How do I solve this making it more efficient? - c++

Question

So, I am trying to solve the following question: https://www.codechef.com/TSTAM15/problems/ACM14AM3
The Mars Orbiter Mission probe lifted-off from the First Launch Pad at Satish Dhawan Space Centre (Sriharikota Range SHAR), Andhra
Pradesh, using a Polar Satellite Launch Vehicle (PSLV) rocket C25 at
09:08 UTC (14:38 IST) on 5 November 2013.
The secret behind this successful launch was the launch pad that ISRO
used. An important part of the launch pad is the launch tower. It is
the long vertical structure which supports the rocket.
ISRO now wants to build a better launch pad for their next mission.
For this, ISRO has acquired a long steel bar, and the launch tower can
be made by cutting a segment from the bar. As part of saving the cost,
the bar they have acquired is not homogeneous.
The bar is made up of several blocks, where the ith block has
durability S[i], which is a number between 0 and 9. A segment is
defined as any contiguous group of one or more blocks.
If they cut out a segment of the bar from ith block to jth block
(i<=j), then the durability of the resultant segment is given by (S[i]*10(j-i) + S[i+1]*10(j-i-1) + S[i+2]*10(j-i-2) + … + S[j] * 10(0)) % M. In other words, if W(i,j) is the base-10 number formed by
concatenating the digits S[i], S[i+1], S[i+2], …, S[j], then
the durability of the segment (i,j) is W(i,j) % M.
For technical reasons that ISRO will not disclose, the durability of
the segment used for building the launch tower should be exactly L.
Given S and M, find the number of ways ISRO can cut out a segment from
the steel bar whose durability is L. Input
The first line contains a string S. The ith character of this string
represents the durability of ith segment. The next line contains a
single integer Q, denoting the number of queries. Each of the next Q
lines contain two space separated integers, denoting M and L. Output
For each query, output the number of ways of cutting the bar on a
separate line. Constraints
1 ≤ |S| ≤ 2 * 10^4
Q ≤ 5
0 < M < 500
0 ≤ L < M
Example
Input:
23128765
3
7 2
9 3
15 5
Output:
9
4
5
Explanation
For M=9, L=3, the substrings whose remainder is 3 when divided by
9 are: 3, 31287, 12 and 876.
Now, what I did was, I initially generate all possible substrings of numbers of the given length, and tried to divide it by the given number to check if it is divisible and added it to the answer. Therefore, my code for the same was,
string s;
cin>>s;
int m,l,ans=0;
for ( i = 0; i < s.length(); i++ )
{
for ( j = i+1; j < s.length(); j++ )
{
string p = s.substr(i,j);
long long num = stoi(p);
if (num%m == l)
ans++;
}
}
cout<<ans<<"\n";
return 0;
But obviously since the input length is upto 10^4, this doesn't work in required time. How can I make it more optimal?

Answer 1

A little advice I can give you is to initialize a variable to s.length() to avoid calling the function each time for each for block.

Answer 2

Ok, here goes, with a working program at the bottom
Major optimization #1
Do not (ever) work with strings when it comes to integer arithmetic. You're converting string => integer over and over and over again (this is an O(n^2) problem), which is painstakingly slow. Besides, it also misses the point.
Solution: first convert your array-of-characters (string) to array-of-numbers. Integer arithmetic is fast.
Major optimization #2
Use a smart conversion from "substring" to number. After transforming the characters to actual integers, they become the factors in the the polynomial a_n * 10^n. To convert a substring of n segments into a number, it is enough to compute sum(a_i * 10^i) for 0 <= i < n.
And nicely enough, if the coefficients a_i are arranged the way they are in the problem's statement, you can use Horner's method (https://en.wikipedia.org/wiki/Horner%27s_method) to very quickly evaluate the numerical value of the substring.
In short: keep a running value of the current substring and growing it by one element is just * 10 + new element
Example: string "128472373".
First substring = "1", value = 1.
For the second substring we need to
add the digit "2" as follows: value = value * 10 + "2", thus: value = 1 * 10 + 2 = 12.
For 3rd substring need to add digit "8": value = value * 10 + "8", thus: value = 12 * 10 + 8 = 128.
Etcetera.
I had some issues with formatting the C++ code inline so I stuck it in IDEone: https://ideone.com/TbJiqK
The gist of the program:
In main loop, loop over all possible start points:
// For all startpoints in the segments array ...
for(int* f=segments; f<segments+n_segments; f++)
// add up the substrings that fullfill the question
n += count_segments(f, segments+n_segments, m, l);
// Output the answer for this question
cout << n << endl;
Implementation of the count_segments() function:
// Find all substrings that % m == l
// Use Horner's algorithm to quickly evaluate sum(a_n*10^n) where
// a_n are the segments' durabilities
int count_segments(int* first, int* last, int m, int l) {
int n = 0, number = 0;
while( first<last ) {
number = number * 10 + *first; // This is Horner's method
if( (number % m)==l ) {
n++;
// If you don't believe - enable this line of output and
// see the numbers matching the combinations of the
//cout << "[" << m << ", " << l << "]: " << number << endl;
}
first++;
}
return n;
}