Compute nCr%p using Lucas Theorem

Given three numbers n, r and p, compute the value of ⁿC_r mod p.

Examples:

Input:  n = 10, r = 2, p = 13
Output: 6
Explanation: 10C2 is 45 and 45 % 13 is 6.

Input:  n = 1000, r = 900, p = 13
Output: 8

We strongly recommend referring below post as a prerequisite of this.
Compute ⁿC_r % p | Set 1 (Introduction and Dynamic Programming Solution)
We have introduced overflow problem and discussed Dynamic Programming based solution in above set 1. The time complexity of the DP-based solution is O(n*r) and it required O(n) space. The time taken and extra space become very high for large values of n, especially values close to 10⁹.
In this post, Lucas Theorem-based solution is discussed. The time complexity of this solution is O(p² * Log_p n) and it requires only O(p) space.

Lucas Theorem:

For non-negative integers n and r and a prime p, the following congruence relation holds: 

where 

and 

Using Lucas Theorem for ⁿC_r % p:

Lucas theorem basically suggests that the value of ⁿC_r can be computed by multiplying results of ^n_iC_ri where n_i and r_i are individual same-positioned digits in base p representations of n and r respectively.

The idea is to one by one compute ^n_iC_ri for individual digits n_i and r_i in base p. We can compute these values DP based solution discussed in previous post. Since these digits are in base p, we would never need more than O(p) space, and time complexity of these individual computations would be bounded by O(p²).

Below is the implementation of the above idea

C++14

// A Lucas Theorem based solution to compute nCr % p
#include<bits/stdc++.h>
using namespace std;

// Returns nCr % p.  In this Lucas Theorem based program,
// this function is only called for n < p and r < p.
int nCrModpDP(int n, int r, int p)
{
    // The array C is going to store last row of
    // pascal triangle at the end. And last entry
    // of last row is nCr
    int C[r+1];
    memset(C, 0, sizeof(C));

    C[0] = 1; // Top row of Pascal Triangle

    // One by constructs remaining rows of Pascal
    // Triangle from top to bottom
    for (int i = 1; i <= n; i++)
    {
        // Fill entries of current row using previous
        // row values
        for (int j = min(i, r); j > 0; j--)

            // nCj = (n-1)Cj + (n-1)C(j-1);
            C[j] = (C[j] + C[j-1])%p;
    }
    return C[r];
}

// Lucas Theorem based function that returns nCr % p
// This function works like decimal to binary conversion
// recursive function.  First we compute last digits of
// n and r in base p, then recur for remaining digits
int nCrModpLucas(int n, int r, int p)
{
   // Base case
   if (r==0)
      return 1;

   // Compute last digits of n and r in base p
   int ni = n%p, ri = r%p;

   // Compute result for last digits computed above, and
   // for remaining digits.  Multiply the two results and
   // compute the result of multiplication in modulo p.
   return (nCrModpLucas(n/p, r/p, p) * // Last digits of n and r
           nCrModpDP(ni, ri, p)) % p;  // Remaining digits
}

// Driver program
int main()
{
    int n = 1000, r = 900, p = 13;
    cout << "Value of nCr % p is " << nCrModpLucas(n, r, p);
    return 0;
}

Java

// A Lucas Theorem based solution to compute nCr % p

class GFG{
// Returns nCr % p. In this Lucas Theorem based program,
// this function is only called for n < p and r < p.
static int nCrModpDP(int n, int r, int p)
{
    // The array C is going to store last row of
    // pascal triangle at the end. And last entry
    // of last row is nCr
    int[] C=new int[r+1];

    C[0] = 1; // Top row of Pascal Triangle

    // One by constructs remaining rows of Pascal
    // Triangle from top to bottom
    for (int i = 1; i <= n; i++)
    {
        // Fill entries of current row using previous
        // row values
        for (int j = Math.min(i, r); j > 0; j--)

            // nCj = (n-1)Cj + (n-1)C(j-1);
            C[j] = (C[j] + C[j-1])%p;
    }
    return C[r];
}

