Segmented Sieve in Python

from bisect import bisect_left

# Segmented Sieve in Python
#
# 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
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 6.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def segmented_sieve(low: int, high: int) -> int:
    alg_require(type(low) is int and -2147483648 <= low <= 2147483647)
    alg_require(type(high) is int and -2147483648 <= high <= 2147483647)
    alg_require(low >= 0 and low <= high and (high <= 1000000))
    limit: int = 1
    while limit <= high // limit:
        limit += 1
    base: list[int] = [0] * alg_size(limit + 1)
    marked: list[int] = [0] * alg_size(high - low + 1)
    p: int = 2
    multiple: int = 0
    count: int = 0
    i: int = 0
    while p < limit:
        if base[p] == 0:
            multiple = p * p
            while multiple < limit:
                base[multiple] = 1
                multiple += p
            multiple = max(p * p, (low + p - 1) // p * p)
            while multiple <= high:
                marked[multiple - low] = 1
                multiple += p
        p += 1
    i = max(2, low)
    while i <= high:
        if marked[i - low] == 0:
            count += 1
        i += 1
    return count

low = 10
high = 30
result = segmented_sieve(low, high)
print(result)