Algorithms
If the input is given as a subroutine for calculating ƒ, the cycle detection problem may be trivially solved using only λ+μ function applications, simply by computing the sequence of values xi and using a data structure such as a hash table to store these values and test whether each subsequent value has already been stored. However, the space complexity of this algorithm is λ+μ, unnecessarily large. Additionally, to implement this method as a pointer algorithm would require applying the equality test to each pair of values, resulting in quadratic time overall. Thus, research in this area has concentrated on two goals: using less space than this naive algorithm, and finding pointer algorithms that use fewer equality tests.
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