diff --git a/OnlineJudges/poj/1523.cpp b/OnlineJudges/poj/1523.cpp new file mode 100644 index 0000000..28928db --- /dev/null +++ b/OnlineJudges/poj/1523.cpp @@ -0,0 +1,60 @@ +#include +#include +#include +#define clz(X) memset(X, 0, sizeof(X)) +inline int max(int a, int b) { return a > b ? a : b; } +inline int min(int a, int b) { return a < b ? a : b; } +inline void read(int &x) +{ + int ch = x = 0; + while (!isdigit(ch = getchar())); + for (; isdigit(ch); ch = getchar()) + x = x * 10 + ch - '0'; +} +int map[1010][1010], range; +int dfn[1010], low[1010], idx; +int son, subnet[1010]; +void tarjan(int u) +{ + dfn[u] = low[u] = ++idx; + for (int v = 1; v <= range; v++) + if (map[u][v]) + if (!dfn[v]) + { + tarjan(v); + low[u] = min(low[u], low[v]); + if (low[v] >= dfn[u]) + (u == 1 ? son : subnet[u])++; + } + else + low[u] = min(low[u], dfn[v]); +} +int main() +{ + int x, y, T = 0; + while (read(x), x) + { + clz(map), clz(dfn), clz(low), clz(subnet), son = idx = 0; + read(y); + map[x][y] = map[y][x] = 1; + range = max(x, y); + while (read(x), x) + { + read(y); + map[x][y] = map[y][x] = 1; + range = max(range, max(x, y)); + } + printf("Network #%d\n", ++T); + tarjan(1); + bool flag = false; + if (son > 1) subnet[1] = son - 1; + for (int i = 1; i <= range; i++) + if (subnet[i]) + printf(" SPF node %d leaves %d subnets\n", i, subnet[i] + 1), + flag = true; + if (!flag) + puts(" No SPF nodes"); + putchar('\n'); + } + return 0; +} \ No newline at end of file diff --git a/OnlineJudges/poj/3177.cpp b/OnlineJudges/poj/3177.cpp new file mode 100644 index 0000000..092d5d8 --- /dev/null +++ b/OnlineJudges/poj/3177.cpp @@ -0,0 +1,45 @@ +#include +inline int min(int a, int b) { return a < b ? a : b; } +int head[5010], to[20010], next[20010], ecnt, map[5010][5010]; +int dfn[5010], low[5010], idx, cnt[5010]; +void addEdge(int f, int t) +{ + ecnt++; + next[ecnt] = head[f]; + head[f] = ecnt; + to[ecnt] = t; +} +void tarjan(int u, int p) +{ + dfn[u] = low[u] = ++idx; + for (int e = head[u]; e; e = next[e]) + if (!dfn[to[e]]) + tarjan(to[e], u), low[u] = min(low[u], low[to[e]]); + else if (to[e] != p) + low[u] = min(low[u], dfn[to[e]]); +} +int main() +{ + int n, m; + scanf("%d%d", &n, &m); + for (int i = 1, x, y; i <= m; i++) + { + scanf("%d%d", &x, &y); + if (!map[x][y]) + { + addEdge(x, y); + addEdge(y, x); + map[x][y] = map[y][x] = true; + } + } + tarjan(1, 0); + for (int i = 1; i <= n; i++) + for (int e = head[i]; e; e = next[e]) + if (low[to[e]] != low[i]) + cnt[low[i]]++; + int ans = 0; + for (int i = 1; i <= n; i++) + ans += cnt[i] == 1; + printf("%d", (ans + 1) >> 1); + return 0; +} diff --git a/省选准备/assets/day1/poj1523.cpp b/省选准备/assets/day1/poj1523.cpp new file mode 100644 index 0000000..28928db --- /dev/null +++ b/省选准备/assets/day1/poj1523.cpp @@ -0,0 +1,60 @@ +#include +#include +#include +#define clz(X) memset(X, 