The document discusses sorting algorithms including bubble sort, selection sort, insertion sort, and merge sort. It provides pseudocode and explanations of how each algorithm works. Bubble sort, selection sort, and insertion sort have O(n2) runtime and are best for small datasets, while merge sort uses a divide-and-conquer approach to sort arrays with O(n log n) runtime, making it more efficient for large datasets. Radix sort is also discussed as an alternative sorting method that is optimized for certain data types.
Sorting algorithms in C++
An introduction to sorting algorithm, with details on bubble sort and merge sort algorithms
Computer science principles course
This document discusses sorting algorithms. It explains that sorting refers to arranging data in a specified order like numerical or alphabetical. Selection sort is described as finding the minimum element and swapping it into the correct position in each pass through the array until sorted. An example of selection sort is provided, showing the steps and swaps to sort an array of numbers.
The document provides information on various sorting and searching algorithms, including bubble sort, insertion sort, selection sort, quick sort, sequential search, and binary search. It includes pseudocode to demonstrate the algorithms and example implementations with sample input data. Key points covered include the time complexity of each algorithm (O(n^2) for bubble/insertion/selection sort, O(n log n) for quick sort, O(n) for sequential search, and O(log n) for binary search) and how they work at a high level.
Data Structures, which is also called as Abstract Data Types (ADT) provide powerful options for programmer. Here is a tutorial which talks about various ADTs - Linked Lists, Stacks, Queues and Sorting Algorithms
The document describes several sorting algorithms:
1) Bubble sort repeatedly compares and swaps adjacent elements, moving the largest values to the end over multiple passes. It has a complexity of O(n^2).
2) Insertion sort inserts elements one by one into the sorted portion of the array by shifting elements and comparing. It is O(n^2) in worst case but O(n) if nearly sorted.
3) Selection sort finds the minimum element and swaps it into the first position in each pass to build the sorted array. It has complexity O(n^2).
4) Merge sort divides the array into halves recursively, then merges the sorted halves to produce the fully sorted array.
The document discusses various sorting algorithms. It begins by explaining the motivation for sorting and providing examples. It then lists some common sorting algorithms like bubble sort, selection sort, and insertion sort. For each algorithm, it provides an informal description, works through examples to show how it sorts a list, and includes Java code implementations. It compares the time complexity of these algorithms, which is O(n2) for bubble sort, selection sort, and insertion sort, and explains why. The document aims to introduce fundamental sorting algorithms and their workings.
A sorting algorithm is an algorithm that arranges items of a list in a certain order like numerical or alphabetical. Common efficient sorting algorithms have average time complexity of O(n log n) by comparing elements, though some have better performance on certain data. Popular general sorting algorithms used in practice are heap sort, merge sort, and quicksort, sometimes with modifications for real-world data.
Sorting algorithms in C++
An introduction to sorting algorithm, with details on bubble sort and merge sort algorithms
Computer science principles course
This document discusses sorting algorithms. It explains that sorting refers to arranging data in a specified order like numerical or alphabetical. Selection sort is described as finding the minimum element and swapping it into the correct position in each pass through the array until sorted. An example of selection sort is provided, showing the steps and swaps to sort an array of numbers.
The document provides information on various sorting and searching algorithms, including bubble sort, insertion sort, selection sort, quick sort, sequential search, and binary search. It includes pseudocode to demonstrate the algorithms and example implementations with sample input data. Key points covered include the time complexity of each algorithm (O(n^2) for bubble/insertion/selection sort, O(n log n) for quick sort, O(n) for sequential search, and O(log n) for binary search) and how they work at a high level.
Data Structures, which is also called as Abstract Data Types (ADT) provide powerful options for programmer. Here is a tutorial which talks about various ADTs - Linked Lists, Stacks, Queues and Sorting Algorithms
The document describes several sorting algorithms:
1) Bubble sort repeatedly compares and swaps adjacent elements, moving the largest values to the end over multiple passes. It has a complexity of O(n^2).
