2533. Number of Good Binary Strings

Problem Description

This problem presents a condition-based combinatorial challenge where we need to count the number of binary strings that adhere to specific grouping rules. The binary strings must:

• Have a length within the inclusive range specified by minLength and maxLength.
• Contain blocks of consecutive 1s where each block's size is a multiple of the given oneGroup.
• Contain blocks of consecutive 0s where each block's size is a multiple of the given zeroGroup.

Furthermore, we are asked to return the count of such "good" binary strings modulo 10^9 + 7 to handle large numbers that could result from the computation.

Intuition

Understanding the pattern of good binary strings can lead us to a dynamic programming approach. Given the conditions, we can incrementally build a binary string of length i by adding a block of 1s or 0s at the end of a string, ensuring the following:

• Blocks must have sizes that are multiples of oneGroup or zeroGroup.
• We only add blocks if they produce strings within the desired length range.

The intuition behind the solution is to use a dynamic programming table to store the number of ways to form a good binary string of length i, which we denote as f[i]. Starting with the base case, f[0] is set to 1, representing the empty string.

For each length i, we look back oneGroup and zeroGroup lengths from i and add up the ways because:

• If we are at length i and we add a block of 1s of oneGroup length, we can use all combinations of length i-oneGroup to form new combinations.
• Similarly, we can do the same with blocks of 0s of zeroGroup length.
• We take care to only add blocks if the resulting length does not exceed our maximum length constraint.

Finally, since we are only interested in strings with lengths between minLength and maxLength, we sum the values stored in our dynamic programming table from f[minLength] to f[maxLength], ensuring we only count the strings of acceptable lengths.

Implementing this approach in a loop that iterates through the possible lengths of binary strings, we can compute the answer. It's important to take the modulo at each step to prevent integer overflow given the constraint on the output.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution to the problem uses a dynamic programming approach, a common pattern in solving combinatorial problems where we build solutions to sub-problems and use those solutions to construct answers to larger problems. Here, the sub-problems involve finding the number of good binary strings of a smaller length, and these are combined to find the number of strings of larger lengths.

The implementation details are as follows:

• Data Structure: We use a list f with length maxLength + 1. The index i in the list represents the number of good binary strings of length i. It is initialized with f[0] = 1 (since the empty string is considered a good string) and 0 for all other lengths.

• Algorithm: We iterate from 1 to maxLength to fill up the dynamic programming table f. For each length i, we can either add a block of 1s or a block of 0s at the end of an existing good string which is shorter by oneGroup or zeroGroup respectively:

  • If i - oneGroup >= 0, we can take all good strings of length i - oneGroup and add a block of 1s to them. Thus, we update f[i] to include these new combinations by adding f[i - oneGroup].

  • If i - zeroGroup >= 0, similarly, we can take all good strings of length i - zeroGroup and add a block of 0s to them, updating f[i] to include these combinations by adding f[i - zeroGroup].

  • We use the modulo operator at each update to ensure that we do not encounter integer overflow, since we are interested in the count modulo 10^9 + 7.

• Final Count: Once the dynamic programming table is complete, the last step is to obtain the sum of all f[i] values for i between minLength and maxLength inclusive, and take the modulo again to get the final count of good binary strings within the given length range.

Here is the core part of the solution, with comments added for clarification:

1mod = 10**9 + 7
2f = [1] + [0] * maxLength
3for i in range(1, len(f)):
4    if i - oneGroup >= 0:
5        f[i] += f[i - oneGroup]
6    if i - zeroGroup >= 0:
7        f[i] += f[i - zeroGroup]
8    f[i] %= mod
9return sum(f[minLength:]) % mod

From the solution, we can see that the algorithm is a bottom-up dynamic programming solution since it builds up the answer iteratively. The space complexity of the algorithm is O(maxLength) because of the array f, and the time complexity is O(maxLength) as well, because we iterate through the array once.

Example Walkthrough

Let us assume minLength is 3, maxLength is 5, oneGroup is 2, and zeroGroup is 3. We need to count the number of binary strings that fit the criteria based on the given conditions.

We initialize our dynamic programming array f with the size of maxLength + 1, which is f[0] to f[5] for this example. We start by setting f[0] = 1 since the empty string is a valid string and zero for all other lengths.

Now, we start populating f from f[1] to f[5]:

1. For i = 1 and i = 2: We cannot add any blocks of 1s or 0s because oneGroup and zeroGroup are larger than i. Thus, f[1] and f[2] remain 0.

2. For i = 3: We can add a block of 0s because i - zeroGroup = 0 and f[0] = 1. Therefore, f[3] becomes 1 after considering the block of 0s. We cannot add a block of 1s since oneGroup is 2 and there's no way to form a 1 block within this length.

3. For i = 4: We can add a block of 1s since i - oneGroup = 2 and f[2] = 0. So, f[4] becomes 0 + 0 = 0. We do not have a way to add consecutive 0s of length zeroGroup to a string of length 2 (since that would exceed the current length i = 4).

4. For i = 5: We can add a block of 1s (f[3] = 1 from prior step), so f[5] is updated to 1. We also check for zero blocks, but 5 - zeroGroup is less than 0 and thus, cannot form a valid string.

To summarize the DP array: f[0] = 1, f[1] = 0, f[2] = 0, f[3] = 1, f[4] = 0, f[5] = 1.

Finally, to count the number of good strings of length between minLength and maxLength, we sum f[3], f[4], and f[5]. The result is 1 + 0 + 1 = 2. Thus, there are two "good" binary strings that satisfy the conditions between the length of 3 and 5, given the groupings of 1s and 0s.

The final answer is 2, and we will return this count modulo 10^9 + 7, which in this simple case remains 2.

Solution Implementation

Time and Space Complexity

The given Python code defines a method within the Solution class that calculates the number of good binary strings with certain properties. The method goodBinaryStrings counts the number of binary strings with a specified minimum and maximum length, where each group of consecutive 1's is at least oneGroup in length and each group of consecutive 0's is at least zeroGroup in length.

Time Complexity

To analyze the time complexity of the method goodBinaryStrings, consider the following points:

• The list f is initialized to have a length of maxLength + 1.
• There is a single loop that iterates maxLength times.
• Within the loop, there are two constant-time conditional checks and arithmetic operations.
• The final summation using sum(f[minLength:]) runs in O(n) time where n is maxLength - minLength + 1.

Combining these factors, the overall time complexity is as follows:

• Initialization of f: O(maxLength)
• Loop operations: O(maxLength)
• Final summation: O(maxLength - minLength)

Since maxLength dominates the runtime as it tends to be larger than maxLength - minLength, the total time complexity is O(maxLength).

Space Complexity

For the space complexity of the method goodBinaryStrings, consider the following points:

• A list f is used to store intermediate results and has a size of maxLength + 1.

Therefore, the space complexity is O(maxLength), as this is the most significant factor in determining the amount of space used by the algorithm.

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