牛客多校第一场补题
难度升序
F:Infinite String Comparision
上手就写，将ab都延长2倍，然后比较到max(a.length,b.length)即可
证法：假设a.length<b.length，然后举例子就行了
#pragma GCC optimize(2)
#include<bits/stdc++.h>
#define ll long long
#define fastio {ios::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL);};
using namespace std;
double pi = acos(-1);
const double eps = 1e-9;
const int inf = 1e9 + 7;
const int maxn = 2e3 + 10;
int judge(string a,string b)
{
int lena = a.length(), lenb = b.length();
int len = max(lena, lenb);
for (int i = 0; i < len; i++)
{
int l = i % lena, r = i % lenb;
if (a[l] < b[r])return -1;
else if (a[l] > b[r])return 1;
}
return 0;
}
int main()
{
fastio;
string a, b;
while (cin >> a >> b)
{
a += a, b += b;
if (judge(a, b) == 1)cout << ">" << endl;
else if(judge(a, b) == 0)cout << "=" << endl;
else cout << "<" << endl;
}
return 0;
}
J-Easy Integration
求这个积分，如果是分数就输出分子乘分母的逆元再模998244353
手残导致输出溢出WA了，我只能爪巴
#pragma GCC optimize(2)
#include<bits/stdc++.h>
#define ll long long
#define fastio {ios::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL);};
using namespace std;
double pi = acos(-1);
const double eps = 1e-9;
const int inf = 1e9 + 7;
const int maxn = 1e6 + 10;
const ll mod = 998244353;
ll f[maxn], fac[maxn], tmp[maxn];
long long qpow(long long x, long long n) {
long long res = 1;
while (n) {
if (n & 1) res = (res * x) % mod;
x = (x * x) % mod;
n /= 2;
}
return res%mod;
}
int main()
{
fastio;
ll n;
f[0] = 1;
fac[0] = 1;
tmp[1] = 2;
for (ll i = 1; i < maxn; i++)
fac[i] = (fac[i - 1] * (2 * i + 1)) % mod;
for (ll i = 2; i < maxn; i++)
tmp[i] = (tmp[i - 1] * 2 * i) % mod;
while (cin >> n)
{
cout << (((qpow(qpow(4, n), mod - 2) % mod) * (tmp[n] % mod))%mod * (qpow(fac[n], mod - 2) % mod)) % mod << endl;
}
}
I-1 or 2
很明显是个网络流，但需要一点建图的技巧
构造源点流量为inf，连接输入点1n，流量为x；然后将1+Nn+N与输入点按照题目要求相连，每条边的流量为1；再将1+N~n+N进行拆点限流，最后汇总到汇点跑dinic检查是否能跑到最大流即可
（代码有点不同，思路一样，板子备注没删）
#pragma GCC optimize(2)
#include<bits/stdc++.h>
#define ll long long
#define fastio {ios::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL);};
using namespace std;
double pi = acos(-1);
const double eps = 1e-9;
const int inf = 1e9 + 7;
const int maxn = 1e6 + 10;
int cnt = 1, head[maxn], now[maxn], deep[maxn];
queue<int>q;
struct edge {
int to;
int cost;
int next;
}e[maxn * 2];
void add(int from, int to, int cost)//前向星
{
e[++cnt].to = to;
e[cnt].cost = cost;
e[cnt].next = head[from];
head[from] = cnt;
}
int n, m, s, t;//n个点，m条边，s为源点，t为汇点
bool bfs()
{
memset(deep, 0, sizeof(deep));//初始化深度
while (!q.empty())q.pop();//初始化队列
q.push(s);
deep[s] = 1;//源点深度赋值1
now[s] = head[s];//将拥有深度的点复制到另一个head数组
while (!q.empty())
{
int from = q.front(), to, cost; q.pop();
for (int i = head[from]; i; i = e[i].next)
{
to = e[i].to;
cost = e[i].cost;
if (cost && !deep[to])//容量>0且未遍历过
{
q.push(to);
deep[to] = deep[from] + 1;
now[to] = head[to];//将拥有深度的点复制
if (to == t)return 1;//找到汇点就return，在deep[t]-1的点早已被复制，所以不用担心直接返回会丢失增广路
}
}
}
return 0;
}
int dfs(int from, int flow)
{
if (from == t)return flow;
int rest = flow, tmp, i;//rest存当前剩余容量，tmp存当前增广路可增广流量，i前向星
//cout << 1 << endl;
for (i = now[from]; i && rest; i = e[i].next)//尝试增广与from相连的所有容量>0且deep[from]+1==deep[to]的点所存在的路径
if (e[i].cost && deep[e[i].to] == deep[from] + 1)
{
tmp = dfs(e[i].to, min(flow, e[i].cost));//取当前路径上最小容量作为flow
if (!tmp)deep[e[i].to] = -2;//to所有边都无法增广，那就炸点，直接把深度赋一个不可能值
e[i].cost -= tmp;//增广完成后正向边减去tmp，反向边加上tmp，剩余容量减去tmp
e[i ^ 1].cost += tmp;//由于存图是从2开始存并且正边和反边相邻，所以正边^1是反边，反之亦然
