Optimal Binary Search Trees: Dynamic Programming Approach and Implementation

Introduction to Optimal BST

An Optimal Binary Search Tree (OBST) is a binary search tree that provides the smallest possible search time for a given sequence of accesses (or probabilities of accesses) to its keys. The goal is to minimize the expected search cost.

Problem Definition

Given:

• A sorted sequence of n distinct keys: K = [k₁, k₂, ..., kₙ]
• Probabilities of searching for each key: P = [p₁, p₂, ..., pₙ]
• Probabilities of searching for values not in the tree (dummy keys representing intervals between real keys): Q = [q₀, q₁, ..., qₙ]

We want to construct a binary search tree with minimal expected search cost.

Expected Search Cost

The expected search cost of a BST is defined as:

E[search cost] = Σ (depth(kᵢ) + 1) * pᵢ + Σ depth(dᵢ) * qᵢ

where:

• depth(kᵢ) is the depth of key kᵢ in the tree
• depth(dᵢ) is the depth of dummy key dᵢ

Dynamic Programming Approach

We use dynamic programming to solve this problem efficiently:

1. Subproblems: For each subtree containing keys kᵢ to k�, compute the optimal cost.
2. Recurrence Relation:

• Let e[i,j] be the expected search cost of the optimal BST containing keys kᵢ to kⱼ
• Let w[i,j] be the sum of probabilities for the subtree containing kᵢ to kⱼ:
  w[i,j] = q_{i-1} + Σ (pₖ + qₖ) for k from i to j
• The recurrence is:
  e[i,j] = min_{i ≤ r ≤ j} (e[i,r-1] + e[r+1,j] + w[i,j])

3. Base Cases:

• e[i,i-1] = q_{i-1} (empty tree with just dummy key d_{i-1})

Algorithm Implementation in C

Here’s the complete implementation of the OBST algorithm in C:

#include 
#include 
#include 

// Function to construct and print the optimal BST
void optimalBST(float p[], float q[], int n) {
    // Create weight, cost, and root tables
    float w[n+2][n+1], e[n+2][n+1];
    int root[n+1][n+1];

    // Initialize tables
    for (int i = 1; i <= n+1; i++) {
        e[i][i-1] = q[i-1];
        w[i][i-1] = q[i-1];
    }

    // Fill tables
    for (int l = 1; l <= n; l++) {
        for (int i = 1; i <= n - l + 1; i++) {
            int j = i + l - 1;
            e[i][j] = INT_MAX;
            w[i][j] = w[i][j-1] + p[j] + q[j];

            for (int r = i; r <= j; r++) {
                float temp = e[i][r-1] + e[r+1][j] + w[i][j];
                if (temp < e[i][j]) {
                    e[i][j] = temp;
                    root[i][j] = r;
                }
            }
        }
    }

    // Print results
    printf("Optimal BST Cost: %.2f\n", e[1][n]);
    printf("Root of the tree: k%d\n", root[1][n]);

    // (Optional: Add code to print the full tree structure)
}

int main() {
    // Example from CLRS book
    float p[] = {0, 0.15, 0.10, 0.05, 0.10, 0.20};
    float q[] = {0.05, 0.10, 0.05, 0.05, 0.05, 0.10};
    int n = sizeof(p)/sizeof(p[0]) - 1;

    optimalBST(p, q, n);

    return 0;
}

Explanation of the Implementation

1. Initialization:

• We create tables w, e, and root to store weights, expected costs, and root information.
• The base cases are initialized where e[i][i-1] represents empty subtrees.

2. Filling the Tables:

• We consider subtrees of increasing lengths (l from 1 to n).
• For each possible subtree (from i to j), we:
  • Calculate the weight w[i][j] as the sum of all probabilities in that range
  • Try every possible root r between i and j
  • Select the root that gives the minimum expected cost

3. Result Extraction:

• The optimal cost is found in e[1][n]
• The root of the entire tree is in root[1][n]

Time and Space Complexity

• Time Complexity: O(n³) due to the triple nested loops
• Space Complexity: O(n²) for storing the DP tables

Example Walkthrough

For the input:

Keys: [ , k1, k2, k3, k4, k5]
p: [0, 0.15, 0.10, 0.05, 0.10, 0.20]
q: [0.05, 0.10, 0.05, 0.05, 0.05, 0.10]

The algorithm will:

1. Compute all possible subtrees
2. Find that the optimal root is k2
3. Calculate the minimal expected search cost as 2.75

Extensions

To actually build the tree (not just compute the cost), you would:

1. Use the root table to recursively construct the tree
2. For each subtree i to j, the root is root[i][j]
3. Left subtree contains keys i to root[i][j]-1
4. Right subtree contains keys root[i][j]+1 to j

This implementation provides the foundation for understanding and implementing optimal BST construction using dynamic programming.