Optimal Binary Search Tree | DP-24

19 Aug, 2021
Given a sorted array key [0.. n-1] of search keys and an array freq[0.. n-1] of frequency counts, where freq[i] is the number of searches for keys[i]. Construct a binary search tree of all keys such that the total cost of all the searches is as small as possible.Let us first define the cost of a BST. The cost of a BST node is the level of that node multiplied by its frequency. The level of the root is 1.
Examples:  
Input:  keys[] = {10, 12}, freq[] = {34, 50}
There can be following two possible BSTs 
        10                       12
          \                     / 
           12                 10
          I                     II
Frequency of searches of 10 and 12 are 34 and 50 respectively.
The cost of tree I is 34*1 + 50*2 = 134
The cost of tree II is 50*1 + 34*2 = 118 


Input:  keys[] = {10, 12, 20}, freq[] = {34, 8, 50}
There can be following possible BSTs
    10                12                 20         10              20
      \             /    \              /             \            /
      12          10     20           12               20         10  
        \                            /                 /           \
         20                        10                12             12  
     I               II             III             IV             V
Among all possible BSTs, cost of the fifth BST is minimum.  
Cost of the fifth BST is 1*50 + 2*34 + 3*8 = 142 
1) Optimal Substructure: The optimal cost for freq[i..j] can be recursively calculated using the following formula. We need to calculate optCost(0, n-1) to find the result. The idea of above formula is simple, we one by one try all nodes as root (r varies from i to j in second term). When we make rth node as root, we recursively calculate optimal cost from i to r-1 and r+1 to j. We add sum of frequencies from i to j (see first term in the above formula)
The reason for adding the sum of frequencies from i to j:
This can be divided into 2 parts one is the freq[r]+sum of frequencies of all elements from i to j except r. The term freq[r] is added because it is going to be root and that means level of 1, so freq[r]*1=freq[r]. Now the actual part comes, we are adding the frequencies of remaining elements because as we take r as root then all the elements other than that are going 1 level down than that is calculated in the subproblem. Let me put it in a more clear way, for calculating optcost(i,j) we assume that the r is taken as root and calculate min of opt(i,r-1)+opt(r+1,j) for all i<=r<=j. Here for every subproblem we are  choosing one node as a root. But in reality the level of subproblem root and all its descendant nodes will be 1 greater than the level of the parent problem root. Therefore the frequency of all the nodes except r should be added which accounts to the descend in their level compared to level assumed in subproblem.2) Overlapping Subproblems Following is recursive implementation that simply follows the recursive structure mentioned above.  
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Java
Python3
C#
Javascript
// A naive recursive implementation of// optimal binary search tree problem#include <bits/stdc++.h>using namespace std; // A utility function to get sum of// array elements freq[i] to freq[j]int sum(int freq[], int i, int j); // A recursive function to calculate// cost of optimal binary search treeint optCost(int freq[], int i, int j){    // Base cases    if (j < i)  // no elements in this subarray        return 0;    if (j == i) // one element in this subarray        return freq[i];         // Get sum of freq[i], freq[i+1], ... freq[j]    int fsum = sum(freq, i, j);         // Initialize minimum value    int min = INT_MAX;         // One by one consider all elements    // as root and recursively find cost    // of the BST, compare the cost with    // min and update min if needed    for (int r = i; r <= j; ++r)    {        int cost = optCost(freq, i, r - 1) +                   optCost(freq, r + 1, j);        if (cost < min)            min = cost;    }         // Return minimum value    return min + fsum;} // The main function that calculates// minimum cost of a Binary Search Tree.// It mainly uses optCost() to find// the optimal cost.int optimalSearchTree(int keys[],                      int freq[], int n){    // Here array keys[] is assumed to be    // sorted in increasing order. If keys[]    // is not sorted, then add code to sort    // keys, and rearrange freq[] accordingly.    return optCost(freq, 0, n - 1);} // A utility function to get sum of// array elements freq[i] to freq[j]int sum(int freq[], int i, int j){    int s = 0;    for (int k = i; k <= j; k++)    s += freq[k];    return s;} // Driver Codeint main(){    int keys[] = {10, 12, 20};    int freq[] = {34, 8, 50};    int n = sizeof(keys) / sizeof(keys[0]);    cout << "Cost of Optimal BST is "         << optimalSearchTree(keys, freq, n);    return 0;} // This is code is contributed// by rathbhupendra
