与图论的邂逅05：最近公共祖先LCA

什么是LCA？

祖先链

对于一棵树T，若它的根节点是r，对于任意一个树上的节点x，从r走到x的路径是唯一的(显然)，那么这条路径上的点都是并且只有这些点是x的祖先。这些点组成的链(或者说路径)就是x的祖先链。

LCA

根据名字来说，最近公共祖先就是两个点最近的相同祖先。实际上也可以理解为：两个点的祖先链深度最大的那个交点。极端的情况下，LCA可以就是两个点之一，或者就是根节点root。

顺便贴下eg:

树中节点8和7的LCA为3，节点4和7的LCA为1，节点5和2的LCA为2。

*可以写成LCA(u,v)=w的形式

---

LCA的求法

问题：给出一棵有n(n≤500000)个节点的树，并给出m(m≤500000)个询问，每个询问给出两个点u,v，请你求出每个u,v的LCA。

1.向上标记法

首先根据定义来。LCA(u,v)是两个点的祖先链的第一个交点(从下往上第一个)。那么我们可以先从u开始往根节点走，那么走到的点都是u的祖先，走出的路径就是u的祖先链。那么我们在走的时候把祖先链上的点都标记一下，再从v开始往根节点走，走到的第一个被标记过的点就是LCA了。最坏的情况下时间复杂度为O(n)，总共就是O(n*m)。显然不能满足题目的需要~

2.树上倍增法

一个一个地跳显然太慢，不如加加速？怎么加速呢？根据一贯套路，慢了就倍增~为什么这里不倍增呢？那就倍增一下咯~

树上倍增的思路不同于标记的思路，因为倍增的时候是无法标记的。反正跳得快，就干脆让u,v都跳一跳，跳到同一个点时就到了LCA了。不过不是一上来就一起跳的，树上倍增的预处理可是很浩大的。

首先预处理出倍增之后跳到的点是谁。设f(i,j)为节点i往上跳2^j步到达的节点，那么

f(i,j)=f(f(i,j-1),j-1)

初始化f(i,0)=fa(i)，即i的父亲。(刚才那里真的是零......)

设d(i)为i的深度，顺便把这货也处理出来(这么简单就不讲了)。

弄了这么多之后还不能直接两个点倍增。首先需要将深度大的那个点(假设是v)往上倍增到和u深度相同为止。基于二进制转化的思想，任何树都可以用二进制表示出来，所以绝对是可以倍增到同一深度的。

接着，两个点同时倍增。经过若干次倍增之后，若fa(u)==fa(v)，那么fa(u)就是LCA了。

放上预处理的代码：

 

 

 

xxxxxxxxxx

 

 

 

 

inline void bfs(){

    q.push(s);

    d[s] = 1;

    while(!q.empty()){

        int u = q.front(); q.pop();

        for(int i = head[u]; ~i; i = e[i].next){

            int v = e[i].to;

            if(d[v]) continue;

            d[v] = d[u] + 1,f[v][0] = u;

            for(int j = 1; j <= t; j++) f[v][j] = f[f[v][j - 1]][j - 1];

            q.push(v);

        }

    }

}

​

//在main函数中

t = (int)(log(n) / log(2)) + 1;

bfs();

 

 

以及倍增的代码：

 

 

 

xxxxxxxxxx

 

 

 

 

inline int lca(int &u, int &v){

    if(d[u] > d[v]) swap(u, v);

    for(int i = t; i >= 0; i--) if(d[f[v][i]] >= d[u]){

        v = f[v][i];

    }

    if(u == v) return u;

    for(int i = t; i >= 0; i--) if(f[u][i] != f[v][i]){

        u = f[u][i];

        v = f[v][i];

    }

    return f[u][0];

}

 

 

*一般数组会开成这样：

 

 

 

xxxxxxxxxx

 

 

 

 

int f[maxn][20];

 

 

