Shell Sort in Python

from bisect import bisect_left

# Shell Sort in Python
#
# PURPOSE
# Shell sort applies insertion sort to elements separated by a gap. Early passes move values
# over long distances before a final gap-one pass.
#
# METHOD
# Start at n/2 and halve the gap. For each gap, insert each value into its gap-separated
# sorted subsequence. Gap one finishes the full ordering.
#
# COMPLEXITY
# O(n squared) worst-case time for this halving gap sequence; O(1) auxiliary space. Not
# stable.
#
# CONTRACT
# Sorts n elements ascending in place; returns 0. Complexity depends on the chosen gap
# sequence.
# 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: 0.
# Expected a after call: [1, 1, 3, 5].
# 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 shell_sort(a: list[int]) -> int:
    alg_require(isinstance(a, list) and len(a) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    gap: int = n // 2
    i: int = 0
    j: int = 0
    key: int = 0
    while gap > 0:
        i = gap
        while i < n:
            key = a[i]
            j = i
            while j >= gap and a[j - gap] > key:
                a[j] = a[j - gap]
                j -= gap
            a[j] = key
            i += 1
        gap = gap // 2
    return 0

a = [5, 1, 3, 1]
result = shell_sort(a)
print(result)
print(a)
# Expected a: [1, 1, 3, 5]