感觉挺不错的一套题,写下部分题目的题解
ZOJ 3789: Gears
题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3789
题意:有N个齿轮,齿轮间只有旋转方向相反才能连接,给出四个操作
L X Y :将齿轮X, Y连接,如果它们已经有已属集合,那集合也要合并
D X:拆掉齿轮X,且X所属集合不会断裂
Q X Y:询问X,Y的旋转方向是否相同
S X:询问X所在集合有多少个齿轮
思路:并查集,维护当前点到根节点的距离及当前点集合的结点个数,方向是否相同可以通过到根节点的距离之差的奇偶性来判断,对于删除结点操作,可以将该点映射到新的结点上,在合并X,Y时要注意更新距离数组
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <climits>
#include <functional>
#include <deque>
#include <ctime>
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
#pragma comment(linker, "/STACK:102400000,102400000")
using namespace std;
const int maxn = 600010;
int fa[maxn], dis[maxn], num[maxn], p[maxn];
void init()
{
for (int i = 0; i < maxn; i++)
{
fa[i] = i;
dis[i] = 0;
num[i] = 1;
p[i] = i;
}
}
int Find(int x)
{
if (x == fa[x]) return x;
int fff = Find(fa[x]);
dis[x] += dis[fa[x]];
return fa[x] = fff;
}
int main()
{
int n, m;
while (~scanf("%d%d", &n, &m))
{
init();
int cnt = n;
for (int i = 0; i < m; i++)
{
char c[5];
scanf("%s", c);
if (c[0] == 'L')
{
int u, v;
scanf("%d%d", &u, &v);
u = p[u], v = p[v];
int fu = Find(u), fv = Find(v);
if (fu == fv) continue;
fa[fu] = fv;
num[fv] += num[fu];
dis[fu] = dis[u] + dis[v] + 1;
}
if (c[0] == 'D')
{
int u;
scanf("%d", &u);
int pu = p[u];
int fu = Find(pu);
num[fu]--;
p[u] = ++cnt;
}
if (c[0] == 'Q')
{
int u, v;
scanf("%d%d", &u, &v);
u = p[u], v = p[v];
int fu = Find(u), fv = Find(v);
if (fu != fv)
puts("Unknown");
else
{
if (abs(dis[u] - dis[v]) % 2 == 0)
puts("Same");
else
puts("Different");
}
}
if (c[0] == 'S')
{
int u;
scanf("%d", &u);
u = p[u];
printf("%d\n", num[Find(u)]);
}
}
}
return 0;
}
ZOJ 3790:Consecutive Blocks
题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3790
题意:给一个长度为N的序列,你可以从该序列中任意删除K个数,问同一个数字最多能连续多长
思路:先离散化处理,然后对于离散化后的每个数字固定起点,枚举终点进行二分判断,找出最长的结果
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <climits>
#include <functional>
#include <deque>
#include <ctime>
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
#pragma comment(linker, "/STACK:102400000,102400000")
using namespace std;
const int maxn = 100010;
map <int, int> ma;
vector <int> v[maxn];
int main()
{
int n, k;
while (~scanf("%d%d", &n, &k))
{
ma.clear();
for (int i = 0; i <= n; i++)
v[i].clear();
int cnt = 0;
for (int i = 0; i < n; i++)
{
int x;
scanf("%d", &x);
int p = ma[x];
if (!p) ma[x] = ++cnt;
v[ma[x]].push_back(i);
}
int ans = 0;
for (int i = 1; i <= cnt; i++)
{
int si = v[i].size();
for (int j = 0; j < si; j++)
{
int l = -1, r = j;
while (r - l > 1)
{
int mid = (l + r) >> 1;
int sr = v[i][j], sl = v[i][mid];
if (sr - sl - (j - mid) > k)
l = mid;
else
r = mid, ans = max(ans, j - mid + 1);
}
ans = max(ans, j - r + 1);
}
}
cout << ans << endl;
}
return 0;
}
ZOJ 3791:An Easy Game
题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3791
题意:给出两个01字符串s1,s2,每次能且只能改变s1上m个位置的字符,问k次之后有多少种方法让s1变作s2
思路:dp[i][j]表示i次之后两串有j个字符不同,经过一次操作后可以将j个字符中的的t个位置变相同,剩余n - j个位置中m - t个位置变不同,故而有转移方程:dp[i + 1][j - t + m - t] += dp[i][j] * C(j, t) * C(n - j, m - t),要注意t的枚举范围需满足j - t + m - t <= n
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <climits>
