Leetcode 318答案详解（基于C++位操作）

随机Pick到Leetcode的318题，选择了C++基于STL的答题模式。
先看题目，

Given a string array words, find the maximum value of length(word[i]) * length(word[j]) where the two words do not share common letters. You may assume that each word will contain only lower case letters. If no such two words exist, return 0.Example 1:
Given ["abcw", "baz", "foo", "bar", "xtfn", "abcdef"]
Return 16
The two words can be "abcw", "xtfn".Example 2:
Given ["a", "ab", "abc", "d", "cd", "bcd", "abcd"]
Return 4
The two words can be "ab", "cd".Example 3:
Given ["a", "aa", "aaa", "aaaa"]
Return 0
No such pair of words.

中文意思就是，算了，既然做到这个题，这题的意思大家应该非常明确了，就不献丑翻译，还是老老实实解题。

简单说来，就是一个字符串数组里面，找到两个不含相同字符的字符串，求出它们的乘积之和。

开始看到以为这么简单的遍历题目，直接循环不就可以了么？

在数组里，每一个元素字符串依次和其他的字符串比较，看是不是有字符是共同的，没有则计算两个字符串的长度之和。

我写的代码如下：

class Solution {
public:int maxProduct(vector<string>& words) {int maxNum = 0;//两层for循环依次比较字符串,剔除自身与自身的比较for(auto i = words.begin();i != words.end()&& i != j; i++){for(auto j = words.begin();j != words.end(); j++){int count = compare(*i,*j);if(maxNum < count){maxNum = count;}}}return maxNum;}int compare(string line1,string line2)//比较两个有没有共同的字符，没有则计算长度积{int flag = 1;int count = 0;for(auto val:line1){if(line2.find(val) != line2.npos){flag = 0;}}if(flag){count = line1.size() * line2.size();}return count;}
};

自定义的测试用例：

[“abcw”,”xyz”,”defg”]
结果是：
16

所以小数据可行。

但是：
提交以后，得到的是时间超时，然后我看到给出的测试用例是：

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终于翻到文字了。。
这个测试，我看着就心虚，这种写法必然通不过啊。。
终于体会到，时间复杂度的重要性了。

但是我没办法了，就去看别人家的代码。

听说用位运算可以搞定，我看了一下，代码都很简洁，但是，我看不懂。。

可怕的是，大家都不解释，美其名为存档备用，或者解释的三言两语，以我的智商，压根搞不明白，好吧，那我就一步一步手动模拟走这个算法，搞出来一堆二进制的数据，可是还是没有懂。
因此，我深刻明白：

你不懂原理的代码，即使你知道输出的结果是正确的，你也不会知道为什么这样。
你也更不可能自己写出来了。
相反，你懂了原理，再稍微想想，转化一下，写出来，非常简单！

我觉得这个过程值得记录一下，也表达下自己的思考。下面进入这个算法的过程。
先上代码，先睹为快：

class Solution {
public:int maxProduct(vector<string>& words) {int maxn = 0;int n = words.size();vector<int> bits(n,0);for(int i = 0; i < words.size(); i++){for(int j = 0; j < words[i].size(); j++){bits[i] |= (1 << (words[i][j] - 'a'));}}for(int i = 0; i < n; i++){for(int j = 0; j < n; j++){if(bits[i] & bits[j]) continue;maxn = max(maxn,(int)(words[i].size() * words[j].size()));}}return maxn;}
};

看着短到不用任何注释，可我相信很多人和我一样，读完后啥感觉都没有。

掰开揉碎体验一下：
这里有一个小的trick:
用二进制来表示26个字母，这里出现的方法其实可以推广到其他的场合。
为什么转换成二进制？
答案是：快。

为什么快？

因为位运算，在底层是超级快的，此处不多加引申，可以读组成原理的书自己展开一下。

好，那就看如何来用二进制表示26个字母。
用的是1，2，4，8，16，32，… 来表示的a,b,c,d,e,f,…z
也就是26位二进制来表示。
我们知道int类型是32位，因此，26位二进制看着很大，其实就是个int可表的数罢了。
既然26位数值，对应26个字母，那么就可以做到每一个位置上用1个1来表示。
这里反复说，知识点有重复，已经看懂的可以略过。
即 : a = 1, b = 10, c = 100, d = 1000, e = 10000,…,z = 100000….000
明白吗？
这样，我们就可以对字符串进行编码。
如果字符串是abc, acb,bca(顺序不重要)那么我们就可以表示为： 111。
如果是def，dfe,efd，可以表示为 111000。我们只管字符是否存在，不管顺序。

1表示这个位置有字符。上面的编码可以这么理解：我们搬来26个凳子，依次排列。如果这个凳子上有人，我们将它记为1，否则就不记为0。

看懂了这个编码，我们再看如何比较是否有相同的字符：
这就牵涉到位运算中的与（&）运算了。
给定两个已经编码的字符串，每位相与，如果为1，则表示这两个位置的字符相等，否则，就是相等。只要出现一个相等字符，则这两个字符串就有共同的字符。

这很好理解，我们给26个凳子编号，a-z对应1-26号凳子。两组凳子对应位置的凳子相比较，如果两个凳子都为空，表示，都没人，那就没什么好比的，只有有人才有比较的意义。也就是说，两组对应的凳子上都有人，就说它们有相同字符出现。

解释完毕。
对应到代码中来，首先解释如何编码：

int n = words.size();
vector<int> bits(n,0);
for(int i = 0; i < words.size(); i++)
{for(int j = 0; j < words[i].size(); j++){bits[i] |= (1 << (words[i][j] - 'a'));}
}

比如一个字符 b, 首先得到它与a之间的差值是1，那么左移1位，得到2，二进制就是10。这是
1 << words[i][j] -'a'
的意思。
然后这个奇怪的语法：
bits[i] |= (1 << words[i][j] -'a')

等价于：

bits[i] = bits[i] | (1 << words[i][j] -'a')

求位或运算。
这是为了求字符串的编码。
比如对abc,
先是a ==> 1,然后拿a的编码去位或b(10),再去位或c(100),其实就是一个加法：a +b + c ==> 111，但是又不是简单的加法，因为这个只记录出现的字符一次。如果是aaa,就不能写成a+a+a = 2 ==> 10，表示的就是b了。这也是为什么这里一定用的是位或运算的原因。

得到一个用数字编码的字符串！

接下来就很好解释了，对这个数字编码的数组进行求与(&)，判断是否相同即可。

for(int i = 0; i < n; i++) { for(int j = 0; j < n; j++) { if(bits[i] & bits[j]) continue; // 若果相同则结束这次循环，开始下一次的循环 maxn = max(maxn,(int)(words[i].size() * words[j].size())); } }
bits数组存的是单词的二进制表示。
我想，我已经解释得很明白了。

由一个小题牵连出来的知识点。
以上。

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