  DeepLLL algorithm (Schnorr, 1994) is a famous variant of LLL lattice basis
reduction algorithm, and PotLLL algorithm (Fontein et al., 2014) and $S^2$LLL
algorithm (Yasuda and Yamaguchi, 2019) are recent polynomial-time variants of
DeepLLL algorithm developed from cryptographic applications. However, the known
polynomial bounds for computational complexity are shown only for parameter
$\delta < 1$; for "optimal" parameter $\delta = 1$ which ensures the best
output quality, no polynomial bounds are known, and except for LLL algorithm,
it is even not formally proved that the algorithm always halts within finitely
many steps. In this paper, we prove that these four algorithms always halt also
with optimal parameter $\delta = 1$, and furthermore give explicit upper bounds
for the numbers of loops executed during the algorithms. Unlike the known bound
(Akhavi, 2003) applicable to LLL algorithm only, our upper bounds are deduced
in a unified way for all of the four algorithms.
