def permutation(swaps, N=None):
    if N is None:
        N = max(swaps)
    perm = list(range(N))
    for s in swaps:
        perm[0], perm[s-1] = perm[s-1], perm[0]
    return perm


def swaptimize(*old_swaps):
    assert old_swaps
    if len(old_swaps) == 1:
        return tuple(old_swaps)

    perm = permutation(old_swaps)

    swaps = []
    for i, x in enumerate(perm):
        # March forward, searching for an un-traversed cycle.
        if x is None or x == i:
            continue
        # Traverse this cycle.
        j = i
        while True:
            if j != 0:
                swaps.append(j + 1)
            if perm[j] is None:
                # Back to the start of the cycle.
                assert j == i
                break
            next_j = perm[j]
            perm[j] = None
            j = next_j

    swaps = tuple(reversed(swaps))

    assert permutation(old_swaps, len(perm)) == permutation(swaps, len(perm))
    assert len(swaps) <= len(old_swaps)

    return swaps

assert swaptimize(1) == (1,) # This should be ()?
assert swaptimize(2) == (2,)
assert swaptimize(3) == (3,)
assert swaptimize(10) == (10,)

assert swaptimize(2, 2, 5, 5) == ()
assert swaptimize(10, 20, 20, 10) == ()
assert swaptimize(2, 3, 4, 3) == (3, 4, 3, 2)

from random import randint

for i in range(100_000):
    for n in range(1, 20):
        swaptimize(*[randint(2, 20) for _ in range(n)])