[RBT] Editorial work in imperative insertion part

liuxinyu95 · Nov 13, 2014 · a53171c · a53171c
1 parent e4c7e06
commit a53171c
Showing 1 changed file with 21 additions and 21 deletions.
diff --git a/datastruct/tree/red-black-tree/rbtree-en.tex b/datastruct/tree/red-black-tree/rbtree-en.tex
@@ -969,23 +969,23 @@ \section{Imperative red-black tree algorithm $\star$}
 \Function{Insert}{$T, k$}
   \State $root \gets T$
   \State $x \gets$ \Call{Create-Leaf}{$k$}
-  \State \Call{Color}{$x$} $\gets RED$
-  \State $parent \gets NIL$
-  \While{$T \neq NIL$}
-    \State $parent \gets T$
+  \State \Call{Color}{$x$} $\gets$ RED
+  \State $p \gets$ NIL
+  \While{$T \neq$ NIL}
+    \State $p \gets T$
     \If{$k <$ \Call{Key}{$T$}}
       \State $T \gets $ \Call{Left}{$T$}
     \Else
       \State $T \gets $ \Call{Right}{$T$}
     \EndIf
   \EndWhile
-  \State \Call{Parent}{$x$} $\gets parent$
-  \If{$parent = NIL$} \Comment{tree $T$ is empty}
+  \State \Call{Parent}{$x$} $\gets p$
+  \If{$p =$ NIL} \Comment{tree $T$ is empty}
     \State \Return $x$
-  \ElsIf{$k <$ \Call{Key}{$parent$}}
-    \State \Call{Left}{$parent$} $\gets x$
+  \ElsIf{$k <$ \Call{Key}{$p$}}
+    \State \Call{Left}{$p$} $\gets x$
   \Else
-    \State \Call{Right}{$parent$} $\gets x$
+    \State \Call{Right}{$p$} $\gets x$
   \EndIf
   \State \Return \Call{Insert-Fix}{$root, x$}
 \EndFunction
@@ -1027,13 +1027,13 @@ \section{Imperative red-black tree algorithm $\star$}
 
 \begin{algorithmic}[1]
 \Function{Insert-Fix}{$T, x$}
-  \While{\Call{Parent}{$x$} $\neq NIL$ and \textproc{Color}(\Call{Parent}{$x$}) $= RED$}
-    \If{\textproc{Color}(\Call{Uncle}{$x$}) $= RED$}
+  \While{\Call{Parent}{$x$} $\neq$ NIL $\land$ \textproc{Color}(\Call{Parent}{$x$}) = RED}
+    \If{\textproc{Color}(\Call{Uncle}{$x$}) $=$ RED}
       \Comment{Case 1, x's uncle is red}
-      \State \textproc{Color}(\Call{Parent}{$x$}) $\gets BLACK$
-      \State \textproc{Color}(\Call{Grand-Parent}{$x$}) $\gets RED$
-      \State \textproc{Color}(\Call{Uncle}{$x$}) $\gets BLACK$
-      \State $x \gets$ \Call{Grandparent}{$x$}
+      \State \textproc{Color}(\Call{Parent}{$x$}) $\gets$ BLACK
+      \State \textproc{Color}(\Call{Grand-Parent}{$x$}) $\gets$ RED
+      \State \textproc{Color}(\Call{Uncle}{$x$}) $\gets$ BLACK
+      \State $x \gets$ \Call{Grand-Parent}{$x$}
     \Else
       \Comment{x's uncle is black}
       \If{\Call{Parent}{$x$} = \textproc{Left}(\Call{Grand-Parent}{$x$})}
@@ -1043,8 +1043,8 @@ \section{Imperative red-black tree algorithm $\star$}
           \State $T \gets$ \Call{Left-Rotate}{$T, x$}
         \EndIf
         \Comment{Case 3, x is a left child}
-        \State \textproc{Color}(\Call{Parent}{$x$}) $\gets BLACK$
-        \State \textproc{Color}(\Call{Grand-Parent}{$x$}) $\gets RED$
+        \State \textproc{Color}(\Call{Parent}{$x$}) $\gets$ BLACK
+        \State \textproc{Color}(\Call{Grand-Parent}{$x$}) $\gets$ RED
         \State $T \gets$ \textproc{Right-Rotate}($T$, \Call{Grand-Parent}{$x$})
       \Else
          \If{ $x =$ \textproc{Left}(\Call{Parent}{$x$})}
@@ -1053,18 +1053,18 @@ \section{Imperative red-black tree algorithm $\star$}
           \State $T \gets$ \Call{Right-Rotate}{$T, x$}
         \EndIf
         \Comment{Case 3, Symmetric}
-        \State \textproc{Color}(\Call{Parent}{$x$}) $\gets BLACK$
-        \State \textproc{Color}(\Call{Grand-Parent}{$x$}) $\gets RED$
+        \State \textproc{Color}(\Call{Parent}{$x$}) $\gets$ BLACK
+        \State \textproc{Color}(\Call{Grand-Parent}{$x$}) $\gets$ RED
         \State $T \gets$ \textproc{Left-Rotate}($T$, \Call{Grand-Parent}{$x$})
       \EndIf
     \EndIf
   \EndWhile
-  \State \Call{Color}{$T$} $\gets BLACK$
+  \State \Call{Color}{$T$} $\gets$ BLACK
   \State \Return $T$
 \EndFunction
 \end{algorithmic}
 
-This program takes $O(\lg N)$ time to insert a new key to the red-black tree.
+This program takes $O(\lg n)$ time to insert a new key to the red-black tree.
 Compare this pseudo code and the $balance$ function we defined in previous
 section, we can see the difference. They differ not only in terms of
 simplicity, but also in logic. Even if we feed the same series of keys to