algorithm - Triple Nested Big O with dependent indices -


i working on finding big o problem, , having difficulty third nested loop. here code

for (int = 1 n)   (int j = n)     (int k = j*j n)       //some constant operation

the i forloop o(n).

the j forloop (n + n-1 + n-2 + ... + 2 + 1) = (n-1)n/2 = o(n^2).

but unsure on how consider k forloop. know 1 full loop of j (1 n), summation (n + n-4 + n-6 + ...) = sum_{k=1}^n (n - k^2), unsure on go there.

any advice on how proceed great!

when j greater sqrt(n), inner loop isn't entered. same true when i greater sqrt(n), because j starts @ i.

therefore, can split work done 2 cases:

  • on iterations i and/or j more sqrt(n): k loop isn't entered, , it's not difficult prove time complexity done outer 2 loops theta(n^2).
  • on iterations both i , j less sqrt(n): first 2 loops run sqrt(n) times each, , last loop runs n times, therefore upper bound of number of total iterations o(sqrt(n) * sqrt(n) * n) = o(n^2).

in both cases, upper bound o(n^2), , first case shows lower bound. therefore, total time complexity theta(n^2).


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