Number of Music Playlists

#include <vector>
    
class PlaylistCalculator {
private:
    static const long long MODULO = 1000000007;  // 10^9 + 7
    std::vector<long long> fact;                // Store factorials
    std::vector<long long> invFact;             // Store inverse factorials
    
    // Function to perform modular exponentiation
    long long modExp(long long base, int exp) {
        long long res = 1;
        while (exp > 0) {
            if (exp % 2 == 1) {
                res = (res * base) % MODULO;
            }
            base = (base * base) % MODULO;
            exp /= 2;
        }
        return res;
    }
    
    // Precompute factorials and their modular inverses
    void prepareFactorials(int maxN) {
        fact.resize(maxN + 1);
        invFact.resize(maxN + 1);
        fact[0] = invFact[0] = 1;
        for (int i = 1; i <= maxN; i++) {
            fact[i] = (fact[i - 1] * i) % MODULO;
            invFact[i] = modExp(fact[i], MODULO - 2);  // Using Fermat's Little Theorem
        }
    }
    
public:
    int countPlaylists(int numSongs, int playlistLength, int repeatLimit) {
        prepareFactorials(numSongs);
        long long result = 0;
        int sign = 1;
        for (int i = numSongs; i >= repeatLimit; i--) {
            long long temp = modExp(i - repeatLimit, playlistLength - repeatLimit);
            temp = temp * invFact[numSongs - i] % MODULO;
            temp = temp * invFact[i - repeatLimit] % MODULO;
            result = (result + sign * temp + MODULO) % MODULO;
            sign *= -1;
        }
        return (fact[numSongs] * result) % MODULO;
    }
};

Calculate the number of music playlists that meet specific conditions using C++, focusing on modular arithmetic for computational efficiency. Here's a detailed rundown of the implemented solution in the provided C++ code:

• The system employs a PlaylistCalculator class that keeps all functional methods encapsulated:

  • fact and invFact vectors store the precomputed factorials and their modular inverses, respectively.

• Key functions in the class:

  • modExp: Handles modular exponentiation, critical for operations under a large prime modulus (MODULO being (10^9 + 7)), ensuring the calculation does not overflow.
  • prepareFactorials: Computes all factorials and their inverses up to the required number maxN using Fermat's Little Theorem for inverses.

• The constructor, PlaylistCalculator:

  • Initializes factorials up to the desired maximum required for the computations, setting up the base for the main function.

• The primary function, countPlaylists:

  • Sets up and calculates the number of valid playlists using combinatorial formulas adjusted for the modulo operations.
  • Implements an inclusion-exclusion principle to accurately calculate the potential playlists by iterating from the total number of songs down to the minimum that may repeat (given by repeatLimit).

The outlined approach ensures:

• Efficient computation using modular arithmetic to handle large numbers.
• Precise calculations with inclusion-exclusion to accommodate for playlists restrictions.
• Use of precomputed values for factorial to optimize multiple factorial calculations, especially within the combinatorial context.

This ensures an optimal implementation to solve for the number of viable playlists adhering to both the length and repetition limitations, leveraging combinatorial mathematics and efficient use of modern C++ data structures and algorithms.

public class PlaylistCalculator {
    private static final long MODULO = 1_000_000_007;
    
    // Arrays for factorials and inverses
    private long[] fact;
    private long[] invFact;
    
    // Calculates number of possible playlists
    public int calculatePlaylists(int songs, int length, int k) {
        // Prepare factorials and their inverses
        setupFactorials(songs);
    
        long total = 0;
        int sig = 1;
    
        // Calculate possibilities
        for (int i = songs; i >= k; i--) {
            long result = modPower(i - k, length - k);
            result = (result * invFact[songs - i]) % MODULO;
            result = (result * invFact[i - k]) % MODULO;
    
            total = (total + sig * result + MODULO) % MODULO;
            sig *= -1;
        }
    
        return (int) ((fact[songs] * total) % MODULO);
    }
    
    // Setup factorial values and their modular inverses
    private void setupFactorials(int upTo) {
        fact = new long[upTo + 1];
        invFact = new long[upTo + 1];
        fact[0] = invFact[0] = 1;
    
        for (int i = 1; i <= upTo; i++) {
            fact[i] = (fact[i - 1] * i) % MODULO;
            invFact[i] = modPower(fact[i], (int) (MODULO - 2));
        }
    }
    
    // Efficient modular exponentiation
    private long modPower(long base, int exp) {
        long res = 1L;
        while (exp > 0) {
            if ((exp & 1) == 1) {
                res = (res * base) % MODULO;
            }
            exp >>= 1;
            base = (base * base) % MODULO;
        }
    
        return res;
    }
}

To solve the problem of calculating the number of music playlists, a Java class named PlaylistCalculator is utilized. This class includes methods for calculating factorial values, their modular inverses, and efficiently computing powers with a modulo operation.

