Sieve of Eratosthenes in Java

// Sieve of Eratosthenes in Java
//
// PURPOSE
// The sieve identifies all primes up to a limit by marking multiples of smaller primes. This
// example returns their count.
//
// METHOD
// Keep a composite marker array. For each unmarked p up to the square root, mark multiples
// starting at p squared; smaller multiples were already handled. Count the remaining
// unmarked values from two onward.
//
// COMPLEXITY
// O(n log log n) time and O(n) auxiliary space.
//
// CONTRACT
// n >= 0. Returns the number of primes <= n. The marker array requires memory proportional
// to n.
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 4.
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

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

    static int sieve_count(int n) {
        alg_require(((n >= 0) && (n <= 1000000)));
        int[] composite = new int[alg_size(Math.addExact(n, 1))];
        int p = 2;
        int multiple = 0;
        int count = 0;
        while ((p <= (n / p))) {
            if ((composite[p] == 0)) {
                multiple = Math.multiplyExact(p, p);
                while ((multiple <= n)) {
                    composite[multiple] = 1;
                    multiple = Math.addExact(multiple, p);
                }
            }
            p = Math.addExact(p, 1);
        }
        p = 2;
        while ((p <= n)) {
            if ((composite[p] == 0)) {
                count = Math.addExact(count, 1);
            }
            p = Math.addExact(p, 1);
        }
        return count;
    }

    public static void main(String[] args) {
        int n = 10;
        int result = sieve_count(n);
        System.out.println(result);
    }
}