Added DP Solution for Optimal BST Problem (#1740)

* Added code to dynamic_programming directory

* Added doctest

* Elaborated BST

* Small tweaks

* Update optimal_bst.py

* Some touchups

* Fixed doctest

* Update optimal_bst.py

* Update optimal_bst.py

* Update optimal_bst.py

* Rename optimal_bst.py to optimal_binary_search_tree.py

Co-authored-by: Christian Clauss <cclauss@me.com>

dynamic_programming/optimal_binary_search_tree.py

@@ -0,0 +1,144 @@
+#!/usr/bin/env python3
+
+# This Python program implements an optimal binary search tree (abbreviated BST)
+# building dynamic programming algorithm that delivers O(n^2) performance.
+#
+# The goal of the optimal BST problem is to build a low-cost BST for a
+# given set of nodes, each with its own key and frequency. The frequency
+# of the node is defined as how many time the node is being searched.
+# The search cost of binary search tree is given by this formula:
+#
+# cost(1, n) = sum{i = 1 to n}((depth(node_i) + 1) * node_i_freq)
+#
+# where n is number of nodes in the BST. The characteristic of low-cost
+# BSTs is having a faster overall search time than other implementations.
+# The reason for their fast search time is that the nodes with high
+# frequencies will be placed near the root of the tree while the nodes
+# with low frequencies will be placed near the leaves of the tree thus
+# reducing search time in the most frequent instances.
+
+import sys
+
+from random import randint
+
+
+class Node:
+    """Binary Search Tree Node"""
+    def __init__(self, key, freq):
+        self.key = key
+        self.freq = freq
+
+    def __str__(self):
+        """
+        >>> str(Node(1, 2))
+        'Node(key=1, freq=2)'
+        """
+        return f"Node(key={self.key}, freq={self.freq})"
+
+
+def print_binary_search_tree(root, key, i, j, parent, is_left):
+    """
+    Recursive function to print a BST from a root table.
+    
+    >>> key = [3, 8, 9, 10, 17, 21]
+    >>> root = [[0, 1, 1, 1, 1, 1], [0, 1, 1, 1, 1, 3], [0, 0, 2, 3, 3, 3], \
+                [0, 0, 0, 3, 3, 3], [0, 0, 0, 0, 4, 5], [0, 0, 0, 0, 0, 5]]
+    >>> print_binary_search_tree(root, key, 0, 5, -1, False)
+    8 is the root of the binary search tree.
+    3 is the left child of key 8.
+    10 is the right child of key 8.
+    9 is the left child of key 10.
+    21 is the right child of key 10.
+    17 is the left child of key 21.
+    """
+    if i > j or i < 0 or j > len(root) - 1:
+        return
+
+    node = root[i][j]
+    if parent == -1:  # root does not have a parent
+        print(f"{key[node]} is the root of the binary search tree.")
+    elif is_left:
+        print(f"{key[node]} is the left child of key {parent}.")
+    else:
+        print(f"{key[node]} is the right child of key {parent}.")
+
+    print_binary_search_tree(root, key, i, node - 1, key[node], True)
+    print_binary_search_tree(root, key, node + 1, j, key[node], False)
+
+
+def find_optimal_binary_search_tree(nodes):
+    """
+    This function calculates and prints the optimal binary search tree.
+    The dynamic programming algorithm below runs in O(n^2) time.
+    Implemented from CLRS (Introduction to Algorithms) book.
+    https://en.wikipedia.org/wiki/Introduction_to_Algorithms
+    
+    >>> find_optimal_binary_search_tree([Node(12, 8), Node(10, 34), Node(20, 50), \
+                                         Node(42, 3), Node(25, 40), Node(37, 30)])
+    Binary search tree nodes:
+    Node(key=10, freq=34)
+    Node(key=12, freq=8)
+    Node(key=20, freq=50)
+    Node(key=25, freq=40)
+    Node(key=37, freq=30)
+    Node(key=42, freq=3)
+    <BLANKLINE>
+    The cost of optimal BST for given tree nodes is 324.
+    20 is the root of the binary search tree.
+    10 is the left child of key 20.
+    12 is the right child of key 10.
+    25 is the right child of key 20.
+    37 is the right child of key 25.
+    42 is the right child of key 37.
+    """
+    # Tree nodes must be sorted first, the code below sorts the keys in
+    # increasing order and rearrange its frequencies accordingly.
+    nodes.sort(key=lambda node: node.key)
+
+    n = len(nodes)
+
+    keys = [nodes[i].key for i in range(n)]
+    freqs = [nodes[i].freq for i in range(n)]
+
+    # This 2D array stores the overall tree cost (which's as minimized as possible);
+    # for a single key, cost is equal to frequency of the key.
+    dp = [[freqs[i] if i == j else 0 for j in range(n)] for i in range(n)]
+    # sum[i][j] stores the sum of key frequencies between i and j inclusive in nodes array
+    sum = [[freqs[i] if i == j else 0 for j in range(n)] for i in range(n)]
+    # stores tree roots that will be used later for constructing binary search tree
+    root = [[i if i == j else 0 for j in range(n)] for i in range(n)]
+
+    for l in range(2, n + 1):  # l is an interval length
+        for i in range(n - l + 1):
+            j = i + l - 1
+
+            dp[i][j] = sys.maxsize  # set the value to "infinity"
+            sum[i][j] = sum[i][j - 1] + freqs[j]
+
+            # Apply Knuth's optimization
+            # Loop without optimization: for r in range(i, j + 1):
+            for r in range(root[i][j - 1], root[i + 1][j] + 1):  # r is a temporal root
+                left = dp[i][r - 1] if r != i else 0  # optimal cost for left subtree
+                right = dp[r + 1][j] if r != j else 0  # optimal cost for right subtree
+                cost = left + sum[i][j] + right
+
+                if dp[i][j] > cost:
+                    dp[i][j] = cost
+                    root[i][j] = r
+
+    print("Binary search tree nodes:")
+    for node in nodes:
+        print(node)
+
+    print(f"\nThe cost of optimal BST for given tree nodes is {dp[0][n - 1]}.")
+    print_binary_search_tree(root, keys, 0, n - 1, -1, False)
+
+
+def main():
+    # A sample binary search tree
+    nodes = [Node(i, randint(1, 50)) for i in range(10, 0, -1)]
+    find_optimal_binary_search_tree(nodes)
+
+
+if __name__ == "__main__":
+    main()