Matrix Chain Multiplication in PHP

<?php
declare(strict_types=1);
// Matrix Chain Multiplication in PHP
//
// 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
// PHP 8.1+ on a 64-bit runtime: packed arrays, strict scalar types, read-only inputs by
// value using copy-on-write, references only for outputs, and checked 32-bit arithmetic.
// Invalid input and overflow throw exceptions. PHP arrays have higher overhead than native
// packed buffers in C/Java.
//
// EXAMPLE
// Expected return: 4500.
// Run: php example.php
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(bool $valid): void { if (!$valid) throw new InvalidArgumentException("invalid algorithm input"); }
function alg_checked(int|float $v): int { if ($v < -2147483648 || $v > 2147483647 || floor($v) != $v) throw new OverflowException("32-bit result overflow"); return (int)$v; }
function alg_add(int $a,int $b): int { return alg_checked($a+$b); }
function alg_sub(int $a,int $b): int { return alg_checked($a-$b); }
function alg_mul(int $a,int $b): int { return alg_checked($a*$b); }
function alg_size(int $n): int { alg_require($n>=0 && $n<=4000000); return $n; }

function matrix_chain(array $dimensions): int {
    alg_require(array_is_list($dimensions));
    $n = alg_sub(count($dimensions), 1);
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((count($dimensions) == alg_add($n, 1)));
    alg_require(($n >= 1));
    for ($alg_i = 0; $alg_i < count($dimensions); $alg_i++) {
        $alg_value = $dimensions[$alg_i];
        alg_require((($alg_value > 0) && ($alg_value <= 1000)));
    }
    $dp = array_fill(0, alg_size(alg_mul($n, $n)), 0);
    $length = 2;
    $left = 0;
    $right = 0;
    $split = 0;
    $candidate = 0;
    while (($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), $right)] = min($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)];
}

$dimensions = [10, 30, 5, 60];
$result = matrix_chain($dimensions);
echo $result, PHP_EOL;