// A naive recursive implementation of optimal binary// search tree problem#include <stdio.h>#include <limits.h> // A utility function to get sum of array elements// freq[i] to freq[j]int sum(int freq[], int i, int j); // A recursive function to calculate cost of optimal// binary search treeint optCost(int freq[], int i, int j){   // Base cases   if (j < i)      // no elements in this subarray     return 0;   if (j == i)     // one element in this subarray     return freq[i];    // Get sum of freq[i], freq[i+1], ... freq[j]   int fsum = sum(freq, i, j);    // Initialize minimum value   int min = INT_MAX;    // One by one consider all elements as root and   // recursively find cost of the BST, compare the   // cost with min and update min if needed   for (int r = i; r <= j; ++r)   {       int cost = optCost(freq, i, r-1) +                  optCost(freq, r+1, j);       if (cost < min)          min = cost;   }    // Return minimum value   return min + fsum;} // The main function that calculates minimum cost of// a Binary Search Tree. It mainly uses optCost() to// find the optimal cost.int optimalSearchTree(int keys[], int freq[], int n){     // Here array keys[] is assumed to be sorted in     // increasing order. If keys[] is not sorted, then     // add code to sort keys, and rearrange freq[]     // accordingly.     return optCost(freq, 0, n-1);} // A utility function to get sum of array elements// freq[i] to freq[j]int sum(int freq[], int i, int j){    int s = 0;    for (int k = i; k <=j; k++)       s += freq[k];    return s;} // Driver program to test above functionsint main(){    int keys[] = {10, 12, 20};    int freq[] = {34, 8, 50};    int n = sizeof(keys)/sizeof(keys[0]);    printf("Cost of Optimal BST is %d ",               optimalSearchTree(keys, freq, n));    return 0;}
// A naive recursive implementation of optimal binary// search tree problempublic class GFG{    // A recursive function to calculate cost of        // optimal binary search tree    static int optCost(int freq[], int i, int j)    {       // Base cases       if (j < i)      // no elements in this subarray         return 0;       if (j == i)     // one element in this subarray         return freq[i];             // Get sum of freq[i], freq[i+1], ... freq[j]       int fsum = sum(freq, i, j);             // Initialize minimum value       int min = Integer.MAX_VALUE;             // One by one consider all elements as root and           // recursively find cost of the BST, compare the           // cost with min and update min if needed       for (int r = i; r <= j; ++r)       {           int cost = optCost(freq, i, r-1) +                          optCost(freq, r+1, j);           if (cost < min)              min = cost;       }             // Return minimum value       return min + fsum;    }         // The main function that calculates minimum cost of        // a Binary Search Tree. It mainly uses optCost() to        // find the optimal cost.    static int optimalSearchTree(int keys[], int freq[], int n)    {         // Here array keys[] is assumed to be sorted in             // increasing order. If keys[] is not sorted, then             // add code to sort keys, and rearrange freq[]             // accordingly.         return optCost(freq, 0, n-1);    }         // A utility function to get sum of array elements        // freq[i] to freq[j]    static int sum(int freq[], int i, int j)    {        int s = 0;        for (int k = i; k <=j; k++)           s += freq[k];        return s;    }         // Driver code    public static void main(String[] args) {        int keys[] = {10, 12, 20};        int freq[] = {34, 8, 50};        int n = keys.length;        System.out.println("Cost of Optimal BST is " +                         optimalSearchTree(keys, freq, n));    }}// This code is contributed by Sumit Ghosh
