# Please help to solve this problem

Ah I see, my bad.

Sample test cases:
1)input : 10
Output: 2
Explanation : As decimal equivalent 10 is 2.Only pairs are (1,2) and (2,1)

2)input: 101
Output: 10
Explanation : As decimal equivalent 101 is 5.Few pairs are (1,4),(1,2) and (2,1)

I think you’re mistaken, a and b can be zero as well, according to the constraints given by Suman, also it makes more sense to include 0.

Sorry, I got a little caught up with something important so couldn’t reply. I think we can solve it with digit dp. I’m attaching the code it’s pretty neat to understand, though feel free to ping me if you need an explanation of my approach.

CODE
#include "bits/stdc++.h"
#define ll long long
using namespace std ;
const int M = 1e9+7,mxN=1e5;
int dp[mxN+1] ;
ll pM(ll b,int p){
ll r=1 ;
for(;p;b=(b*b)%M,p/=2)
if(p&1)
r=(r*b)%M ;
return r ;
}
int main(){
ios_base::sync_with_stdio(false);
cin.tie(NULL);
string s ; cin >> s ;
int n = s.size() ;
dp[n]=1;
for(int i=n-1,j=0;~i;--i,j++)
dp[i]=(s[i]=='0'?dp[i+1]:(2*dp[i+1]+pM(3,j))%M)  ;
cout << dp[0] ;
}


@anon41318104

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Please explain the approach .

hey @cubefreak777 how do you put your code in a drop-down way(▼)? using some latex code?

Summary

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Thanks

That’s a really cool thing. thanks a lot @cubefreak777 . btw got to know about your final exams from a thread. All the very best!

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Thanks!

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If the i’th bit is set in the string then if you keep that bit unset you have 3 combinations for all remaining bits , those are {{0,1},{1,0},{0,0}}.
You cannot have {1,1} as its xor will be 0 whereas sum will be more than 0.
Where as to keep that bit set you have 2 combination available {{1,0},{0,1}}

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What is the use of j in the function

Same question

### NOTATION:

1. S is the binary string given.
2. N is length of S.
3. (A,B) is any valid pair such that conditions given hold true.
4. X = A+B=A \oplus B
5. D(S) is the decimal representation of binary string S.
6. C(i) is the number of set bits in i.
7. S[i \dots N] is the suffix of S after index i.

We can reduce the problem to find the following summation, since it is given in the problem that A+B \le D \implies X \le D. So we just have sum up the number of pairs for all possible values of X. i.e. the following summation.

R= \sum_{X=0}^{D(S)} 2^{C(X)}

But how to find this? obviously, we can’t naively iterate overall all values from 0 to 2^N where N\le 10^5.
So we’ll use digit dp to find R.

### DIGIT DP STATES AND TRANSITIONS:

DP[i]\displaystyle \sum_{k=0}^{D(S[i \dots N])}2^{C(k)} i.e. Sum of 2^{C(k)} for all 0 \le k \le D(S[i \dots N]) in the suffix of S after index i.
Transitions are pretty straightforword.

#### IF S[i]=0:

Then it’d be the same as the answer we obtained at index i+1 (Since we have no option but to put a zero in here otherwise it’ll be greater than D(S[i \dots N])

\boxed{DP[i]=DP[i+1]}

#### IF S[i]=1:

Now we can put either a zero or a 1 at index i.
If we happen to put a 0 then for the entire suffix from i+1 to N we can have any number of set bits we want. So the number of numbers with k set bits would just be \displaystyle \binom{N-1-i}{k} but since the contribution of each number with k set bits is 2^k so total contribution would be \displaystyle \sum_{k=0}^{N-1-i}\binom{N-1-i}{k}2^k
But what if we put a 1 at index i then it is clearly evident that it is the sub subproblem for suffix from i+1 that we solved already i.e. DP[i+1] but with one extra set bit so the contribution from this subpart will be 2 \times DP[i+1]. So the transition will look like.

DP[i]=2 \times DP[i+1]+\sum_{k=0}^{N-1-i}\binom{N-1-i}{k}2^k \\ \sum_{k=0}^{N-1-i}\binom{N-1-i}{k}2^k=(1+2)^{N-1-i}=3^{N-1-i} \\ \boxed{DP[i]=2 \times DP[i+1]+3^{N-1-i}}

Base case is DP[N]=1 and final answer R would just be DP[0].

#### ASYMPOTICS.

Time & Space complexity for this solution is \mathcal{O}(N)

CODE FOR REFERENCE
#include "bits/stdc++.h"
#define ll long long
using namespace std ;
const int M = 1e9+7,mxN=1e7;
int dp[mxN+1] ;
ll pM(ll b,int p){
ll r=1 ;
for(;p;b=(b*b)%M,p/=2)
if(p&1)
r=(r*b)%M ;
return r ;
}
int main(){
string s ; cin >> s ;
int n = s.size() ;
dp[n]=1;
for(int i=n-1;~i;--i)
dp[i]=(s[i]=='0'?dp[i+1]:(2*dp[i+1]+pM(3,n-1-i))%M)  ;
cout << dp[0] ;
}



Solutions that involve converting the entire string to integer as presented by @suman_18733097, won’t work for strings greater than length 10^4(thanks to python, C++ would’ve failed to handle strings of length 60 ) it is because you’re essentially storing 2^{10^5} in int data type which is way way too much!!.

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You don’t need to have j, that was just for simplicity. j=N-1-i

Check this
#include "bits/stdc++.h"
#define ll long long
using namespace std ;
const int M = 1e9+7,mxN=1e7;
int dp[mxN+1] ;
ll pM(ll b,int p){
ll r=1 ;
for(;p;b=(b*b)%M,p/=2)
if(p&1)
r=(r*b)%M ;
return r ;
}
int main(){
string s ; cin >> s ;
int n = s.size() ;
dp[n]=1;
for(int i=n-1;~i;--i)
dp[i]=(s[i]=='0'?dp[i+1]:(2*dp[i+1]+pM(3,n-1-i))%M)  ;
cout << dp[0] ;
}

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Thank You for clearing doubt

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@cubefreak777 thanks a lot for helping

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@cubefreak777 could share some best resources to become good in dp

Did you fully solve CSES DP section & Atcoder Educational DP Contest ? If not then start from here. Then practice specific dp types individually, like digit dp, dp on graphs etc. from codeforces.

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@cubefreak777 No i haven’t solved any .Thanks for sharing

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hi @cubefreak777 I have one more bitwise question , I have posted it here