2) Insertion sort inserts elements one by one into the sorted portion of the array by shifting elements and comparing. It is O(n^2) in worst case but O(n) if nearly sorted.
3) Selection sort finds the minimum element and swaps it into the first position in each pass to build the sorted array. It has complexity O(n^2).
4) Merge sort divides the array into halves recursively, then merges the sorted halves to produce the fully sorted array.
The document discusses various sorting algorithms. It begins by explaining the motivation for sorting and providing examples. It then lists some common sorting algorithms like bubble sort, selection sort, and insertion sort. For each algorithm, it provides an informal description, works through examples to show how it sorts a list, and includes Java code implementations. It compares the time complexity of these algorithms, which is O(n2) for bubble sort, selection sort, and insertion sort, and explains why. The document aims to introduce fundamental sorting algorithms and their workings.
A sorting algorithm is an algorithm that arranges items of a list in a certain order like numerical or alphabetical. Common efficient sorting algorithms have average time complexity of O(n log n) by comparing elements, though some have better performance on certain data. Popular general sorting algorithms used in practice are heap sort, merge sort, and quicksort, sometimes with modifications for real-world data.
This document discusses algorithms for sorting and searching. It introduces algorithm analysis and why it is important to analyze algorithms to compare different solutions, predict performance, and set parameter values. It discusses analyzing running time by counting primitive operations as a function of input size. Common growth rates like linear, quadratic, and cubic functions are introduced. Big-O notation is explained as a way to classify algorithms by their worst-case running time. Several specific sorting algorithms are described, including bubble sort, insertion sort, merge sort, and quicksort. Their pseudocode and running times are analyzed. Graph traversal algorithms like depth-first search and breadth-first search are also introduced.
This document provides an overview and comparison of insertion sort and shellsort sorting algorithms. It describes how insertion sort works by repeatedly inserting elements into a sorted left portion of the array. Shellsort improves on insertion sort by making passes with larger increments to shift values into approximate positions before final sorting. The document discusses the time complexities of both algorithms and provides examples to illustrate how they work.
It is a presentation on some Searching and Sorting Techniques for Computer Science.
It consists of the following techniques:
Sequential Search
Binary Search
Selection Sort
Bubble Sort
Insertion Sort
The document discusses several sorting algorithms: selection sort, bubble sort, quicksort, and merge sort. Selection sort has linear time complexity for swaps but quadratic time for comparisons. Bubble sort is quadratic time for both swaps and comparisons, making it the least efficient. Quicksort and merge sort are the fastest algorithms, both with logarithmic time complexity of O(n log n) for swaps and comparisons. Quicksort risks quadratic behavior in the worst case if pivots are chosen poorly, while merge sort requires more data copying between temporary and full lists.
The document discusses various sorting algorithms. It describes how sorting algorithms arrange elements of a list in a certain order. Efficient sorting is important as a subroutine for algorithms that require sorted input, such as search and merge algorithms. Common sorting algorithms covered include insertion sort, selection sort, bubble sort, merge sort, and quicksort. Quicksort is highlighted as an efficient divide and conquer algorithm that recursively partitions elements around a pivot point.
The document discusses various sorting algorithms. It begins by defining a sorting algorithm as arranging elements of a list in a certain order, such as numerical or alphabetical order. It then discusses popular sorting algorithms like insertion sort, bubble sort, merge sort, quicksort, selection sort, and heap sort. For each algorithm, it provides examples to illustrate how the algorithm works step-by-step to sort a list of numbers. Code snippets are also included for insertion sort and bubble sort.
The document discusses various searching and sorting algorithms. It begins by defining search algorithms as methods for finding items within a collection. It then covers linear search, which has O(n) complexity, and binary search, which has O(log n) complexity but requires a sorted list. The document also discusses sorting algorithms like bubble sort, selection sort, and merge sort. Merge sort uses a divide-and-conquer approach and has O(n log n) complexity, making it one of the most efficient common sorting algorithms. Code implementations and complexity analyses are provided for many of the algorithms.
Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages:
This document provides an overview of the Data Structures I course. It outlines the course objectives of becoming familiar with problem solving, algorithms, data structures, and tracing algorithms. The course will cover fundamentals of data structures and algorithms, static and dynamic data structures, searching and sorting algorithms, recursion, abstract data types, stacks, queues and trees. Exams, labs, participation and quizzes will be used for grading. Pseudo-code is introduced as a way to express algorithms independent of a programming language. Examples of algorithms for determining even/odd numbers and computing weekly wages are provided.
The document discusses insertion sort, a simple sorting algorithm that builds a sorted output list from an input one element at a time. It is less efficient on large lists than more advanced algorithms. Insertion sort iterates through the input, at each step removing an element and inserting it into the correct position in the sorted output list. The best case for insertion sort is an already sorted array, while the worst is a reverse sorted array.
The document summarizes various sorting algorithms:
- Bubble sort works by repeatedly swapping adjacent elements that are in the wrong order until the list is fully sorted. It requires O(n^2) time.
- Insertion sort iterates through the list and inserts each element into its sorted position. It is an adaptive algorithm with O(n) time for nearly sorted inputs.
- Quicksort uses a divide and conquer approach, recursively partitioning the list around a pivot element and sorting the sublists. It has average case performance of O(nlogn) time.
This document discusses various sorting algorithms including merge sort. It begins with an introduction to sorting and searching. It then provides pseudocode for the merge sort algorithm which works by dividing the array into halves, recursively sorting the halves, and then merging the sorted halves back together. An example is provided to illustrate the merge sort process. Key steps include dividing, conquering by recursively sorting subarrays, and combining through merging. The overall time complexity of merge sort is O(n log n).
Different types of Shoring Algorithms with AnimationZakaria Hossain
This presentation discusses several sorting algorithms: bubble sort, insertion sort, selection sort, merge sort, and quick sort. Bubble sort, insertion sort, and selection sort have average and worst-case complexities of O(n^2), making them unsuitable for large data sets. Merge sort has a worst-case complexity of O(n log n), making it efficient. Quick sort also has average and worst-case complexities of O(n log n), making it suitable for large data sets. The presentation provides descriptions and animations of the sorting processes for each algorithm.
This document provides a 90-minute discussion on algorithms including quicksort, order statistics, searching, and substring searching. It begins with an overview of the topics and then provides details on quicksort, including the divide and conquer approach and partitioning elements around a pivot. It also describes algorithms for order statistics to find the kth smallest element, binary search, and a basic substring searching approach. Special cases and better solutions like Boyer-Moore are also mentioned.
The document describes insertion sort, a sorting algorithm. It lists the group members who researched insertion sort and provides an introduction. It then explains how insertion sort works by example, showing how it iterates through an array and inserts elements into the sorted portion. Pseudocode and analysis of insertion sort's runtime is provided. Comparisons are made between insertion sort and other algorithms like bubble sort, selection sort, and merge sort, analyzing their time complexities in best, average, and worst cases.
Array implementation and linked list as datat structureTushar Aneyrao
The document discusses arrays and linked lists. It defines arrays as collections of elements of the same data type stored in contiguous memory. Linked lists store elements in memory locations that are not contiguous. Each element of a linked list contains a data field and a pointer to the next node. This allows for efficient insertion and deletion of nodes compared to arrays. The document provides examples of implementing common linked list operations like insertion at the head, tail, and deletion.
This document discusses various sorting algorithms and their complexities. It begins by defining an algorithm and complexity measures like time and space complexity. It then defines sorting and common sorting algorithms like bubble sort, selection sort, insertion sort, quicksort, and mergesort. For each algorithm, it provides a high-level overview of the approach and time complexity. It also covers sorting algorithm concepts like stable and unstable sorting. The document concludes by discussing future directions for sorting algorithms and their applications.
This document provides a summary of a lecture on sorting algorithms:
1) It introduces sorting concepts and sorting problems. Common sorting algorithms discussed include selection sort, insertion sort, merge sort, and quick sort.
2) Selection sort and insertion sort algorithms are explained in detail with examples. Selection sort has a running time of O(n^2) in all cases.
