基础算法训练题单之二分（从入门到入土）

A 【深基13.例1】查找

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

int n, t;
int q[N];

int serch(int x) {
	int l = 0, r = n+1;

	while (l + 1 < r) {
		int mid = l + r >> 1;
		if (q[mid] < x) l = mid;
		else r = mid;
		//printf("%d %d\n",l,r);
	}

	if (q[r] != x) return -1;
	else return r;

}

signed main(){
	import;
	cin >> n >> t;

	for (int i = 1; i <= n; ++i) cin >> q[i];

	while (t--) {
		int x; cin >> x;
		
		int idx = serch(x);

		printf("%d ", idx);
	}

	return 0;
}

B, A-B 数对

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

LL n, c;
LL q[N];

int serch1(int x) {
	int l = -1, r = n;
	while (l + 1 < r) {
		int mid = l + r >> 1;
		if (q[mid] < x) l = mid;
		else r = mid;
	}

	if (q[r] != x) return -1;
	else return r;
}

int serch2(int x) {
	int l = -1, r = n;
	while (l + 1 < r) {
		int mid = l + r >> 1;
		if (q[mid] > x) r = mid;
		else l = mid;
	}

	if (q[l] != x) return -1;
	else return l;
}

signed main(){
	import;
	cin >> n >> c;

	LL ans = 0;

	for (int i = 0; i < n; ++i) cin >> q[i];

	sort(q, q + n);

	for (int i = 0; i < n; ++i) {
		LL x = q[i] + c;

		LL idx1 = serch1(x);
		LL idx2 = serch2(x);

		if (idx1 != -1) {
			if (idx1 == idx2) ans++;
			else ans += idx2 - idx1 + 1;
		}
	}

	cout << ans << endl;

	
	return 0;
}

C [COCI 2011/2012 #5] EKO / 砍树

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

LL n, x,ans;
LL q[N];

LL check(LL mid) {
	LL sum = 0;
	for (int i = 1; i <= n; ++i) {
		sum += max(0LL, q[i] - mid);
	}
	if (sum >= x) return 1;
	else return 0;
}

signed main() {
	
	import;
	scanf("%lld %lld", &n, &x);

	LL heightest = 0;
	for (int i = 1; i <= n; ++i)
	{
		scanf("%lld", &q[i]);
		heightest = max(heightest, q[i]);
	}

	LL l = 0, r = heightest + 1;

	while (l + 1 < r) {
		LL mid = l + r >> 1;
		if (check(mid)) l = mid;
		else r = mid;
	}

	printf("%lld", l);

	return 0;
}

D [NOIP2001 提高组] 一元三次方程求解

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

double a, b, c, d;
double l, r;

double  fun(double x) {
	return a * x * x * x + b * x * x + c * x + d;
}

signed main() {
	
	import;
	
	cin >> a >> b >> c >> d;
	int sum = 0;
	for (int i = -100; i < 100; ++i) {
		l = i, r = i + 1;
		if (fun(l)==0) {
			printf("%.2lf ", l);
			sum++;
			continue;
		}

		if (fun(l) * fun(r) < 0) {
			while (r - l > 0.001) {
				double mid = (l + r) / 2;
				if (fun(l) * fun(mid) < 0) r = mid;
				else l = mid;
			}
			printf("%.2lf ", r);
			sum++;
		}
		if (sum == 3) break;
	}

	return 0;
}

E 烦恼的高考志愿

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

int n, m;
int q[N];

signed main() {
	
	import;
	cin >> n >> m;

	for (int i = 1; i <= n; ++i) cin >> q[i];

	sort(q + 1, q + n + 1);

	LL sum = 0;
	for (int i = 0; i < m; ++i) {
		int x; cin >> x;
		int l = 1, r = n;
		while (l + 1 < r) {
			int mid = (l + r) >> 1;
			if (q[mid]<=x) l = mid;
			else r = mid;
		}
		sum += min(abs(q[l] - x), abs(q[r] - x));
	}

	cout << sum << endl;

	return 0;
}

F 木材加工

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

int n, x;
int q[N];

bool check(int mid) {
	int sum = 0;
	for (int i = 1; i <= n; ++i) {
		sum += q[i] / mid;
		if (sum >= x) return true;
	}
	return false;
}

signed main() {
	
	import;
	cin >> n >> x;

	int heightest = 0;
	for (int i = 1; i <= n; ++i) {
		cin >> q[i];
		heightest = max(heightest, q[i]);
	}

	int l = 0, r = heightest + 1;

	while (l + 1 < r) {
		int mid = l + r >> 1;
		if (check(mid)) l = mid;
		else r = mid;
	}

	cout << l << endl;

