二分查找与二分答案

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原创 2022-04-04 18:26:23 438 阅读 AI 写同款· GEO 优化 ›

学习笔记 - 二分查找与答案（附归并模版、前缀和、差分）二分也就是二分查找，又叫折半查找。这种算法正如其名，每一次都要分一半。二分算法可以分为二分查找和二分答案。以在一个升序数组中查找一个数为例，每次考察数组当前部分的中间元素，如果中间元素刚好是要找的，就结束搜索过程；如果中间元素小于所查找的值，那么左侧的只会更小，不会有所查找的元素，只需到右侧查找；如果中间元素大于所查找的值同理，只需到左侧查找。二分查找和二分答案有什么区别？二分查找：在一个已知的有序数据集上进行二分地查找二分答案：答案有一个区间，在这个区间中二分，直到找到最优答案。阅读详情

壹·二分查找

一、基础

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cmath>
#include <cstring>
using namespace std;

int x, n, mid, high, low = 1;

int main() {
	scanf("%d %d", &x, &n);
	high = n;
	
	while(low < high) {
		mid = (high+low)/2;
		if(mid < x) 
			low = mid+1;
		else 
			high = mid;
	}
	
	printf("%d right", x);
	return 0;
}

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cmath>
#include <cstring>
using namespace std;

int x, n, mid, high, low = 1;

int main() {
	scanf("%d %d", &x, &n);
	high = n;
	
	while(low < high) {
		mid = (high+low+1)/2;
		if(mid > x) 
			high = mid-1;
		else 
			low = mid;
	}
	
	printf("%d right", x);
	return 0;
}

二、进阶

1. 查找第一个大于 $x$ 的数

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cmath>
#include <cstring>
using namespace std;

int l = 1, r, mid, n, s;
int a[20005];

void bs(){
	while(l<r) {
		mid= (l+r)/2;
		if(a[mid] > s)
			r = mid;
		else l = mid+1;
	}
}

int main() {
	scanf("%d", &n);
	r = n+1;
	for(int i = 1; i <= n; i++) {
		scanf("%d", &a[i]);
	}
	scanf("%d", &s);
	
	sort(a+1, a+1+n);
	bs();
	
	(l > r)? printf("No\n"): printf("%d", a[l]);
	return 0;
}

2. 查找第一个大于等于 $x$ 的数

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cmath>
#include <cstring>
using namespace std;

int l = 1, r, mid, n, s;
int a[20005];

void bs(){
	while(l<r) {
		mid= (l+r)/2;
		if(a[mid] >= s)
			r = mid;
		else l = mid+1;
	}
}

int main() {
	scanf("%d", &n);
	r = n+1;
	for(int i = 1; i <= n; i++) {
		scanf("%d", &a[i]);
	}
	scanf("%d", &s);
	
	sort(a+1, a+1+n);
	bs();
	
	(l > r)? printf("No\n"): printf("%d", a[l]);
	return 0;
}

3. 查找第一个小于 $x$ 的数

// 第一个小于 
#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cmath>
#include <cstring>
using namespace std;

int l = 1, r, mid, n, s;
int a[20005];

void bs(){
	while(l<r) {
		mid= (l+r+1)/2;
		if(a[mid] >= s)
			r = mid-1;
		else l = mid;
	}
}

int main() {
	scanf("%d", &n);
	r = n+1;
	for(int i = 1; i <= n; i++) {
		scanf("%d", &a[i]);
	}
	scanf("%d", &s);
	
	sort(a+1, a+1+n);
	bs();
	
	(l < 1)? printf("No\n"): printf("%d", a[l]);
	return 0;
}

4. 查找第一个小于等于 $x$ 的数

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cmath>
#include <cstring>
using namespace std;

int l = 1, r, mid, n, s;
int a[20005];

void bs(){
	while(l<r) {
		mid= (l+r+1)/2;
		if(a[mid] <= s)
			r = mid-1;
		else l = mid;
	}
}

int main() {
	scanf("%d", &n);
	r = n+1;
	for(int i = 1; i <= n; i++) {
		scanf("%d", &a[i]);
	}
	scanf("%d", &s);
	
	sort(a+1, a+1+n);
	bs();
	
	(l < 1)? printf("No\n"): printf("%d", a[l]);
	return 0;
}