# A naive recursive implementation of# optimal binary search tree problem # A recursive function to calculate# cost of optimal binary search treedef optCost(freq, i, j):         # Base cases    if j < i:     # no elements in this subarray        return 0    if j == i:     # one element in this subarray        return freq[i]         # Get sum of freq[i], freq[i+1], ... freq[j]    fsum = Sum(freq, i, j)         # Initialize minimum value    Min = 999999999999         # One by one consider all elements as    # root and recursively find cost of    # the BST, compare the cost with min    # and update min if needed    for r in range(i, j + 1):        cost = (optCost(freq, i, r - 1) +                optCost(freq, r + 1, j))        if cost < Min:            Min = cost         # Return minimum value    return Min + fsum # The main function that calculates minimum# cost of a Binary Search Tree. It mainly# uses optCost() to find the optimal cost.def optimalSearchTree(keys, freq, n):         # Here array keys[] is assumed to be    # sorted in increasing order. If keys[]    # is not sorted, then add code to sort     # keys, and rearrange freq[] accordingly.    return optCost(freq, 0, n - 1) # A utility function to get sum of# array elements freq[i] to freq[j]def Sum(freq, i, j):    s = 0    for k in range(i, j + 1):        s += freq[k]    return s # Driver Codeif __name__ == '__main__':    keys = [10, 12, 20]    freq = [34, 8, 50]    n = len(keys)    print("Cost of Optimal BST is",           optimalSearchTree(keys, freq, n))     # This code is contributed by PranchalK
// A naive recursive implementation of optimal binary// search tree problemusing System; class GFG{    // A recursive function to calculate cost of    // optimal binary search tree    static int optCost(int []freq, int i, int j)    {             // Base cases    // no elements in this subarray    if (j < i)            return 0;         // one element in this subarray       if (j == i)            return freq[i];         // Get sum of freq[i], freq[i+1], ... freq[j]    int fsum = sum(freq, i, j);         // Initialize minimum value    int min = int.MaxValue;         // One by one consider all elements as root and    // recursively find cost of the BST, compare the    // cost with min and update min if needed    for (int r = i; r <= j; ++r)    {        int cost = optCost(freq, i, r-1) +                        optCost(freq, r+1, j);        if (cost < min)            min = cost;    }         // Return minimum value    return min + fsum;    }         // The main function that calculates minimum cost of    // a Binary Search Tree. It mainly uses optCost() to    // find the optimal cost.    static int optimalSearchTree(int []keys, int []freq, int n)    {        // Here array keys[] is assumed to be sorted in        // increasing order. If keys[] is not sorted, then        // add code to sort keys, and rearrange freq[]        // accordingly.        return optCost(freq, 0, n-1);    }         // A utility function to get sum of array elements    // freq[i] to freq[j]    static int sum(int []freq, int i, int j)    {        int s = 0;        for (int k = i; k <=j; k++)        s += freq[k];        return s;    }         // Driver code    public static void Main()    {        int []keys = {10, 12, 20};        int []freq = {34, 8, 50};        int n = keys.Length;        Console.Write("Cost of Optimal BST is " +                        optimalSearchTree(keys, freq, n));    }} // This code is contributed by Sam007
<script>//Javascript Implementation  // A recursive function to calculate// cost of optimal binary search treefunction optCost(freq, i, j){    // Base cases    if (j < i)  // no elements in this subarray        return 0;    if (j == i) // one element in this subarray        return freq[i];          // Get sum of freq[i], freq[i+1], ... freq[j]    var fsum = sum(freq, i, j);          // Initialize minimum value    var min = Number. MAX_SAFE_INTEGER;          // One by one consider all elements    // as root and recursively find cost    // of the BST, compare the cost with    // min and update min if needed    for (var r = i; r <= j; ++r)    {        var cost = optCost(freq, i, r - 1) +                   optCost(freq, r + 1, j);        if (cost < min)            min = cost;    }          // Return minimum value    return min + fsum;}  // The main function that calculates// minimum cost of a Binary Search Tree.// It mainly uses optCost() to find// the optimal cost.function optimalSearchTree(keys, freq, n){    // Here array keys[] is assumed to be    // sorted in increasing order. If keys[]    // is not sorted, then add code to sort    // keys, and rearrange freq[] accordingly.    return optCost(freq, 0, n - 1);}  // A utility function to get sum of// array elements freq[i] to freq[j]function sum(freq, i, j){    var s = 0;    for (var k = i; k <= j; k++)    s += freq[k];    return s;}   // Driver Code var keys = [10, 12, 20];var freq = [34, 8, 50];var n = keys.length;document.write("Cost of Optimal BST is " +    optimalSearchTree(keys, freq, n));     // This code is contributed by shubhamsingh10</script>
Output: 
Cost of Optimal BST is 142
Time complexity of the above naive recursive approach is exponential. It should be noted that the above function computes the same subproblems again and again. We can see many subproblems being repeated in the following recursion tree for freq[1..4].  