// Lucas Theorem based function that returns nCr % p
// This function works like decimal to binary conversion
// recursive function. First we compute last digits of
// n and r in base p, then recur for remaining digits
static int nCrModpLucas(int n, int r, int p)
{
// Base case
if (r==0)
    return 1;

// Compute last digits of n and r in base p
int ni = n%p;
int ri = r%p;

// Compute result for last digits computed above, and
// for remaining digits. Multiply the two results and
// compute the result of multiplication in modulo p.
return (nCrModpLucas(n/p, r/p, p) * // Last digits of n and r
        nCrModpDP(ni, ri, p)) % p; // Remaining digits
}

// Driver program
public static void main(String[] args)
{
    int n = 1000, r = 900, p = 13;
    System.out.println("Value of nCr % p is "+nCrModpLucas(n, r, p));
}
}
// This code is contributed by mits

Python3

# A Lucas Theorem based solution
# to compute nCr % p

# Returns nCr % p. In this Lucas
# Theorem based program, this
# function is only called for
# n < p and r < p.
def nCrModpDP(n, r, p):

    # The array C is going to store
    # last row of pascal triangle
    # at the end. And last entry
    # of last row is nCr
    C = [0] * (n + 1);

    # Top row of Pascal Triangle
    C[0] = 1;

    # One by constructs remaining
    # rows of Pascal Triangle from
    # top to bottom
    for i in range(1, (n + 1)):

        # Fill entries of current
        # row using previous row
        # values
        j = min(i, r);
        while(j > 0):
            C[j] = (C[j] + C[j - 1]) % p;
            j -= 1;
    return C[r];

# Lucas Theorem based function that
# returns nCr % p. This function
# works like decimal to binary
# conversion recursive function.
# First we compute last digits of
# n and r in base p, then recur
# for remaining digits
def nCrModpLucas(n, r, p):

    # Base case
    if (r == 0):
        return 1;

    # Compute last digits of n
    # and r in base p
    ni = int(n % p);
    ri = int(r % p);

    # Compute result for last digits
    # computed above, and for remaining
    # digits. Multiply the two results
    # and compute the result of
    # multiplication in modulo p.
    # Last digits of n and r
    return (nCrModpLucas(int(n / p), int(r / p), p) *
            nCrModpDP(ni, ri, p)) % p; # Remaining digits

# Driver Code
n = 1000;
r = 900;
p = 13;
print("Value of nCr % p is",
       nCrModpLucas(n, r, p));

# This code is contributed by mits

C#

// A Lucas Theorem based solution
// to compute nCr % p
using System;

class GFG
{
// Returns nCr % p. In this Lucas
// Theorem based program, this
// function is only called for
// n < p and r < p.
static int nCrModpDP(int n, int r, int p)
{
    // The array C is going to store
    // last row of pascal triangle
    // at the end. And last entry
    // of last row is nCr
    int[] C = new int[r + 1];

    C[0] = 1; // Top row of Pascal Triangle

    // One by constructs remaining
    // rows of Pascal Triangle
    // from top to bottom
    for (int i = 1; i <= n; i++)
    {
        // Fill entries of current row
        // using previous row values
        for (int j = Math.Min(i, r); j > 0; j--)

            // nCj = (n-1)Cj + (n-1)C(j-1);
            C[j] = (C[j] + C[j - 1]) % p;
    }
    return C[r];
}

// Lucas Theorem based function that
// returns nCr % p. This function works
// like decimal to binary conversion
// recursive function. First we compute
// last digits of n and r in base p,
// then recur for remaining digits
static int nCrModpLucas(int n, int r, int p)
{
// Base case
if (r == 0)
    return 1;

// Compute last digits of n
// and r in base p
int ni = n % p;
int ri = r % p;

// Compute result for last digits
// computed above, and for remaining
// digits. Multiply the two results
// and compute the result of
// multiplication in modulo p.
return (nCrModpLucas(n / p, r / p, p) * // Last digits of n and r
        nCrModpDP(ni, ri, p)) % p; // Remaining digits
}

// Driver Code
public static void Main()
{
    int n = 1000, r = 900, p = 13;
    Console.Write("Value of nCr % p is " +
                   nCrModpLucas(n, r, p));
}
}

