Question

Create the optimal binary search tree for the following items, where the probability occurrence of each word is given in parentheses: CASE (.05), ELSE (.15), END (.05), IF (.35), OF (.05), THEN (.35)

Short answer

The constructed optimal binary search tree based on the given probabilities for each word will look like this: root[IF], left child [ELSE], right child [THEN], left child's left child [CASE], left child's right child [END], right child's left child [OF]. Please note, the tree structure could have different shapes as there are multiple optimal solutions for given probabilities.

Step-by-step solution

Sort the Words

The first step involves sorting the words (items) in ascending order based on their probabilities. It looks like this: CASE (.05), END (.05), OF (.05), ELSE(.15), THEN (.35), IF (.35)

Setup the Cost Table

Create a cost table of size n*n (n= number of items). Populate the principal diagonal elements of the cost table with the probability values. Diagonal elements represent trees with a single node.

Compute Optimal Cost for Binary Trees

For each subtree size from 2 to n, fill the cost table diagonally from top to bottom and from left to right. Each time, compute the cost of each binary tree following the formula: cost(subtree) = min [cost(leftsubtree) + cost(rightsubtree)] + sum of probability nodes of the tree.

Construct Optimal Binary Tree

To construct the optimal binary tree, follow the populated cost table. The root is chosen from the row that contains the optimal cost for subtree size n. The items to the left of the root form the left subtree, and the items to the right, the right subtree. For each subtree, choose the root the same way until each node has no more than two children.

Key concepts

Probability Occurrence

In the context of an Optimal Binary Search Tree (OBST), the probability occurrence refers to the likelihood that a particular item will be accessed or searched for in the tree. The OBST algorithm takes these probabilities into account to arrange the tree in a way that minimizes the average search time. Essential to predicting the performance of search operations, probability occurrence can drastically change how a binary search tree is structured.

For instance, in a list of words each with an associated probability occurrence, the OBST must be carefully constructed to ensure that more frequently searched items are placed closer to the tree's root, as seen with the words 'IF' and 'THEN', each with a high probability of .35. This strategic placement reduces the number of comparisons needed to find these popular elements, thereby optimizing the overall search operation in the binary search tree.

Dynamic Programming

Dynamic Programming (DP) is a method for solving complex problems by breaking them down into simpler subproblems. It is widely used in algorithm design, especially when the problem can be divided into overlapping subproblems that can be solved independently. The main feature of dynamic programming is the use of previously solved subproblems to build up solutions to larger problems.

In the creation of an OBST, dynamic programming is utilized to compute the optimal structure. Instead of evaluating an exponential number of tree configurations, dynamic programming approaches the problem by constructing a cost table that keeps track of optimal subtree solutions. This table progressively stores solutions to smaller instances and uses them to address larger ones, significantly reducing the computational burden while ensuring that the search tree is as efficient as possible.

Algorithm Analysis

Algorithm Analysis is the study of the performance and efficiency of algorithms, which is crucial in determining the most effective means to solve a given problem. In terms of an OBST, algorithm analysis helps us to understand the time complexities involved in constructing the tree and the efficiency of search operations within it.

When evaluating the OBST construction process, not only does one need to account for the various probabilities of occurrence, but also how quickly and efficiently the tree can be constructed and utilized. The OBST algorithm aims to minimize the total expected search cost, and by performing algorithmic analysis, one can ensure that the time and space complexities are within acceptable bounds for the intended application. This analysis becomes especially important when dealing with large datasets where the overhead of building and searching the tree can have substantial impacts on performance.

Data Structures

Data structures are a pivotal concept in computer science used to store, organize, and manage data efficiently. The binary search tree (BST) is one such data structure that allows for fast data retrieval, insertion, and deletion, given that the tree remains balanced.

The optimal binary search tree is a specialized variant of the BST, specifically optimized for situations where the frequency of access for each element is known. The goal of the OBST is to reduce the average search time, which is achieved by structuring the tree based upon the probability occurrence of each element, ensuring those with higher probabilities are closer to the root. Therefore, an OBST is an excellent example of how a data structure can be adapted to suit the particular needs of an application, making it a powerful tool in the arsenal of a programmer or data scientist.