The concept of a Binary Search Tree is already an optimised version of traversing the nodes of a regular Binary Tree by managing the aspect of time consumption.
With the use of Optimal BST i.e Optimal Binary Search Tree, you can not only reduce the time complexity but, you can also manage the cost efficiency of searching the nodes of a Binary Tree.
This method is more commonly known as Weight-Balanced Binary Tree. The Optimal Binary Tree provides an expected search time for travelling each node so that the programmer can calculate the cost of designing a software using the elements from the BST.
But, in order to perform these functions, the Optimal Tree structure requires a few criterias to be fulfilled.
Learn about these criterias in this blog.
What is meant by OBST in terms of a Binary Search Tree?
The concept of Binary Search Trees is based on the fact that the tree consists of nodes that hold a certain value.
Now, these values (also referred to as key values) are used for determining the cost of searching the nodes.
Now, in order to optimize the cost and function of the software programs, it is important to reduce the cost of searching these nodes in the Binary Search Trees.
This action of cutting or reducing costs of searching a Binary Tree is in other terms referred to as OBST or Optimal Binary Search Tree.
In the field of computer science, the Optimal BST is also referred to as a weight-balanced Binary Tree.
This sort of binary search tree essentially facilitates the least possible search time for any given sequence of data.
The OBST is generally categorized in two types:
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Static Optimality
This form of OBST cannot be tampered with or modified once it is implemented in a program. In such a scenario, the layout of the nodes are formed in a manner that essentially decides the least amount of time for the nodes to be analysed and traversed.
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Dynamic Optimality
On the other hand we have the dynamic Optimality function in Optimal BST where the nodes of the Binary Search Tree can be typically modified by implementing tree rotations.
The dynamic optimal search tree carries a cursor situated at the node of the tree which allows the programmers to perform modifications within the tree itself.
One can imagine that the process of optimising the searching of nodes in a Binary Tree definitely requires a few conditions to be fulfilled for reducing the cost of traversing the tree.
Find a detailed discussion on reducing the cost function along with an OBST problem in the following sections.
What is the best condition for an OBST in a Binary Search Tree?
So, what condition must the Optimal BST fulfill in order to use its potential to its fullest advantage?
- The main criteria that needs to be fulfilled is that we should absolutely avoid any sort of modification of the Binary Search Tree.
- Also, it is important to figure out how oftenly the keys are accessed within the program.
- Hence, by cutting the cost modification and limited access to the nodes of the BST, we can ensure to establish the Optimal BST for any given Binary Tree structure.
Did you know?
Optimal Binary Tree structures can be used for solving array based problems such as find first and last position of element in sorted Arrays?
Find a detailed discussion on the implementation of the optimal binary tree for an array based problem in the section below.
How to solve problem statements related to OBST in a Binary Search Tree?
Sorted array based problem statement:
You have been given a sorted array consisting of search keys and a specified frequency for the array. You are required to design a BST for all the keys while keeping in mind to reduce the total cost to a minimum.
Answer Key
Before starting off with finding the solution to this problem. Firstly, you are required to define a particular cost for the BST.
Effectively, the cost can be defined by multiplying the level of the specific node with its frequency.
Starting at the root level, this value would be considered as 1.
Now, we can define the solutions to this problem by implementing three different approaches with different time complexities.
Method 1: Optimising the structure
In order to optimise the structure of the nodes effectively, you can use the following algorithm:
- Start by calculating the optimal cost of the frequency of nodes.
The formula for this can be written as follows:
optCost(O, n-1)
- We implement this formula for all the nodes of the tree and make the root node as “r”.
- Finally, we will add the value of all the frequencies.
Method 2: Using Overlapping Subproblems
The Recursive implementation for this solution simply follows the algorithm for recursive approach:
- A Naive recursive algorithm is implemented first
- The sum of the elements of the array from “i” to the “j” elements are calculated (these are the elements from the first and the last nodes)
- Thus, by considering all the nodes of the BST one at a time we try to define the cost of the root.
- Finally, we use the optCost() function for defining the optimal cost.
Time Complexity for this approach will be exponential.
Wrapping Up
If you are a software developer, the first thing you would consider for implementing a computer program is its cost of execution and time complexity.
The higher costs of implementation can be discouraging for both the clients and the supplier for investing in the program.
Such is the problem of finding the cost of nodes in a BST that we have discussed in the blog.
Using Optimal Search we can reduce this cost to a significant amount and, what’s more beneficial is that it can also be applied to other areas of DSA such as find first and last position of element in sorted array.
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