3) Insertion sort's running time depends on how sorted the input array is - it is O(n) for a presorted array but O(n^2) for a reverse sorted array.
4) The time complexity of both algorithms is analyzed mathematically. Selection sort is always O(n^2) while
Data Structures- Part3 arrays and searching algorithmsAbdullah Al-hazmy
This document discusses arrays and searching algorithms. It begins by defining data structures and describing different types, including arrays. Arrays are introduced as a way to store multiple values of the same type in contiguous memory locations. Sequential and binary search algorithms are then described. Sequential search has linear time complexity, while binary search has logarithmic time complexity when used on a sorted array. Key concepts of arrays like indexing, dimensionality, and operations are also covered. The document concludes by looking ahead to sorting algorithms to be discussed in the next week.
The document discusses various sorting algorithms. It provides an overview of bubble sort, selection sort, and insertion sort. For bubble sort, it explains the process of "bubbling up" the largest element to the end of the array through successive passes. For selection sort, it illustrates the process of finding the largest element and swapping it into the correct position in each pass to sort the array. For insertion sort, it notes that elements are inserted into the sorted portion of the array in the proper position.
Selection sort is an in-place comparison sorting algorithm that works as follows: (1) Find the minimum value in the list, (2) Swap it with the value in the first position, (3) Repeat for the remainder of the list. It has a time complexity of O(n2), making it inefficient for large lists. While simple, it has advantages over more complex algorithms when auxiliary memory is limited. Variants include heapsort, which improves efficiency, and bingo sort, which is more efficient for lists with duplicate values.
Bubble sort is a simple sorting algorithm that compares adjacent elements and swaps them if they are in the wrong order. This is repeated for each pair of adjacent elements with at least one swap happening per iteration until the list is fully sorted. Selection sort works by finding the minimum element in the unsorted section and swapping it with the leftmost element to build up the sorted section from left to right. Insertion sort maintains a sorted sub-list and inserts new elements into the correct position in the sub-list by shifting other elements over as needed.
This document discusses algorithms for sorting and searching. It introduces algorithm analysis and why it is important to analyze algorithms to compare different solutions, predict performance, and set parameter values. It discusses analyzing running time by counting primitive operations as a function of input size. Common growth rates like linear, quadratic, and cubic functions are introduced. Big-O notation is explained as a way to classify algorithms by their worst-case running time. Several specific sorting algorithms are described, including bubble sort, insertion sort, merge sort, and quicksort. Their pseudocode and running times are analyzed. Graph traversal algorithms like depth-first search and breadth-first search are also introduced.
This document provides an overview and comparison of insertion sort and shellsort sorting algorithms. It describes how insertion sort works by repeatedly inserting elements into a sorted left portion of the array. Shellsort improves on insertion sort by making passes with larger increments to shift values into approximate positions before final sorting. The document discusses the time complexities of both algorithms and provides examples to illustrate how they work.
It is a presentation on some Searching and Sorting Techniques for Computer Science.
It consists of the following techniques:
Sequential Search
Binary Search
Selection Sort
Bubble Sort
Insertion Sort
The document discusses several sorting algorithms: selection sort, bubble sort, quicksort, and merge sort. Selection sort has linear time complexity for swaps but quadratic time for comparisons. Bubble sort is quadratic time for both swaps and comparisons, making it the least efficient. Quicksort and merge sort are the fastest algorithms, both with logarithmic time complexity of O(n log n) for swaps and comparisons. Quicksort risks quadratic behavior in the worst case if pivots are chosen poorly, while merge sort requires more data copying between temporary and full lists.
The document discusses various sorting algorithms. It describes how sorting algorithms arrange elements of a list in a certain order. Efficient sorting is important as a subroutine for algorithms that require sorted input, such as search and merge algorithms. Common sorting algorithms covered include insertion sort, selection sort, bubble sort, merge sort, and quicksort. Quicksort is highlighted as an efficient divide and conquer algorithm that recursively partitions elements around a pivot point.
