Jump Search in Python

from bisect import bisect_left

# Jump Search in Python
#
# PURPOSE
# Jump search skips fixed-size blocks in a sorted sequence, then scans the candidate block.
#
# METHOD
# Choose a block width near square root of n. Jump until the block endpoint is at least the
# target, then scan the block.
#
# COMPLEXITY
# O(sqrt(n)) comparisons and O(1) auxiliary space. Binary search is usually faster on
# random-access arrays.
#
# CONTRACT
# Ascending input; returns first matching index 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
# 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: 1.
# 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 jump_search(a: list[int], target: int) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    alg_require(type(target) is int and -2147483648 <= target <= 2147483647)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    step: int = 1
    while step <= n // step:
        step += 1
    start: int = 0
    end: int = min(step, n)
    while start < n and a[end - 1] < target:
        start = end
        end = min(end + step, n)
    while start < end:
        if a[start] == target:
            return start
        start += 1
    return -1

a = [1, 3, 3, 7]
target = 3
result = jump_search(a, target)
print(result)