Problem Description

Given a list of words and a list of puzzles, the aim is to figure out how many words from the word list are valid according to a specific pair of conditions for each puzzle. A word is valid with respect to a puzzle if:

1. The word contains the first letter of the puzzle.
2. Every letter of the word is also in the given puzzle.

It's important to note that it is not necessary for all letters of the puzzle to be in the word. For example, for the puzzle "abcdefg", words such as "faced", "cabbage", and "baggage" are valid because they include the first letter 'a' of the puzzle and all other letters in these words are contained within the puzzle. On the other hand, "beefed" and "based" are invalid because "beefed" lacks the letter 'a', and "based" contains the letter 's', which is not present in the puzzle.

The output is an array where each element corresponds to the number of valid words for each puzzle.

Intuition

The intuition behind the solution is based on converting both puzzles and words into binary representations where each bit corresponds to a letter's presence. To tackle this kind of problem, we use bitwise operations to perform checks and counts efficiently.

To represent a set of letters as a binary number, we assign each letter a bit position based on its order in the alphabet. For instance, 'a' corresponds to the least significant bit, and 'z' to the most significant bit of a 26-bit integer, since there are 26 letters in the English alphabet.

We start by creating masks for the words - a mask is a binary number where a bit is set (i.e., 1) if the corresponding letter is in the word. We use a counter to keep track of how many times every possible combination (mask) appears in the list of words. This preprocessing is helpful because we need to compare each word with multiple puzzles, and using masks allows us to do these comparisons quickly using bitwise operations.

For each puzzle, we also create a mask. We then need to count all valid word masks with respect to the puzzle mask. A word mask is valid if it includes the first letter of the puzzle, and all bits set in the word mask are also set in the puzzle mask. To find these valid word masks, we iterate over every submask (a mask with some bits possibly turned off, but still containing the first letter of the puzzle) of the puzzle mask and sum their counts in the counter.

This way, we leverage the precomputed frequencies of each word mask to efficiently calculate the number of valid words for each puzzle.

Learn more about Trie patterns.

Solution Approach

The implementation of the solution involves understanding how bitwise operations can effectively represent unique sets of characters and allow for fast subset enumeration. Here's the step-by-step approach, highlighted by key algorithms, data structures, and patterns:

1. Counter for masks of words: We create a Counter from the collections module in Python to keep track of all the unique masks created from the words list. To generate a mask, the solution goes through each character in a word and performs an OR | operation between the mask (initialized to zero) and a bit shifted by the alphabetical position of the character (1 << (ord(c) - ord("a"))). This results in a number where the bits corresponding to letters in the word are set to 1.

2. Processing the puzzles: For each puzzle in puzzles, we:

   • Compute the puzzle's mask in the same way as the word's mask.
   • Store the first letter's position in i because it must be present in each valid word.
   • Initialize x, which will accumulate the count of valid words for this puzzle.

3. Subset enumeration: Enumerate submasks of the puzzle's mask:

   • The inner while loop does the following: Starting with j as the puzzle mask, we repeatedly find submasks by decrementing j. The expression j = (j - 1) & mask turns off one bit at a time in the submask that is also on in the puzzle's mask.
   • If the submask contains the first letter of the puzzle (j >> i & 1), we add the count of this mask from our precomputed Countertox`.

4. Edge case handling: For puzzles that don't even have one word matching, the count would simply be added as zero.

5. Result assembly: Finally, the counts of valid words (stored in x) for each puzzle are appended to the ans list, which represents the number of valid words for each puzzle.

Through this application of bitwise operations, subsets enumeration and using a counter to map set bits to frequencies, we achieve an efficient solution to a problem that might seem combinatorial and complex at first glance.

Mathematically, if |W| is the length of the word list, and |P| is the length of the puzzle list, the algorithm efficiently compresses the O(|W| * |P|) brute-force approach into a faster one by using bit masks and counting common patterns.

Example Walkthrough

Let's use a small example to illustrate the solution approach. Say our word list is words = ["echo", "hold", "chef"] and our puzzle list puzzles = ["chloe", "dolph"].