The document discusses various sorting algorithms. It begins by defining a sorting algorithm as arranging elements of a list in a certain order, such as numerical or alphabetical order. It then discusses popular sorting algorithms like insertion sort, bubble sort, merge sort, quicksort, selection sort, and heap sort. For each algorithm, it provides examples to illustrate how the algorithm works step-by-step to sort a list of numbers. Code snippets are also included for insertion sort and bubble sort.
The document discusses various searching and sorting algorithms. It begins by defining search algorithms as methods for finding items within a collection. It then covers linear search, which has O(n) complexity, and binary search, which has O(log n) complexity but requires a sorted list. The document also discusses sorting algorithms like bubble sort, selection sort, and merge sort. Merge sort uses a divide-and-conquer approach and has O(n log n) complexity, making it one of the most efficient common sorting algorithms. Code implementations and complexity analyses are provided for many of the algorithms.
Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages:
This document provides an overview of the Data Structures I course. It outlines the course objectives of becoming familiar with problem solving, algorithms, data structures, and tracing algorithms. The course will cover fundamentals of data structures and algorithms, static and dynamic data structures, searching and sorting algorithms, recursion, abstract data types, stacks, queues and trees. Exams, labs, participation and quizzes will be used for grading. Pseudo-code is introduced as a way to express algorithms independent of a programming language. Examples of algorithms for determining even/odd numbers and computing weekly wages are provided.
The document discusses insertion sort, a simple sorting algorithm that builds a sorted output list from an input one element at a time. It is less efficient on large lists than more advanced algorithms. Insertion sort iterates through the input, at each step removing an element and inserting it into the correct position in the sorted output list. The best case for insertion sort is an already sorted array, while the worst is a reverse sorted array.
The document summarizes various sorting algorithms:
- Bubble sort works by repeatedly swapping adjacent elements that are in the wrong order until the list is fully sorted. It requires O(n^2) time.
- Insertion sort iterates through the list and inserts each element into its sorted position. It is an adaptive algorithm with O(n) time for nearly sorted inputs.
- Quicksort uses a divide and conquer approach, recursively partitioning the list around a pivot element and sorting the sublists. It has average case performance of O(nlogn) time.
This document discusses various sorting algorithms including merge sort. It begins with an introduction to sorting and searching. It then provides pseudocode for the merge sort algorithm which works by dividing the array into halves, recursively sorting the halves, and then merging the sorted halves back together. An example is provided to illustrate the merge sort process. Key steps include dividing, conquering by recursively sorting subarrays, and combining through merging. The overall time complexity of merge sort is O(n log n).
Different types of Shoring Algorithms with AnimationZakaria Hossain
This presentation discusses several sorting algorithms: bubble sort, insertion sort, selection sort, merge sort, and quick sort. Bubble sort, insertion sort, and selection sort have average and worst-case complexities of O(n^2), making them unsuitable for large data sets. Merge sort has a worst-case complexity of O(n log n), making it efficient. Quick sort also has average and worst-case complexities of O(n log n), making it suitable for large data sets. The presentation provides descriptions and animations of the sorting processes for each algorithm.
This document provides a 90-minute discussion on algorithms including quicksort, order statistics, searching, and substring searching. It begins with an overview of the topics and then provides details on quicksort, including the divide and conquer approach and partitioning elements around a pivot. It also describes algorithms for order statistics to find the kth smallest element, binary search, and a basic substring searching approach. Special cases and better solutions like Boyer-Moore are also mentioned.
The document describes insertion sort, a sorting algorithm. It lists the group members who researched insertion sort and provides an introduction. It then explains how insertion sort works by example, showing how it iterates through an array and inserts elements into the sorted portion. Pseudocode and analysis of insertion sort's runtime is provided. Comparisons are made between insertion sort and other algorithms like bubble sort, selection sort, and merge sort, analyzing their time complexities in best, average, and worst cases.
