Algorithms: Recurrences

using System;
// Josephus Survivor in C#
//
// PURPOSE
// The Josephus recurrence finds the last survivor when every kth person in a circle is
// eliminated.
//
// METHOD
// Start with survivor index zero for a one-person circle. Grow the circle one person at a
// time using survivor=(survivor+k) modulo size.
//
// COMPLEXITY
// O(n) time, O(1) space.
//
// CONTRACT
// 1 <= n <= 1000000, 1 <= k <= 1000000. Returns a zero-based survivor index.
// 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# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 3.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

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

    static int josephus(int n, int k) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((n >= 1) && (n <= 1000000) && (k >= 1) && (k <= 1000000)));
            int survivor = 0;
            int size = 2;
            while ((size <= n)) {
                survivor = ((survivor + k) % size);
                size = (size + 1);
            }
            return survivor;
        }
    }

    static void Main() {
        int n = 7;
        int k = 3;
        int result = josephus(n, k);
        Console.WriteLine(result);
    }
}

<?php
declare(strict_types=1);
// Josephus Survivor in PHP
//
// PURPOSE
// The Josephus recurrence finds the last survivor when every kth person in a circle is
// eliminated.
//
// METHOD
// Start with survivor index zero for a one-person circle. Grow the circle one person at a
// time using survivor=(survivor+k) modulo size.
//
// COMPLEXITY
// O(n) time, O(1) space.
//
// CONTRACT
// 1 <= n <= 1000000, 1 <= k <= 1000000. Returns a zero-based survivor index.
// 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: 3.
// 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 josephus(int $n, int $k): int {
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((($n >= 1) && ($n <= 1000000) && ($k >= 1) && ($k <= 1000000)));
    $survivor = 0;
    $size = 2;
    while (($size <= $n)) {
        $survivor = (alg_add($survivor, $k) % $size);
        $size = alg_add($size, 1);
    }
    return $survivor;
}

$n = 7;
$k = 3;
$result = josephus($n, $k);
echo $result, PHP_EOL;