0, sizeof(X)) +inline int max(int a, int b) { return a > b ? a : b; } +inline int min(int a, int b) { return a < b ? a : b; } +inline void read(int &x) +{ + int ch = x = 0; + while (!isdigit(ch = getchar())); + for (; isdigit(ch); ch = getchar()) + x = x * 10 + ch - '0'; +} +int map[1010][1010], range; +int dfn[1010], low[1010], idx; +int son, subnet[1010]; +void tarjan(int u) +{ + dfn[u] = low[u] = ++idx; + for (int v = 1; v <= range; v++) + if (map[u][v]) + if (!dfn[v]) + { + tarjan(v); + low[u] = min(low[u], low[v]); + if (low[v] >= dfn[u]) + (u == 1 ? son : subnet[u])++; + } + else + low[u] = min(low[u], dfn[v]); +} +int main() +{ + int x, y, T = 0; + while (read(x), x) + { + clz(map), clz(dfn), clz(low), clz(subnet), son = idx = 0; + read(y); + map[x][y] = map[y][x] = 1; + range = max(x, y); + while (read(x), x) + { + read(y); + map[x][y] = map[y][x] = 1; + range = max(range, max(x, y)); + } + printf("Network #%d\n", ++T); + tarjan(1); + bool flag = false; + if (son > 1) subnet[1] = son - 1; + for (int i = 1; i <= range; i++) + if (subnet[i]) + printf(" SPF node %d leaves %d subnets\n", i, subnet[i] + 1), + flag = true; + if (!flag) + puts(" No SPF nodes"); + putchar('\n'); + } + return 0; +} \ No newline at end of file diff --git a/省选准备/assets/day1/poj3177.cpp b/省选准备/assets/day1/poj3177.cpp new file mode 100644 index 0000000..092d5d8 --- /dev/null +++ b/省选准备/assets/day1/poj3177.cpp @@ -0,0 +1,45 @@ +#include +inline int min(int a, int b) { return a < b ? a : b; } +int head[5010], to[20010], next[20010], ecnt, map[5010][5010]; +int dfn[5010], low[5010], idx, cnt[5010]; +void addEdge(int f, int t) +{ + ecnt++; + next[ecnt] = head[f]; + head[f] = ecnt; + to[ecnt] = t; +} +void tarjan(int u, int p) +{ + dfn[u] = low[u] = ++idx; + for (int e = head[u]; e; e = next[e]) + if (!dfn[to[e]]) + tarjan(to[e], u), low[u] = min(low[u], low[to[e]]); + else if (to[e] != p) + low[u] = min(low[u], dfn[to[e]]); +} +int main() +{ + int n, m; + scanf("%d%d", &n, &m); + for (int i = 1, x, y; i <= m; i++) + { + scanf("%d%d", &x, &y); + if (!map[x][y]) + { + addEdge(x, y); + addEdge(y, x); + map[x][y] = map[y][x] = true; + } + } + tarjan(1, 0); + for (int i = 1; i <= n; i++) + for (int e = head[i]; e; e = next[e]) + if (low[to[e]] != low[i]) + cnt[low[i]]++; + int ans = 0; + for (int i = 1; i <= n; i++) + ans += cnt[i] == 1; + printf("%d", (ans + 1) >> 1); + return 0; +} diff --git a/省选准备/assets/day1/tarjan-bcc.cpp b/省选准备/assets/day1/tarjan-bcc.cpp new file mode 100644 index 0000000..11fb6e9 --- /dev/null +++ b/省选准备/assets/day1/tarjan-bcc.cpp @@ -0,0 +1,9 @@ +void tarjan(int u, int p) +{ + dfn[u] = low[u] = ++idx; + for (int e = head[u]; e; e = next[e]) + if (!dfn[to[e]]) + tarjan(to[e], u), low[u] = min(low[u], low[to[e]]); + else if (to[e] != p) + low[u] = min(low[u], dfn[to[e]]); +} \ No newline at end of file diff --git