computer notes - Data Structures - 25
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The document describes the build heap algorithm for creating a min heap data structure from an unordered array of keys. The algorithm iterates from the last parent node to the first, calling a percolateDown function to shift nodes and maintain the min heap property as it builds the heap structure from the initial array. It also provides code snippets in C++ for implementing common heap operations like insertion, deletion, and accessing the minimum element.
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![Heap code in C++ #include "Heap.h“ template <class eType> Heap<eType>::Heap( int capacity ) { array = new etype[capacity + 1]; currentSize=0; } http://ecomputernotes.com](https://image.slidesharecdn.com/10993919-120112104900-phpapp02/85/computer-notes-Data-Structures-25-15-320.jpg)
![Heap code in C++ // Insert item x into the heap, maintaining heap // order. Duplicates are allowed. template <class eType> bool Heap<eType>::insert( const eType & x ) { if( isFull( ) ) { cout << "insert - Heap is full." << endl; return 0; } // Percolate up int hole = ++currentSize; for(; hole > 1 && x < array[hole/2 ]; hole /= 2) array[ hole ] = array[ hole / 2 ]; array[hole] = x; } http://ecomputernotes.com](https://image.slidesharecdn.com/10993919-120112104900-phpapp02/85/computer-notes-Data-Structures-25-16-320.jpg)
![Heap code in C++ template <class eType> void Heap<eType>::deleteMin( eType & minItem ) { if( isEmpty( ) ) { cout << " heap is empty. “ << endl ; return; } minItem = array[ 1 ]; array[ 1 ] = array[ currentSize-- ]; percolateDown( 1 ); } http://ecomputernotes.com](https://image.slidesharecdn.com/10993919-120112104900-phpapp02/85/computer-notes-Data-Structures-25-17-320.jpg)
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computer notes - Data Structures - 18ecomputernotes The document discusses reference variables and how objects stored in data structures by reference can cause issues if the objects are transient. It provides an example where objects created on the stack disappear after the function returns. To prevent this, the document recommends allocating objects on the heap using new, so the objects persist in memory even after the function returns. It also discusses using delete to remove objects from the heap when no longer needed.
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computer notes - Data Structures - 9ecomputernotes The document discusses data structures like stacks and queues. It provides examples of implementing a queue using both a linked list and an array. It describes how a queue is a FIFO structure where elements are inserted at the rear and removed from the front. It also discusses some uses of queues, providing a bank simulation as an example where a queue models customers waiting to be served by tellers.
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computer notes - Data Structures - 1ecomputernotes The document discusses data structures and their implementation in C++. It covers topics like the need for data structures to organize data efficiently, commonly used data structures like arrays, linked lists, stacks and queues, selecting appropriate data structures based on algorithm requirements, and implementing dynamic arrays in C++ using pointers and the new operator.
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HeapsIIUM The document discusses priority queues and binary heaps. It explains that priority queues store tasks based on priority level and ensure the highest priority task is at the head of the queue. Binary heaps are the underlying data structure used to implement priority queues. The key operations on a binary heap are insert and deleteMin. Insert involves adding an element and percolating it up the heap, while deleteMin removes the minimum element and percolates the replacement down. Both operations have O(log n) time complexity. The document provides examples and pseudocode for building a heap from a list of elements in O(n) time using a buildHeap method.
Lec10-Binary-Heaps-19122022-113509am.pptx
Lec10-Binary-Heaps-19122022-113509am.pptxIqraHanif27 The document provides an overview of binary heaps, explaining the minheap and maxheap properties along with structural constraints. It details algorithms for insertion, deletion, and maintaining heap properties, showcasing practical examples and methods for implementing these operations. The document also includes a class definition for a minheap, demonstrating its methods such as insert, extract, decrease, and delete key functionalities.
Data Structures and Agorithm: DS 18 Heap.pptx
Data Structures and Agorithm: DS 18 Heap.pptxRashidFaridChishti The document provides an in-depth lecture on heaps, specifically focusing on the min-heap data structure, its properties, and operations such as insertion and retrieval of the minimum element. It details the binary tree representation of heaps and includes examples of operations using a sample code implementation in C++. The document also highlights key functions for inserting elements into the heap, finding the minimum value, and the internal structure of the minimum heap class.
05 heap 20161110_jintaeks
05 heap 20161110_jintaeksJinTaek Seo The document discusses heap data structures and their implementation using arrays. It defines a heap as a complete binary tree that satisfies the heap property - a node's value is greater than or equal to its children's values. Heaps can be implemented using arrays by numbering nodes from top to bottom and storing each node at its number index. Common heap operations like building a heap from an array, inserting/extracting elements, and deleting elements are described along with their time complexities of O(n) or O(log n).
