Problem 612: Friend numbers

(see projecteuler.net/problem=612)

Let's call two numbers friend numbers if their representation in base 10 has at least one common digit.
E.g. 1123 and 3981 are friend numbers.

Let f(n) be the number of pairs (p,q) with 1 <= p < q < n such that p and q are friend numbers.
f(100)=1539

Find f(10^18) mod 1000267129.

Not solved yet

My code works for the "small" example but can't find the solution to the "big" problem.

What I've Done So Far

I came up with two brute force algorithms. Given enough time they could solve the problem - but you should be ready to wait a few years until you see the result ...

Both algorithms rely on the function fingerprint with returns a bitmask: if its parameter x contains the digit n at least once, then the n-th bit of the result is set.
For example fingerprint(1305) is 00101011 in binary (or 43 decimal) and fingerprint(52) is 00100100 in binary (= 36 in decimal).
If two numbers are friend numbers then their fingerprints share at least one bit and (fingerprint(a) & fingerprint(b)) != 0. Note: it's a single & because it's the binary AND, not the logical AND.

Let's check whether 1305 and 52 are friends numbers:
(fingerprint(1305) & fingerprint(52)) = 00101011b & 00100100b = 00100000b != 0 → yes, they are friend numbers

The first algorithm bruteForce is an implementation of the obvious O(n^2) strategy:
compare each number q with all smaller numbers p.
It takes about 0.5 seconds to solve f(10^4) and 50 seconds to solve f(10^5).
Runtime for f(10^n) seems to be approximately dfrac{10^{2n}}{2 * 10^8} seconds. So f(10^18) needs approx 5 * 10^27 seconds which is 10^10 longer than the universe existed. Oops.

A significant speed-up can be achieved if switching from that basic O(n^2) algorithm to a slightly smarter O(n) algorithm:

It takes one second to solve f(10^6), ten seconds for f(10^7) and about 103 seconds for f(10^8), a nice linear behaviour.
Runtime for f(10^n) seems to be approximately 10^n * 10^-6 seconds. So f(10^18) needs approx 10^12 seconds which is close to 110000 years.

It's getting late now and I'm too sleepy to figure out a proper way to solve that problem ...

Interactive test

This feature is not available for the current problem.

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
 
// taken from problem statement
const unsigned int Modulo = 1000267129;
 
// set the n-th bit if x contains digit n
unsigned int fingerprint(unsigned long long x)
{
unsigned int result = 0;
while (x > 0)
{
auto digit = x % 10;
result |= 1 << digit;
x /= 10;
}
return result;
}
 
// O(n^2) algorithm, compare each number q with all smaller numbers p
unsigned long long bruteForce(unsigned long long limit)
{
unsigned long long result = 0;
for (unsigned long long q = 1; q < limit; q++)
{
// bitmask of used digits
auto maskQ = fingerprint(q);
 
for (unsigned long long p = 1; p < q; p++)
{
auto maskP = fingerprint(p);
// share at least one digit
if ((maskQ & maskP) != 0)
{
result++;
// much faster than %-operator
if (result == Modulo)
result = 0;
}
}
}
 
return result;
}
 
// O(n) algorithm, much faster than bruteForce() but still too slow to handle limit = 10^18
// but still useful up to 10^8
unsigned long long slow(unsigned long long limit)
{
unsigned long long result = 0;
 
// setup counters for each bitmask (10 digits => 2^10 bitmasks
const auto NumCounters = 1 << 10;
unsigned long long maskCount[NumCounters] = { 0 };
 
// iterate over all numbers
for (unsigned long long q = 1; q < limit; q++)
{
// get current bitmask
auto maskQ = fingerprint(q);
 
// add all counters that share at least one bit with the current bitmask
for (auto i = 1; i < NumCounters; i++)
if (i & maskQ)
{
result += maskCount[i];
if (result >= Modulo)
result %= Modulo;
}
 
// add current number to its bitmask counter
maskCount[maskQ]++;
// avoid overflows
if (maskCount[maskQ] == Modulo)
maskCount[maskQ] = 0;
}
 
return result;
}
 
int main()
{
// friends( 100) = 1539
// friends( 1000) = 289665
// friends( 10000) = 39235977
// friends( 100000) = 4528635021 % 1000267129 = 527566505
// friends(1000000) = 481858831665 % 1000267129 = 730342616
// friends( 10^7) = 49369806187101 % 1000267129 = 621768177
// friends( 10^8) = 4979780329300065 % 1000267129 = 440930015
unsigned long long limit = 1000000ULL;
std::cin >> limit;
//std::cout << bruteForce(limit) << std::endl;
std::cout << slow(limit) << std::endl;
return 0;
}

This solution contains 12 empty lines, 22 comments and 1 preprocessor command.

Changelog

December 14, 2017 solve up to 10^8

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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

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