LeetCode Meditations: Word Break

The description for this problem is:

Given a string s and a dictionary of strings wordDict, return true if s can be segmented into a space-separated sequence of one or more dictionary words.

Note that the same word in the dictionary may be reused multiple times in the segmentation.

For example:

Input: s = "leetcode", wordDict = ["leet", "code"]
Output: true

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

Or:

Input: s = "applepenapple", wordDict = ["apple", "pen"]
Output: true

Explanation: Return true because "applepenapple" can be segmented as "apple pen apple".
Note that you are allowed to reuse a dictionary word.

Or:

Input: s = "catsandog", wordDict = ["cats", "dog", "sand", "and", "cat"]
Output: false

Also, our constraints indicate that all the strings of wordDict are **unique**, and:

• 1 <= s.length <= 300
• 1 <= wordDict.length <= 1000
• 1 <= wordDict[i].length <= 20
• s and wordDict[i] consist of only lowercase English letters.

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Continuing with dynamic programming solutions, we can look at a popular bottom-up approach where we build a dp array to track whether it is possible to break s into words in wordDict at each index.

Each index $i$ in the dp array will indicate whether it is possible to break the entire string into words starting at index $i$ .

Note
dp needs to be of size s.length + 1 to hold the edge case of an empty string, in other words, when we're out of bounds.

Let's create it with initially false values:

let dp = Array.from({ length: s.length + 1 }, () => false); // +1 for the base case, out of bounds

The last index is the empty string, which can be considered breakable, or in other words, valid:

dp[s.length] = true; // base case

Going backwards, for each index of s, we can check if any word in wordDict can be reached from that index onwards:

for (let i = s.length - 1; i >= 0; i--) {
  for (const word of wordDict) {
    /* ... */
  }
}

If we're still within bounds of s (i + word.length <= s.length) and we find the word (s.slice(i, i + word.length) === word), we'll mark that slot as the truth value of the "next position" that we can break the string, which will be i + word.length:

for (let i = s.length - 1; i >= 0; i--) {
  for (const word of wordDict) {
    if (i + word.length <= s.length && s.slice(i, i + word.length) === word) {
      dp[i] = dp[i + word.length];
    }
    /* ... */
  }
}

If we can break it into any word in wordDict, we don't have to keep looking at other words, so we can just break out of the loop:

for (let i = s.length - 1; i >= 0; i--) {
  for (const word of wordDict) {
    if (i + word.length <= s.length && s.slice(i, i + word.length) === word) {
      dp[i] = dp[i + word.length];
    }
    if (dp[i]) {
      break;
    }
  }
}

Finally, we return dp[0] — if the whole string is breakable into words in wordDict, its value will store true, otherwise false:

function wordBreak(s: string, wordDict: string[]): boolean {
  /* ... */
  return dp[0];
}

And, here is the final solution:

function wordBreak(s: string, wordDict: string[]): boolean {
  let dp = Array.from({ length: s.length + 1 }, () => false); // +1 for the base case, out of bounds
  dp[s.length] = true; // base case

  for (let i = s.length - 1; i >= 0; i--) {
    for (const word of wordDict) {
      if (i + word.length <= s.length && s.slice(i, i + word.length) === word) {
        dp[i] = dp[i + word.length];
      }
      if (dp[i]) {
        break;
      }
    }
  }

  return dp[0];
}

Time and space complexity

The time complexity is $O(n * m * t)$ where $n$ is the string s, $m$ is the number of words in wordDict, and $t$ is the maximum length word in wordDict — as we have a nested loop that runs through each word in wordDict with a slice operation that uses word.length for each character in s.

The space complexity is $O(n)$ because of the dp array we store for each index of s.

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The last dynamic programming problem in the series will be Longest Increasing Subsequence. Until then, happy coding.