这样一般就够用了。

预处理的复杂度为O(nlogn)，每个询问的复杂度为O(logn)，一共就是O((n+m)logn)，可以跑过题目的数据。

---

3.Tarjan

不同于倍增，Tarjan是一种离线算法，并且很优秀很优秀(就是难写)。考虑到刚接触LCA的OIer可能难以理解Tarjan，这里我用几种不同的方式来讲。

本人的方式：你首先在心里构造出一棵树来......标记祖先链的方式其实可以多个询问一起进行。首先依照Tarjan的流程，遍历这棵树。假设现在遍历到了点u，并且开始回溯，那么你可以想象在回溯的过程中实际上是从节点u往上拉出了一条祖先链。当拉链子拉到了一处分叉，并且分叉的另一边还没去过时，就停止拉链，往下走。假设这个过程中走到了节点v并开始回溯，并且询问里有问u和v的LCA，那么在从v回溯的过程中相当于从v开始也拉上去了一条祖先链，会一直拉到之前的分叉处，此时u和v的祖先链便第一次相交，交于分叉处，这个分叉处便是LCA(u,v)。也就是说，当我们遍历到一个点v，发现询问里有LCA(u,v)这个询问，并且从u已经拉出了祖先链，那么LCA(u,v)就是此时u的祖先链的顶端。其实我们不该只局限于这一个询问，从宏观上看，在遍历的过程中我们其实从每个回溯过的点都拉出了一条祖先链来。如果你觉得拉的链子太多，我们可以剪掉一些：只留下询问里涉及的点的祖先链。那么这些祖先链的交点就是遍历过程中走到的那些分叉口，所以一次遍历就可以求出所有询问的LCA。

还是附个流程图吧......

 

#mermaidChart139 { color: rgba(184, 191, 198, 1); font: normal normal 400 normal 14.4px / 26px Monaco, Consolas, "Andale Mono", "DejaVu Sans Mono", monospace }

我们先在要求：LCA(9,10),LCA(9,5),LCA(8,7),LCA(9,8)。

然后我们从左到右遍历下去，遍历到了9号节点，便往回拉祖先链：

 

#mermaidChart140 { color: rgba(184, 191, 198, 1); font: normal normal 400 normal 14.4px / 26px Monaco, Consolas, "Andale Mono", "DejaVu Sans Mono", monospace }

(箭头的边是祖先链)现在拉到了4号节点，发现有分叉，往下走到10号节点，发现询问里有LCA(9,10)这个询问，并且从9已经拉出了祖先链，那么LCA(9,10)就是此时9的祖先链的顶端：4号节点。(由于关于10的询问都处理完了，10的祖先链就不画了)此时继续往上拉链：

此时发现又有一个分叉，便往分叉走，走到5的位置，并发现询问里有LCA(9,5)，那么LCA(9,5)就是此时9的祖先链的顶端：2号节点。(5的祖先链也不画了)继续往上拉链。

发现有分叉，往下走到8的位置，发现询问里有LCA(9,8)，那么LCA(9,8)就是此时9的祖先链的顶端：1号节点。接着从8往上拉祖先链：

发现有分叉，往下走到7的位置，发现询问里有LCA(8,7)，那么LCA(8,7)就是此时8的祖先链的顶端：3号节点。

这样应该就可以理解了......整个遍历的复杂度为O(n)，加上回答询问的复杂度总共也才O(n+m)。像上面题目里的数据范围可以随便跑。

下面是教练的方式：拉祖先链的过程可以想象成灌水......当你往下灌到底部时，水就会慢慢往上涨，当涨到一个分岔口时就会往另一边流。其实也差不多啦~

附上存询问的代码(用邻接表来存，方便查询)：

 

 

 

xxxxxxxxxx

 

 

 

 

struct edge{

    int to, next, lca;

    edge(){}

    edge(register const int &_to, register const int &_next){

        to = _to,next = _next;

    }

}qe[maxm << 1];

int qhead[maxn], qk;

​

inline void qadd(register const int &u, register const int &v){

    qe[qk] = edge(v, qhead[u]);

    qhead[u] = qk++;

}

​

//main函数中

    memset(qhead, -1, sizeof qhead);

    for(register int i = 1; i <= m; i++){

        scanf("%d%d", &u, &v);

        qadd(u, v),qadd(v, u);

    }

 

 

然后是Tarjan函数：

 

 

 

xxxxxxxxxx

 

 

 

 

inline int find(register const int &x){

    if(fa[x] == x) return x;

    return fa[x] = find(fa[x]);

}//用并查集的方式查找祖先链的顶端

​

void LCA_tarjan(int u, int pre){

    vis[u] = true;

    for(register int i = head[u]; ~i; i = e[i].next){

        int v = e[i].to;

        if(v != pre){

            LCA_tarjan(v, u);

            fa[v] = u;//拉祖先链

        }

    }

    for(register int i = qhead[u]; ~i; i = qe[i].next){

        v = qe[i].to;

        //如果v已经拉出了祖先链，就回答询问

        if(vis[v]) qe[i].lca = qe[i ^ 1].lca = find(v);

    }

}

​

//main函数中

    for(register int i = 1; i <= n; i++) fa[i] = i;

    LCA_tarjan(s, 0);//s为根

 

 

以及回答询问：

 

 

 

xxxxxxxxxx

 

 

 

 

    for(int i = 0; i < qk; i += 2) printf("%d\n", qe[i].lca);