Since same subproblems are called again, this problem has Overlapping Subproblems property. So optimal BST problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array cost[][] in bottom up manner.Dynamic Programming Solution Following is C/C++ implementation for optimal BST problem using Dynamic Programming. We use an auxiliary array cost[n][n] to store the solutions of subproblems. cost[0][n-1] will hold the final result. The challenge in implementation is, all diagonal values must be filled first, then the values which lie on the line just above the diagonal. In other words, we must first fill all cost[i][i] values, then all cost[i][i+1] values, then all cost[i][i+2] values. So how to fill the 2D array in such manner> The idea used in the implementation is same as Matrix Chain Multiplication problem, we use a variable ‘L’ for chain length and increment ‘L’, one by one. We calculate column number ‘j’ using the values of ‘i’ and ‘L’.  
C++
C
Java
Python3
C#
Javascript
// Dynamic Programming code for Optimal Binary Search// Tree Problem#include <bits/stdc++.h>using namespace std; // A utility function to get sum of array elements// freq[i] to freq[j]int sum(int freq[], int i, int j); /* A Dynamic Programming based function that calculatesminimum cost of a Binary Search Tree. */int optimalSearchTree(int keys[], int freq[], int n){    /* Create an auxiliary 2D matrix to store results    of subproblems */    int cost[n][n];     /* cost[i][j] = Optimal cost of binary search tree    that can be formed from keys[i] to keys[j].    cost[0][n-1] will store the resultant cost */     // For a single key, cost is equal to frequency of the key    for (int i = 0; i < n; i++)        cost[i][i] = freq[i];     // Now we need to consider chains of length 2, 3, ... .    // L is chain length.    for (int L = 2; L <= n; L++)    {        // i is row number in cost[][]        for (int i = 0; i <= n-L+1; i++)        {            // Get column number j from row number i and            // chain length L            int j = i+L-1;            cost[i][j] = INT_MAX;             // Try making all keys in interval keys[i..j] as root            for (int r = i; r <= j; r++)            {            // c = cost when keys[r] becomes root of this subtree            int c = ((r > i)? cost[i][r-1]:0) +                    ((r < j)? cost[r+1][j]:0) +                    sum(freq, i, j);            if (c < cost[i][j])                cost[i][j] = c;            }        }    }    return cost[0][n-1];} // A utility function to get sum of array elements// freq[i] to freq[j]int sum(int freq[], int i, int j){    int s = 0;    for (int k = i; k <= j; k++)    s += freq[k];    return s;} // Driver codeint main(){    int keys[] = {10, 12, 20};    int freq[] = {34, 8, 50};    int n = sizeof(keys)/sizeof(keys[0]);    cout << "Cost of Optimal BST is " << optimalSearchTree(keys, freq, n);    return 0;} // This code is contributed by rathbhupendra
// Dynamic Programming code for Optimal Binary Search// Tree Problem#include <stdio.h>#include <limits.h> // A utility function to get sum of array elements// freq[i] to freq[j]int sum(int freq[], int i, int j); /* A Dynamic Programming based function that calculates  minimum cost of a Binary Search Tree. */int optimalSearchTree(int keys[], int freq[], int n){    /* Create an auxiliary 2D matrix to store results      of subproblems */    int cost[n][n];     /* cost[i][j] = Optimal cost of binary search tree       that can be  formed from keys[i] to keys[j].       cost[0][n-1] will store the resultant cost */     // For a single key, cost is equal to frequency of the key    for (int i = 0; i < n; i++)        cost[i][i] = freq[i];     // Now we need to consider chains of length 2, 3, ... .    // L is chain length.    for (int L=2; L<=n; L++)    {        // i is row number in cost[][]        for (int i=0; i<=n-L+1; i++)        {            // Get column number j from row number i and            // chain length L            int j = i+L-1;            cost[i][j] = INT_MAX;             // Try making all keys in interval keys[i..j] as root            for (int r=i; r<=j; r++)            {               // c = cost when keys[r] becomes root of this subtree               int c = ((r > i)? cost[i][r-1]:0) +                       ((r < j)? cost[r+1][j]:0) +                       sum(freq, i, j);               if (c < cost[i][j])                  cost[i][j] = c;            }        }    }    return cost[0][n-1];} // A utility function to get sum of array elements// freq[i] to freq[j]int sum(int freq[], int i, int j){    int s = 0;    for (int k = i; k <=j; k++)       s += freq[k];    return s;} // Driver program to test above functionsint main(){    int keys[] = {10, 12, 20};    int freq[] = {34, 8, 50};    int n = sizeof(keys)/sizeof(keys[0]);    printf("Cost of Optimal BST is %d ",                 optimalSearchTree(keys, freq, n));    return 0;}