Within the calculatePlaylists method:

• First, factorial arrays are initialized by invoking the setupFactorials method for a given number of songs.
• The total number of combinations is calculated using a formula derived from combinatorial mathematics and properties of permutations and combinatorial arrangements. Alternating signs are managed using a variable sig that flips between each iteration to handle inclusion-exclusion principles.
• Each iteration computes possible configurations based on available songs and spots in the playlist, adjusting for songs that cannot immediately repeat within k positions using modular arithmetic to ensure computations stay within bounds of the given MODULO (1,000,000,007).

The array fact is used to store precomputed factorial values, and invFact holds the modular inverses of these factorial values. The inverses aid in efficient division under modulo properties.

The setupFactorials method sets up the factorial and inverse arrays. Each factorial is computed sequentially, and inverses are calculated using the modPower method:

• The modPower method performs modular exponentiation which is essential for finding factorial inverses via Fermat's Little Theorem. This process helps manage large numbers effectively without overflow, crucial for handling large factorial values within the modulo.

This solution ensures a linear-time operation for setting up factorials and a potentially O(L) complexity for playlist calculation, where L is the playlist length, making it efficient for the computation of very large numbers in combinatorial scenarios.

class Solution:
    MODULUS = 1000000007
    
    def exponentiation(self, base, exp):
        result = 1
        while exp > 0:
            if exp % 2 == 1:
                result = (result * base) % self.MODULUS
            exp //= 2
            base = (base * base) % self.MODULUS
        return result
    
    def precompute_factorials(self, n):
        self.fact = [1] * (n + 1)
        self.inv_fact = [1] * (n + 1)
        for i in range(1, n + 1):
            self.fact[i] = (self.fact[i - 1] * i) % self.MODULUS
            self.inv_fact[i] = self.exponentiation(self.fact[i], self.MODULUS - 2)
    
    def number_of_playlists(self, n, goal, k):
        self.precompute_factorials(n)
        sign = 1
        total = 0
        for i in range(n, k - 1, -1):
            temp_result = self.exponentiation(i - k, goal - k)
            temp_result = (temp_result * self.inv_fact[n - i]) % self.MODULUS
            temp_result = (temp_result * self.inv_fact[i - k]) % self.MODULUS
            total = (total + sign * temp_result + self.MODULUS) % self.MODULUS
            sign *= -1
        return (self.fact[n] * total) % self.MODULUS

The Python program provided solves the problem of determining the number of unique playlists that can be generated given specific constraints, using a modular arithmetic approach to address issues related to large numbers.

• The class Solution includes methods for modular exponentiation, precomputation of factorials and their modular inverses, and the main method number_of_playlists that calculates the number of ways to create a playlist.
• Modular exponentiation, implemented in the exponentiation method, efficiently calculates powers of numbers under a modulus, a key operation given constraints often involve large values.
• Factorials and their inverses modulo a large number (10^9 + 7) are precomputed in the precompute_factorials method. This precomputation speeds up the calculation in the main function as factorials are used repetitively.
• number_of_playlists uses a dynamic programming approach with inclusion-exclusion to calculate the result.
• It iterates backwards calculating contribution of each possible number of distinct songs to total playlists, adjusting for overcounts using alternating signs (managed by sign), mimicking the inclusion-exclusion principle.

The roles of major variables:

• n is the number of distinct songs available.
• goal represents the target number of positions (length of the playlist) to be filled.
• k sets a condition that a song cannot be repeated within 'k' subsequent positions once it is played.

Through these algorithms and number theory principles, the program efficiently handles calculating permutations with constraints by modular arithmetic, avoiding overflow problems and delivering results for large inputs expediently. This solution leverages mathematical properties to handle potentially vast numbers that could result from combinatorial calculations, crucial in cases where direct computation is not feasible due to time or space constraints.