Word Break

Question

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Given a string s and a dictionary of words dict, determine if s can be
segmented into a space-separated sequence of one or more dictionary words.

For example, given
s = "leetcode",
dict = ["leet", "code"].

Return true because "leetcode" can be segmented as "leet code".

题解

单序列(DP_Sequence) DP 题,由单序列动态规划的四要素可大致写出:

  1. State: f[i] 表示前i个字符能否根据词典中的词被成功分词。
  2. Function: f[i] = or{f[j], j < i, letter in [j+1, i] can be found in dict}, 含义为小于i的索引j中只要有一个f[j]为真且j+1i中组成的字符能在词典中找到时,f[i]即为真,否则为假。具体实现可分为自顶向下或者自底向上。
  3. Initialization: f[0] = true, 数组长度为字符串长度 + 1,便于处理。
  4. Answer: f[s.length]

考虑到单词长度通常不会太长,故在s较长时使用自底向上效率更高。

Python

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class Solution:
# @param s, a string
# @param wordDict, a set<string>
# @return a boolean
def wordBreak(self, s, wordDict):
if not s:
return True
if not wordDict:
return False

max_word_len = max([len(w) for w in wordDict])
can_break = [True]
for i in xrange(len(s)):
can_break.append(False)
for j in xrange(i, -1, -1):
# optimize for too long interval
if i - j + 1 > max_word_len:
break
if can_break[j] and s[j:i + 1] in wordDict:
can_break[i + 1] = True
break
return can_break[-1]

C++

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class Solution {
public:
bool wordBreak(string s, unordered_set<string>& wordDict) {
if (s.empty()) return true;
if (wordDict.empty()) return false;

// get the max word length of wordDict
int max_word_len = 0;
for (unordered_set<string>::iterator it = wordDict.begin();
it != wordDict.end(); ++it) {

max_word_len = max(max_word_len, (*it).size());
}

vector<bool> can_break(s.size() + 1, false);
can_break[0] = true;
for (int i = 1; i <= s.size(); ++i) {
for (int j = i - 1; j >= 0; --j) {
// optimize for too long interval
if (i - j > max_word_len) break;

if (can_break[j] &&
wordDict.find(s.substr(j, i - j)) != wordDict.end()) {

can_break[i] = true;
break;
}
}
}

return can_break[s.size()];
}
};

Java

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public class Solution {
public boolean wordBreak(String s, Set<String> wordDict) {
if (s == null || s.length() == 0) return true;
if (wordDict == null || wordDict.isEmpty()) return false;

// get the max word length of wordDict
int max_word_len = 0;
for (String word : wordDict) {
max_word_len = Math.max(max_word_len, word.length());
}

boolean[] can_break = new boolean[s.length() + 1];
can_break[0] = true;
for (int i = 1; i <= s.length(); i++) {
for (int j = i - 1; j >= 0; j--) {
// optimize for too long interval
if (i - j > max_word_len) break;

String word = s.substring(j, i);
if (can_break[j] && wordDict.contains(word)) {
can_break[i] = true;
break;
}
}
}

return can_break[s.length()];
}
}

源码分析

Python 之类的动态语言无需初始化指定大小的数组,使用时下标i比 C++和 Java 版的程序少1。使用自底向上的方法求解状态转移,首先遍历一次词典求得单词最大长度以便后续优化。

复杂度分析

  1. 求解词典中最大单词长度,时间复杂度为词典长度乘上最大单词长度 $$O(L_D \cdot L_w)$$
  2. 词典中找单词的时间复杂度为 $$O(1)$$(哈希表结构)
  3. 两重 for 循环,内循环在超出最大单词长度时退出,故最坏情况下两重 for 循环的时间复杂度为 $$O(n L_w)$$.
  4. 故总的时间复杂度近似为 $$O(n L_w)$$.
  5. 使用了与字符串长度几乎等长的布尔数组和临时单词word,空间复杂度近似为 $$O(n)$$.



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