I have been given a binary string of length n and i need to find the minimum numbers of operations to perform such that string does not contain more than k consecutive equal characters.
Only kind of operation I am allowed to perform is to flip any ith character of the string. flipping a character means changing a '1' to '0' or a '0' to '1'.
for example:
if n = 4 , k = 1 and string = 1001
then Answer:
string = 1010 and minimum operations = 2
I need to also find the new string.
can anyone tell me an efficient algorithm for solving problem considering n <=10^5
There's one way:
if k>1:
if k+1 matching characters are found:
if a[k+1]==a[k+2]:
flip a[k+1]
else if a[k+1]!=a[k+2]:
flip a[k]
for k=1 you can do it!
Here flipping means from 1 to 0 and vice-versa
For k=1 there are only two possible output strings - the one beginning with 0 and the one beginning with 1. You can check which of them is closer to the input string.
For larger k, you can just look at every sequence of k+1 identical characters, and fix it internally - without changing the characters at either end. For a sequence of k' > k you would need floor(k'/(k+1)) flips. It should not be hard to show that this is optimal.
Running time is linear and extra space is constant.
There are 2 cases:
1)For k>1
We have 2 possibilities.
a)one that is starting with 0:
eg:0101010101
b)one that is starrting with 1
eg:10101010.....
We should now calculate the distance(the number of different elements between the 2 strings)for each possiblity.Then the ans will be the one that has minimum changes.
2)for k>1
res2=0;res1=1;
c1=A[i];//it represents the last elemnet
i=1;
while(A[i]!='\0'){
if(A[i]==c1){
res1++;//the no of consecutive elements
if(res1>k){
if(A[i]==A[i+1])
flip(i);//it flips the ith element
else
flip(i-1);
res2++;//it counts the no of changes
res1=1;
}
}
else
res1=1;
c1=A[i];
i++;
}
Related
Given a string I need to remove the smallest character and return the sum of indices of removed charecter.
Suppose the string is 'abcab' I need to remove first a at index 1.
We are left with 'bcab'. Now remove again a which is smallest in remaining string and is at index 3
We are left with 'bcb'.
In the same way remove b at index 1,then remove again b from 'cb' at index 2 and finally remove c
Total of all indices is 1+3+1+2+1=8
Question is simple but we need to do it in O(n). for that I need to remove kth element in O(1). In python del list[index] has time complexity O(n).
How can I delete in constant time using python
Edit
This is the exact question
You are given a string S of size N. Assume that count is equal to 0.
Your task is the remove all the N elements of string S by performing the following operation N times
• In a single operation, select an alphabetically smallest character in S, for example, Remove from S and add its index to count. If multiple characters such as c exist, then select that has the smallest index.
Print the value of count.
Note Consider 1-based indexing
Solve the problem for T test cases
Input format
The first line of the input contains an integer T denoting the number of test cases • The first line of each test case contains an integer N denoting the size of string S
• The second line of each test case contains a string S
Output format
For each test case print a single line containing one integer denoting the value of count
1<T, N < 10^5
• S contains only lowercase English alphabets
Sum of N over all test cases does not exceed 10
Sample input 1
5
abcab
Sample Output1
8
Explanation
The operations occur in the following order
Current string S= abcab', The alphabetically smallest character of s is 'a As there are 2 occurrences of a, we choose the first occurrence. Its Index 1 will be added to the count and a will be removed. Therefore, S becomes bcab
a will.be removed from 5 (bcab) and 3 will.be added to count
The first occurrence of b will be removed from (bcb) and 1 will be added to count.
b will be removed from s (cb) and 2 will be added to count
c will be removed from 5 (c) and 1 will be added to count
If you follow your procedure of repeatedly removing the first occurrence of the smallest character, then each character's index -- when you remove it -- is the number of preceding larger characters in the original string plus one.
So what you really need to do is find, for each character, the number of preceding larger characters, and then add up all those counts.
There are only 26 characters, so you can do this as you go with 26 counters.
Please link to the original problem statement, or copy/paste exactly what it says, without trying to explain it. As is, what you're asking for is impossible.
Forget deleting: if what you're asking for was possible, sorting would be worse-case O(n) (remove the minimum remaining n times, at O(1) cost for each), but it's well known that comparison-based sorting cannot do better than worst case O(n log n).
One bet: the original problem statement doesn't require that you delete anything - but instead that you return the result as if you had deleted.