Complete binary tree and heap
Complete binary tree and heapNazir Ahmed - A complete binary tree is a tree that is completely filled, with the possible exception of the bottom level, which is filled from left to right.
- Such a tree can be stored efficiently in an array, with the left and right children and parent of each node calculable from its array position.
- A heap is a complete binary tree that follows the heap property - a parent node's key is always greater than or equal to its children's keys. Heaps allow efficient insertion and deletion of minimum/maximum elements.
Heaps and tries power point this is an educational material
Heaps and tries power point this is an educational materialAymericTaylor This document provides information about heaps and tries data structures. It discusses min heaps, max heaps, and min-max heaps, including their implementations and operations like insertion, deletion, and finding the minimum/maximum elements. It also covers tries data structures and their applications in auto-complete, spell checking, and their tree structure for storing prefixes of words.
thisisheapsortpptfilewhichyoucanuseanywhereanytim
thisisheapsortpptfilewhichyoucanuseanywhereanytimhamdardhamdard864 heap sort ppt for university studetn
Heap sort
Heap sortMohd Arif The document discusses heap sort, which is a sorting algorithm that uses a heap data structure. It works in two phases: first, it transforms the input array into a max heap using the insert heap procedure; second, it repeatedly extracts the maximum element from the heap and places it at the end of the sorted array, reheapifying the remaining elements. The key steps are building the heap, processing the heap by removing the root element and allowing the heap to reorder, and doing this repeatedly until the array is fully sorted.
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lecture 5sajinsc The document discusses heapsort, an efficient sorting algorithm that uses a heap data structure. Heapsort first builds a max heap from the input array in O(n) time using a heapify procedure. It then extracts elements from the heap into the sorted output array, maintaining the heap property, resulting in an overall time complexity of O(n log n).
Heapsortokkay
HeapsortokkayRemesha Heapsort is an O(n log n) sorting algorithm that uses a heap data structure. It works by first converting the input array into a max heap, where the largest element is stored at the root. It then repeatedly removes the largest element from the heap and inserts it into the sorted end of the array. This process continues until the heap is empty, resulting in a fully sorted array. The document provides detailed explanations and examples of how to construct a max heap from an array and how to remove and re-heapify elements during the sorting process.
Lec20
Lec20Anjneya Varshney The document discusses priority queues and their implementation using binary heaps. A priority queue supports insertion, finding the minimum element, and deletion of the minimum element. Binary heaps can be used to implement priority queues efficiently with insertion and finding the minimum element taking O(log n) time due to the heap structure. Heapify may be needed to restore the heap property after inserting, which also takes O(log n) time.
10 chapter6 heaps_priority_queues
10 chapter6 heaps_priority_queuesSSE_AndyLi Priority queues are data structures that allow efficient retrieval of the maximum or minimum element based on some ordering criteria. A binary heap implementation provides efficient insert and delete-min operations in O(log N) time on average. Elements are stored in a complete binary tree structure in array form. The insert operation bubbles the element up the tree until the heap property is satisfied. Delete-min removes the root and sinks the last element into its place, maintaining the heap property. Building the initial heap from a unsorted list takes O(N) time using the buildHeap algorithm.
Binary heap (Priority Queue).ppt
Binary heap (Priority Queue).pptMano Arun The document discusses priority queues and binary heaps, which are data structures that support efficient insertion and retrieval of minimum elements. It explains how a binary heap can be implemented using a complete binary tree stored in an array, and describes the insertion and deletion algorithms which maintain the heap ordering property by bubbling elements up or down as needed. Priority queues and binary heaps support various algorithms by allowing efficient minimum access.
Heaps
HeapsHoang Nguyen The document discusses different types of heaps and heap algorithms. It describes binary min-heaps and max-heaps, including their properties and implementations using arrays. Basic heap operations like insert, delete, and build heap are explained along with their time complexities. Applications of heaps like priority queues and selection algorithms are covered. More advanced heap types like leftist heaps, skew heaps and binomial queues are also mentioned.
Complete the classes shown below 1. The MinHeap Class Write necessa.pdf
Complete the classes shown below 1. The MinHeap Class Write necessa.pdfamericanopticalscbe The document outlines the implementation of a MinHeap class in Java, detailing methods like buildMinHeap, downHeap, and upHeap necessary for heap operations. It provides a structure for managing a min-heap, including the insertion of new elements and removal of the minimum elements. The code must be completed and submitted via Gradescope, with specific input and output requirements for testing the functionality.
