Quantum span program algorithms for function evaluation commonly have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these speed-ups persist even without having a promise ahead of time, and we extend this approach to the more general problem of state conversion. For example, there is a span program algorithm that decides whether two vertices are connected in an $n$-vertex graph with $O(n^{3/2})$ queries in general, but with $O(sqrt{k}n)$ queries if promised that, if there is a path, there is one with at most $k$ edges. Our algorithm uses $tilde{O}(sqrt{k}n)$ queries to solve this problem if there is a path with at most $k$ edges, without knowing $k$ ahead of time.
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