2020-12-30, 09:50 | #1 |
"Sam"
Nov 2016
2·163 Posts |
Computation of P-Recursive Sequences in Polynomial Time
Is it possible for centered trinomial coefficients to be computed in O(log n) time, analogous to how the time complexity of modular exponentiation is O(log n)?
I am well aware that factorials cannot be computed in logarithmic time, making Wilson's theorem impractical for large numbers. On the other hand, raising an integer to a power can easily be done in logarithmic time, making Fermat's test practical for large numbers. Where does computing centered trinomial coefficient (and similar sequences), fall into? To be clear, central trinomial coefficients are the coefficient of x^n in (x^2 + x + 1)^n for n ≥ 0. Meanwhile, centered binomial coefficients binomial(2*n, n) are the coefficient of x^n in (x^2 + 2x + 1)^n for n ≥ 0. Catalan numbers have the form binomial(2*n, n)/(n + 1) for n ≥ 0. I can't name a single algorithm for computing either central binomial coefficients or Catalan numbers in O(log n) time. Is the same true for centered trinomial coefficients, and if so, why is it impossible to compute any particular term in O(log n) time? |
2021-01-03, 22:16 | #2 |
"Sam"
Nov 2016
2·163 Posts |
The examples involving Trinomial Coefficients and Catalan numbers (also see here) are examples of P-recursive sequences, which is what I am generally after.
After doing a bit of research, I was able to find this article, which explicitly states that the N-th term of a P-recursive sequence can be computed in no more than O(N^(1/2)) operations. This paper also seems to agree. Unfortunately, that's not practical for (primality) testing large numbers. That being said, it's not impossible to design an algorithm that can reduce the complexity from almost linear to polynomial time, as is the case of C-recursive sequences (Fibonacci, Lucas, Mersenne,..). My idea is to reduce the baby-step giant step techniques with something analogous to square-multiply for P-recursive sequences, which I'm sure exists. If anyone know of a paper that proves otherwise, or some better ideas, please let me know. Last fiddled with by carpetpool on 2021-01-03 at 22:20 |
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