Merge Based Inversion Count in Python

from bisect import bisect_left

# Merge Based Inversion Count in Python
#
# PURPOSE
# An inversion is a pair i<j with a[i]>a[j]. Counting them during merging avoids testing
# every pair.
#
# METHOD
# Merge sorted runs bottom up. Whenever a right-run value precedes a larger left-run value,
# add the number of remaining left-run values.
#
# COMPLEXITY
# O(n log n) time, O(n) auxiliary space.
#
# CONTRACT
# Mutates input into sorted order. n <= 65536 so the inversion count fits signed 32-bit.
# 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: 2.
# Expected a after call: [1, 2, 3].
# 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 inversion_count(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)
    alg_require(n <= 65536)
    buffer: list[int] = [0] * alg_size(n + 1)
    width: int = 1
    left: int = 0
    mid: int = 0
    right: int = 0
    i: int = 0
    j: int = 0
    out: int = 0
    count: int = 0
    while width < n:
        left = 0
        while left < n:
            mid = min(left + width, n)
            right = min(left + 2 * width, n)
            i = left
            j = mid
            out = left
            while out < right:
                if i < mid and (j >= right or a[i] <= a[j]):
                    buffer[out] = a[i]
                    i += 1
                else:
                    buffer[out] = a[j]
                    j += 1
                    count += mid - i
                out += 1
            left += 2 * width
        i = 0
        while i < n:
            a[i] = buffer[i]
            i += 1
        width *= 2
    return count

a = [3, 1, 2]
result = inversion_count(a)
print(result)
# Expected a: [1, 2, 3]
print(a)