本文探讨了分治法中的主定理，它用于分析递归问题的时间复杂度。主定理适用于a=b的情况，如快速排序和归并排序等算法，通过解决规模为n/b的子问题来解决问题。根据不同的情况，主定理可以得出时间复杂度为O(nlogn)，O(n)和O(n^2)。最后，强调了分治法在算法设计中的重要性，如在归并排序和快速排序中的应用。
分治法——主定理(Divide and Conquer - The Master Theorem)
Divide-and-Conquer Recurrences
What is the time required to solve a problem of size n by divide-and-conquer?
Generally, we split the problem into b small problems. The size of each small problem is n/b. There are a small problems needed to be solved(In general, a=b). So, we can express the recurrence as
T(n) = aT(n/b) + f(n).
f(n) represents the time spent on dividing a problem into b sub-problems and combining the a results.
For example, T(n) = 2T(n/2) + n.
The Mater Theorem
For integer constants a ≥ 1 and b > 1, and the function f with f(n) ∈ Θ(n^d ), d ≥ 0, the recurrence
T(n) = aT(n/b) + f(n)
(with T(1) = c) has solutions, and
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