Lecture3
Question
\[ T(n) = 4T(n / \sqrt{2}) + O(n^3) \]
\(\log_b a = 4\)，这是主定理的第一种情况，\(T(n) = n^4\)
Divide and conquer¶
Big integers multiplication¶
Divide and conquer¶
\[ T(n) = 3T(n / 2) + \Theta(n) \]
Case1：\(T(n) = \Theta(n^{\log_b a}) = \Theta(n^{\log_2 3})\)
Matrix multiplication¶
Strassen's Divide and Conquer¶
\[ T(n) = \Theta(n^{\log_2 7}) \]
Closest pair of points¶
One dimension example¶
\[ S_1 = \lbrace x \in S \mid x \leq m $$
$$ S_2 = \lbrace x \in S \mid x > m \]
答案为 \(S_1\) 中 的最近点对， \(S_2\) 中最近点对，\(S_1\) 最小减 \(S_2\) 最大
Two dimension example¶
Randomized Algorithms¶
Hiring Problem¶
Uniform Random Permutation¶
PERMUTE-BY-SORTING
for i = 1 to n
P[i] = Random(1, n^3)
sort A, using P as sort keys.
Lemma
Procedure PERMUTE-BY-SORTING produces a uniform random permutation of the input, assuming that all priorities are distinct.
A Better Random Permutation¶
for i = 1 to n
swap A[i], A(Random(i, n))