Breadth First Search Shortest Path in PHP

<?php
declare(strict_types=1);
// Breadth First Search Shortest Path in PHP
//
// PURPOSE
// BFS explores an unweighted graph one distance layer at a time. The first discovery of a
// vertex gives its shortest number of edges from the source.
//
// METHOD
// Put the source into a FIFO queue and mark its distance zero. Remove vertices in order and
// enqueue each undiscovered neighbor with distance one greater. Mark on enqueue so each
// vertex enters at most once.
//
// COMPLEXITY
// O(n squared) time with this adjacency matrix and O(n) auxiliary space. An adjacency-list
// implementation is O(V+E).
//
// CONTRACT
// graph is a flattened n by n 0/1 adjacency matrix, n >= 1, source and target are valid
// vertices. Directed edges are supported. Returns distance or -1.
// 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: 2.
// 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 bfs_distance(array $graph, int $n, int $source, int $target): int {
    alg_require(array_is_list($graph));
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((count($graph) == alg_mul($n, $n)));
    alg_require((($n >= 1) && ($source >= 0) && ($source < $n)));
    alg_require((($target >= 0) && ($target < $n)));
    $distance = array_fill(0, alg_size($n), 0);
    $queue = array_fill(0, alg_size($n), 0);
    $v = 0;
    $u = 0;
    $head = 0;
    $tail = 1;
    while (($v < $n)) {
        $distance[$v] = alg_sub(0, 1);
        $v = alg_add($v, 1);
    }
    $distance[$source] = 0;
    $queue[0] = $source;
    while (($head < $tail)) {
        $u = $queue[$head];
        $head = alg_add($head, 1);
        $v = 0;
        while (($v < $n)) {
            if ((($graph[alg_add(alg_mul($u, $n), $v)] != 0) && ($distance[$v] == alg_sub(0, 1)))) {
                $distance[$v] = alg_add($distance[$u], 1);
                $queue[$tail] = $v;
                $tail = alg_add($tail, 1);
            }
            $v = alg_add($v, 1);
        }
    }
    return $distance[$target];
}

$graph = [0, 1, 0, 0, 0, 1, 0, 0, 0];
$n = 3;
$source = 0;
$target = 2;
$result = bfs_distance($graph, $n, $source, $target);
echo $result, PHP_EOL;