With one pass over the input
Putting together various ideas, the final index of a character is one more than the number of larger characters seen before it. So it's possible to do this in one left-to-right pass over the input, using O(1) storage and O(n) time, while deleting nothing:
def crunch(s):
neq = [0] * 26
result = 0
orda = ord('a')
for ch in map(ord, s):
ch -= orda
result += sum(neq[i] for i in range(ch + 1, 26)) + 1
neq[ch] += 1
return result
For your original:
>>> crunch('abcab')
8
But it's also possible to process arbitary iterables one character at a time:
>>> from itertools import repeat, chain
>>> crunch(chain(repeat('y', 1000000), 'xz'))
2000002
x is originally at (1-based) index 1000001, which accounts for half the result. Then each of a million 'y's is conceptually deleted, each at index 1. Finally 'z' is at index 1, for a grand total of 2000002.
Looks like you're only interested in the resulting sum of indices and don't need to simulate this algorithm step by step.
In which case you could compute the result in the following way:
For each letter from a to z:
Have a counter of already removed letters set to 0
Iterate over the string and if you encounter the current letter add current_index - already_removed_counter to the result.
2a. If you encounter current or earlier (smaller) letter increase the counter as it already has been removed
The time complexity is 26 * O{n} which is O{n}.
Since there are only 26 distinct chatacters in the string, we can take each character separately and linearly traverse the string to find all its occurences. Keep a counter of how many chacters were found. Each time an occurence of a given character is found display its index decreased by the counter. Before switching to a new character, remove all the occurences of the previous one - this can be done in linear time.
res = 0
for c in 'a' .. 'z'
cnt = 0
for idx = 1 .. len(s)
if s[idx] = c
print idx - cnt
res += idx - cnt
cnt++
removeAll(s, c)
return res
where
removeAll(s,c):
i = 1
cnt = 0
n = len(s)
while (i < n)
if s[i + cnt] = c
cnt++
n--
else
s[i] = s[i + cnt]
i++
len(s) = n
It prints the elements of the sum to better illustrate what's going on.
Edit:
An updated version based on Igor's answer, that does not require actually removing elements. The complexity is the same i.e. O(n).
res = 0
for c in 'a' .. 'z'
cnt = 0
for idx = 1 .. len(s)
if s[idx] <= c
if s[idx] = c
print idx - cnt
res += idx - cnt
cnt++
return res
I was solving a problem on HackerRank. It required me to see if it is possible to convert string s to string t by performing k operations.
https://www.hackerrank.com/challenges/append-and-delete/problem
The operations we can perform are: appending a lowercase letter to the end of s or removing a lowercase letter from the end of s. For example Ash Ashley 2 would return No since we need 3 operations, not 2.
I tried solving the problem as follows:
def appendAndDelete(s, t, k):
if len(s) > len(t):
maxs = [s,t]
else:
maxs = [t,s]
maximum = maxs[0]
minimum = maxs[1]
k -= len(maximum) - len(minimum)
substr = maximum[len(minimum): len(maximum)]
maximum = maximum.replace(substr, '')
i = 0
while i < len(maximum):
if maximum[i] != minimum[i]:
k -= (len(maximum)-i)*2
break
i += 1
if k < 0:
return 'No'
else:
return 'Yes'
However, it fails at this weird test case. y yu 2. The expected answer is No but according to my code, it would return Yes since only one operation was required. Is there something I do not understand?
Since you don't explain your idea, it's difficult for us to understand
what you mean in your code and debug it to tell you where you went wrong.
However, I would like to share my idea(I solved this on the website too)-
len1 => Length of first string s.
len2 => Length of second/target string t.
Exactly K makes it a bit tricky. So, if len1 + len2 <= k, you can blindly assume it can be accomplished and return true since we can delete empty string many times to get an empty string(as it says) and we can delete characters of one string entirely and keep appending new letters to get the another.
When we start matching s with t from left to right, this looks more like longest common prefix but this is NOT the case. Let's take an example -
aaaaaaaaa (source)
aaaa (target)
7 (k)
Here, up till aaaa it's common and looks like there are additional 5 a's in the source. So, we can delete those 5 a's and get the target but 5 != 7, hence it appears to be a No. But this ain't the case since we can delete an a from the source just like that and append it again(2 operations) just to satisfy k. So, it need not be longest common prefix all the time, however it gets us closer to the solution.