	

	return 0;
}

G [NOIP2015 提高组]跳石头

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

int d,n, m;
int q[N];

bool check(int x) {
	int cnt = 0;
	int now = 0;
	for (int i = 1; i <= n+1;++i) {
		if (q[i] - q[now] < x) {
			cnt++;
		}
		else {
			now = i;
		}
	}
	if (cnt <= m) return true;
	else return false;
}

signed main() {
	
	import;
	cin >> d >> n >> m;

	for (int i = 1; i <= n; ++i) cin >> q[i];

	q[n + 1] = d;

	int l = 0, r = d + 1;

	while (l + 1 < r) {
		int mid = (l + r) >> 1;
		if (check(mid)) l = mid;
		else r = mid;
	}

	cout << l << endl;

	return 0;
}

H [TJOI2007] 路标设置

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

int d, n, m;
int q[N], p[N];

bool check(int x) {
	int cnt = 0; //记录操作次数
	int now = 0; //当前的位置
	// i 为要跳跃的位置
	for (int i = 1; i <= n + 1; i++) {
		if (p[i] > x) { //如果距离大于二分值,则要插入路标使得距离变小
			int num = p[i] - x;
			cnt++; //操作次数加一
			while (num > x) { //记录操作次数，使得距离小于二分值
				num -= x;
				cnt++;
			}
		}
		else {
			now = i; //小于等于二分值就直接跳过
		}
	}
	if (cnt <= m) return true; //操作次数小于等于m，说明操作次数少了，最小空旷指数不是最小的，r = mid,即下次二分距离变小
	else return false; //操作次数大于m，不符合，则需要将最小空旷指数往上调大，l = mid,即下次二分距离变大 
}

signed main() {

import;
	cin >> d >> n >> m; //公路的长度，原有路标的数量，以及最多可增设的路标数量

	for (int i = 1; i <= n; ++i) {
		cin >> q[i];
		p[i] = q[i] - q[i - 1]; //p数组记录相邻两个路标的距离
	}

	q[n + 1] = d; //将最后一个路标加入数组
	p[n + 1] = q[n + 1] - q[n]; //最后一个与终点的距离

	int l = 0, r = d + 1; //二分答案

	while (l + 1 < r) {
		int mid = (l + r) >> 1;
		if (check(mid)) r = mid; //说明操作次数少了，换句话说，是最小空旷指数比较大，还可以小
		else l = mid; //这个则是操作次数多了了，不符合

	}

	//这里有个方法，如果你分不清边界，就去检查一下再输出,实际这题输出 r
	if (check(r)) cout << r << endl;
	else cout << l << endl;

	return 0;
}

I 数列分段 Section II

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

int n, m;
int q[N], p[N];

bool check(int x) {
	int cnt = 1;
	int sum = 0;
	for (int i = 0; i < n; ++i) {
		if (sum + q[i] <= x ) {
			sum += q[i];
		}
		else {
			sum = q[i];
			cnt++;
		}
	}

    return cnt <= m ;

}

signed main() {

	import;
	cin >> n >> m;
	int l = 0, r = 0;
	for (int i = 0; i < n; ++i) {
		cin >> q[i];
		l = max(l, q[i]);
		r += q[i];
	}

	 while (l < r) //二分模板
    {
        int mid = l + r >> 1;
        if (check(mid))  r = mid;
        else l = mid + 1 ;
    }
	
    cout << r << endl;
    
	return 0;
}

J 银行贷款

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

double w,pay,month;
int q[N], p[N];

bool check(double x) {
	double t = w;
	for (int i = 0; i < month; ++i) {
		t = t * (1 + x) - pay;
	}
	if (t <= 0) return true;
	else return false;
}

signed main() {

	import;
	cin >> w >> pay >> month;

	double l = 0, r = 5;

	while (r - l > 0.0001) {
		double mid = (l + r) / 2;
		if (check(mid))  l = mid;
		else r = mid;
	}
	
	printf("%.1lf", l*100);
	
	return 0;
}

K 小鸟的设备

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#define _CRT_SECURE_NO_WARNINGS 1

#define all(C) C.begin(),C.end()
#define S second
#define F first
#define PB push_back
#define mod 1000000007
#define import ios_base::sync_with_stdio(false);cin.tie(NULL)

#include <cstring>
#include <iostream>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <string>
#include <vector>
#include <map>

using namespace std;
typedef long long LL;

const int N = 1e6 + 10;

double n, p, a[N], b[N];

bool check(double mid)
{
    double sum = 0;
    for (int i = 1; i <= n; i++) {
        if (a[i] * mid > b[i])
            sum += a[i] * mid - b[i];
    }
    return sum <= p * mid;
}

signed main() {
    cin >> n >> p;

    for (int i = 1; i <= n; i++)
        cin >> a[i] >> b[i];

    double tmp = 0;
    for (int i = 1; i <= n; i++) {
        tmp += a[i];
    }
    if (tmp <= p) {
        cout << "-1";
        return 0;
    }
    double l = 0, r = 1e10 + 1;

    while (r - l >= 0.00001) {

        double mid = (l + r) / 2;

        if (check(mid)) l = mid;
        else r = mid;
    }
    cout << l << endl;
    return 0;
}