Longest Common Subsequence in Python

from bisect import bisect_left

# Longest Common Subsequence in Python
#
# PURPOSE
# LCS finds the longest sequence appearing in two inputs in the same order, with gaps
# allowed. It is a foundation for comparison tools.
#
# METHOD
# Use one row of prefix lengths and retain the previous diagonal in a scalar. Equal tokens
# extend the diagonal; otherwise choose the best result from above or left.
#
# COMPLEXITY
# O(nm) time and O(min(n,m)) auxiliary space. The public function automatically orients the
# shorter input as its work row.
#
# CONTRACT
# Inputs are integer token sequences. Returns length only. Empty inputs are valid. Only one
# row of the dynamic program is retained.
# 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.
# 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 lcs_length(a: list[int], b: list[int]) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    alg_require(isinstance(b, (list, tuple)) and len(b) <= 1000000)
    n: int = len(a)
    m: int = len(b)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(m >= 0 and m <= 1000000)
    alg_require(len(a) == n)
    alg_require(len(b) == m)
    if n < m:
        return lcs_length(b, a)
    dp: list[int] = [0] * alg_size(m + 1)
    i: int = 0
    j: int = 0
    diagonal: int = 0
    old: int = 0
    while i < n:
        diagonal = 0
        j = 1
        while j <= m:
            old = dp[j]
            if a[i] == b[j - 1]:
                dp[j] = diagonal + 1
            else:
                dp[j] = max(dp[j], dp[j - 1])
            diagonal = old
            j += 1
        i += 1
    return dp[m]

a = [1, 2, 3]
b = [2, 3, 4]
result = lcs_length(a, b)
print(result)