 

 

考虑到并查集的存在，Tarjan的复杂度其实为：O(n+mlogn)，只不过实际远远达不到这个程度而已。

---

---

只有倍增和Tarjan两种算法可以跑LCA？

4.树链剖分

*声明：如果你不会树链剖分你可以不看这一块。

树上倍增嫌慢？Tarjan嫌内存太大操作太麻烦？树链剖分求LCA，你值得拥有！类似于树上倍增的思想，只不过加快了往上跳的速度而已。树链剖分的方法是，一条链子一条链子地往上跳！当跳到两个点所在的链子为同一条时，浅的那个点就是LCA。

 

 

 

xxxxxxxxxx

 

 

 

 

#include <stdio.h>

#include <string.h>

#define maxn 500010

#define maxm 500010

​

struct graph{

    struct edge{

        int to, next;

        edge(){}

        edge(const int &_to, const int &_next){

            to = _to;

            next = _next;

        }

    }e[maxm << 1];

    int head[maxn], k;

    inline void init(){

        memset(head, -1, sizeof head);

        k = 0;

    }

    inline void add(const int &u, const int &v){

        e[k] = edge(v, head[u]);

        head[u] = k++;

    }

}g;

​

int fa[maxn], son[maxn], size[maxn], dep[maxn];

int dfn[maxn], id[maxn], top[maxn], cnt[maxn], tot;

int n, m, s;

​

inline void swap(int &x, int &y){int t = x; x = y; y = t;}

​

inline void dfs_getson(int u){

    size[u] = 1;

    for(int i = g.head[u]; ~i; i = g.e[i].next){

        int v = g.e[i].to;

        if(v == fa[u]) continue;

        dep[v] = dep[u] + 1;

        fa[v] = u;

        dfs_getson(v);

        size[u] += size[v];

        if(size[v] > size[son[u]]) son[u] = v;

    }

}

​

inline void dfs_rewrite(int u, int tp){

    top[u] = tp;

    dfn[u] = ++tot;

    id[tot] = u;

    if(son[u]) dfs_rewrite(son[u], tp);

    for(int i = g.head[u]; ~i; i = g.e[i].next){

        int v = g.e[i].to;

        if(v != son[u] && v != fa[u]) dfs_rewrite(v, v);

    }

    cnt[u] = tot;

}

​

inline int lca(int u, int v){

    while(top[u] != top[v]){

        if(dep[top[u]] > dep[top[v]]) swap(u, v);

        v = fa[top[v]];

    }

    if(dep[u] > dep[v]) swap(u, v);

    return u;

}

​

int main(){

    g.init();

    scanf("%d%d%d", &n, &m, &s);

    for(int i = 1; i < n; i++){

        int u, v;

        scanf("%d%d", &u, &v);

        g.add(u, v);

        g.add(v, u);

    }

    dfs_getson(s);

    dfs_rewrite(s, s);

    for(int i = 1; i <= n; i++){

        int u, v;

        scanf("%d%d", &u, &v);

        printf("%d\n", lca(u, v));

    }

    return 0;

}

 

 

#滑稽

经过亲测，放出三种算法的成绩：

树上倍增——用时2287ms,内存50.66MB

Tarjan——用时1272ms,内存29.76MB

树链剖分——用时1538ms,内存25.49MB

综合一下时间和内存的开销，Tarjan和树剖完爆树上倍增......不过树剖的过程中可以很容易得实现更复杂的操作：区间修改、子树修改之类的，个人认为树剖比Tarjan更优。