#include <functional>
#include <deque>
#include <ctime>
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
#pragma comment(linker, "/STACK:102400000,102400000")
using namespace std;
const int maxn = 200;
const int mod = 1e9 + 9;
typedef long long ll;
ll dp[maxn][maxn], c[maxn][maxn];
char s[maxn], ss[maxn];
void init()
{
c[0][0] = 1;
for (int i = 1; i <= 100; i++)
{
c[i][0] = 1;
for (int j = 1; j <= i; j++)
c[i][j] = (c[i - 1][j - 1] + c[i - 1][j]) % mod;
}
}
int main()
{
init();
int n, m, k;
while (~scanf("%d%d%d", &n, &k, &m))
{
memset(dp, 0, sizeof(dp));
scanf("%s", s);
scanf("%s", ss);
int num = 0;
for (int i = 0; i < n; i++)
if (s[i] != ss[i]) num++;
dp[0][num] = 1;
for (int i = 0; i < k; i++)
for (int j = 0; j <= n; j++)
{
if (dp[i][j] == 0) continue;
for (int t = max((j + m - n) / 2, 0); t <= j && t <= m; t++)
dp[i + 1][j - t + m - t] = (dp[i + 1][j - t + m - t] + dp[i][j] * c[j][t] % mod * c[n - j][m - t] % mod) % mod;
}
cout << dp[k][0] << endl;
}
return 0;
}
ZOJ 3792:Romantic Value
题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3792
题意:给一张无向图,给出起点和终点,要删除一些边使起点到终点不连通,在删除边权值最小的情况下保证删除边的数目最少,求剩余边权值和 / 删除边数的这一比值
思路:可以构造每条边的边权为w * maxn + 1,这样就不会出现多种最小割的方案了,注意maxn要较大使得边数不会影响网络流的流向,最终跑出来的结果ans / maxn为最小割,ans % maxn为删除边数
也可以在原图跑完最大流的基础上,找出那些满流的边将其流量改为1,其余边流量改为INF,再跑一次最大流也可以找出删除的最少边数
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <climits>
#include <functional>
#include <deque>
#include <ctime>
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
#pragma comment(linker, "/STACK:102400000,102400000")
using namespace std;
const int MAXN = 1010;
const int MAXM = 50010;
const int INF = 0x3f3f3f3f;
typedef long long ll;
struct Edge
{
int to, next, cap, flow;
} edge[MAXM];
int tol;
int head[MAXN];
int gap[MAXN], dep[MAXN], cur[MAXN];
void init()
{
tol = 0;
memset(head, -1, sizeof(head));
}
void addedge(int u, int v, int w, int rw = 0)
{
edge[tol].to = v;
edge[tol].cap = w;
edge[tol].flow = 0;
edge[tol].next = head[u];
head[u] = tol++;
edge[tol].to = u;
edge[tol].cap = rw;
edge[tol].flow = 0;
edge[tol].next = head[v];
head[v] = tol++;
}
int Q[MAXN];
void BFS(int start, int end)
{
memset(dep, -1, sizeof(dep));
memset(gap, 0, sizeof(gap));
gap[0] = 1;
int front = 0, rear = 0;
dep[end] = 0;
Q[rear++] = end;
while (front != rear)
{
int u = Q[front++];
for (int i = head[u]; i != -1; i = edge[i].next)
{
int v = edge[i].to;
if (dep[v] != -1)continue;
Q[rear++] = v;
dep[v] = dep[u] + 1;
gap[dep[v]]++;
}
}
}
int S[MAXN];
int sap(int start, int end, int N)
{
BFS(start, end);
memcpy(cur, head, sizeof(head));
int top = 0;
int u = start;
int ans = 0;
while (dep[start] < N)
{
if (u == end)
{
int Min = INF;
int inser;
for (int i = 0; i < top; i++)
if (Min > edge[S[i]].cap - edge[S[i]].flow)
{
Min = edge[S[i]].cap - edge[S[i]].flow;
inser = i;
}
for (int i = 0; i < top; i++)
{
edge[S[i]].flow += Min;
edge[S[i] ^ 1].flow -= Min;
}
ans += Min;
top = inser;
u = edge[S[top] ^ 1].to;
continue;
}
bool flag = false;
int v;
for (int i = cur[u]; i != -1; i = edge[i].next)
{
v = edge[i].to;