// This code is contributed
// by ChitraNayal

PHP

<?php
// A Lucas Theorem based
// solution to compute
// nCr % p

// Returns nCr % p. In this
// Lucas Theorem based program,
// this function is only called
// for n < p and r < p.
function nCrModpDP($n, $r, $p)
{
    // The array C is going to
    // store last row of pascal
    // triangle at the end. And
    // last entry of last row is nCr
    $C = array_fill(0, $n + 1,
                       false);

    // Top row of
    // Pascal Triangle
    $C[0] = 1;

    // One by constructs remaining
    // rows of Pascal Triangle from
    // top to bottom
    for ($i = 1; $i <= $n; $i++)
    {
        // Fill entries of current
        // row using previous row
        // values
        for ($j = min($i, $r);
             $j > 0; $j--)

            $C[$j] = ($C[$j] +
                      $C[$j - 1]) % $p;
    }
    return $C[$r];
}

// Lucas Theorem based function
// that returns nCr % p. This
// function works like decimal
// to binary conversion recursive
// function. First we compute last
// digits of n and r in base p,
// then recur for remaining digits
function nCrModpLucas($n, $r, $p)
{

// Base case
if ($r == 0)
    return 1;

// Compute last digits
// of n and r in base p
$ni = $n % $p;
$ri = $r % $p;

// Compute result for last
// digits computed above,
// and for remaining digits.
// Multiply the two results
// and compute the result of
// multiplication in modulo p.
return (nCrModpLucas($n / $p,
                     $r / $p, $p) * // Last digits of n and r
        nCrModpDP($ni, $ri, $p)) % $p; // Remaining digits
}

// Driver Code
$n = 1000; $r = 900; $p = 13;
echo "Value of nCr % p is " ,
    nCrModpLucas($n, $r, $p);

// This code is contributed by ajit
?>

Javascript

<script>

// A Lucas Theorem based solution to compute nCr % p

// Returns nCr % p. In this Lucas Theorem based program,
// this function is only called for n < p and r < p.
function nCrModpDP(n, r, p)
{
    // The array C is going to store last row of
    // pascal triangle at the end. And last entry
    // of last row is nCr
    C = Array(r+1).fill(0);


    C[0] = 1; // Top row of Pascal Triangle

    // One by constructs remaining rows of Pascal
    // Triangle from top to bottom
    for (var i = 1; i <= n; i++)
    {
        // Fill entries of current row using previous
        // row values
        for (var j = Math.min(i, r); j > 0; j--)

            // nCj = (n-1)Cj + (n-1)C(j-1);
            C[j] = (C[j] + C[j-1])%p;
    }
    return C[r];
}

// Lucas Theorem based function that returns nCr % p
// This function works like decimal to binary conversion
// recursive function. First we compute last digits of
// n and r in base p, then recur for remaining digits
function nCrModpLucas(n, r, p)
{
// Base case
if (r==0)
    return 1;

// Compute last digits of n and r in base p
var ni = n%p, ri = r%p;

// Compute result for last digits computed above, and
// for remaining digits. Multiply the two results and
// compute the result of multiplication in modulo p.
return (nCrModpLucas(parseInt(n/p), parseInt(r/p), p) * // Last digits of n and r
        nCrModpDP(ni, ri, p)) % p; // Remaining digits
}

// Driver program
var n = 1000, r = 900, p = 13;
document.write("Value of nCr % p is " + nCrModpLucas(n, r, p));


</script>

Output

Value of nCr % p is 9

Time Complexity: Time complexity of this solution is O(p² * Log_p n). There are O(Log_p n) digits in base p representation of n. Each of these digits is smaller than p, therefore, computations for individual digits take O(p²). Note that these computations are done using DP method which takes O(n*r) time.
Auxiliary Space: O(r) as extra space for array C is being used

Alternate Implementation with O(p² + Log_p n) time and O(p²) space:
The idea is to precompute Pascal triangle for size p x p and store it in 2D array. All values needed would now take O(1) time. Therefore overall time complexity becomes O(p² + Log_p n).
This article is contributed by Ruchir Garg. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above