Given a binary string S of length N, the task is to find the number of sub-sequences of non-zero length which are divisible by 3. Leading zeros in the sub-sequences are allowed.
Input: S = “1001”
“11”, “1001”, “0”, “0” and “00” are
the only subsequences divisible by 3.
Input: S = “1”
Naive approach: Generate all the possible sub-sequences and check if they are divisible by 3. Time complexity for this will be O((2N) * N).
Better approach: Dynamic programming can be used to solve this problem. Let’s look at the states of the DP.
DP[i][r] will store the number of sub-sequences of the substring S[i…N-1] such that they give a remainder of (3 – r) % 3 when divided by 3.
Let’s write the recurrence relation now.
DP[i][r] = DP[i + 1][(r * 2 + s[i]) % 3] + DP[i + 1][r]
The recurrence is derived because of the two choices below:
- Include the current index i in the sub-sequence. Thus, the r will be updated as r = (r * 2 + s[i]) % 3.
- Don’t include current index in the sub-sequence.
Below is the implementation of the above approach:
Time Complexity: O(n)
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