  Session types are a type discipline for describing and specifying
communication behaviours of concurrent processes. Session subtyping, firstly
introduced by Gay and Hole, is widely used for enlarging typability of session
programs. This paper gives the complexity analysis of three algorithms for
subtyping of synchronous binary session types. First, we analyse the complexity
of the algorithm from the original paper, which is based on an inductive tree
search. We then introduce its optimised version, which improves the complexity,
but is still exponential against the size of the two types. Finally, we propose
a new quadratic algorithm based on a graph search using the concept of
$\mathcal{XYZW}$-simulation, recently introduced by Silva et al.