So, let's match both strings from left to right and stop when there is a mismatch. Let's assume we got this index in a variable called first_unmatched. Initialize first_unmatched = min(len(s),len(t)) at the beginning of your method itself.
Let
rem1 = len1 - first_unmatched
rem2 = len2 - first_unmatched
where rem1 is remaining substring of s and rem2 is the remaining substring of t.
Now, comes the conditions.
if(rem1 + rem2 == k) return true-
This is because rem1 characters to delete and rem2 characters to add. If both sum up to k then it's possible.
if(rem1 + rem2 > k) return false-
This is because rem1 characters to delete and rem2 characters to add. If both sum greater than k then it's not possible.
if(rem1 + rem2 < k) return (k - (rem1 + rem2)) % 2 == 0-
This is because rem1 characters to delete and rem2 characters to add. If both sum less than k, then it depends.
Here, (k - (rem1 + rem2)) will give you the extra in k. This extra can or cannot depends upon whether it's divisible by 2 or not. Here, we do %2 because we have 2 operations in our question - delete and append. If the extra k falls short of any operation, then the answer is No, else it's a Yes.
You can cross check this with above example.
I'm working on a problem where I need to check how many words in a dictionary can be combined to match a single word.
For example:
Given the string "hellogoodsir", and the dictionary: {hello, good, sir, go, od, e, l}, the goal is to find all the possible combinations to form the string.
In this case, the result would be hello + good + sir, and hello + go + od + sir, resulting in 3 + 4 = 7 words used, or 1 + 1 = 2 combinations.
What I've come up with is simply to put all the words starting with the first character ("h" in this instance) in one hashmap (startH), and the rest in another hashmap (endH). I then go through every single word in the startH hashmap, and check if "hellogoodsir" contains the new word (start + end), where end is every word in the endH hashmap. If it does, I check if it equals the word to match, and then increments the counter with the value of the number for each word used. If it contains it, but doesn't equal it, I call the same method (recursion) using the new word (i.e. start + end), and proceed to try to append any word in the end hashmap to the new word to get a match.
This is obviously very slow for large number of words (and a long string to match). Is there a more efficient way to solve this problem?
As far as I know, this is an O(n^2) algorithm, but I'm sure this can be done faster.
Let's start with your solution. It is no linear nor quadric time, it's actually exponential time. A counter example that shows that is:
word = "aaa...a"
dictionary = {"a", "aa", "aaa", ..., "aa...a"}
Since your solution is going through each possible matching, and there is exponential number of such in this example - the solution is exponential time.
However, that can be done more efficiently (quadric time worst case), with Dynamic Programming, by following the recursive formula:
D[0] = 1 #
D[i] = sum { D[j] | word.Substring(i,j) is in the dictionary | 0 <= j < i }
Calculating each D[i] (given the previous ones are already known) is done in O(i)
This sums to total O(n^2) time, with O(n) extra space.
Quick note: By iterating the dictionary instead of all (i,j) pairs for each D[i], you can achieve O(k) time for each D[i], which ends up as O(n*k), where k is the dictionary size. This can be optimized for some cases by traversing only potentially valid strings - but for the same counter example as above, it will result in O(n*k).
Example:
dictionary = {hello, good, sir, go, od, e, l}
string = "hellogoodsir"
D[0] = 1
D[1] = 0 (no substring h)
D[2] = 0 (no substring he, d[1] = 0 for e)
...
D[5] = 1 (hello is the only valid string in dictionary)
D[6] = 0 (no dictionary string ending with g)
D[7] = D[5], because string.substring(5,7)="go" is in dictionary
D[8] = 0, no substring ending with "oo"
D[9] = 2: D[7] for "od", and D[5] for "good"
D[10] = D[11] = 0 (no strings in dictionary ending with "si" or "s")
D[12] = D[7] = 2 for substring "sir"
My suggestion would be to use a prefix tree. The nodes beneath the root would be h, g, s, o, e, and l. You will need nodes for terminating characters as well, to differentiate between go and good.
To find all matches, use a Breadth-first-search approach. The state you will want to keep track of is a composition of: the current index in the search-string, the current node in the tree, and the list of words used so far.
The initial state should be 0, root, []
While the list of states is not empty, dequeue the next state, and see if the index matches any of the keys of the children of the node. If so, modify a copy of the state and enqueue it. Also, if any of the children are the terminating character, do the same, adding the word to the list in the state.