加上常数优化的树剖：用时1102ms,内存25.55MB

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sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgba(85, 85, 85, 1); border-radius: 4px; cursor: pointer }
sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit }
#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit }
figure { overflow-x: auto; margin: 1.2em 0; max-width: calc(100% + 16px); padding: 0 }
figure>table { margin: 0 !important }
tr { break-inside: avoid; break-after: auto }
thead { display: table-header-group }
table { border-collapse: collapse; border-spacing: 0; width: 100%; overflow: auto; break-inside: auto; text-align: left }
table.md-table td { min-width: 32px }
.CodeMirror-gutters { border-right: 0; background-color: inherit }
.CodeMirror-linenumber { user-select: none }
.CodeMirror { text-align: left }
.CodeMirror-placeholder { opacity: 0.3 }
.CodeMirror pre { padding: 0 4px }
.CodeMirror-lines { padding: 0 }
div.hr:focus { cursor: none }
#write pre { white-space: pre-wrap }
#write.fences-no-line-wrapping pre { white-space: pre }
#write pre.ty-contain-cm { white-space: normal }
.CodeMirror-gutters { margin-right: 4px }
.md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; position: relative !important }
.md-diagram-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0; padding-bottom: 8px; overflow-x: auto }
#write .md-fences.mock-cm { white-space: pre-wrap }
.md-fences.md-fences-with-lineno { padding-left: 0 }
#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto }
.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px }
.CodeMirror-line, twitterwidget { break-inside: avoid }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em }
.footnotes+.footnotes { margin-top: 0 }
.md-reset { margin: 0; padding: 0; border: 0; outline: 0; vertical-align: top; background: left top; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr }
li div { padding-top: 0 }
blockquote { margin: 1rem 0 }
li .mathjax-block, li p { margin: 0.5rem 0 }
li { margin: 0; position: relative }
blockquote>:last-child { margin-bottom: 0 }
blockquote>:first-child, li>:first-child { margin-top: 0 }
.footnotes-area { color: rgba(136, 136, 136, 1); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal }
#write .footnote-line { white-space: pre-wrap }
@media print { body, html { border: 1px solid rgba(0, 0, 0, 0); height: 99%; break-after: avoid; break-before: avoid } #write { margin-top: 0; padding-top: 0; border-color: rgba(0, 0, 0, 0) !important } .typora-export * { -webkit-print-color-adjust: exact } html.blink-to-pdf { font-size: 13px } .typora-export #write { padding-left: 32px; padding-right: 32px; padding-bottom: 0; break-after: avoid } .typora-export #write::after { height: 0 } @page { margin-top: 20mm margin-right: 0 margin-bottom: 20mm margin-left: 0 } }
.footnote-line { margin-top: 0.714em; font-size: 0.7em }
a img, img a { cursor: pointer }
pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgba(204, 204, 204, 1); display: block; overflow-x: hidden }
p>.md-image:only-child:not(.md-img-error) img, p>img:only-child { display: block; margin: auto }
p>.md-image:only-child { display: inline-block; width: 100% }
#write .MathJax_Display { margin: 0.8em 0 0 }
.md-math-block { width: 100% }
.md-math-block:not(:empty)::after { display: none }
[contenteditable="true"]:active, [contenteditable="true"]:focus { outline: 0; box-shadow: none }
.md-task-list-item { position: relative; list-style-type: none }
.task-list-item.md-task-list-item { padding-left: 0 }
.md-task-list-item>input { position: absolute; top: 0; left: 0; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none }
.math { font-size: 1rem }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px }
.md-toc-content { position: relative; margin-left: 0 }
.md-toc-content::after, .md-toc::after { display: none }
.md-toc-item { display: block; color: rgba(65, 131, 196, 1) }
.md-toc-item a { text-decoration: none }
.md-toc-inner:hover { text-decoration: underline }
.md-toc-inner { display: inline-block; cursor: pointer }
