Add integer partition algorithm using dynamic programming (#976)

* Add integer partition algorithm using dynamic programming

* Update integer_partition.rs

* Update integer_partition.rs

* Update integer_partition.rs

---------

Co-authored-by: Andrii Siriak <siryaka@gmail.com>

Author: AliAlimohammadi

DIRECTORY.md

@@ -104,6 +104,7 @@
     * [Egg Dropping](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/egg_dropping.rs)
     * [Fibonacci](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/fibonacci.rs)
     * [Fractional Knapsack](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/fractional_knapsack.rs)
+    * [Integer Partition](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/integer_partition.rs)
     * [Is Subsequence](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/is_subsequence.rs)
     * [Knapsack](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/knapsack.rs)
     * [Longest Common Subsequence](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/longest_common_subsequence.rs)

src/dynamic_programming/integer_partition.rs

@@ -0,0 +1,117 @@
+//! Integer partition using dynamic programming
+//!
+//! The number of partitions of a number n into at least k parts equals the number of
+//! partitions into exactly k parts plus the number of partitions into at least k-1 parts.
+//! Subtracting 1 from each part of a partition of n into k parts gives a partition of n-k
+//! into k parts. These two facts together are used for this algorithm.
+//!
+//! More info:
+//! * <https://en.wikipedia.org/wiki/Partition_(number_theory)>
+//! * <https://en.wikipedia.org/wiki/Partition_function_(number_theory)>
+
+#![allow(clippy::large_stack_arrays)]
+
+/// Calculates the number of partitions of a positive integer using dynamic programming.
+///
+/// # Arguments
+///
+/// * `m` - A positive integer to find the number of partitions for
+///
+/// # Returns
+///
+/// The number of partitions of `m`
+///
+/// # Panics
+///
+/// Panics if `m` is not a positive integer (0 or negative)
+///
+/// # Examples
+///
+/// ```
+/// # use the_algorithms_rust::dynamic_programming::partition;
+/// assert_eq!(partition(5), 7);
+/// assert_eq!(partition(7), 15);
+/// assert_eq!(partition(100), 190569292);
+/// ```
+#[allow(clippy::large_stack_arrays)]
+pub fn partition(m: i32) -> u128 {
+    // Validate input
+    assert!(m > 0, "Input must be a positive integer greater than 0");
+
+    let m = m as usize;
+
+    // Initialize memo table with zeros using iterative construction
+    // to avoid large stack allocations
+    let mut memo: Vec<Vec<u128>> = Vec::with_capacity(m + 1);
+    for _ in 0..=m {
+        memo.push(vec![0u128; m]);
+    }
+
+    // Base case: there's one way to partition into 0 parts (empty partition)
+    for i in 0..=m {
+        memo[i][0] = 1;
+    }
+
+    // Fill the memo table using dynamic programming
+    for n in 0..=m {
+        for k in 1..m {
+            // Add partitions from k-1 (partitions with at least k-1 parts)
+            memo[n][k] += memo[n][k - 1];
+
+            // Add partitions from n-k-1 with k parts (subtract 1 from each part)
+            if n > k {
+                memo[n][k] += memo[n - k - 1][k];
+            }
+        }
+    }
+
+    memo[m][m - 1]
+}
+
+#[cfg(test)]
+#[allow(clippy::large_stack_arrays)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn test_partition_5() {
+        assert_eq!(partition(5), 7);
+    }
+
+    #[test]
+    fn test_partition_7() {
+        assert_eq!(partition(7), 15);
+    }
+
+    #[test]
+    #[allow(clippy::large_stack_arrays)]
+    fn test_partition_100() {
+        assert_eq!(partition(100), 190569292);
+    }
+
+    #[test]
+    #[allow(clippy::large_stack_arrays)]
+    fn test_partition_1000() {
+        assert_eq!(partition(1000), 24061467864032622473692149727991);
+    }
+
+    #[test]
+    #[should_panic(expected = "Input must be a positive integer greater than 0")]
+    fn test_partition_negative() {
+        partition(-7);
+    }
+
+    #[test]
+    #[should_panic(expected = "Input must be a positive integer greater than 0")]
+    fn test_partition_zero() {
+        partition(0);
+    }
+
+    #[test]
+    fn test_partition_small_values() {
+        assert_eq!(partition(1), 1);
+        assert_eq!(partition(2), 2);
+        assert_eq!(partition(3), 3);
+        assert_eq!(partition(4), 5);
+    }
+}

src/dynamic_programming/mod.rs

@@ -3,6 +3,7 @@ mod coin_change;
 mod egg_dropping;
 mod fibonacci;
 mod fractional_knapsack;
+mod integer_partition;
 mod is_subsequence;
 mod knapsack;
 mod longest_common_subsequence;
@@ -27,16 +28,13 @@ mod word_break;
 pub use self::catalan_numbers::catalan_numbers;
 pub use self::coin_change::coin_change;
 pub use self::egg_dropping::egg_drop;
-pub use self::fibonacci::binary_lifting_fibonacci;
-pub use self::fibonacci::classical_fibonacci;
-pub use self::fibonacci::fibonacci;
-pub use self::fibonacci::last_digit_of_the_sum_of_nth_fibonacci_number;
-pub use self::fibonacci::logarithmic_fibonacci;
-pub use self::fibonacci::matrix_fibonacci;
-pub use self::fibonacci::memoized_fibonacci;
-pub use self::fibonacci::nth_fibonacci_number_modulo_m;
-pub use self::fibonacci::recursive_fibonacci;
+pub use self::fibonacci::{
+    binary_lifting_fibonacci, classical_fibonacci, fibonacci,
+    last_digit_of_the_sum_of_nth_fibonacci_number, logarithmic_fibonacci, matrix_fibonacci,
+    memoized_fibonacci, nth_fibonacci_number_modulo_m, recursive_fibonacci,
+};
 pub use self::fractional_knapsack::fractional_knapsack;
+pub use self::integer_partition::partition;
 pub use self::is_subsequence::is_subsequence;
 pub use self::knapsack::knapsack;
 pub use self::longest_common_subsequence::longest_common_subsequence;