a/省选准备/assets/day1/tarjan.cpp b/省选准备/assets/day1/tarjan-scc.cpp similarity index 100% rename from 省选准备/assets/day1/tarjan.cpp rename to 省选准备/assets/day1/tarjan-scc.cpp diff --git a/省选准备/省选基础算法.pdf b/省选准备/省选基础算法.pdf index f60ecfd..d4037dc 100644 Binary files a/省选准备/省选基础算法.pdf and b/省选准备/省选基础算法.pdf differ diff --git a/省选准备/省选基础算法.tex b/省选准备/省选基础算法.tex index efc698b..9e57ef2 100644 --- a/省选准备/省选基础算法.tex +++ b/省选准备/省选基础算法.tex @@ -24,12 +24,10 @@ \\$\}$\\ 当结点$u$的搜索过程结束后，若$dfn(u)=low(u)$，则以$u$为根的搜索子树上所有还在栈中的结点是一个强连通分量。 \subparagraph{代码}$\\$ - \codeinput[tarjan - SCC]{assets/day1/tarjan.cpp} + \codeinput[tarjan - SCC]{assets/day1/tarjan-scc.cpp} \paragraph{练习题} \subparagraph{ - \href{http://poj.org/problem?id=2186}{POJ2186}/ - \href{http://www.lydsy.com/JudgeOnline/problem.php?id=1051}{BZOJ1051} - - Popular Cows} 双倍的快乐 + \href{http://poj.org/problem?id=2186}{POJ2186}/\href{http://www.lydsy.com/JudgeOnline/problem.php?id=1051}{BZOJ1051} - Popular Cows} 双倍的快乐 \codeinput[Popular Cows]{assets/day1/poj2186.cpp} \subparagraph{\href{http://poj.org/problem?id=3180}{POJ3180} - The Cow Prom} \verb|The N (2 <= N <= 10,000) cows are so excited.| @@ -37,33 +35,32 @@ \subparagraph{\href{http://poj.org/problem?id=3180}{POJ1236} - Network of Schools} 强连通分量缩点求出度为0的和入度为0的分量个数 \codeinput[Network of Schools]{assets/day1/poj1236.cpp} - \newpage - \section{day2 } - \newpage - \section{day3 } - \newpage - \section{day4 } - \newpage - \section{day5 } - \newpage - \section{day6 } - \newpage - \section{day7 } - \newpage - \section{day8 } - \newpage - \section{day9 } - \newpage - \section{day10 } - \newpage - \section{day11 } - \newpage - \section{day12 } - \newpage - \section{day13 } - \newpage - \section{day14 } - \newpage - \section{day15 } - \newpage + \subsection{图的割点、桥与双连通分量} + \paragraph{定义} + \subparagraph{点连通度与边连通度} + 在一个\textbf{无向连通图}中，如果有一个顶点集合$V$，删除顶点集合$V$，以及与$V$中顶点相连（至少有一端在$V$中）的所有边后，原图\textbf{不连通}，就称这个点集$V$为\textbf{割点集合}。\\ + 一个图的\textbf{点连通度}的定义为：最小割点集合中的顶点数。\\ + 类似的，如果有一个边集合，删除这个边集合以后，原图不连通，就称这个点集为\textbf{割边集合}。 + \subparagraph{双连通图、割点与桥} + 如果一个无向连通图的\textbf{点连通度大于$1$}，则称该图是\textbf{点双连通的(point biconnected)}，简称双连通或重连通。一个图有\textbf{割点}，当且仅当这个图的点连通度为$1$，则割点集合的唯一元素被称为\textbf{割点(cut point)}，又叫关节点(articulation point)。一个图可能有多个割点。\\ + 如果一个无向连通图的\textbf{边连通度大于$1$}，则称该图是\textbf{边双连通的(edge biconnected)}，简称双连通或重连通。一个图有\textbf{桥}，当且仅当这个图的边连通度为$1$，则割边集合的唯一元素被称为\textbf{桥(bridge)}，又叫关节边(articulation edge)。一个图可能有多个桥。\\ + 可以看出，点双连通与边双连通都可以简称为双连通，它们之间是有着某种联系的，下文中提到的双连通，均既可指点双连通，又可指边双连通。（但这并不意味着它们等价）\\ + 双连通分量（分支）：在图 G 的所有子图 G'中，如果 G'是双连通的，则称 G'为双连通子图。如果一个双连通子图 G'它不是任何一个双连通子图的真子集，则 G'为极大双连通子图。双连通分量(biconnected component)，或重连通分量，就是图的极大双连通子图。特殊的，点双连通分量又叫做块。 + \paragraph{Tarjan 算法} + 给出函数$low(u)$使得\\ + $low(u) = min$ + \\$\{$\\ + \verb| |$dfn(u),$\\ + \verb| |$low(v),$ \quad $(u,v)$为树枝边(父子边)\\ + \verb| |$dfn(v)\;$ \quad $(u,v)$为后向边(返祖边) 等价于$dfn(v)