.md-toc-h1 .md-toc-inner { margin-left: 0; font-weight: 700 }
.md-toc-h2 .md-toc-inner { margin-left: 2em }
.md-toc-h3 .md-toc-inner { margin-left: 4em }
.md-toc-h4 .md-toc-inner { margin-left: 6em }
.md-toc-h5 .md-toc-inner { margin-left: 8em }
.md-toc-h6 .md-toc-inner { margin-left: 10em }
@media screen and (max-width: 48em) { .md-toc-h3 .md-toc-inner { margin-left: 3.5em } .md-toc-h4 .md-toc-inner { margin-left: 5em } .md-toc-h5 .md-toc-inner { margin-left: 6.5em } .md-toc-h6 .md-toc-inner { margin-left: 8em } }
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit }
.footnote-line a:not(.reversefootnote) { color: inherit }
.md-attr { display: none }
.md-fn-count::after { content: "." }
code, pre, samp, tt { font-family: var(--monospace) }
kbd { margin: 0 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgba(36, 39, 41, 1); background: rgba(255, 255, 255, 1); border: 1px solid rgba(173, 179, 185, 1); border-radius: 3px; box-shadow: 0 1px rgba(12, 13, 14, 0.2), inset 0 0 2px rgba(255, 255, 255, 1); white-space: nowrap; vertical-align: middle }
.md-comment { color: rgba(162, 127, 3, 1); opacity: 0.8; font-family: var(--monospace) }
code { text-align: left; vertical-align: initial }
a.md-print-anchor { white-space: pre !important; border-style: none !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0 !important; outline: 0 !important; background: left top !important; text-shadow: initial !important }
.md-inline-math .MathJax_SVG .noError { display: none !important }
.html-for-mac .inline-math-svg .MathJax_SVG { vertical-align: 0.2px }
.md-math-block .MathJax_SVG_Display { text-align: center; margin: 0; position: relative; text-indent: 0; max-width: none; max-height: none; min-height: 0; min-width: 100%; width: auto; overflow-y: hidden; display: block !important }
.MathJax_SVG_Display, .md-inline-math .MathJax_SVG_Display { width: auto; display: inline-block !important }
.MathJax_SVG .MJX-monospace { font-family: var(--monospace) }
.MathJax_SVG .MJX-sans-serif { font-family: sans-serif }
.MathJax_SVG { display: inline; font-style: normal; font-weight: 400; line-height: normal; zoom: 90%; text-indent: 0; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; padding: 0; margin: 0 }
.MathJax_SVG * { }
.MathJax_SVG_Display svg { vertical-align: middle !important; margin-bottom: 0 !important }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif }
.md-diagram-panel>svg { max-width: 100% }
[lang="mermaid"] svg, [lang="flow"] svg { max-width: 100% }
[lang="mermaid"] .node text { font-size: 1rem }
table tr th { border-bottom: 0 }
video { max-width: 100%; display: block; margin: 0 auto }
iframe { max-width: 100%; width: 100%; border: none }
.highlight td, .highlight tr { border: 0 }
.CodeMirror { height: auto }
.CodeMirror.cm-s-inner { }
.CodeMirror-scroll { overflow-y: hidden; overflow-x: auto; z-index: 3 }
.CodeMirror-gutter-filler, .CodeMirror-scrollbar-filler { background-color: rgba(255, 255, 255, 1) }
.CodeMirror-gutters { border-right: 1px solid rgba(221, 221, 221, 1); white-space: nowrap }
.CodeMirror-linenumber { padding: 0 3px 0 5px; text-align: right; color: rgba(153, 153, 153, 1) }
.cm-s-inner .cm-keyword { color: rgba(119, 0, 136, 1) }
.cm-s-inner .cm-atom, .cm-s-inner.cm-atom { color: rgba(34, 17, 153, 1) }
.cm-s-inner .cm-number { color: rgba(17, 102, 68, 1) }
.cm-s-inner .cm-def { color: rgba(0, 0, 255, 1) }
.cm-s-inner .cm-variable { color: rgba(0, 0, 0, 1) }
.cm-s-inner .cm-variable-2 { color: rgba(0, 85, 170, 1) }
.cm-s-inner .cm-variable-3 { color: rgba(0, 136, 85, 1) }
.cm-s-inner .cm-string { color: rgba(170, 17, 17, 1) }
.cm-s-inner .cm-property { color: rgba(0, 0, 0, 1) }
.cm-s-inner .cm-operator { color: rgba(152, 26, 26, 1) }
.cm-s-inner .cm-comment, .cm-s-inner.cm-comment { color: rgba(170, 85, 0, 1) }
.cm-s-inner .cm-string-2 { color: rgba(255, 85, 0, 1) }
.cm-s-inner .cm-meta { color: rgba(85, 85, 85, 1) }
.cm-s-inner .cm-qualifier { color: rgba(85, 85, 85, 1) }
.cm-s-inner .cm-builtin { color: rgba(51, 0, 170, 1) }