I'm not sure on the O(n) time on this algorithm, but it should be much faster.
We are given a string which consists of digits 0-9. We have to count number of sub-strings divisible by a number k. One way is to generate all the sub-strings and check if it is divisible by k but this will take O(n^2) time. I want to solve this problem in O(n*k) time.
1 <= n <= 100000 and 2 <= k <= 1000.
I saw a similar question here. But k was fixed as 4 in that question. So, I used the property of divisibility by 4 to solve the problem.
Here is my solution to that problem:
int main()
{
string s;
vector<int> v[5];
int i;
int x;
long long int cnt = 0;
cin>>s;
x = 0;
for(i = 0; i < s.size(); i++) {
if((s[i]-'0') % 4 == 0) {
cnt++;
}
}
for(i = 1; i < s.size(); i++) {
int f = s[i-1]-'0';
int s1 = s[i] - '0';
if((10*f+s1)%4 == 0) {
cnt = cnt + (long long)(i);
}
}
cout<<cnt;
}
But I wanted a general algorithm for any value of k.
This is a really interesting problem. Rather than jumping into the final overall algorithm, I thought I'd start with a reasonable algorithm that doesn't quite cut it, then make a series of modifications to it to end up with the final, O(nk)-time algorithm.
This approach combines together a number of different techniques. The major technique is the idea of computing a rolling remainder over the digits. For example, let's suppose we want to find all prefixes of the string that are multiples of k. We could do this by listing off all the prefixes and checking whether each one is a multiple of k, but that would take time at least Θ(n2) since there are Θ(n2) different prefixes. However, we can do this in time Θ(n) by being a bit more clever. Suppose we know that we've read the first h characters of the string and we know the remainder of the number formed that way. We can use this to say something about the remainder of the first h+1 characters of the string as well, since by appending that digit we're taking the existing number, multiplying it by ten, and then adding in the next digit. This means that if we had a remainder of r, then our new remainder is (10r + d) mod k, where d is the digit that we uncovered.
Here's quick pseudocode to count up the number of prefixes of a string that are multiples of k. It runs in time Θ(n):
remainder = 0
numMultiples = 0
for i = 1 to n: // n is the length of the string
remainder = (10 * remainder + str[i]) % k
if remainder == 0
numMultiples++
return numMultiples
We're going to use this initial approach as a building block for the overall algorithm.
So right now we have an algorithm that can find the number of prefixes of our string that are multiples of k. How might we convert this into an algorithm that finds the number of substrings that are multiples of k? Let's start with an approach that doesn't quite work. What if we count all the prefixes of the original string that are multiples of k, then drop off the first character of the string and count the prefixes of what's left, then drop off the second character and count the prefixes of what's left, etc? This will eventually find every substring, since each substring of the original string is a prefix of some suffix of the string.
Here's some rough pseudocode:
numMultiples = 0
for i = 1 to n:
remainder = 0
for j = i to n:
remainder = (10 * remainder + str[j]) % k
if remainder == 0
numMultiples++
return numMultiples
For example, running this approach on the string 14917 looking for multiples of 7 will turn up these strings:
String 14917: Finds 14, 1491, 14917
String 4917: Finds 49,
String 917: Finds 91, 917
String 17: Finds nothing
String 7: Finds 7
The good news about this approach is that it will find all the substrings that work. The bad news is that it runs in time Θ(n2).
But let's take a look at the strings we're seeing in this example. Look, for example, at the substrings found by searching for prefixes of the entire string. We found three of them: 14, 1491, and 14917. Now, look at the "differences" between those strings:
The difference between 14 and 14917 is 917.
The difference between 14 and 1491 is 91
The difference between 1491 and 14917 is 7.
Notice that the difference of each of these strings is itself a substring of 14917 that's a multiple of 7, and indeed if you look at the other strings that we've matched later on in the run of the algorithm we'll find these other strings as well.
This isn't a coincidence. If you have two numbers with a common prefix that are multiples of the same number k, then the "difference" between them will also be a multiple of k. (It's a good exercise to check the math on this.)
So this suggests another route we can take. Suppose that we find all prefixes of the original string that are multiples of k. If we can find all of them, we can then figure out how many pairwise differences there are among those prefixes and potentially avoid rescanning things multiple times. This won't find everything, necessarily, but it will find all substrings that can be formed by computing the difference of two prefixes. Repeating this over all suffixes - and being careful not to double-count things - could really speed things up.
