Merge Based Inversion Count in C++

#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;
}