if (edge[i].cap - edge[i].flow && dep[v] + 1 == dep[u])
{
flag = true;
cur[u] = i;
break;
}
}
if (flag)
{
S[top++] = cur[u];
u = v;
continue;
}
int Min = N;
for (int i = head[u]; i != -1; i = edge[i].next)
if (edge[i].cap - edge[i].flow && dep[edge[i].to] < Min)
{
Min = dep[edge[i].to];
cur[u] = i;
}
gap[dep[u]]--;
if (!gap[dep[u]])return ans;
dep[u] = Min + 1;
gap[dep[u]]++;
if (u != start)u = edge[S[--top] ^ 1].to;
}
return ans;
}
int main()
{
int tt;
cin >> tt;
while (tt--)
{
init();
int n, m, p, q;
cin >> n >> m >> p >> q;
int sum = 0;
for (int i = 0; i < m; i++)
{
int u, v, w;
scanf("%d%d%d", &u, &v, &w);
sum += w;
addedge(u, v, w * MAXN + 1);
addedge(v, u, w * MAXN + 1);
}
int ans = sap(p, q, n + 1);
if (ans == 0)
puts("Inf");
else
printf("%.2f\n", 1.0 * (sum - ans / MAXN) / (ans % MAXN));
}
return 0;
}
ZOJ 3793:First Digit
题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3793
题意:本福特定律的介绍
思路:数字1是满足那个概率范围的,所以输出1即可
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <climits>
#include <functional>
#include <deque>
#include <ctime>
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
#pragma comment(linker, "/STACK:102400000,102400000")
using namespace std;
const int maxn = 100010;
typedef long long ll;
int a[maxn];
int main()
{
int t;
cin >> t;
while (t--)
{
int b, e;
cin >> b >> e;
cout << 1 << endl;
}
return 0;
}
ZOJ 3794:Greedy Driver
题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3794
题意:一个司机要从1走到N,他可以在任意有加油站的地点加油,走每条道路需要消耗相应的油量,有些城市可以通过卖油赚钱,但只能卖一次油,如果司机不能走到N,输出-1,如果能走到输出最多能赚多少钱
思路:从起点跑一次spfa算出到每个点最多能剩多少油,再从起点反向跑一次spfa算出每个点到终点的最短距离,即最少需要多少油到终点,然后枚举每个可以卖油的城市更新答案,在跑spfa的过程中要注意对加油站的处理
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <climits>
#include <functional>
#include <deque>
#include <ctime>
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
#pragma comment(linker, "/STACK:102400000,102400000")
using namespace std;
const int maxn = 1010;
const int maxm = 100010;
const int inf = 0x3f3f3f3f;
int n, m, c;
int cnt, head[maxn];
int f[maxn], vis[maxn], dis[maxn];
int p[maxn], q[maxn];
struct edge
{
int to, w, nxt, id;
} e[maxm << 1];
void init()
{
cnt = 0;
memset(head, -1, sizeof(head));
}
void add(int u, int v, int w, int id)
{
e[cnt].to = v;
e[cnt].w = w;
e[cnt].id = id;
e[cnt].nxt = head[u];
head[u] = cnt++;
}
void spfa(int s)
{
memset(vis, 0, sizeof(vis));
memset(f, -1, sizeof(f));
queue <int> que;
que.push(s);
f[s] = c, vis[s] = 1;
while (!que.empty())
{
int u = que.front();
que.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = e[i].nxt)
{
if (e[i].id == 1) continue;
int v = e[i].to;
if (f[u] >= e[i].w)
{
int tmp = f[u] - e[i].w;
if (p[v]) tmp = c;
if (tmp > f[v])
{
f[v] = tmp;
if (!vis[v])
{
vis[v] = 1;
que.push(v);
}
}
}
}
}
}
void rspfa(int s)
{
memset(vis, 0, sizeof(vis));
memset(dis, inf, sizeof(dis));
queue <int> que;
que.push(s);
dis[s] = 0, vis[s] = 1;
while (!que.empty())
{
int u = que.front();
que.pop();
vis[u] = 0;
for (int i = head[u]; ~i; i = e[i].nxt)
{
if (e[i].id == 0) continue;
int v = e[i].to;
if (dis[v] > dis[u] + e[i].w)
{
dis[v] = dis[u] + e[i].w;