Heaps
Heapspratmash A heap data structure is a binary tree that satisfies two properties: it is a complete binary tree where each level is filled from left to right, and the value stored at each node is greater than or equal to the values of its children (the heap property). Heaps can be implemented using arrays where the root is at index 0, left child at 2i+1, and right child at 2i+2. The basic heap operations like building a heap from an array and heapifying subtrees run in O(log n) time, allowing priority queues and other applications to be efficiently implemented using heaps.
filesHeap.h#ifndef HEAP_H#define HEAP_H#includ.docx
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computer notes - Data Structures - 30
computer notes - Data Structures - 30ecomputernotes The document discusses different techniques for improving the efficiency of union-find algorithms, including union-by-size, path compression, and union-by-height. Union-by-size works by making the root of the smaller tree the child of the larger tree during a union operation, keeping tree heights small. Path compression further optimizes find operations by updating the parent pointers along the search path. Together these optimizations allow union-find algorithms to run in almost linear time for practical purposes.
computer notes - Data Structures - 39
computer notes - Data Structures - 39ecomputernotes The document discusses divide and conquer algorithms for sorting, specifically mergesort and quicksort. It explains that mergesort works by recursively splitting a list in half, sorting the halves, and then merging the sorted halves together. Quicksort works by picking a pivot value and partitioning the list into elements less than and greater than the pivot, before recursively sorting the sublists. Both algorithms run in O(n log n) time.
computer notes - Data Structures - 28
computer notes - Data Structures - 28ecomputernotes The document discusses equivalence relations and the union-find algorithm. It defines what makes a binary relation an equivalence relation by having the properties of reflexivity, symmetry, and transitivity. It gives examples like electrical connectivity being an equivalence relation. The union-find algorithm can be used to dynamically determine if elements are in the same equivalence class based on the given relations, by performing find and union operations in time proportional to m+n for m finds and n-1 unions.
computer notes - Data Structures - 19
computer notes - Data Structures - 19ecomputernotes The document discusses various aspects of balanced binary search trees, including:
1) Const keyword can be used to mark parameters and return values as constant to prevent unintended modification.
2) AVL trees are binary search trees where the heights of left and right subtrees differ by at most 1.
3) For a binary search tree to be balanced, the heights of left and right subtrees should be close to equal to avoid a skewed or degenerate tree structure.
computer notes - Data Structures - 4
computer notes - Data Structures - 4ecomputernotes The document discusses different types of linked lists, including singly linked lists, doubly linked lists, and circularly linked lists. It provides code examples for implementing linked lists in C++ and compares the time complexity of different linked list operations. It also describes how a circularly linked list can be used to solve the Josephus problem of eliminating people seated in a circle.
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computer notes - Data Structures - 16ecomputernotes The document discusses deleting nodes from a binary search tree (BST). There are three cases to consider when deleting a node: 1) if the node is a leaf, it can be deleted immediately, 2) if the node has one child, its parent pointer is redirected to the child node, 3) if the node has two children, it is replaced with its inorder successor. The algorithm and C++ code for deleting nodes from a BST is presented.
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computer notes - Data Structures - 35ecomputernotes The document discusses various data structures including skip lists, AVL trees, and hashing. It explains that skip lists allow for logarithmic-time operations and are simple to implement. Hashing provides constant-time operations by mapping keys to array indices via a hash function, but collisions must be handled. Common hash functions discussed include summing character codes or converting to a number in a prime base.
computer notes - Data Structures - 36
computer notes - Data Structures - 36ecomputernotes The document discusses different methods for handling collisions in hash tables, which occur when two keys hash to the same slot. It describes linear probing, where the next empty slot is used to store the colliding key; quadratic probing which uses a quadratic function to determine subsequent slots; and chaining, where each slot contains a linked list of colliding keys. It notes the advantages and disadvantages of each approach.
computer notes - Data Structures - 14
computer notes - Data Structures - 14ecomputernotes The document discusses recursion and different traversal methods for binary trees. It explains how the call stack is used during recursive function calls and shows examples of preorder and inorder recursive tree traversals. It then describes how to perform non-recursive inorder traversal using an explicit stack. Finally, it introduces level-order traversal, which visits all nodes at each level from left to right before proceeding to the next level.