Step 1: Creating masks for words and initializing Counter:

We iterate through each word in words to create a binary mask. For instance, for "echo" the binary mask would be:

• e: 1 << (4) => 00010000
• c: 1 << (2) => 00000100
• h: 1 << (7) => 10000000
• o: 1 << (14) => 0100000000000000

OR all the above: mask for echo = 11010100

Perform this process for each word:

• echo: 11010100
• hold: 1000010010010
• chef: 10011000

Our Counter would contain the count of the unique masks. From the example words, we have unique masks so each would have a count of 1 in the Counter.

Step 2: Processing the puzzles:

For "chloe", the binary mask would be computed similarly to words. So:

• mask for "chloe" => 11011101

Step 3 & 4: Enumerating subsets and counting the valid words:

For puzzle "chloe", we find all the subset masks of the puzzle's mask. The first letter 'c' must be included, so the submask must have the bit for 'c' set. Then we enumerate by repeatedly removing one bit until 'c' is the only bit left.

The submasks of "chloe" we consider will be (represented as strings for simplicity): "11011101", "11011100", "11011001", ... , "00000100" (which represents just 'c').

For each submask, we check if our word masks match.

• The mask of "echo" (11010100) is a subset of "chloe" mask, it's valid.
• The mask of "hold" (1000010010010) is not a subset since 'd' is not in "chloe".
• The mask of "chef" (10011000) is also a valid subset since all bits match and the first letter 'c' is present.

So, count for "chloe" is 2.

For "dolph", we would do the same. The word "hold" would be the only match since it's the only word containing 'd' and all its letters are in the puzzle "dolph". So, count for "dolph" is 1.

Step 5: Result assembly:

Finally, we would assemble our results into an array with [2, 1], which indicates the number of valid words for the puzzles "chloe" and "dolph", respectively.

Solution Implementation

Time and Space Complexity

Time Complexity

The given algorithm involves two main parts, constructing a bitmask for each word and puzzle and then solving the puzzles.

1. For each word in the list of words:

   • We iterate through all the characters in the word and create a bitmask representing the unique characters. Assuming the average length of the words is L, this operation is O(L).
   • However, we perform this operation for all N words. So, the time complexity for this part is O(N * L).

2. For each puzzle in the list of puzzles:

   • Similar to words, we create a bitmask for the puzzle. Assuming the average length of puzzles is P, this operation is O(P).
   • The novel part is trying out different combinations of the puzzle's letters represented by the bitmask and checking those combinations in the word counter. In the worst-case scenario, there could be 2^P combinations if each puzzle has distinct letters.
   • Since we need to check each of these combinations against our counter, and the first letter of the puzzle must be included, we're looking at O(2^(P-1)) operations for each puzzle. This is because we iterate through all the subsets of the bitmask that includes the first letter.
   • Performing this for all M puzzles, the total time complexity for solving puzzles is O(M * 2^(P-1)).

Combining the two parts, the total time complexity is O(N * L + M * 2^(P-1)). Note that L and P usually have upper limits (since the question constrains are such that the length of the words and puzzles would not exceed a certain length e.g., 7 for puzzles), so for very large N and M the dominant term could be either depending on the values of N, M, L, and P.

Space Complexity

1. Counter for words (cnt): This will store the frequency of each unique bitmask for the words. The space used by the counter will be proportional to the number of unique bitmasks rather than the number of words themselves. Since there are at most 26 characters, and each character could be present or absent, we theoretically can have at most 2^26 entries, though in practice, this will be much less due to the constraints of the problem (e.g., if words are of limited length). Therefore, the space complexity for cnt can be considered O(26) assuming the length of the word is limited, but in the unconstrained case, it would be O(2^26).

2. The space complexity for storing the answer (ans) is O(M), where M is the number of puzzles.

The total space complexity of the algorithm is, therefore, O(2^26 + M), but again, assuming a practical limit to the number of unique bitmasks due to constraints on the word length, the space would be more accurately described as O(26 + M).

Learn more about how to find time and space complexity quickly using problem constraints.