rest -= tmp;
//cout << rest << endl;
}
now[from] = i;//当前弧优化，下次遍历到from时会直接从i(flow完了i就不一定是0了)开始
return flow - rest;//返回一个增广消耗的值
}
ll dinic()
{
long long maxflow = 0;
long long flow = 0;
while (bfs())
{
//for (int i = 1; i <= n; i++)
// cout << deep[i] << " ";
while (flow = dfs(s, inf))//dfs没能找到增广路就重新进行分层
maxflow += flow;
}
return maxflow;
}
int main()
{
fastio;
int n, m;
s = 200, t = 301;
while (cin >> n >> m)
{
cnt = 1;
memset(head, 0, sizeof(head));
int sum = 0;
for (int i = 1; i <= n; i++)
{
int x;
cin >> x;
sum += x;
add(i, i + 100, 0);
add(i + 100, i, x);//限流器
add(i, t, inf);
add(t, i, 0);
add(s, i + 200, x);
add(i + 200, s, x);
}
for (int i = 1; i <= m; i++)
{
int a, b;
cin >> a >> b;
add(a + 200, b + 100, 1);
add(b + 100, a + 200, 0);
add(b + 200, a + 100, 1);
add(a + 100, b + 200, 0);
}
if (dinic() == sum)cout << "Yes" << endl;
else cout << "No" << endl;
}
return 0;
}
H-Minimum-cost Flow
最小费用流，题目直接告诉做法了，直接跑就完事了（我板子好像常数有点问题，cin直接原地爆炸）
还是有丶做法：首先跑流量无限，所有边容量为1的最小费用流，记录每一条增广路的花费(每一条都是满流)；
然后对于每一对uv，让每条边的容量都变成u，那么每一条增广路的花费也会变成u倍；
由于源点是单位流，边权原本是\(u/v\)，现在变成了\(u\)，那么源点的流量就变成了\(v\)，因此最大流（或者说答案需求流）必然只能是v
设当前已增广的流量为flownow
flownow=0
这样遍历每一条增广路(已知必然满流)，若\(flownow+u<=v\),则\(flownow += u, cost += tracost[i] * u\);
如果\(flownow+u>v\)，但\(flownow<v\)，这一条就跑不满，\(cost += tracost[i] * (v - tmp), flownow = v\);
如果跑完之后\(flownow<v\)，则跑不满单位流，输出\(NaN\);
如果跑满了，就把\(cost\)和\(v\)求个\(gcd\)，然后输出就完事了
#pragma GCC optimize(2)
#include<bits/stdc++.h>
#define ll long long
using namespace std;
double pi = acos(-1);
const double eps = 1e-9;
const int inf = 1e9 + 7;
const int maxn = 60;
int cnt = 1, head[maxn], pre[maxn], last[maxn];
int dis[maxn], flow[maxn], maxflow;
int vis[maxn];
vector<ll>tracost;
queue<int>q;
int n, m, S, T;
struct edge {
int to;
int cost;
int vue;
int next;
}e[maxn * 4];
inline void add(int from, int to, int cost, int vue)//前向星
{
e[++cnt].to = to;
e[cnt].cost = cost;
e[cnt].vue = vue;
e[cnt].next = head[from];
head[from] = cnt;
}
ll __gcd(ll a, ll b)
{
while (b)
{
ll tmp = b;
b = a % b;
a = tmp;
}
return a;
}
ll SPFA(int s, int t) {
memset(dis, 0x3f, sizeof(dis));
memset(vis, 0, sizeof(vis));
q.push(s);
dis[s] = 0, vis[s] = 1, pre[t] = -1, flow[s] = inf;
while (!q.empty()) {
int now = q.front();
q.pop();
vis[now] = 0;
for (int i = head[now]; i != -1; i = e[i].next) {
if (e[i].cost > 0) {
int po = e[i].to;
long long lo = e[i].vue;
if (dis[po] > dis[now] + lo) {
dis[po] = dis[now] + lo;
flow[po] = min(flow[now], e[i].cost);
last[po] = i;
pre[po] = now;
if (!vis[po]) {
vis[po] = 1;
q.push(po);
}
}
}
}
}
return pre[t] != -1;
}
void MCMF(int s, int t) {
maxflow = 0;
while (SPFA(s, t)) {
maxflow += flow[t];
tracost.push_back(dis[t] * flow[t]);
for (int i = t; i != s; i = pre[i]) {
e[last[i]].cost -= flow[t],
e[last[i] ^ 1].cost += flow[t];
}
}
}
int main()
{
S = 1;
while (~scanf("%d %d", &n, &m))
{
memset(head, -1, sizeof(head));
tracost.clear();
cnt = 1;
T = n;
for (int i = 1; i <= m; ++i)
{
ll a, b, c;
scanf("%d%d%d", &a, &b, &c);
add(a, b, 1, c);
add(b, a, 0, -c);
}
MCMF(S, T);
int q;
scanf("%d", &q);
int u, v;
while (q--)
{
scanf("%d%d", &u, &v);
ll tmp = 0, cost = 0;
int size = tracost.size();
for (int i = 0; i < size; i++)
{
if (tmp + u <= v)
tmp += u, cost += tracost[i] * u;
else if (tmp < v)
cost += tracost[i] * (v - tmp), tmp = v;
else
break;
}
if (tmp == v)
{
ll gcd = __gcd(cost, v);
printf("%lld/%lld\n", cost / gcd, v / gcd);
}
else printf("NaN\n");
}
}
return 0;
}
