Asked in: GOOGLE
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All Testcases Passed ✔
Solution
def getCount(binary):
# Write your code here
n = len(binary)
st = set()
// ... rest of solution available after purchaseExplanation
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To solve the problem of determining how many unique decimal values can be represented by all possible non-empty subsequences of a binary string, it's essential to first fully understand what a subsequence is, how binary numbers work, and how to efficiently track unique values without being overwhelmed by the large number of possibilities.
A binary string is composed solely of the characters '0' and '1'. Each character contributes to the numeric value of a binary number based on its position. A subsequence, by definition, is formed by deleting zero or more characters from the original string without changing the order of the remaining characters. This means that any combination of characters that maintains the left-to-right order from the original string is considered a valid subsequence.
Given that, for a string of length `n`, there can be up to `2^n - 1` non-empty subsequences. This is because each character has two choices: to be included or not, and we exclude the empty set. However, many of these subsequences will convert to the same decimal value when interpreted as binary. The task, then, is not to count all subsequences, but to count how many **distinct** decimal values arise from interpreting those subsequences as binary numbers.
Now, the key challenge lies in the potential size of the input. Since the number of possible subsequences grows exponentially with the length of the string, generating all of them explicitly and converting each to a decimal would be computationally infeasible for large strings (especially if the length goes beyond 20 or 30).
To approach the problem efficiently, you must think about ways to avoid redundant computations and avoid explicitly generating every subsequence. This can be done by using a form of **memoization or dynamic programming**, or by cleverly managing sets of binary strings that represent values you've already encountered.
A natural approach is to simulate the process of generating subsequences, but with the constraint of avoiding recomputation. One way to conceptualize this is by using a set to keep track of all the unique numeric values formed. As you iterate through the binary string, for each character, you consider all current binary numbers that have been formed so far and extend them by adding this new character at the end. You also consider starting a new subsequence with this character.
This way, you're progressively building up all the binary numbers that could be created through subsequences, but you're doing it iteratively and tracking them by their values, not by the exact string form. Every time a new value is formed, you store it in a set. At the end, the size of the set gives you the number of unique decimal values.
However, care must be taken to interpret the binary strings properly. Leading zeros are allowed in binary representations of subsequences, but in decimal form, '0001' and '1' are the same. This implies that while multiple subsequences may look different (e.g., '001' and '1'), they can represent the same value, so it's the decimal value that matters, not the binary string itself.
Also, it's important to remember that for binary strings, the values can grow large quickly, especially if the subsequences are long. So the algorithm should be mindful of integer representations and possibly rely on native large integer support if necessary.
Now let’s look at a small example to solidify the idea. Suppose the string is "010". The non-empty subsequences are:
- '0' → 0
- '1' → 1
- '01' → 1
- '10' → 2
- '010' → 2
From the above, the decimal values obtained are 0, 1, and 2 — three unique values.
An efficient solution doesn't need to store the actual binary strings but only the decimal numbers they represent. For this, each new bit can be appended to existing numbers by shifting left and adding the new bit (in integer terms, multiply by 2 and add the bit). This can be efficiently tracked using sets.
To ensure efficiency, avoid string concatenation for binary values and instead rely on mathematical manipulation of integer values. This approach scales better for longer strings and is less memory-intensive.
In summary, the approach involves:
1. Understanding and tracking how binary subsequences evolve.
2. Using an efficient structure (like a set) to store unique decimal values.
3. Iterating through the binary string and updating current possible values.
4. Avoiding redundant recomputation by building on existing values.
5. Returning the count of unique values stored.
This ensures that the problem is handled efficiently even for larger input sizes, leveraging mathematical operations over brute-force generation.
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