问题
Example T(n)=T(n/3)+T(n/4)+3n is this solvable with iterative master theorem or recursion tree.Can someone solve it analytically to show how it's done ?
回答1:
We can expand T(n) with a binomial summation:
(after some steps - can be proven by induction
For some depth of expansion / recursion k.
Where do we terminate? When the parameters to all instances of f(n) reach a certain threshold C. Thus the maximum depth of expansion:
We choose the smallest between a, b because the parameter with only powers of min(a, b) decreases at the slowest rate:
Thus the general expression for T(n) is:
The existence of a closed form analytical solution depends on the form of f(n). For the example provided:
The inner summation is the expansion of a binomial bracket raised to power j:
This is a geometric series, and equals (using standard formula):
Now since 7/12 is less than 1, the power term in the above result becomes vanishingly small for large values of k (and thus n). Therefore in the limit of large n:
Truth be told the above example could have been done more straightforwardly with a recursion tree; but the same does not go for e.g. other powers of n, e.g. f(n) = Cn^2, which can be trivially incorporated into the general formula.
来源：https://stackoverflow.com/questions/45424921/tn-c1tn-a-c2t-b-fn