Reduction from non-injective hidden shift problem to injective hidden shift problem

Quantum Pattern Matching Fast on Average

“…First we show that random strings can be made injective by only considering a small number of subsequent positions. A similar result was previously shown in a more general context by Gharibi [16], using a different notion of injectivisation. Recall that we write q = |Σ|.…”

“…To use the algorithm, we first make the pattern and text injective. This is similar to the "injectivisation" idea used by Gharibi [16] in the context of quantum algorithms for abelian hidden shift problems, but here we need a slightly different notion, as used by Knuth [21], to ensure we preserve matching after injectivisation. We then apply the hidden shift algorithm by guessing an offset where the pattern matches the text.…”

“…To convert non-injective strings into injective strings, we concatenate adjacent symbols within the string. A similar idea was recently used by Gharibi to convert non-injective hidden shift problems over an arbitrary group into injective hidden shift problems [16]. The general framework takes a function f : G → X and a k-tuple V ∈ G k , and defines a new function f…”