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- 1. Class No.25 Data Structures http://ecomputernotes.com
- 2. BuildHeap The general algorithm is to place the N keys in an array and consider it to be an unordered binary tree. The following algorithm will build a heap out of N keys. for( i = N/2; i > 0; i-- ) percolateDown(i); http://ecomputernotes.com
- 3. BuildHeap i = 15/2 = 7 65 19 21 14 24 26 13 15 68 16 65 21 19 26 14 16 68 13 24 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 31 31 11 32 32 12 15 4 5 13 70 14 12 15 5 70 12 i i Why I=n/2? http://ecomputernotes.com
- 4. BuildHeap i = 15/2 = 7 65 19 21 14 24 26 13 15 12 16 65 21 19 26 14 16 12 13 24 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 31 31 11 32 32 12 15 4 5 13 70 14 68 15 5 70 68 i i http://ecomputernotes.com
- 5. BuildHeap i = 6 65 19 21 14 24 26 13 15 12 16 65 21 19 26 14 16 12 13 24 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 31 31 11 32 32 12 15 4 5 13 70 14 68 15 5 70 68 i i http://ecomputernotes.com
- 6. BuildHeap i = 5 65 5 21 14 24 26 13 15 12 16 65 21 5 26 14 16 12 13 24 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 31 31 11 32 32 12 15 4 19 13 70 14 68 15 19 70 68 i i http://ecomputernotes.com
- 7. BuildHeap i = 4 65 5 14 21 24 26 13 15 12 16 65 14 5 26 21 16 12 13 24 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 31 31 11 32 32 12 15 4 19 13 70 14 68 15 19 70 68 i i http://ecomputernotes.com
- 8. BuildHeap i = 3 65 5 14 21 24 13 26 15 12 16 65 14 5 13 21 16 12 26 24 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 31 31 11 32 32 12 15 4 19 13 70 14 68 15 19 70 68 i i http://ecomputernotes.com
- 9. BuildHeap i = 2 65 16 14 21 24 13 26 15 12 32 65 14 16 13 21 32 12 26 24 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 31 31 11 5 5 12 15 4 19 13 70 14 68 15 19 70 68 i i http://ecomputernotes.com
- 10. BuildHeap i = 1 65 16 14 21 31 24 26 15 12 32 65 14 16 24 21 32 12 26 31 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 13 13 11 5 5 12 15 4 19 13 70 14 68 15 19 70 68 i i http://ecomputernotes.com
- 11. BuildHeap Min heap 5 16 14 21 31 24 26 15 65 32 5 14 16 24 21 32 65 26 31 15 1 2 3 5 6 7 8 9 10 11 12 13 14 0 1 2 3 7 6 5 4 8 9 10 13 13 11 12 12 12 15 4 19 13 70 14 68 15 19 70 68 http://ecomputernotes.com
- 12. Other Heap Operations decreaseKey(p, delta) : lowers the value of the key at position ‘p’ by the amount ‘delta’. Since this might violate the heap order, the heap must be reorganized with percolate up (in min heap) or down (in max heap). increaseKey(p, delta) : opposite of decreaseKey. remove(p) : removes the node at position p from the heap. This is done by first decreaseKey(p, ) and then performing deleteMin(). http://ecomputernotes.com
- 13. Heap code in C++ template <class eType> class Heap { public: Heap( int capacity = 100 ); void insert( const eType & x ); void deleteMin( eType & minItem ); const eType & getMin( ); bool isEmpty( ); bool isFull( ); int Heap<eType>::getSize( ); http://ecomputernotes.com
- 14. Heap code in C++ private: int currentSize; // Number of elements in heap eType* array; // The heap array int capacity; void percolateDown( int hole ); }; http://ecomputernotes.com
- 15. Heap code in C++ #include "Heap.h“ template <class eType> Heap<eType>::Heap( int capacity ) { array = new etype[capacity + 1]; currentSize=0; } http://ecomputernotes.com
- 16. Heap code in C++ // Insert item x into the heap, maintaining heap // order. Duplicates are allowed. template <class eType> bool Heap<eType>::insert( const eType & x ) { if( isFull( ) ) { cout << "insert - Heap is full." << endl; return 0; } // Percolate up int hole = ++currentSize; for(; hole > 1 && x < array[hole/2 ]; hole /= 2) array[ hole ] = array[ hole / 2 ]; array[hole] = x; } http://ecomputernotes.com
- 17. Heap code in C++ template <class eType> void Heap<eType>::deleteMin( eType & minItem ) { if( isEmpty( ) ) { cout << " heap is empty. “ << endl ; return; } minItem = array[ 1 ]; array[ 1 ] = array[ currentSize-- ]; percolateDown( 1 ); } http://ecomputernotes.com