// Dynamic Programming Java code for Optimal Binary Search// Tree Problempublic class Optimal_BST2 {         /* A Dynamic Programming based function that calculates        minimum cost of a Binary Search Tree.  */    static int optimalSearchTree(int keys[], int freq[], int n) {         /* Create an auxiliary 2D matrix to store results of           subproblems */        int cost[][] = new int[n + 1][n + 1];         /* cost[i][j] = Optimal cost of binary search tree that           can be formed from keys[i] to keys[j]. cost[0][n-1]           will store the resultant cost */         // For a single key, cost is equal to frequency of the key        for (int i = 0; i < n; i++)            cost[i][i] = freq[i];         // Now we need to consider chains of length 2, 3, ... .        // L is chain length.        for (int L = 2; L <= n; L++) {             // i is row number in cost[][]            for (int i = 0; i <= n - L + 1; i++) {                 // Get column number j from row number i and                // chain length L                int j = i + L - 1;                cost[i][j] = Integer.MAX_VALUE;                 // Try making all keys in interval keys[i..j] as root                for (int r = i; r <= j; r++) {                     // c = cost when keys[r] becomes root of this subtree                    int c = ((r > i) ? cost[i][r - 1] : 0)                            + ((r < j) ? cost[r + 1][j] : 0) + sum(freq, i, j);                    if (c < cost[i][j])                        cost[i][j] = c;                }            }        }        return cost[0][n - 1];    }     // A utility function to get sum of array elements    // freq[i] to freq[j]    static int sum(int freq[], int i, int j) {        int s = 0;        for (int k = i; k <= j; k++) {            if (k >= freq.length)                continue;            s += freq[k];        }        return s;    }     public static void main(String[] args) {                 int keys[] = { 10, 12, 20 };        int freq[] = { 34, 8, 50 };        int n = keys.length;        System.out.println("Cost of Optimal BST is "                + optimalSearchTree(keys, freq, n));    } }//This code is contributed by Sumit Ghosh
# Dynamic Programming code for Optimal Binary Search# Tree Problem INT_MAX = 2147483647 """ A Dynamic Programming based function thatcalculates minimum cost of a Binary Search Tree. """def optimalSearchTree(keys, freq, n):     """ Create an auxiliary 2D matrix to store        results of subproblems """    cost = [[0 for x in range(n)]               for y in range(n)]     """ cost[i][j] = Optimal cost of binary search    tree that can be formed from keys[i] to keys[j].    cost[0][n-1] will store the resultant cost """     # For a single key, cost is equal to    # frequency of the key    for i in range(n):        cost[i][i] = freq[i]     # Now we need to consider chains of    # length 2, 3, ... . L is chain length.    for L in range(2, n + 1):             # i is row number in cost        for i in range(n - L + 2):                         # Get column number j from row number            # i and chain length L            j = i + L - 1            if i >= n or j >= n:                break            cost[i][j] = INT_MAX                         # Try making all keys in interval            # keys[i..j] as root            for r in range(i, j + 1):                                 # c = cost when keys[r] becomes root                # of this subtree                c = 0                if (r > i):                    c += cost[i][r - 1]                if (r < j):                    c += cost[r + 1][j]                c += sum(freq, i, j)                if (c < cost[i][j]):                    cost[i][j] = c    return cost[0][n - 1]  # A utility function to get sum of# array elements freq[i] to freq[j]def sum(freq, i, j):     s = 0    for k in range(i, j + 1):        s += freq[k]    return s     # Driver Codeif __name__ == '__main__':    keys = [10, 12, 20]    freq = [34, 8, 50]    n = len(keys)    print("Cost of Optimal BST is",           optimalSearchTree(keys, freq, n))     # This code is contributed by SHUBHAMSINGH10
// Dynamic Programming C# code for Optimal Binary Search// Tree Problemusing System; class GFG{    /* A Dynamic Programming based function that calculates    minimum cost of a Binary Search Tree. */    static int optimalSearchTree(int []keys, int []freq, int n) {         /* Create an auxiliary 2D matrix to store results of        subproblems */        int [,]cost = new int[n + 1,n + 1];         /* cost[i][j] = Optimal cost of binary search tree that        can be formed from keys[i] to keys[j]. cost[0][n-1]        will store the resultant cost */         // For a single key, cost is equal to frequency of the key        for (int i = 0; i < n; i++)            cost[i,i] = freq[i];         // Now we need to consider chains of length 2, 3, ... .        // L is chain length.        for (int L = 2; L <= n; L++) {             // i is row number in cost[][]            for (int i = 0; i <= n - L + 1; i++) {                 // Get column number j from row number i and                // chain length L                int j = i + L - 1;                cost[i,j] = int.MaxValue;                 // Try making all keys in interval keys[i..j] as root                for (int r = i; r <= j; r++) {                     // c = cost when keys[r] becomes root of this subtree                    int c = ((r > i) ? cost[i,r - 1] : 0)                            + ((r < j) ? cost[r + 1,j] : 0) + sum(freq, i, j);                    if (c < cost[i,j])                        cost[i,j] = c;                }            }        }        return cost[0,n - 1];    }     // A utility function to get sum of array elements    // freq[i] to freq[j]    static int sum(int []freq, int i, int j) {        int s = 0;        for (int k = i; k <= j; k++) {            if (k >= freq.Length)                continue;            s += freq[k];        }        return s;    }     public static void Main() {                 int []keys = { 10, 12, 20 };        int []freq = { 34, 8, 50 };        int n = keys.Length;        Console.Write("Cost of Optimal BST is "                + optimalSearchTree(keys, freq, n));    }}// This code is contributed by Sam007
<script>// Dynamic Programming code for Optimal Binary Search// Tree Problem /* A Dynamic Programming based function that calculatesminimum cost of a Binary Search Tree. */function optimalSearchTree(keys, freq, n){    /* Create an auxiliary 2D matrix to store results    of subproblems */    var cost = new Array(n);    for (var i = 0; i < n; i++)        cost[i] = new Array(n);      /* cost[i][j] = Optimal cost of binary search tree    that can be formed from keys[i] to keys[j].    cost[0][n-1] will store the resultant cost */      // For a single key, cost is equal to frequency of the key    for (var i = 0; i < n; i++)        cost[i][i] = freq[i];      // Now we need to consider chains of length 2, 3, ... .    // L is chain length.    for (var L = 2; L <= n; L++)    {        // i is row number in cost[][]        for (var i = 0; i <= n-L+1; i++)        {            // Get column number j from row number i and            // chain length L            var j = i+L-1;            if ( i >= n || j >= n)                break            cost[i][j] = Number. MAX_SAFE_INTEGER;              // Try making all keys in interval keys[i..j] as root            for (var r = i; r <= j; r++)            {            // c = cost when keys[r] becomes root of this subtree            var c = 0;            if (r > i)                c += cost[i][r-1]            if (r < j)                c += cost[r+1][j]            c += sum(freq, i, j);            if (c < cost[i][j])                cost[i][j] = c;            }        }    }    return cost[0][n-1];}  // A utility function to get sum of array elements// freq[i] to freq[j]function sum(freq, i, j){    var s = 0;    for (var k = i; k <= j; k++)        s += freq[k];    return s;}var keys = [10, 12, 20];var freq = [34, 8, 50];var n = keys.length;document.write("Cost of Optimal BST is " +    optimalSearchTree(keys, freq, n));  // This code contributed by shubhamsingh10</script>
Output:  
Cost of Optimal BST is 142
Notes 1) The time complexity of the above solution is O(n^4). The time complexity can be easily reduced to O(n^3) by pre-calculating sum of frequencies instead of calling sum() again and again.2) In the above solutions, we have computed optimal cost only. The solutions can be easily modified to store the structure of BSTs also. We can create another auxiliary array of size n to store the structure of tree. All we need to do is, store the chosen ‘r’ in the innermost loop.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
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