# tarjan algorithm board

## Undirected graph

• concept

• time stamp
\(dfn[x] \), in depth first traversal, integer marks are made in the order in which each node is first accessed

• Retroactive value
\(low[x] \), the minimum timestamp that can be reached by a non searching edge

• ### Edge cutting criterion

• An undirected edge \ ((x,y) \) is a cut edge / bridge if and only if there is a child node of X satisfying \ (DFN [x] < low [y] \)
After deleting the undirected edge ((x,y)), the graph is broken into two parts

• board

``````int dfn[N], low[N], dfcnt;
bool g[M];
void tarjan(int x, int ei) {
dfn[x] = low[x] = ++dfcnt;
for(int i = head[x]; i; i = e[i].next) {
int y = e[i].t;
if (!dfn[y]) {
tarjan(y, i);
low[x] = min(low[x], low[y]);
if (dfn[x] < low[y]) g[i] = g[i^1] = 1;
}
else if (i != (ei^1)) low[x] = min(low[x], dfn[y]);
}
}
``````
• ### Cut point decision rule

• If x is not a root node, then x is a cut point if and only if there is a child node y satisfying \ (DFN [x] \ \ Leq low [y] \)
If x is the root node, then x is the cut point if and only if there are at least two child nodes (y)_ 1,y_ 2)

• board

``````int dfn[N], low[N], dfcnt, rt;
bool g[N];
void tarjan(int x) {
dfn[x] = low[x] = ++dfcnt;
int son = 0;
for (int i = head[x]; i; i = e[i].next) {
int y = e[i].t;
if (!dfn[y]) {
tarjan(y);
low[x] = min(low[x], low[y]);
if (dfn[x] <= low[y]) {
son++;
if (x != rt || son > 1) g[x] = 1;
}
}
else low[x] = min(low[x], dfn[y]);
}
}
``````
• ### Point double connected component

• For every double center, there is no cut point in the graph

• board

``````int dfn[N], low[N], dfcnt, sta[N], top, cnt;
vector<int> dcc[N];
bool  g[N];
void tarjan(int x, int rt) {
dfn[x] = low[x] = ++dfcnt;
sta[++top] = x;
int son = 0;
for (int i = head[x]; i; i = e[i].next) {
int y = e[i].t;
if (!dfn[y]) {
tarjan(y);
low[x] = min(low[x], low[y]);
if (dfn[x] <= low[y]) {
son++;
if (x != rt || son > 1)  g[x] = 1;
dcc[++cnt].clear();
while (1) {
int z = sta[top--];
dcc[cnt].push_back(z);
if (y == z) break;
}
dcc[cnt].push_back(x);
}
}
else low[x] = min(low[x], dfn[y]);
}
}
``````
• ### Side double connected component

• For an edge double, any two points have two non coincident paths

• board

``````int dfn[N], low[N], dfcnt;
bool g[M];
void tarjan(int x, int ei) {
dfn[x] = low[x] = ++dfcnt;
for(int i = head[x]; i; i = e[i].next) {
int y = e[i].t;
if (!dfn[y]) {
tarjan(y, i);
low[x] = min(low[x], low[y]);
if (dfn[x] < low[y]) g[i] = g[i^1] = 1;
}
else if (i != (ei^1)) low[x] = min(low[x], dfn[y]);
}
}
int n, m, d[N], b[N], cnt, ans;
void dfs(int x) {
b[x] = cnt;
for(int i = head[x]; i; i = e[i].next) {
int y = e[i].t;
if (b[y] || g[i]) continue;
dfs(y);
}
}
int main() {
//~~~
for(int i = 1; i <= n; i++)
if (!dfn[i]) tarjan(i, 0);
for(int i = 1; i <= n; i++)
if (!b[i]) cnt++, dfs(i);
//~~~
return 0;
}
``````

# Digraph

• ## Strong connected components of Digraphs

• In a strong connectivity component, if there is a path from x to y, there is a path from y to x

• board

``````void tarjan(int x) {
dfn[x] = low[x] = ++dfcnt;
s[++top] = x;
for(int i = head[x]; i; i = e[i].next) {
int y = e[i].t;
if (!dfn[y]) tarjan(y), low[x] = min(low[x], low[y]);
else if (!b[y]) low[x] = min(low[x], dfn[y]);
}
if (dfn[x] == low[x]) {
cnt++;
while(1) {
int y = s[top--];
b[y] = cnt;
size[cnt]++;
if (x == y) break;
}
}
}
``````

### Example

``````#include <queue>
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
const int N = 1e4+5, M = 1e5+5;
struct side { int t, next; } e[M][2];
void add(int x, int y, int k) {
e[tot[k]][k].t = y;
}
int n, m, w[N], r[N], d[N], ans;
int dfn[N], low[N], dfcnt, sta[N], top, cnt, bel[N], sum[N];
void tarjan(int x) {
dfn[x] = low[x] = ++dfcnt;
sta[++top] = x;
for (int i = head[x][0]; i; i = e[i][0].next) {
int y = e[i][0].t;
if (!dfn[y]) tarjan(y), low[x] = min(low[x], low[y]);
else if (!bel[y]) low[x] = min(low[x], dfn[y]);
}
if (dfn[x] == low[x]) {
cnt++;
while (1) {
int y = sta[top--];
bel[y] = cnt;
sum[cnt] += w[y];
if (x == y) break;
}
}
}
queue<int> q;
int tuopu() {
for (int i = 1; i <= cnt; i++)
if (!r[i]) q.push(i), d[i] = sum[i];
while (!q.empty()) {
int x = q.front(); q.pop();
for (int i = head[x][1]; i; i = e[i][1].next) {
int y = e[i][1].t;
d[y] = max(d[y], d[x] + sum[y]);
if (--r[y] == 0) q.push(y);
}
}
for (int i = 1; i <= cnt; i++)
ans = max(ans, d[i]);
return ans;
}
int main() {
scanf("%d%d", &n, &m);
for (int i = 1; i <= n; i++)
scanf("%d", &w[i]);
for (int i = 1; i <= m; i++) {
int x, y;
scanf("%d%d", &x, &y);
}
for (int i = 1; i <= n; i++)
if (!dfn[i]) tarjan(i);
for (int x = 1; x <= n; x++)
for (int i = head[x][0]; i; i = e[i][0].next) {
int y = e[i][0].t;
if (bel[x] != bel[y])