- Design and Analysis of Algorithms
- Basics of Algorithms
- DAA - Introduction
- DAA - Analysis of Algorithms
- DAA - Methodology of Analysis
- Asymptotic Notations & Apriori Analysis
- DAA - Space Complexities
- Design Strategies
- DAA - Divide & Conquer
- DAA - Max-Min Problem
- DAA - Merge Sort
- DAA - Binary Search
- Strassen’s Matrix Multiplication
- DAA - Greedy Method
- DAA - Fractional Knapsack
- DAA - Job Sequencing with Deadline
- DAA - Optimal Merge Pattern
- DAA - Dynamic Programming
- DAA - 0-1 Knapsack
- Longest Common Subsequence
- Graph Theory
- DAA - Spanning Tree
- DAA - Shortest Paths
- DAA - Multistage Graph
- Travelling Salesman Problem
- Optimal Cost Binary Search Trees
- Sorting Methods
- DAA - Bubble Sort
- DAA - Insertion Sort
- DAA - Selection Sort
- DAA - Quick Sort
- DAA - Radix Sort
- Complexity Theory
- Deterministic vs. Nondeterministic Computations
- DAA - Max Cliques
- DAA - Vertex Cover
- DAA - P and NP Class
- DAA - Cook’s Theorem
- NP Hard & NP-Complete Classes
- DAA - Hill Climbing Algorithm
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Design and Analysis Binary Heap
There are several types of heaps, however in this chapter, we are going to discuss binary heap. A binary heap is a data structure, which looks similar to a complete binary tree. Heap data structure obeys ordering properties discussed below. Generally, a Heap is represented by an array. In this chapter, we are representing a heap by H.
As the elements of a heap is stored in an array, considering the starting index as 1, the position of the parent node of ith element can be found at ⌊ i/2 ⌋ . Left child and right child of ith node is at position 2i and 2i + 1.
A binary heap can be classified further as either a max-heap or a min-heap based on the ordering property.
In this heap, the key value of a node is greater than or equal to the key value of the highest child.
Hence, H[Parent(i)] ≥ H[i]
In mean-heap, the key value of a node is lesser than or equal to the key value of the lowest child.
Hence, H[Parent(i)] ≤ H[i]
In this context, basic operations are shown below with respect to Max-Heap. Insertion and deletion of elements in and from heaps need rearrangement of elements. Hence, Heapify function needs to be called.
A complete binary tree can be represented by an array, storing its elements using level order traversal.
Let us consider a heap (as shown below) which will be represented by an array H.
Considering the starting index as 0, using level order traversal, the elements are being kept in an array as follows.
In this context, operations on heap are being represented with respect to Max-Heap.
To find the index of the parent of an element at index i, the following algorithm Parent (numbers, i) is used.
Algorithm: Parent (numbers, i) if i == 1 return NULL else [i / 2]
The index of the left child of an element at index i can be found using the following algorithm, Left-Child (numbers, i).
Algorithm: Left-Child (numbers, i) If 2 * i ≤ heapsize return [2 * i] else return NULL
The index of the right child of an element at index i can be found using the following algorithm, Right-Child(numbers, i).
Algorithm: Right-Child (numbers, i) if 2 * i < heapsize return [2 * i + 1] else return NULL