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栈的反转问题

John Waken posted @ Dec 31, 2009 11:07:10 PM in Algorithm with tags stack , 13445 阅读

作者：John Waken
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给你一个有n个元素的栈，问要经过多少次pop和push操作才能把这个栈反转。如下，1 2 3 4 5 变为 5 4 3 2 1

 栈反转例子

解决这个问题还是用递归思想。记住，要让递归融入你的血液，嵌入你的灵魂。

定义符号$$(a_{n}, b_{n})$$，它表示是反转大小为n的栈时pop操作的次数是$$a_{n}$$，push操作的次数是 $$b_{n}$$

现在要构造第推式了。要反转n个数，就是把栈顶的元素放到底部去，然后再反转上面n－1个数。于是步骤出来了：
       0. 先定义栈反转的递归终止条件：反转空栈时，什么也不做。

       1. 执行 pop 操作，将 pop 出来的数保存到 temp 中
       2. 反转剩下来的 n-1 个数
       3. 把 temp 这个数 push 进去
       4. 再反转这 n 个数（这样temp就到最底层去了，而上面 n－1 个数没有变）
       5. 保持栈底的元素不动，反转最上面 n-1 个元素

OK，大功告成，这就是反转一个栈的算法。如果你真的这样认为，那么你还没有理解递归。递归是把大问题化为基础问题和能用同样方法解决的小问题，但是请看步骤4，这里出现了和原问题同样规模的子问题，所以违背了递归的本意。

没办法，重新试探一些构造方法吧。没错，是“ 试探 ”，虽然演绎推理诠释了数学的精妙，但数学的美丽与震撼还是来自于经验和直觉。

       0. 递归终止条件还是碰到空栈就什么也不做

       1. 执行 pop 操作，将弹出来的数存在 temp1 中
       2. 反转栈。此时栈中有 n-1 个数
       3. 如果栈中还有元素，就将它弹出来，保存在 temp2 中
       4. 反转栈。此时栈中有 n-1 个数
       5. 执行push，将 temp1 压入栈
       6. 反转栈。此时栈中有 n-1 个数
       7. 如果temp2保存来弹出来的数，则将它压入栈中

这才真正OK，真正地“ 大事化小” 了。这个算法的基本思想是反转一个序列可以先将首尾互换，再将中间的序列反转。当n为奇数，最后一次递归时，temp2肯定没用。

然后我们再从算法中提取有关pop和push操作次数的信息：

             $$a_{n} = 3a_{n-1} + 2, \quad n \geq 2

             $$a_{n} = 3a_{n-1} + 1, \quad n=1

             起始条件 $$a_{0} = 0 $$

将上面式子调整一下，得到：

             $$a_{n} = 3a_{n-1} + 2, \quad n \geq 2

             起始条件是 $$a_{0}=0, a_{1} = 1 $$

同理可求得push操作的第推式：

             $$b_{n} = 3b_{n-1} + 2, \quad n \geq 2

              起始条件是  $$b_{0} = 0, b_{1} = 1

综上所述得到：
             $$(a_{n}, b_{n})=(3a_{n-1}+2,3b_{n-1}+2), \quad n \geq 2$$

              起始条件 $$(a_{0},b_{0})=(0,0),\quad (a_{1},b_{1})=(1,1) $$


思考 1：这个算法只用到了两个辅助记录，temp1和temp2，那么它的空间复杂度是O(1)吗？ 不是，因为这是一个递归过程，在每次递归时都会用到新的temp1，temp2。所以空间复杂度是O(n)。再看他的时间复杂度，根据第推公式可知，通项肯定是指数形式的。

思考 2：由以上论述可知借助栈来实现栈的反转是很复杂的（当然如果你借助很多栈的话就像当于借助足够的变量了，这样就意思了）；试着用一个同样大小的队列来实现栈的反转，这就简单高效的多了。

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