First, let's imagine that we find r different prefixes of the string that are multiples of k. How many total substrings did we just find if we include differences? Well, we've found k strings, plus one extra string for each (unordered) pair of elements, which works out to k + k(k-1)/2 = k(k+1)/2 total substrings discovered. We still need to make sure we don't double-count things, though.
To see whether we're double-counting something, we can use the following technique. As we compute the rolling remainders along the string, we'll store the remainders we find after each entry. If in the course of computing a rolling remainder we rediscover a remainder we've already computed at some point, we know that the work we're doing is redundant; some previous scan over the string will have already computed this remainder and anything we've discovered from this point forward will have already been found.
Putting these ideas together gives us this pseudocode:
numMultiples = 0
seenRemainders = array of n sets, all initially empty
for i = 1 to n:
remainder = 0
prefixesFound = 0
for j = i to n:
remainder = (10 * remainder + str[j]) % k
if seenRemainders[j] contains remainder:
break
add remainder to seenRemainders[j]
if remainder == 0
prefixesFound++
numMultiples += prefixesFound * (prefixesFound + 1) / 2
return numMultiples
So how efficient is this? At first glance, this looks like it runs in time O(n2) because of the outer loops, but that's not a tight bound. Notice that each element can only be passed over in the inner loop at most k times, since after that there aren't any remainders that are still free. Therefore, since each element is visited at most O(k) times and there are n total elements, the runtime is O(nk), which meets your runtime requirements.
If we have string A of length N and string B of length M, where M < N, can I quickly compute the minimum number of letters I have to remove from string A so that string B does not occur as a substring in A?
If we have tiny string lengths, this problem is pretty easy to brute force: you just iterate a bitmask from 0 to 2^N and see if B occurs as a substring in this subsequence of A. However, when N can go up to 10,000 and M can go up to 1,000, this algorithm obviously falls apart quickly. Is there a faster way to do this?
Example: A=ababaa B=aba. Answer=1.Removing the second a in A will result in abbaa, which does not contain B.
Edit: User n.m. posted a great counter example: aabcc and abc. We want to remove the single b, because removing any a or c will create another instance of the string abc.
Solve it with dynamic programming. Let dp[i][j] the minimum operator to make A[0...i-1] have a suffix of B[0...j-1] as well as A[0...i] doesn't contain B, dp[i][j] = Infinite to index the operator is impossible. Then
if(A[i-1]=B[i-1])
dp[i][j] = min(dp[i-1][j-1], dp[i-1][j])
else dp[i][j]=dp[i-1][j]`,
return min(A[N][0],A[N][1],...,A[N][M-1]);`
Can you do a graph search on the string A. This is probably too slow for large N and special input but it should work better than an exponential brute force algorithm. Maybe a BFS.
I'm not sure this question is still of someone interest, but I have an idea that maybe could work.
once we decided that the problem is not to find the substring, is to decide which letter is more convenient to remove from string A, the solution to me appears pretty simple: if you find an occurrence of B string into A, the best thing you can do is just remove a char that is inside the string, closed to the right bondary...let say the one previous the last. That's why if you have a substring that actually end how it starts, if you remove a char at the beginning you just remove one of the B occurencies, while you can actually remove two at once.
Algorithm in pseudo cose:
String A, B;
int occ_posit = 0;
N = B.length();
occ_posit = A.getOccurrencePosition(B); // pseudo function that get the first occurence of B into A and returns the offset (1° byte = 1), or 0 if no occurence present.
while (occ_posit > 0) // while there are B into A
{
if (B.firstchar == B.lastchar) // if B starts as it ends
{
if (A.charat[occ_posit] == A.charat[occ_posit+1])
A.remove[occ_posit - 1]; // no reason to remove A[occ_posit] here
else
A.remove[occ_posit]; // here we remove the last char, so we could remove 2 occurencies at the same time
}
else
{
int x = occ_posit + N - 1;
while (A.charat[x + 1] == A.charat[x])
x--; // find the first char different from the last one
A.remove[x]; // B does not ends as it starts, so if there are overlapping instances they overlap by more than one char. Removing the first that is not equal to the char following B instance, we kill both occurrencies at once.
