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111 changes: 111 additions & 0 deletions
src/dynamic_programming/catalan_numbers.rs
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| //! Catalan Numbers using Dynamic Programming | ||
| //! | ||
| //! The Catalan numbers are a sequence of positive integers that appear in many | ||
| //! counting problems in combinatorics. Such problems include counting: | ||
| //! - The number of Dyck words of length 2n | ||
| //! - The number of well-formed expressions with n pairs of parentheses | ||
| //! (e.g., `()()` is valid but `())(` is not) | ||
| //! - The number of different ways n + 1 factors can be completely parenthesized | ||
| //! (e.g., for n = 2, C(n) = 2 and (ab)c and a(bc) are the two valid ways) | ||
| //! - The number of full binary trees with n + 1 leaves | ||
| //! | ||
| //! A Catalan number satisfies the following recurrence relation: | ||
| //! - C(0) = C(1) = 1 | ||
| //! - C(n) = sum(C(i) * C(n-i-1)), from i = 0 to n-1 | ||
| //! | ||
| //! Sources: | ||
| //! - [Brilliant.org](https://brilliant.org/wiki/catalan-numbers/) | ||
| //! - [Wikipedia](https://en.wikipedia.org/wiki/Catalan_number) | ||
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| /// Computes the Catalan number sequence from 0 through `upper_limit`. | ||
| /// | ||
| /// # Arguments | ||
| /// | ||
| /// * `upper_limit` - The upper limit for the Catalan sequence (must be ≥ 0) | ||
| /// | ||
| /// # Returns | ||
| /// | ||
| /// A vector containing Catalan numbers from C(0) to C(upper_limit) | ||
| /// | ||
| /// # Complexity | ||
| /// | ||
| /// * Time complexity: O(n2) | ||
| /// * Space complexity: O(n) | ||
| /// | ||
| /// where `n` is the `upper_limit`. | ||
| /// | ||
| /// # Examples | ||
| /// | ||
| /// ``` | ||
| /// use the_algorithms_rust::dynamic_programming::catalan_numbers; | ||
| /// | ||
| /// assert_eq!(catalan_numbers(5), vec![1, 1, 2, 5, 14, 42]); | ||
| /// assert_eq!(catalan_numbers(2), vec![1, 1, 2]); | ||
| /// assert_eq!(catalan_numbers(0), vec![1]); | ||
| /// ``` | ||
| /// | ||
| /// # Warning | ||
| /// | ||
| /// This will overflow the 64-bit unsigned integer for large values of `upper_limit`. | ||
| /// For example, C(21) and beyond will cause overflow. | ||
| pub fn catalan_numbers(upper_limit: usize) -> Vec<u64> { | ||
| let mut catalan_list = vec![0u64; upper_limit + 1]; | ||
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| // Base case: C(0) = 1 | ||
| catalan_list[0] = 1; | ||
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| // Base case: C(1) = 1 | ||
| if upper_limit > 0 { | ||
| catalan_list[1] = 1; | ||
| } | ||
|
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| // Recurrence relation: C(i) = sum(C(j) * C(i-j-1)), from j = 0 to i-1 | ||
| for i in 2..=upper_limit { | ||
| for j in 0..i { | ||
| catalan_list[i] += catalan_list[j] * catalan_list[i - j - 1]; | ||
| } | ||
| } | ||
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| catalan_list | ||
| } | ||
|
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| #[cfg(test)] | ||
| mod tests { | ||
| use super::*; | ||
|
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| #[test] | ||
| fn test_catalan_numbers_basic() { | ||
| assert_eq!(catalan_numbers(5), vec![1, 1, 2, 5, 14, 42]); | ||
| assert_eq!(catalan_numbers(2), vec![1, 1, 2]); | ||
| assert_eq!(catalan_numbers(0), vec![1]); | ||
| } | ||
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| #[test] | ||
| fn test_catalan_numbers_single() { | ||
| assert_eq!(catalan_numbers(1), vec![1, 1]); | ||
| } | ||
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| #[test] | ||
| fn test_catalan_numbers_extended() { | ||
| let result = catalan_numbers(10); | ||
| assert_eq!(result.len(), 11); | ||
| assert_eq!(result[0], 1); | ||
| assert_eq!(result[1], 1); | ||
| assert_eq!(result[2], 2); | ||
| assert_eq!(result[3], 5); | ||
| assert_eq!(result[4], 14); | ||
| assert_eq!(result[5], 42); | ||
| assert_eq!(result[6], 132); | ||
| assert_eq!(result[7], 429); | ||
| assert_eq!(result[8], 1430); | ||
| assert_eq!(result[9], 4862); | ||
| assert_eq!(result[10], 16796); | ||
| } | ||
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| #[test] | ||
| fn test_catalan_first_few() { | ||
| // Verify the first few Catalan numbers match known values | ||
| assert_eq!(catalan_numbers(3), vec![1, 1, 2, 5]); | ||
| assert_eq!(catalan_numbers(4), vec![1, 1, 2, 5, 14]); | ||
| } | ||
| } |
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