Array implementation and linked list as datat structureTushar Aneyrao
The document discusses arrays and linked lists. It defines arrays as collections of elements of the same data type stored in contiguous memory. Linked lists store elements in memory locations that are not contiguous. Each element of a linked list contains a data field and a pointer to the next node. This allows for efficient insertion and deletion of nodes compared to arrays. The document provides examples of implementing common linked list operations like insertion at the head, tail, and deletion.
This document discusses various sorting algorithms and their complexities. It begins by defining an algorithm and complexity measures like time and space complexity. It then defines sorting and common sorting algorithms like bubble sort, selection sort, insertion sort, quicksort, and mergesort. For each algorithm, it provides a high-level overview of the approach and time complexity. It also covers sorting algorithm concepts like stable and unstable sorting. The document concludes by discussing future directions for sorting algorithms and their applications.
This document provides a summary of a lecture on sorting algorithms:
1) It introduces sorting concepts and sorting problems. Common sorting algorithms discussed include selection sort, insertion sort, merge sort, and quick sort.
2) Selection sort and insertion sort algorithms are explained in detail with examples. Selection sort has a running time of O(n^2) in all cases.
3) Insertion sort's running time depends on how sorted the input array is - it is O(n) for a presorted array but O(n^2) for a reverse sorted array.
4) The time complexity of both algorithms is analyzed mathematically. Selection sort is always O(n^2) while
Data Structures- Part3 arrays and searching algorithmsAbdullah Al-hazmy
This document discusses arrays and searching algorithms. It begins by defining data structures and describing different types, including arrays. Arrays are introduced as a way to store multiple values of the same type in contiguous memory locations. Sequential and binary search algorithms are then described. Sequential search has linear time complexity, while binary search has logarithmic time complexity when used on a sorted array. Key concepts of arrays like indexing, dimensionality, and operations are also covered. The document concludes by looking ahead to sorting algorithms to be discussed in the next week.
The document discusses various sorting algorithms. It provides an overview of bubble sort, selection sort, and insertion sort. For bubble sort, it explains the process of "bubbling up" the largest element to the end of the array through successive passes. For selection sort, it illustrates the process of finding the largest element and swapping it into the correct position in each pass to sort the array. For insertion sort, it notes that elements are inserted into the sorted portion of the array in the proper position.
Selection sort is an in-place comparison sorting algorithm that works as follows: (1) Find the minimum value in the list, (2) Swap it with the value in the first position, (3) Repeat for the remainder of the list. It has a time complexity of O(n2), making it inefficient for large lists. While simple, it has advantages over more complex algorithms when auxiliary memory is limited. Variants include heapsort, which improves efficiency, and bingo sort, which is more efficient for lists with duplicate values.
Bubble sort is a simple sorting algorithm that compares adjacent elements and swaps them if they are in the wrong order. This is repeated for each pair of adjacent elements with at least one swap happening per iteration until the list is fully sorted. Selection sort works by finding the minimum element in the unsorted section and swapping it with the leftmost element to build up the sorted section from left to right. Insertion sort maintains a sorted sub-list and inserts new elements into the correct position in the sub-list by shifting other elements over as needed.
The document provides an overview of several sorting algorithms, including insertion sort, bubble sort, selection sort, and radix sort. It describes the basic approach for each algorithm through examples and pseudocode. Analysis of the time complexity is also provided, with insertion sort, bubble sort, and selection sort having worst-case performance of O(n^2) and radix sort having performance of O(nk) where k is the number of passes.
This document provides an overview of sorting algorithms. It defines sorting as arranging data in a particular order like ascending or descending. Common sorting algorithms discussed include bubble sort, selection sort, insertion sort, merge sort, and quick sort. For each algorithm, the working method, implementation in C, time and space complexity is explained. The document also covers sorting terminology like stable vs unstable sorting and adaptive vs non-adaptive algorithms. Overall, the document serves as a comprehensive introduction to sorting and different sorting techniques.
Bubble sort, selection sort, and insertion sort are O(n^2) sorting algorithms discussed in the document. Bubble sort compares and swaps adjacent elements, selection sort finds the minimum element and swaps it into place each iteration, and insertion sort inserts each new element into the sorted portion of the array. Merge sort is more efficient at O(n log n) time by dividing the array into halves, sorting them, and merging the results. It is well-suited for large datasets that do not fit into memory.