}
}
Let's explain with an example:
A = "123456789000987654321"
B = "890"
read this as a table:
occ_posit: 123456789012345678901
A = "123456789000987654321"
B = "890"
first occurrence is at occ_posit = 8. B does not end as it starts, so it get into the second loop:
int x = 8 + 3 - 1 = 10;
while (A.charat[x + 1] == A.charat[x])
x--; // find the first char different from the last one
A.remove[x];
the while find that A.charat11 matches A.charat[10] (="0"), so x become 9 and then while exits as A.charat[10] does not match A.charat9. A then become:
A = "12345678000987654321"
with no more occurencies in it.
Let's try with another:
A = "abccccccccc"
B = "abc"
first occurrence is at occ_posit = 1. B does not end as it starts, so it get into the second loop:
int x = 1 + 3 - 1 = 3;
while (A.charat[x + 1] == A.charat[x])
x--; // find the first char different from the last one
A.remove[x];
the while find that A.charat4 matches A.charat[3] (="c"), so x become 2 and then while exits as A.charat[3] does not match A.charat2. A then become:
A = "accccccccc"
let's try with overlapping:
A = "abcdabcdabff"
B = "abcdab"
the algorithm results in: A = "abcdacdabff" that has no more occurencies.
finally, one letter overlap:
A = "abbabbabbabba"
B = "abba"
B end as it starts, so it enters the first if:
if (A.charat[occ_posit] == A.charat[occ_posit+1])
A.remove[occ_posit - 1]; // no reason to remove A[occ_posit] here
else
A.remove[occ_posit]; // here we remove the last char, so we could remove 2 occurencies at the same time
that lets the last "a" of B instance to be removed. So:
1° step: A= "abbbbabbabba"
2° step: A= "abbbbabbbba" and we are done.
Hope this helps
EDIT: pls note that the algotirhm must be corrected a little not to give error when you are close to the A end with your search, but this is just an easy programming issue.
Here's a sketch I've come up with.
First, if A contains any symbols that are not found in B, split up A into a bunch of smaller strings containing only those characters found in B. Apply the algorithm on each of the smaller strings, then glue them back together to get the total result. This really functions as an optimization.
Next, check if A contains any of B. If there isn't, you're done. If A = B, then delete all of them.
I think a relatively greedy algorithm may work.
First, mark all of the symbols in A which belong to at least one occurrence of B. Let A = aabcbccabcaa, B = abc. Bolding indicates these marked characters:
a abc bcc abc aa. If there's an overlap, mark all possible. This operation is naively approximately (A-B) operations, but I believe it can be done in around (A/B) operations.
Consider the deletion of each marked letter in A: a abc bcc abc aa.
Check whether the deletion of that marked letter decreases the number of marked letters. You only need to check the substrings which could possibly be affected by the deletion of the letter. If B has a length of 4, only the substrings starting at the following locations would need to be deleted if x were being checked:
-------x------
^^^^
Any further left or right will exist regardless of the presence of x.
For instance:
Marking the [a] in the following string: a [a]bc bcc abc aa.
Its deletion yields abcbccabcaa, which when marked produces abc bcc abc aa, which has an equal number of marked characters. Since only the relative number is required for this operation, it can be done in approximately 2B time for each selected letter. For each, assign the relative difference between the two. Pick an arbitrary one which is maximal and delete it. Repeat until done. Each pass is roughly up to 2AB operations, for a maximum of A passes, giving a total time of about 2A^2 B.
In the above example, these values are assigned:
aabcbccabcaa
033 333
So arbitrarily deleting the first marked b gives you: aacbccabcaa. If you repeat the process, you get:
aacbccabcaa
333
The final result is done.
I believe the algorithm is correctly minimal. I think it is true that whenever A requires only one deletion, the algorithm must be optimal. In that case, the letter which reduces the most possible matches (ie: all of them) should be best. I can come up with no such proof, though. I'd be interested in finding any counter-examples to optimality.
Find the indeces of each substring in the main string.
Then using a dynamic programming algorithm (so memoize intermediate values), remove each letter that is part of a substring from the main string, add 1 to the count, and repeat.
You can find the letters, because they are within the indeces of each match index + length of B.
A = ababaa
B = aba
count = 0
indeces = (0, 2)
A = babaa, aabaa, abbaa, abbaa, abaaa, ababa
B = aba
count = 1
(2nd abbaa is memoized)
indeces = (1), (1), (), (), (0), (0, 2)
answer = 1
You can take it a step further, and try to memoize the substring match indeces of substrings, but that might not actually be a performance gain.
Not sure on the exact bounds, but shouldn't take too long computationally.