.cm-s-inner .cm-bracket { color: rgba(153, 153, 119, 1) }
.cm-s-inner .cm-tag { color: rgba(17, 119, 0, 1) }
.cm-s-inner .cm-attribute { color: rgba(0, 0, 204, 1) }
.cm-s-inner .cm-header, .cm-s-inner.cm-header { color: rgba(0, 0, 255, 1) }
.cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgba(0, 153, 0, 1) }
.cm-s-inner .cm-hr, .cm-s-inner.cm-hr { color: rgba(153, 153, 153, 1) }
.cm-s-inner .cm-link, .cm-s-inner.cm-link { color: rgba(0, 0, 204, 1) }
.cm-negative { color: rgba(221, 68, 68, 1) }
.cm-positive { color: rgba(34, 153, 34, 1) }
.cm-header, .cm-strong { font-weight: 700 }
.cm-del { text-decoration: line-through }
.cm-em { font-style: italic }
.cm-link { text-decoration: underline }
.cm-error { color: rgba(255, 0, 0, 1) }
.cm-invalidchar { color: rgba(255, 0, 0, 1) }
.cm-constant { color: rgba(38, 139, 210, 1) }
.cm-defined { color: rgba(181, 137, 0, 1) }
div.CodeMirror span.CodeMirror-matchingbracket { color: rgba(0, 255, 0, 1) }
div.CodeMirror span.CodeMirror-nonmatchingbracket { color: rgba(255, 34, 34, 1) }
.cm-s-inner .CodeMirror-activeline-background { }
.CodeMirror { position: relative; overflow: hidden }
.CodeMirror-scroll { height: 100%; outline: 0; position: relative; box-sizing: content-box }
.CodeMirror-sizer { position: relative }
.CodeMirror-gutter-filler, .CodeMirror-hscrollbar, .CodeMirror-scrollbar-filler, .CodeMirror-vscrollbar { position: absolute; z-index: 6; display: none }
.CodeMirror-vscrollbar { right: 0; top: 0; overflow: hidden }
.CodeMirror-hscrollbar { bottom: 0; left: 0; overflow: hidden }
.CodeMirror-scrollbar-filler { right: 0; bottom: 0 }
.CodeMirror-gutter-filler { left: 0; bottom: 0 }
.CodeMirror-gutters { position: absolute; left: 0; top: 0; padding-bottom: 30px; z-index: 3 }
.CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block }
.CodeMirror-gutter-wrapper { position: absolute; z-index: 4; background: left top !important; border: none !important }
.CodeMirror-gutter-background { position: absolute; top: 0; bottom: 0; z-index: 4 }
.CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4 }
.CodeMirror-lines { cursor: text }
.CodeMirror pre { border-radius: 0; border-width: 0; background: left top; font-family: inherit; font-size: inherit; margin: 0; white-space: pre; word-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible }
.CodeMirror-wrap pre { word-wrap: break-word; white-space: pre-wrap; word-break: normal }
.CodeMirror-code pre { border-right: 30px solid rgba(0, 0, 0, 0) }
.CodeMirror-wrap .CodeMirror-code pre { border-right: none; width: auto }
.CodeMirror-linebackground { position: absolute; left: 0; right: 0; top: 0; bottom: 0; z-index: 0 }
.CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto }
.CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden }
.CodeMirror-measure { position: absolute; width: 100%; height: 0; overflow: hidden; visibility: hidden }
.CodeMirror-measure pre { position: static }
.CodeMirror div.CodeMirror-cursor { position: absolute; visibility: hidden; border-right: none; width: 0 }
.CodeMirror div.CodeMirror-cursor { visibility: hidden }
.CodeMirror-focused div.CodeMirror-cursor { visibility: inherit }
.cm-searching { background: rgba(255, 255, 0, 0.4) }
@media print { .CodeMirror div.CodeMirror-cursor { visibility: hidden } }
.mermaid .label { color: rgba(50, 61, 71, 1) }
g.label { color: rgba(51, 51, 51, 1) }
.node rect, .node circle, .node ellipse, .node polygon { fill: rgb(189, 213, 234); stroke: rgba(129, 177, 219, 1); stroke-width: 1px }
.edgePath .path { stroke: rgba(211, 211, 211, 1) }
.edgeLabel { background-color: rgba(232, 232, 232, 1) }
.cluster rect { fill: rgb(109, 109, 101) !important; rx: 4 !important; stroke: rgba(255, 255, 255, 0.25) !important; stroke-width: 1px !important }
.cluster text { fill: rgb(249, 255, 254) }
.actor { stroke: rgba(129, 177, 219, 1); fill: rgb(189, 213, 234) }
text.actor { fill: black; stroke: none }
.actor-line { stroke: rgba(211, 211, 211, 1) }
.messageLine0 { stroke-width: 1.5; stroke: rgba(211, 211, 211, 1) }
.messageLine1 { stroke-width: 1.5; stroke: rgba(211, 211, 211, 1) }