The document discusses simple sorting and searching algorithms. It describes selection sort, bubble sort, and insertion sort, which are all O(n^2) elementary sorting algorithms best for small lists. It also covers linear/sequential search, which has O(n) complexity, and binary search, which has optimal O(log n) complexity but requires a sorted list. Pseudocode and examples are provided for each algorithm.
This document discusses different sorting algorithms. It defines sorting as rearranging elements in a list or array based on a comparison operator. Three sorting algorithms are described: selection sort works by selecting the minimum element and placing it at the front of the sorted subarray; insertion sort works by inserting elements into the sorted position one by one; and bubble sort works by repeatedly swapping adjacent elements if they are in the wrong order. Examples are provided to illustrate how each algorithm sorts an array.
The document discusses various searching and sorting algorithms. It describes linear search, binary search, selection sort, bubble sort, and heapsort. For each algorithm, it provides pseudocode examples and analyzes their performance in terms of number of comparisons required in the worst case. Linear search requires N comparisons in the worst case, while binary search requires log N comparisons. Selection sort and bubble sort both require approximately N^2 comparisons, while heapsort requires 1.5NlogN comparisons.
The document discusses various searching and sorting algorithms including linear/sequential search, binary search, selection sort, bubble sort, insertion sort, quick sort, and merge sort. It provides descriptions of each algorithm and examples to illustrate how they work on sample data sets. Key steps and properties of each algorithm are outlined such as complexity, how elements are compared and swapped during sorting, and dividing arrays during searching.
The document discusses various sorting and searching algorithms. It begins by introducing selection sort, insertion sort, and bubble sort. It then covers merge sort and explains how it works by dividing the list, sorting sublists recursively, and merging the results. Finally, it discusses linear/sequential search and binary search, noting that sequential search checks every element while binary search repeatedly halves the search space.
This document provides an overview of several advanced sorting algorithms: Shell sort, Quick sort, Heap sort, and Merge sort. It describes the key ideas, time complexities, and provides examples of implementing each algorithm to sort sample data sets. Shell sort improves on insertion sort by sorting elements in a two-dimensional array. Quick sort uses a pivot element and partitions elements into left and right subsets. Heap sort uses a heap data structure and sorts by swapping elements. Merge sort divides the list recursively and then merges the sorted halves.
This document discusses three simple sorting algorithms: bubble sort, selection sort, and insertion sort. It provides pseudocode for each algorithm and analyzes their time complexities. Bubble sort and selection sort are analyzed to be O(n^2) time due to nested loops of size n. Insertion sort is also O(n^2) as it iterates through the array once and shifts elements in the inner loop approximately n/2 times on average. Loop invariants are developed to explain the sorting process for each algorithm.
MYSQL DATABASE MYSQL DATABASE MYSQL DATABASE BUBLESORT.pptxArjayBalberan1
The document describes the bubble sort algorithm. Bubble sort works by repeatedly comparing adjacent elements and swapping them if they are in the wrong order, causing the largest elements to "bubble" to the top of the list. This process is repeated, each time allowing the next largest element to bubble to the top, until the entire list is sorted after N-1 passes through the list. The algorithm is simple but inefficient for large lists, as it compares and swaps elements multiple times.
This document discusses various sorting and searching algorithms. It begins by listing sorting algorithms like selection sort, insertion sort, bubble sort, merge sort, and radix sort. It then discusses searching algorithms like linear/sequential search and binary search. It provides details on the implementation and time complexity of linear search, binary search, bubble sort, insertion sort, selection sort, and merge sort. Key points covered include the divide and conquer approach of merge sort and how binary search recursively halves the search space.
