Word Break

Background Knowledge

The Word Break problem is a classic example of a dynamic programming problem. Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems, solving each subproblem only once, and storing the solutions to subproblems to avoid redundant computation. In the context of the Word Break problem, we need to understand how to break down a string into smaller substrings and check if each substring can be formed using the given word dictionary.

To approach this problem, it's essential to have a good understanding of string manipulation and dictionary lookup. We need to be able to iterate over the input string, extract substrings, and check if each substring is present in the word dictionary. Additionally, we need to understand how to use boolean arrays or tables to store the results of subproblems and avoid redundant computation.

The Word Break problem also involves overlapping subproblems, which is a key characteristic of dynamic programming problems. This means that the solution to the overall problem depends on the solutions to smaller subproblems, and some subproblems may be identical or have similar solutions. By using dynamic programming, we can avoid recomputing the solutions to these subproblems and reduce the overall time complexity of the algorithm.

Algorithm/Approach

The general approach to solving the Word Break problem involves using a bottom-up dynamic programming approach. We start by initializing a boolean array or table to store the results of subproblems, where each entry represents whether a substring can be segmented into dictionary words. We then iterate over the input string, filling in the table by checking if each substring can be formed using the given word dictionary.

Step-by-Step Strategy

To implement the solution, follow these steps:

• Initialize a boolean array dp of size n+1, where n is the length of the input string s. dp[i] will be True if the substring s[0:i] can be segmented into dictionary words.
• Set dp = True, since an empty string can always be segmented.
• Iterate over the input string s from i = 1 to n.
• For each i, iterate over the substring s[0:i] and check if any prefix of the substring is in the word dictionary.
• If a prefix is found in the dictionary and the remaining substring can be segmented, set dp[i] = True.
• Finally, return dp[n], which represents whether the entire input string can be segmented into dictionary words.

Common Pitfalls

When implementing the solution, watch out for the following common pitfalls:

• Not initializing the dp array correctly, leading to incorrect results.
• Not handling edge cases, such as an empty input string or an empty word dictionary.
• Not using a efficient data structure for the word dictionary, leading to slow lookup times.

Time & Space Complexity

The expected time complexity of the solution is $O(n^2)$, where $n$ is the length of the input string s. This is because we are iterating over the input string and checking each substring against the word dictionary. The space complexity is $O(n)$, since we need to store the dp array of size n+1.