Update 22.3.md

docs/Chap22/22.3.md

@@ -6,35 +6,35 @@ According to Theorem 22.7, there are 3 cases of relationship between interval of
 
 - For Directed Graph:
 
-    1. $\text{BLACK}$ to $\text{BLACK}$ and $\text{WHITE}$ to $\text{WHITE}$ works with everything, we omit this two later.
-    2. For Cross Edge $(u, v)$, we must have $v.d < v.f < u.d < u.f$, then it could be $\text{WHITE}$ to $\text{BLACK}$, $\text{WHITE}$ to $\text{GRAY}$ and $\text{GRAY}$ to $\text{BLACK}$.
-    3. For Tree Edge and Forward Edge $(u, v)$, we must have $u.d < v.d < v.f < u.f$, then it could be $\text{GRAY}$ to $\text{WHITE}$, $\text{GRAY}$ to $\text{GRAY}$ and $\text{GRAY}$ to $\text{BLACK}$.
-    4. For Back Edge $(u, v)$, we must have $v.d < u.d < u.f < v.f$, then it could be $\text{WHITE}$ to $\text{GRAY}$, $\text{GRAY}$ to $\text{GRAY}$ and $\text{BLACK}$ to $\text{GRAY}$.
-
-    $$
-    \begin{array}{c|ccc}
-    from\backslash to & \text{BLACK}                & \text{GRAY}                & \text{WHITE} \\\\
-    \hline
-    \text{BLACK}      & \text{Allkinds}             & \text{Back}                & - \\\\
-    \text{GRAY}       & \text{Tree, Forward, Cross} & \text{Tree, Forward, Back} & \text{Tree, Forward} \\\\
-    \text{WHITE}      & \text{Cross}                & \text{Cross, Back}         & \text{Allkinds}
-    \end{array}
-    $$
+  1. $\text{BLACK}$ to $\text{BLACK}$ and $\text{WHITE}$ to $\text{WHITE}$ works with everything, we omit this two later.
+  2. For Cross Edge $(u, v)$, we must have $v.d < v.f < u.d < u.f$, then it could be $\text{WHITE}$ to $\text{BLACK}$, $\text{WHITE}$ to $\text{GRAY}$ and $\text{GRAY}$ to $\text{BLACK}$.
+  3. For Tree Edge and Forward Edge $(u, v)$, we must have $u.d < v.d < v.f < u.f$, then it could be $\text{GRAY}$ to $\text{WHITE}$, $\text{GRAY}$ to $\text{GRAY}$ and $\text{GRAY}$ to $\text{BLACK}$.
+  4. For Back Edge $(u, v)$, we must have $v.d < u.d < u.f < v.f$, then it could be $\text{WHITE}$ to $\text{GRAY}$, $\text{GRAY}$ to $\text{GRAY}$ and $\text{BLACK}$ to $\text{GRAY}$.
+
+  $$
+  \begin{array}{c|ccc}
+  from\backslash to & \text{BLACK}                & \text{GRAY}                & \text{WHITE} \\\\
+  \hline
+  \text{BLACK}      & \text{Allkinds}             & \text{Back}                & - \\\\
+  \text{GRAY}       & \text{Tree, Forward, Cross} & \text{Tree, Forward, Back} & \text{Tree, Forward} \\\\
+  \text{WHITE}      & \text{Cross}                & \text{Cross, Back}         & \text{Allkinds}
+  \end{array}
+  $$
 
 - For Undirected Graph, starting from Directed Chart, we remove the Forward Edge and Cross Edge, and when a Back Edge exist, we add Tree Edge; when a Tree Edge exist, we add Back Edge. This is correct for following reason:
 
-    1. Theorem 22.10: In a depth-first search of an undirected graph $G$, every edge of $G$ is either a tree edge or a back edge. So Tree edge and back edge only.
-    2. If $(u, v)$ is a Tree edge from $u$'s perspective, $(u, v)$ is also a Back Edge from $v$'s perspective.
+  1. Theorem 22.10: In a depth-first search of an undirected graph $G$, every edge of $G$ is either a tree edge or a back edge. So Tree edge and back edge only.
+  2. If $(u, v)$ is a Tree edge from $u$'s perspective, $(u, v)$ is also a Back Edge from $v$'s perspective.
 
-    $$
-    \begin{array}{c|ccc}
-    from\backslash to & \text{BLACK}      & \text{GRAY}       & \text{WHITE} \\\\
-    \hline
-    \text{BLACK}      & \text{Tree, Back} & \text{Tree, Back} & - \\\\
-    \text{GRAY}       & \text{Tree, Back} & \text{Tree, Back} & \text{Tree, Back} \\\\
-    \text{WHITE}      & -                 & \text{Tree, Back} & \text{Tree, Back}
-    \end{array}
-    $$
+  $$
+  \begin{array}{c|ccc}
+  from\backslash to & \text{BLACK}      & \text{GRAY}       & \text{WHITE} \\\\
+  \hline
+  \text{BLACK}      & \text{Tree, Back} & \text{Tree, Back} & - \\\\
+  \text{GRAY}       & \text{Tree, Back} & \text{Tree, Back} & \text{Tree, Back} \\\\
+  \text{WHITE}      & -                 & \text{Tree, Back} & \text{Tree, Back}
+  \end{array}
+  $$
 
 ## 22.3-2
 
@@ -98,7 +98,34 @@ By Theorem 22.10, every edge of an undirected graph is either a tree edge or a b
 
 > Rewrite the procedure $\text{DFS}$, using a stack to eliminate recursion.
 
-See the algorithm $\text{DFS-STACK}(G)$. Note that by a similar justification to 22.2-3, we may remove line 8 from the original $\text{DFS-VISIT}$ algorithm without changing the final result of the program, that is just working with the colors white and gray.
+```cpp
+DFS(G, s)
+    for each vertex u ∈ G.V
+        u.color = WHITE
+        u.π = NIL
+    time = 0
+    S = Ø
+    for each vertex u ∈ G.V
+        if u.color == WHITE
+            S.PUSH(u)
+            while !S.EMPTY()
+                u = S.TOP()
+                if u.color == WHITE
+                    u.color = GRAY
+                    time = time + 1
+                    u.d = time
+                    for each v ∈ G.Adj[u].REVERSE()
+                        if v.color == WHITE
+                            v.π = u
+                            S.PUSH(v)
+                else if u.color == GRAY
+                    u.color = BLACK
+                    time = time + 1
+                    u.f = time
+                    S.POP()
+                else if u.color == BLACK
+                    S.POP()
+```
 
 ## 22.3-8