if (p[v]) dis[v] = 0;
if (!vis[v])
{
vis[v] = 1;
que.push(v);
}
}
}
}
}
int main()
{
while (~scanf("%d%d%d", &n, &m, &c))
{
init();
for (int i = 0; i < m; i++)
{
int u, v, w;
scanf("%d%d%d", &u, &v, &w);
add(u, v, w, 0);
add(v, u, w, 1);
}
memset(p, 0, sizeof(p));
memset(q, 0, sizeof(q));
int P, Q;
scanf("%d", &P);
while (P--)
{
int x;
scanf("%d", &x);
p[x] = 1;
}
scanf("%d", &Q);
while (Q--)
{
int x, v;
scanf("%d%d", &x, &v);
q[x] = v;
}
spfa(1);
if (f[n] == -1)
{
puts("-1");
continue;
}
rspfa(n);
// for (int i = 1; i <= n; i++)
// printf("************%d %d\n", f[i], dis[i]);
int ans = 0;
for (int i = 1; i <= n; i++)
{
if (q[i] && f[i] >= dis[i])
ans = max(ans, (f[i] - dis[i]) * q[i]);
}
cout << ans << endl;
}
return 0;
}
ZOJ 3795:Grouping
题目链接:http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=3795
题意:给定N个点M条单向边,问最少需要分多少个集合,使得集合内的点都不能达到其他点
思路:对于强连通分量内的点肯定不能在一个集合,另外该分量所在的那一条路径上的点也不能在同一个集合,故缩点后处理出一条点权和最大的路径即是答案
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
#include <vector>
#include <utility>
#include <cmath>
#include <queue>
#include <set>
#include <map>
#include <climits>
#include <functional>
#include <deque>
#include <ctime>
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
#pragma comment(linker, "/STACK:102400000,102400000")
using namespace std;
const int maxn = 101000;
const int maxm = 300010;
const int inf = 0x3f3f3f3f;
int cnt, head[maxn];
int low[maxn], dfn[maxn], sta[maxn], bel[maxn];
int idx, top, scc;
bool insta[maxn];
int num[maxn];
struct edge
{
int to, w, nxt;
} e[maxm];
void init()
{
cnt = 0;
memset(head, -1, sizeof(head));
}
void add(int u, int v)
{
e[cnt].to = v;
e[cnt].nxt = head[u];
head[u] = cnt++;
}
void tarjan(int u)
{
int v;
low[u] = dfn[u] = ++idx;
sta[top++] = u;
insta[u] = true;
for (int i = head[u]; ~i; i = e[i].nxt)
{
int v = e[i].to;
if (!dfn[v])
{
tarjan(v);
if (low[u] > low[v])
low[u] = low[v];
}
else if (insta[v] && low[u] > dfn[v])
low[u] = dfn[v];
}
if (low[u] == dfn[u])
{
scc++;
do
{
v = sta[--top];
insta[v] = false;
bel[v] = scc;
num[scc]++;
} while (v != u);
}
}
void solve(int n)
{
idx = scc = top = 0;
memset(dfn, 0, sizeof(dfn));
memset(insta, 0, sizeof(insta));
memset(num, 0, sizeof(num));
for (int i = 1; i <= n; i++)
if (!dfn[i])
tarjan(i);
}
vector <int> g[maxn];
int dp[maxn];
int dfs(int u)
{
if (dp[u] != -1) return dp[u];
int res = 0;
for (int i = 0; i < g[u].size(); i++)
{
int v = g[u][i];
res = max(res, dfs(v));
}
dp[u] = res + num[u];
return dp[u];
}
int main()
{
int n, m;
while (~scanf("%d%d", &n, &m))
{
init();
for (int i = 0; i < m; i++)
{
int u, v;
scanf("%d%d", &u, &v);
add(u, v);
}
solve(n);
memset(dp, -1, sizeof(dp));
for (int i = 0; i <= scc; i++)
g[i].clear();
for (int u = 1; u <= n; u++)
{
for (int i = head[u]; ~i; i = e[i].nxt)
{
int v = e[i].to;
if (bel[u] != bel[v])
g[bel[u]].push_back(bel[v]);
}
}
// for (int i = 1; i <= scc; i++)
// for (int j = 0; j < g[i].size(); j++)
// printf("********** %d\n", g[i][j]);
int ans = 0;
for (int i = 1; i <= scc; i++)
ans = max(ans, dfs(i));
cout << ans << endl;
}
return 0;
}