
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            i = alg_add(i, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        width = alg_mul(width, 2, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = count;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(buffer);
    return alg_result;
}

int main(void) {
    int a[] = {3, 1, 2};
    int result = inversion_count(a, 3);
    printf("%d\n", result);
    // Expected a: [1, 2, 3]
    for (int alg_print = 0; alg_print < 3; alg_print++) printf("%d ", a[alg_print]);
    printf("\n");
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}
Nov 15, 2026 C
Algorithms: Divide and Conquer

Merge Based Inversion Count in C++

An inversion is a pair i<j with a[i]>a[j].

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#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Merge Based Inversion Count in C++
//
// PURPOSE
// An inversion is a pair i<j with a[i]>a[j]. Counting them during merging avoids testing
// every pair.
//
// METHOD
// Merge sorted runs bottom up. Whenever a right-run value precedes a larger left-run value,
// add the number of remaining left-run values.
//
// COMPLEXITY
// O(n log n) time, O(n) auxiliary space.
//
// CONTRACT
// Mutates input into sorted order. n <= 65536 so the inversion count fits signed 32-bit.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 2.
// Expected a after call: [1, 2, 3].
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int inversion_count(std::vector<int>& a) {
    alg_require(a.size() <= 1000000);
    int n = (int)a.size();
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)a.size() == n));
    alg_require((n <= 65536));
    std::vector<int> buffer(alg_size(alg_add(n, 1)), 0);
    int width = 1;
    int left = 0;
    int mid = 0;
    int right = 0;
    int i = 0;
    int j = 0;
    int out = 0;
    int count = 0;
    while ((width < n)) {
        left = 0;
        while ((left < n)) {
            mid = alg_min(alg_add(left, width), n);
            right = alg_min(alg_add(left, alg_mul(2, width)), n);
            i = left;
            j = mid;
            out = left;
            while ((out < right)) {
                if (((i < mid) && ((j >= right) || (a[i] <= a[j])))) {
                    buffer[out] = a[i];
                    i = alg_add(i, 1);
                } else {
                    buffer[out] = a[j];
                    j = alg_add(j, 1);
                    count = alg_add(count, alg_sub(mid, i));
                }
                out = alg_add(out, 1);
            }
            left = alg_add(left, alg_mul(2, width));
        }
        i = 0;
        while ((i < n)) {
            a[i] = buffer[i];
            i = alg_add(i, 1);
        }
        width = alg_mul(width, 2);
    }
    return count;
}

int main(void) {
    std::vector<int> a = {3, 1, 2};
    int result = inversion_count(a);
    printf("%d\n", result);
    // Expected a: [1, 2, 3]
    for (int alg_print = 0; alg_print < 3; alg_print++) printf("%d ", a[alg_print]);
    printf("\n");
    return 0;
}
Nov 15, 2026 C++