I encountered the 10^9+7 problem but I can't understand the relation between the distributive properties of mod and my problem

Question

Given 3 numbers a b c get a^b , b^a , c^x where x is abs diff between b and a cout each one but mod 10^9+7 in ascending order. well I searched web for how to use the distributive property but didn't understand it since I am beginner,

I use very simple for loops so understanding this problem is a bit hard for me so how can I relate these mod rules with powers too in loops? If anyone can help me I would be so happy. note time limit is 1 second which makes it harder

I tried to mod the result every time in the loop then times it by the original number. for example if 2^3 then 1st loop given variables cin>>a,a would be 2, num =a would be like this a = (a % 10^9 + 7) * num this works for very small inputs but large ones it exceed time

#include <iostream>
#include <cmath>

using namespace std;
int main ()
{
    long long a,b,c,one,two,thr;
    long long x;
    long long mod = 1e9+7;
    cin>>a>>b>>c;
    one = a;
    two = b;
    thr = c;
    if (a>=b)
    x = a - b;
    else
    x = b - a;

    for(int i = 0; i < b-1;i++)
    {
        a = ((a % mod) * (one%mod))%mod;
    }
    for(int j = 0; j < a-1;j++)
    {
        b = ((b % mod) * (two%mod))%mod;
    }
    for(int k = 0; k < x-1;k++)
    {
        c = ((c % mod) * (thr%mod))%mod;
    }
}

Answer 1

I use very simple for loops [...] this works for very small inputs, but large ones it exceeds time.

There is an algorithm called "exponentiation by squaring" that has a logarithmic time complexity, rather then a linear one.

It works breaking down the power exponent while increasing the base.

Consider, e.g. x³⁵⁵. Instead of multiplying x 354 times, we can observe that

x³⁵⁵ = x·x³⁵⁴ = x·(x²)¹⁷⁷ = x·x²·(x²)¹⁷⁶ = x·x²·(x⁴)⁸⁸ = x·x²·(x⁸)⁴⁴ = x·x²·(x¹⁶)²² = x·x²·(x³²)¹¹ = x·x²·x³²·(x³²)¹⁰ = x·x²·x³²·(x⁶⁴)⁵ = x·x²·x³²·x⁶⁴·(x⁶⁴)⁴ = x·x²·x³²·x⁶⁴·(x¹²⁸)² = x¹·x²·x³²·x⁶⁴·x²⁵⁶

That took "only" 12 steps.

To implement it, we only need to be able to perform modular multiplications safely, without overflowing. Given the value of the modulus, a type like std::int64_t is wide enough.

#include <iostream>
#include <cstdint>
#include <limits>
#include <cassert>

namespace modular
{
  auto exponentiation(std::int64_t base, std::int64_t exponent) -> std::int64_t;
}

int main()
{
  std::int64_t a, b, c; 
  std::cin >> a >> b >> c;  

  auto const x{ b < a ? a - b : b - a };

  std::cout << modular::exponentiation(a, b) << '\n'
            << modular::exponentiation(b, a) << '\n'
            << modular::exponentiation(c, x) << '\n';

  return 0;
}

namespace modular
{
  constexpr std::int64_t M{ 1'000'000'007 };

  // We need the mathematical modulo
  auto from(std::int64_t x)
  {
    static_assert(M > 0);
    x %= M;
    return x < 0 ? x + M : x;
  }

  // It assumes that both a and b are already mod M
  auto multiplication_(std::int64_t a, std::int64_t b)
  {
    assert( 0 <= a  and a < M  and  0 <= b  and  b < M );
    assert( b == 0  or  a <= std::numeric_limits<int64_t>::max() / b );

    return (a * b) % M;
  }

  // Implements exponentiation by squaring
  auto exponentiation(std::int64_t base, std::int64_t exponent) -> std::int64_t
  {
    assert( exponent >= 0 );
    
    auto b{ from(base) };
    std::int64_t x{ 1 };
    while ( exponent > 1 )
    {
      if ( exponent % 2 != 0 )
      {
        x = multiplication_(x, b);
        --exponent;
      }
      b = multiplication_(b, b);
      exponent /= 2;
    }
    return multiplication_(b, x);
  }
}