#arrowhead { fill: lightgrey !important }
#crosshead path { fill: lightgrey !important; stroke: rgba(211, 211, 211, 1) !important }
.messageText { fill: lightgrey; stroke: none }
.labelBox { stroke: rgba(129, 177, 219, 1); fill: rgb(189, 213, 234) }
.labelText { fill: rgb(50, 61, 71); stroke: none }
.loopText { fill: lightgrey; stroke: none }
text>tspan { fill: inherit }
.loopLine { stroke-width: 2; stroke: rgba(129, 177, 219, 1) }
.note { stroke: rgba(255, 255, 255, 0.25); fill: rgb(255, 245, 173) }
.noteText { fill: black; stroke: none; font-family: "trebuchet ms", verdana, arial; font-size: 14px }
.section { stroke: none; opacity: 0.2 }
.section0 { fill: rgba(255, 255, 255, 0.3) }
.section2 { fill: rgb(234, 232, 185) }
.section1, .section3 { fill: white; opacity: 0.2 }
.sectionTitle0 { fill: rgb(249, 255, 254) }
.sectionTitle1 { fill: rgb(249, 255, 254) }
.sectionTitle2 { fill: rgb(249, 255, 254) }
.sectionTitle3 { fill: rgb(249, 255, 254) }
.sectionTitle { text-anchor: start; font-size: 11px }
.grid .tick { stroke: rgba(255, 255, 255, 0.3); opacity: 0.3; shape-rendering: crispEdges }
.grid .tick text { fill: lightgrey; opacity: 0.5 }
.grid path { stroke-width: 0 }
.today { fill: none; stroke: rgba(219, 87, 87, 1); stroke-width: 2px }
.task { stroke-width: 1 }
.taskText { text-anchor: middle; font-size: 11px }
.taskTextOutsideRight { fill: rgb(50, 61, 71); text-anchor: start; font-size: 11px }
.taskTextOutsideLeft { fill: rgb(50, 61, 71); text-anchor: end; font-size: 11px }
.taskText0, .taskText1, .taskText2, .taskText3 { fill: rgb(50, 61, 71) }
.task0, .task1, .task2, .task3 { fill: rgb(189, 213, 234); stroke: rgba(255, 255, 255, 0.5) }
.taskTextOutside0, .taskTextOutside2 { fill: lightgrey }
.taskTextOutside1, .taskTextOutside3 { fill: lightgrey }
.active0, .active1, .active2, .active3 { fill: rgb(129, 177, 219); stroke: rgba(255, 255, 255, 0.5) }
.activeText0, .activeText1, .activeText2, .activeText3 { fill: rgb(50, 61, 71) !important }
.done0, .done1, .done2, .done3 { fill: lightgrey }
.doneText0, .doneText1, .doneText2, .doneText3 { fill: rgb(50, 61, 71) !important }
.crit0, .crit1, .crit2, .crit3 { stroke: rgba(232, 55, 55, 1); fill: rgb(232, 55, 55); stroke-width: 2 }
.activeCrit0, .activeCrit1, .activeCrit2, .activeCrit3 { stroke: rgba(232, 55, 55, 1); fill: rgb(129, 177, 219); stroke-width: 2 }
.doneCrit0, .doneCrit1, .doneCrit2, .doneCrit3 { stroke: rgba(232, 55, 55, 1); fill: lightgrey; stroke-width: 1; cursor: pointer; shape-rendering: crispEdges }
.doneCritText0, .doneCritText1, .doneCritText2, .doneCritText3 { fill: lightgrey !important }
.activeCritText0, .activeCritText1, .activeCritText2, .activeCritText3 { fill: rgb(50, 61, 71) !important }
.titleText { text-anchor: middle; font-size: 18px; fill: lightgrey }
.node text { font-family: "trebuchet ms", verdana, arial; font-size: 14px }
div.mermaidTooltip { position: absolute; text-align: center; max-width: 200px; padding: 2px; font-family: "trebuchet ms", verdana, arial; font-size: 12px; background: rgba(109, 109, 101, 1); border: 1px solid rgba(255, 255, 255, 0.25); border-radius: 2px; pointer-events: none; z-index: 100 }
.cm-s-inner .cm-variable, .cm-s-inner .cm-operator, .cm-s-inner .cm-property { color: rgba(184, 191, 198, 1) }
.cm-s-inner .cm-keyword { color: rgba(200, 143, 208, 1) }
.cm-s-inner .cm-tag { color: rgba(125, 244, 106, 1) }
.cm-s-inner .cm-attribute { color: rgba(117, 117, 228, 1) }
.CodeMirror div.CodeMirror-cursor { border-left: 1px solid rgba(184, 191, 198, 1); z-index: 3 }
.cm-s-inner .cm-string { color: rgba(210, 107, 107, 1) }
.cm-s-inner .cm-comment, .cm-s-inner.cm-comment { color: rgba(218, 146, 74, 1) }
.cm-s-inner .cm-header, .cm-s-inner .cm-def, .cm-s-inner.cm-header, .cm-s-inner.cm-def { color: rgba(141, 141, 240, 1) }
.cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgba(87, 172, 87, 1) }
.cm-s-inner .cm-hr { color: rgba(216, 213, 213, 1) }
.cm-s-inner .cm-link { color: rgba(211, 211, 239, 1) }
.cm-s-inner .cm-negative { color: rgba(217, 80, 80, 1) }
