Matrix Chain Multiplication in C++

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Matrix Chain Multiplication in C++
//
// PURPOSE
// Matrix-chain DP chooses the parenthesization requiring the fewest scalar multiplications
// without changing matrix order.
//
// METHOD
// For each interval length, try every split and combine both subchain costs with the
// multiplication cost of their results.
//
// COMPLEXITY
// O(n cubed) time and O(n squared) space for n matrices.
//
// CONTRACT
// dimensions has n+1 positive entries; matrix i has dimensions[i] by dimensions[i+1]. n>=1.
// All candidate costs must be < 1000000000.
// 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: 4500.
// 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 matrix_chain(const std::vector<int>& dimensions) {
    alg_require(dimensions.size() <= 1000000);
    int n = alg_sub((int)dimensions.size(), 1);
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require(((int)dimensions.size() == alg_add(n, 1)));
    alg_require((n >= 1));
    for (int alg_i = 0; alg_i < (int)dimensions.size(); alg_i++) {
        int alg_value = dimensions[alg_i];
        alg_require(((alg_value > 0) && (alg_value <= 1000)));
    }
    std::vector<int> dp(alg_size(alg_mul(n, n)), 0);
    int length = 2;
    int left = 0;
    int right = 0;
    int split = 0;
    int candidate = 0;
    while ((length <= n)) {
        left = 0;
        while ((alg_add(left, length) <= n)) {
            right = alg_sub(alg_add(left, length), 1);
            dp[alg_add(alg_mul(left, n), right)] = 1000000000;
            split = left;
            while ((split < right)) {
                candidate = alg_add(alg_add(dp[alg_add(alg_mul(left, n), split)], dp[alg_add(alg_mul(alg_add(split, 1), n), right)]), alg_mul(alg_mul(dimensions[left], dimensions[alg_add(split, 1)]), dimensions[alg_add(right, 1)]));
                dp[alg_add(alg_mul(left, n), right)] = alg_min(dp[alg_add(alg_mul(left, n), right)], candidate);
                split = alg_add(split, 1);
            }
            left = alg_add(left, 1);
        }
        length = alg_add(length, 1);
    }
    return dp[alg_sub(n, 1)];
}

int main(void) {
    std::vector<int> dimensions = {10, 30, 5, 60};
    int result = matrix_chain(dimensions);
    printf("%d\n", result);
    return 0;
}