Recently, I tried to solve some recursive relations from CLRS, and I noticed a strange nuance when solving these equations. I don’t know if you noticed any of you or not, or maybe members of the theory can throw more light on this. (I also have a degree in CS, but not in Theory!). Solving recursion for the main theorem:
T (n) = a T (n / b) + f (n)
I noticed that the reasoning goes something like this:
i) expand the a-arry recursion tree, and we get the nodes log b n where the work done for the node is Θ (1), which gives Θ (n log b a ) for all leaf nodes
ii) for all non-leaf nodes, g (n) = Σ a j f (b / n j ), where j sums from 0 to the floor (log b n - 1), where the height of the tree is log b n
iii) Now take the leap of faith: make the statement that f (n) is really limited to O (n log b a - epsilon;) for some & epsilon; > 0
iv) Now solve g (n) via f (n) and solve T (n) via g (n). As mentioned in step i, T (n) really is Θ (n log b a ) + g (n), so when you have some g (n), combine with another term to come up with T (n)
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