| Summary: | Session types are a type discipline for describing and specifying communication behaviours of concurrent processes. <i>Session subtyping</i>, 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 <i>XYZW</i>-simulation, recently introduced by Silva et al.
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