optimal binary search tree added in dynamic_programming (TheAlgorithms#890)

* optimal search binary tree added

* obst link in DIRECTORY.md

Author: butterpaneermasala

DIRECTORY.md

@@ -97,6 +97,7 @@
     * [Maximal Square](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/maximal_square.rs)
     * [Maximum Subarray](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/maximum_subarray.rs)
     * [Minimum Cost Path](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/minimum_cost_path.rs)
+    * [Optimal Binary Seacrh Tree](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/optimal_bst.rs)
     * [Rod Cutting](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/rod_cutting.rs)
     * [Snail](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/snail.rs)
     * [Subset Generation](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/subset_generation.rs)

src/dynamic_programming/mod.rs

@@ -12,6 +12,7 @@ mod matrix_chain_multiply;
 mod maximal_square;
 mod maximum_subarray;
 mod minimum_cost_path;
+mod optimal_bst;
 mod rod_cutting;
 mod snail;
 mod subset_generation;
@@ -40,6 +41,7 @@ pub use self::matrix_chain_multiply::matrix_chain_multiply;
 pub use self::maximal_square::maximal_square;
 pub use self::maximum_subarray::maximum_subarray;
 pub use self::minimum_cost_path::minimum_cost_path;
+pub use self::optimal_bst::optimal_search_tree;
 pub use self::rod_cutting::rod_cut;
 pub use self::snail::snail;
 pub use self::subset_generation::list_subset;

src/dynamic_programming/optimal_bst.rs

@@ -0,0 +1,93 @@
+// Optimal Binary Search Tree Algorithm in Rust
+// Time Complexity: O(n^3) with prefix sum optimization
+// Space Complexity: O(n^2) for the dp table and prefix sum array
+
+/// Constructs an Optimal Binary Search Tree from a list of key frequencies.
+/// The goal is to minimize the expected search cost given key access frequencies.
+///
+/// # Arguments
+/// * `freq` - A slice of integers representing the frequency of key access
+///
+/// # Returns
+/// * An integer representing the minimum cost of the optimal BST
+pub fn optimal_search_tree(freq: &[i32]) -> i32 {
+    let n = freq.len();
+    if n == 0 {
+        return 0;
+    }
+
+    // dp[i][j] stores the cost of optimal BST that can be formed from keys[i..=j]
+    let mut dp = vec![vec![0; n]; n];
+
+    // prefix_sum[i] stores sum of freq[0..i]
+    let mut prefix_sum = vec![0; n + 1];
+    for i in 0..n {
+        prefix_sum[i + 1] = prefix_sum[i] + freq[i];
+    }
+
+    // Base case: Trees with only one key
+    for i in 0..n {
+        dp[i][i] = freq[i];
+    }
+
+    // Build chains of increasing length l (from 2 to n)
+    for l in 2..=n {
+        for i in 0..=n - l {
+            let j = i + l - 1;
+            dp[i][j] = i32::MAX;
+
+            // Compute the total frequency sum in the range [i..=j] using prefix sum
+            let fsum = prefix_sum[j + 1] - prefix_sum[i];
+
+            // Try making each key in freq[i..=j] the root of the tree
+            for r in i..=j {
+                // Cost of left subtree
+                let left = if r > i { dp[i][r - 1] } else { 0 };
+                // Cost of right subtree
+                let right = if r < j { dp[r + 1][j] } else { 0 };
+
+                // Total cost = left + right + sum of frequencies (fsum)
+                let cost = left + right + fsum;
+
+                // Choose the minimum among all possible roots
+                if cost < dp[i][j] {
+                    dp[i][j] = cost;
+                }
+            }
+        }
+    }
+
+    // Minimum cost of the optimal BST storing all keys
+    dp[0][n - 1]
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    // Macro to generate multiple test cases for the optimal_search_tree function
+    macro_rules! optimal_bst_tests {
+        ($($name:ident: $input:expr => $expected:expr,)*) => {
+            $(
+                #[test]
+                fn $name() {
+                    let freq = $input;
+                    assert_eq!(optimal_search_tree(freq), $expected);
+                }
+            )*
+        };
+    }
+
+    optimal_bst_tests! {
+        // Common test cases
+        test_case_1: &[34, 10, 8, 50] => 180,
+        test_case_2: &[10, 12] => 32,
+        test_case_3: &[10, 12, 20] => 72,
+        test_case_4: &[25, 10, 20] => 95,
+        test_case_5: &[4, 2, 6, 3] => 26,
+
+        // Edge test cases
+        test_case_single: &[42] => 42,
+        test_case_empty: &[] => 0,
+    }
+}