The input size of a certain algorithm is n^2+n*m. Its running time is O(m*n^3). Can the running time be considered polynomial in the input size?
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The input size of a certain algorithm is n^2+n*m. Its running time is O(m*n^3). Can the running time be considered polynomial in the input size?
Run time T(n,m) is said to be polynomial in the input size S(n,m) = n^2+n*m if there is a polynomial in S that is an upper bound on T(n,m).
Consider the polynomial S^2(n,m) = (n^2+n*m)^2 = n^4 + 2(n^2)n*m + (n^2)(m^2). Since n^4 and (n^2)(m^2) are squares of positive integers they are positive, so S^2(n,m) > 2(n^2)n*m > n^3 * m.
Since T(n,m) is O(n^3 * m) and S^2(n,m) > n^3 * m we have T(n,m) is O(S^2(n,m)) hence run time is bounded by a polynomial in input size.