Leetcode刷题笔记_二分法部分

二分法

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• 二分法的精髓：函数单调性+计算内容重复
• 不同于分治
• 求答案->求判定
  
  • 传统二分法
    
    • 给大小找下标or给下标找大小？
  • 二分答案法
  • Leetcode 69,410,363,378

69. Sqrt(x)

Implement int sqrt(int x).

Compute and return the square root of x.

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题目分析：
实现 int sqrt(int x) 函数，计算并返回 x 的平方根。
样例
sqrt(3) = 1
sqrt(4) = 2
sqrt(5) = 2
sqrt(10) = 3

N =	123	45678	91011	12131415	16171819	2021222	24	252627	282930
sqrt(N)=	1	2	3	3	4	4	4	5	5
N/4=	0	1	2	3	4	5	6	6	7
sqrt(N/4)	0	1	1	1	2	2	2	2	2
2*sqrt(N/4)	0	2	2	2	4	4	4	4	4

如果(2*sqrt(N/4)+1)*(2*sqrt(N/4)+1)<=N，则sqrt(N)=2*sqrt(N/4)+1;否则sqrt(N)=2*sqrt(N/4)

java:

public class Solution {
    public int mySqrt(int x) {

        if(x<4) return x==0?0:1;
        int res = 2*mySqrt(x/4);
        return ((res+1)*(res+1)<=x&&(res+1)*(res+1)>=0)?(res+1):res;
    }
}

410. Split Array Largest Sum

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Given an array which consists of non-negative integers and an integer m, you can split the array into m non-empty continuous subarrays. Write an algorithm to minimize the largest sum among these m subarrays.

Note:
If n is the length of array, assume the following constraints are satisfied:

1 ≤ n ≤ 1000
1 ≤ m ≤ min(50, n)

Examples:

Input:
nums = [7,2,5,10,8]
m = 2

Output:
18

Explanation:
There are four ways to split nums into two subarrays.
The best way is to split it into [7,2,5] and [10,8],
where the largest sum among the two subarrays is only 18.

363. Max Sum of Rectangle No Larger Than K

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Given a non-empty 2D matrix matrix and an integer k, find the max sum of a rectangle in the matrix such that its sum is no larger than k.

Example:

Given matrix = [
  [1,  0, 1],
  [0, -2, 3]
]
k = 2

The answer is 2. Because the sum of rectangle [[0, 1], [-2, 3]] is 2 and 2 is the max number no larger than k (k = 2).

Note:

• The rectangle inside the matrix must have an area > 0.
• What if the number of rows is much larger than the number of columns?

378. Kth Smallest Element in a Sorted Matrix

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Given a n x n matrix where each of the rows and columns are sorted in ascending order, find the kth smallest element in the matrix.

Note that it is the kth smallest element in the sorted order, not the kth distinct element.

Example:

matrix = [
   [ 1,  5,  9],
   [10, 11, 13],
   [12, 13, 15]
],
k = 8,

return 13.

Note:
You may assume k is always valid, 1 ≤ k ≤ n2.