From 74801a2c4d72fd610257345748265092aa0cabe7 Mon Sep 17 00:00:00 2001 From: Shine wOng <1551885@tongji.edu.cn> Date: Sun, 2 Jun 2019 13:26:57 +0800 Subject: [PATCH] update traversal in conclusion of chp5. Add in-order traversal version #2 , with all tests passed. --- thu_dsa/chp5/binNode.h | 28 ++++- thu_dsa/chp5/chp5.md | 191 +++++++++++++++++++++++++++++++ thu_dsa/chp5/images/inOrder.png | Bin 0 -> 189519 bytes thu_dsa/chp5/images/preOrder.png | Bin 0 -> 329675 bytes thu_dsa/chp5/testBinTree.cpp | 4 +- 5 files changed, 220 insertions(+), 3 deletions(-) create mode 100644 thu_dsa/chp5/images/inOrder.png create mode 100644 thu_dsa/chp5/images/preOrder.png diff --git a/thu_dsa/chp5/binNode.h b/thu_dsa/chp5/binNode.h index 35d7308..9ba252a 100644 --- a/thu_dsa/chp5/binNode.h +++ b/thu_dsa/chp5/binNode.h @@ -37,7 +37,8 @@ public: template void preOrder_It1(VST &visit); template void preOrder_It2(VST &visit); template void inOrder_Re(VST &visit); - template void inOrder_It(VST &visit); + template void inOrder_It1(VST &visit); + template void inOrder_It2(VST &visit); template void postOrder_Re(VST &visit); template void postOrder_It(VST &visit); template void levelOrder(VST &visit); @@ -145,7 +146,7 @@ void BinNode::inOrder_Re(VST &visit){ } template template -void BinNode::inOrder_It(VST &visit){ +void BinNode::inOrder_It1(VST &visit){ Stack S; BinNodePosi(T) curr = this; while(true){ @@ -158,6 +159,29 @@ void BinNode::inOrder_It(VST &visit){ } } +template template +void BinNode::inOrder_It2(VST &visit) { + BinNodePosi(T) curr = this; + bool backTrack = false; + while(curr){ + if (!backTrack) { + while (curr->leftChild) curr = curr->leftChild; + backTrack = true; + } + else{ + visit(curr->data); + if (curr->rightChild) { + curr = curr->rightChild; + backTrack = false; + } + else{ + while (curr->parent && curr == curr->parent->rightChild) curr = curr->parent; + curr = curr->parent; + } + } + } +} + template template void BinNode::postOrder_Re(VST &visit){ if (this->leftChild) this->leftChild->postOrder_Re(visit); diff --git a/thu_dsa/chp5/chp5.md b/thu_dsa/chp5/chp5.md index 0bc6744..225e90d 100644 --- a/thu_dsa/chp5/chp5.md +++ b/thu_dsa/chp5/chp5.md @@ -111,3 +111,194 @@ wald(T) = wald(T^{'}) + W_a + W_b $$ 假设$T$并不是最优带权编码树,则存在一个比$T$更优的编码树$T_1$,满足$wald(T_1) < wald(T)$,那么将$T_1$中`x`和`y`结点合并成为`z`结点,对应的编码树$T_{1}^{'}$满足$wald(T_{1}^{'}) = wald(T_1) - W_a - W_b < wald(T^{'})$,所以$T_{1}^{'}$是一棵比$T^{'}$更优的带权编码树,与假设矛盾。 +## 二叉树的遍历 + +### 先序遍历 + +> 先序遍历递归算法。 + +这个就不多说了。关键还是体会它的思想,即首先访问根节点,再访问左子树,最后再访问右子树,这个顺序递归地进行。根据这样的思想,可以构思出一种迭代进行先序遍历的方法。 + +> 先序遍历迭代算法。 + +在递归算法中,两个递归调用都处于函数中的最后的操作,类似于尾递归,因此不难写出迭代版本的遍历算法。由于左子树总是先于右子树访问,即使左子树中具有较高深度的结点,也会先于右子树中具有较小深度的结点,所以首先需要将右子树结点延迟缓冲起来,待访问了左子树中所有结点,再来访问右子树。因此,这里考虑使用一个栈。 + +每步迭代,都是首先访问当前结点。然后将右子树和左子树依次进栈。由于栈后进先出的性质,在下一次迭代中,左子树首先被弹出作为当前结点,将迭代过程继续下去。在某一次弹栈时,如果栈为空,则标志着当前结点访问完毕(左右子树都为空),并且所有祖先结点都访问完毕(栈为空),则循环结束。其具体的代码如下: + +```cpp +template template +void preOrder_It1(BinNodePosi(T) x, VST& viist){ + Stack S; + S.push(x); + while(!S.empty()){ + x = S.pop(); + visit(x->data); + if(x->rightChild) S.push(x->rightChild); + if(x->leftChild) S.push(x->leftChild); + } +} +``` + +可以用数学归纳法证明,该算法是可以实现二叉树的先序遍历的(这里懒得写了,打公式真烦)。 + +> 先序遍历第二版迭代算法。 + +上面的算法固然是可以工作,但是它其实就是从递归来的,看它的代码,其实和递归差别不是很大,并不能揭示先序遍历的本质--访问每个结点的直接后继是谁?让人感觉有些美中不足。 + +为了写第二版算法,首先需要探究二叉树的先序遍历究竟是以怎样的先后次序进行。 + +从根节点开始,不断向左子树访问,直到当前结点x不拥有左子树,这也相当于左子树已经访问完毕,此时转入当前结点x的右子树。右子树访问完毕后,即以x为根的子树已经访问完毕,此时应该转入x父亲的右子树,随后是x父亲的父亲的右子树,然后是x父亲的父亲的...整个访问过程可以如下面这张图所示: + +![先序遍历访问顺序](images/preOrder.png) + +从上面的归纳以及这张图可以看出,先序遍历的过程可以总结为:自上而下对左侧分支进行访问,随后自下而上对右侧分支进行访问。由于右子树的访问是自下而上的,因此需要一个栈来实现延迟缓冲的功能。整体的代码如下: + +```cpp +template template +void visitAlongLeftBranch(BinNodePosi(T) x, VST &visit, Stack &S){ + while(x){ + visit(x->data); + S.push(x->rightChild); + x = x->leftChild; + } +} + +template template +void preOrder_It2(BinNodePosi(T) x, VST &visit){ + Stack S; + S.push(x); + while(!S.empty()){ + x = S.pop(); + visitAlongLeftBranch(x, visit, S); + } +} +``` + +这里的分析思路其实挺重要的,会一直延续到后面的中序遍历和后续遍历迭代版算法。 + +### 中序遍历 + +> 中序遍历递归版算法。 + +这个也直接略过了。反正就是说有限访问左子树,再访问根节点,最后访问右子树。通过这样的思路,要能设计出迭代版的中序遍历算法。 + +> 中序遍历迭代版算法。 + +仿照先序遍历的思路,先分析中序遍历中各个节点访问的先后次序--谁是第一个被访问的结点,以及每个结点的直接后继又是谁? + +同样是从根节点开始,不断地沿着左子树向下走。不同的是,这里向下行进的过程中不能访问当前结点,只有俟到当前结点的左子树完成访问时,才能轮到当前结点,因此想到引入一个栈来实现延迟缓冲的功能。走到最左侧的第一个没有左子树的叶子结点时,没有左子树也相当于已经完成了左子树的访问,于是随后便访问当前结点x,然后转入到x的右子树。 + +当x的右子树完成访问时,即标志着以x为根的字数访问完毕,随机访问x的父亲结点,然后访问x的父亲的右字结点。x的右兄弟结点访问完毕时,即标志着以x的父亲的根的子树访问完毕,随机访问x父亲的父亲,然后是x父亲的父亲的父亲...整个访问过程如下图所示: + +![中序遍历的访问次序](images/inOrder.png) + +所以中序遍历迭代版的算法如下: + +``` +template +void goAlongLeftBranch(BinNodePosi(T) x, StackleftChild; + } +} + +template template +void inOrder_It1(BinNodePosi(T) x, VST &visit){ + Stack S; + while(true){ + goAlongLeftBranch(x, S); + if(S.empty()) break; + x = S.pop(); + visit(x->data); + x = x->rightChild; + } +} +``` + +> 中序遍历非递归算法的优化。 + +上面的代码固然已经甩递归算法几条街了。但是这里还是用到了一个栈,栈的最大高度等于二叉树的最大高度,即需要额外的$O(h)$的空间复杂度,在最坏的情况下与n相当。现在考虑在空间上对该算法进行优化。 + +由于前面二叉树结点的定义中,还定义了一个指向`parent`的`BinNodePosi(T)`变量。是否可以利用这个变量来实现访问的回溯,从而规避栈的使用。具体的代码如下: + +```cpp +template template +void BinNode::inOrder_It2(VST &visit) { + BinNodePosi(T) curr = this; + bool backTrack = false; + while(curr){ + if (!backTrack) { + while (curr->leftChild) curr = curr->leftChild; + backTrack = true; + } + else{ + visit(curr->data); + if (curr->rightChild) { + curr = curr->rightChild; + backTrack = false; + } + else{ + while (curr->parent && curr == curr->parent->rightChild) curr = curr->parent; + curr = curr->parent; + } + } + } +} +``` + +上面的算法主要讨论了两种情况:回溯与非回溯。 + ++ 非回溯代表了一开始的情形,以及从某个子树的根节点刚转入其右孩子的情形,对于非回溯的情况,不断地沿着左边路径向下移动,直到当前结点不再具有左子树。前面说过,没有左子树相当于左子树访问完毕,即左子树访问完毕后回溯到当前结点,因此设回溯标志`backTrack`为`true`。 + ++ 回溯表示某个结点的左子树或者右子树访问完毕后,回溯到当前结点的情形。如果当前结点有右子树,则去访问右子树,并且将回溯标识`backTrack`置`false`。而如果当前结点没有右子树,也代表了右子树访问完毕,亦即以当前结点为根的子树访问完毕。如果当前结点是右边结点的话,则继续上溯,代表路径上的各个祖先结点都已经访问完毕。直到当前结点是左边结点,代表当前结点父亲的左子树访问完毕,接下来应该访问父亲结点。由于仍然是回溯的情形,这里不修改回溯标志位。需要注意的是,通过这种方式回溯得到的结点,其右子树一定是尚未访问过的。 + +### 后序遍历 + +> 后序遍历递归算法。 + +和前面一样。所谓后续遍历,就是先访问左子树,再访问右子树,最后访问根节点,递归的进行下去。同样地,需要设计出相应的非递归算法。 + +> 后序遍历非递归算法。 + +首要的还是分析后续遍历各个节点之间的先后次序--哪个结点首先被访问,每个结点的直接后继是谁? + +从根节点出发,不断沿着左子树向下进行,直到当前结点不具有左子树。此时应该进入右子树,然后重复上面的过程。这样,第一个被访问的结点,应该是最高左侧可见结点,即从左边向右看过去,未被遮挡的最高叶节点,该结点既可以是左节点,也可是右节点。从而,显而易见,这里也需要引入一个栈来实现延迟缓冲的作用。根据上面的分析,应该是当前结点首先入栈,其次是右孩子,最后是左孩子。 + +当一个结点访问完毕时,代表着以该结点为根的子树访问完毕,此时应该弹栈以访问下一个结点。此时弹栈无非两种情况: + ++ 当前结点是左结点。则弹栈的结点是当前结点的右结点。对右结点的处理应该继续迭代地访问右结点的最高左侧可见结点。 ++ 当前结点是右结点。则弹栈的结点是当前结点的父结点。此时应该访问当前结点,然后直接进入下一步的迭代,再次进行弹栈操作。 + +综上所述,后序遍历的非递归算法如下: + +```cpp +//Highest Leaf Visible From Left +template +void goHLVFL(Stack S){ + BinNodePosi(T) x; + while(x = S.top()){ + if(x->leftChild){ + if(x->rightChild) S.push(x->rightChild); + S.push(x->leftChild); + } + else S.push(x->rightChild); + } + S.pop(); +} + +template template +void postOrder_It(BinNodePosi(T) x, VST &visit){ + Stack S; + S.push(x); + while(!S.empty()){ + if(S.top() != x->parent) goHLVFL(S); + x = S.pop(); + visit(x->data); + } +} +``` + +### 层序遍历 + +层序遍历是按照自上而下,自左而右的次序遍历,不涉及任何的逆序,延迟缓冲之类的操作。因此就不使用栈了,转而使用队列。代码略过不表。 diff --git a/thu_dsa/chp5/images/inOrder.png b/thu_dsa/chp5/images/inOrder.png 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