.cm-s-inner .cm-positive { color: rgba(80, 230, 80, 1) }
.cm-s-inner .cm-string-2 { color: rgba(255, 85, 0, 1) }
.cm-s-inner .cm-meta, .cm-s-inner .cm-qualifier { color: rgba(183, 179, 179, 1) }
.cm-s-inner .cm-builtin { color: rgba(243, 179, 248, 1) }
.cm-s-inner .cm-bracket { color: rgba(153, 153, 119, 1) }
.cm-s-inner .cm-atom, .cm-s-inner.cm-atom { color: rgba(132, 182, 203, 1) }
.cm-s-inner .cm-number { color: rgba(100, 171, 143, 1) }
.cm-s-inner .cm-variable { color: rgba(184, 191, 198, 1) }
.cm-s-inner .cm-variable-2 { color: rgba(159, 186, 213, 1) }
.cm-s-inner .cm-variable-3 { color: rgba(28, 198, 133, 1) }
.CodeMirror-selectedtext, .CodeMirror-selected { background: rgba(74, 137, 220, 1); text-shadow: none; color: rgba(255, 255, 255, 1) !important }
.CodeMirror-gutters { border-right: none }
:root { --bg-color: #363B40; --side-bar-bg-color: #2E3033; --text-color: #b8bfc6; --select-text-bg-color: #4a89dc; --control-text-color: #b7b7b7; --control-text-hover-color: #eee; --window-border: 1px solid #555; --active-file-bg-color: rgb(34, 34, 34); --active-file-border-color: #8d8df0; --active-file-text-color: white; --item-hover-bg-color: #70717d; --item-hover-text-color: white; --primary-color: #6dc1e7; --rawblock-edit-panel-bd: #4B535A }
html { font-size: 16px }
html, body { text-size-adjust: 100%; background: var(--bg-color); fill: currentcolor }
#write { max-width: 914px }
html, body, button, input, select, textarea, div.code-tooltip-content { color: rgba(184, 191, 198, 1); border-color: rgba(0, 0, 0, 0) }
div.code-tooltip, .md-hover-tip .md-arrow::after { background: rgba(75, 83, 90, 1) }
.popover.bottom>.arrow::after { border-bottom-color: rgba(75, 83, 90, 1) }
html, body, button, input, select, textarea { font-style: normal; line-height: 1.625rem; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif }
hr { height: 2px; border: 0; margin: 24px 0 !important }
h1, h2, h3, h4, h5, h6 { font-family: "Lucida Grande", Corbel, sans-serif; font-weight: normal; clear: both; word-wrap: break-word; margin: 0; padding: 0; color: rgba(222, 222, 222, 1) }
h1 { font-size: 2.5rem; line-height: 2.75rem; margin-bottom: 1.5rem; letter-spacing: -1.5px }
h2 { font-size: 1.63rem; line-height: 1.875rem; margin-bottom: 1.5rem; letter-spacing: -1px; font-weight: bold }
h3 { font-size: 1.17rem; line-height: 1.5rem; margin-bottom: 1.5rem; letter-spacing: -1px; font-weight: bold }
h4 { font-size: 1.12rem; line-height: 1.375rem; margin-bottom: 1.5rem; color: rgba(255, 255, 255, 1) }
h5 { font-size: 0.97rem; line-height: 1.25rem; margin-bottom: 1.5rem; font-weight: bold }
h6 { font-size: 0.93rem; line-height: 1rem; margin-bottom: 0.75rem; color: rgba(255, 255, 255, 1) }
@media (min-width: 980px) { h3.md-focus::before, h4.md-focus::before, h5.md-focus::before, h6.md-focus::before { color: rgba(221, 221, 221, 1); border: 1px solid rgba(221, 221, 221, 1); border-radius: 3px; position: absolute; left: -1.64286rem; top: 0.357143rem; float: left; font-size: 9px; padding-left: 2px; padding-right: 2px; vertical-align: bottom; font-weight: normal; line-height: normal } h3.md-focus::before { content: "h3" } h4.md-focus::before { content: "h4" } h5.md-focus::before { content: "h5"; top: 0 } h6.md-focus::before { content: "h6"; top: 0 } }
a { text-decoration: none; outline: 0 }
a:hover { outline: 0 }
a:focus { outline: 1px dotted }
sup.md-footnote { background-color: rgba(85, 85, 85, 1); color: rgba(221, 221, 221, 1) }
p { word-wrap: break-word }
p, ul, dd, ol, hr, address, pre, table, iframe, .wp-caption, .wp-audio-shortcode, .wp-video-shortcode { margin-top: 0; margin-bottom: 1.5rem }
li>blockquote { margin-bottom: 0 }
audio:not([controls]) { display: none }
[hidden] { display: none }
.in-text-selection, ::selection { background: rgba(74, 137, 220, 1); color: rgba(255, 255, 255, 1); text-shadow: none }
ul, ol { padding: 0 0 0 1.875rem }
ul { list-style: square }
ol { list-style: decimal }
ul ul, ol ol, ul ol, ol ul { margin: 0 }
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