Segmented Sieve in Java

// Segmented Sieve in Java
//
// PURPOSE
// A segmented sieve marks composites only inside a requested interval, reducing memory
// compared with sieving all the way to its upper endpoint.
//
// METHOD
// Sieve base primes up to square root of high. For each prime, mark its multiples in
// [low,high], starting no earlier than p squared. Count unmarked values at least two.
//
// COMPLEXITY
// O(sqrt(high) log log high + (high-low+1) log log high) time, O(sqrt(high)+high-low+1)
// space.
//
// CONTRACT
// 0 <= low <= high <= 1000000. Returns number of primes in the inclusive interval.
// 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: 6.
// 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 segmented_sieve(int low, int high) {
        alg_require(((low >= 0) && (low <= high) && (high <= 1000000)));
        int limit = 1;
        while ((limit <= (high / limit))) {
            limit = Math.addExact(limit, 1);
        }
        int[] base = new int[alg_size(Math.addExact(limit, 1))];
        int[] marked = new int[alg_size(Math.addExact(Math.subtractExact(high, low), 1))];
        int p = 2;
        int multiple = 0;
        int count = 0;
        int i = 0;
        while ((p < limit)) {
            if ((base[p] == 0)) {
                multiple = Math.multiplyExact(p, p);
                while ((multiple < limit)) {
                    base[multiple] = 1;
                    multiple = Math.addExact(multiple, p);
                }
                multiple = Math.max(Math.multiplyExact(p, p), Math.multiplyExact((Math.subtractExact(Math.addExact(low, p), 1) / p), p));
                while ((multiple <= high)) {
                    marked[Math.subtractExact(multiple, low)] = 1;
                    multiple = Math.addExact(multiple, p);
                }
            }
            p = Math.addExact(p, 1);
        }
        i = Math.max(2, low);
        while ((i <= high)) {
            if ((marked[Math.subtractExact(i, low)] == 0)) {
                count = Math.addExact(count, 1);
            }
            i = Math.addExact(i, 1);
        }
        return count;
    }

    public static void main(String[] args) {
        int low = 10;
        int high = 30;
        int result = segmented_sieve(low, high);
        System.out.println(result);
    }
}