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion should be inserted
while some elements unsorted:
Using linear search, find the location in the sorted portion where the 1st element of the unsorted portion
The document discusses various sorting algorithms including exchange sorts like bubble sort and quicksort, selection sorts like straight selection sort, and tree sorts like heap sort. For each algorithm, it provides an overview of the approach, pseudocode, analysis of time complexity, and examples. Key algorithms covered are bubble sort (O(n2)), quicksort (average O(n log n)), selection sort (O(n2)), and heap sort (O(n log n)).
Selection sort is an in-place comparison sorting algorithm where the minimum element from the unsorted section of the list is found and swapped into the sorted position in each pass. It has a time complexity of O(n^2), making it inefficient for large lists, but it is simple to implement and has advantages over more complex algorithms when auxiliary memory is limited. The algorithm divides the input list into a sorted sublist on the left and unsorted sublist on the right. It iterates through the unsorted sublist to find the minimum element and swap it into the sorted position, shrinking the unsorted sublist by one element in each pass until the entire list is sorted.
The document outlines and compares three simple sorting algorithms: bubble sort, selection sort, and insertion sort. It provides pseudocode and examples for each algorithm. Bubble sort and selection sort are analyzed and shown to have an O(n2) running time, as does insertion sort. While all three algorithms have quadratic runtime, insertion sort is generally the fastest for small arrays.
Similar to Sorting algorithums > Data Structures & Algorithums (20)
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The document describes the merge sort algorithm. Merge sort is a sorting algorithm that works by dividing an unsorted list into halves, recursively sorting each half, and then merging the sorted halves into one sorted list. It has the following steps:
1. Divide: Partition the input list into equal halves.
2. Conquer: Recursively sort each half by repeating steps 1 and 2.
3. Combine: Merge the sorted halves into one sorted output list.
The algorithm has a runtime of O(n log n), making it an efficient sorting algorithm for large data sets. An example execution is shown step-by-step to demonstrate how merge sort partitions, recursively sorts, and merges sublists to produce
Huffman coding is a data compression technique that uses variable-length code words to encode characters based on their frequency of occurrence. It involves building a Huffman tree from character frequencies, assigning code words by traversing the tree, and encoding the text. This results in more common characters having shorter code words, reducing the number of bits needed and allowing for smaller file sizes compared to fixed-length encodings like ASCII.
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The document describes Prim's algorithm for finding a minimum spanning tree (MST) in a weighted, undirected graph. Prim's algorithm is a greedy algorithm that grows the MST by iteratively adding the closest vertex to the tree. It starts with a single vertex and adds the neighboring vertex with the lowest weight edge until all vertices are included in the tree. Pseudocode is provided to illustrate the steps of Prim's algorithm, which initializes all vertex keys to infinity except the starting vertex, then extracts the minimum key vertex and updates neighboring vertex keys if a lower weight edge is found. Examples are shown applying Prim's algorithm to find the MST of a sample graph.
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Sorting algorithums > Data Structures & Algorithums
1.
2. Why Sort?
A classic problem in computer science!
Data requested in sorted order
e.g.,
find students in increasing gpa order
Sorting is first step in bulk loading B+ tree index.
Sorting useful for eliminating duplicate copies in a
collection of records
Sorting is useful for summarizing related groups of
tuples
Sort-merge join algorithm involves sorting.
Problem: sort 100Gb of data with 1Gb of RAM.
why
not virtual memory?
3. Bubble sort
Compare each element (except the last one) with its
neighbor to the right
If they are out of order, swap them
This puts the largest element at the very end
The last element is now in the correct and final place
Compare each element (except the last two) with its
neighbor to the right
If they are out of order, swap them
This puts the second largest element next to last
The last two elements are now in their correct and final places
Compare each element (except the last three) with its
neighbor to the right
Continue as above until you have no unsorted elements on the
left
5. Sorting - Bubble
From the first element
Exchange pairs if they’re out of order
Last one must now be the largest
Repeat from the first to n-1
Stop when you have only one element to check
6. Bubble Sort
/* Bubble sort for integers */
#define SWAP(a,b)
{ int t; t=a; a=b; b=t; }
void bubble( int a[], int n ) {
int i, j;
for(i=0;i<n;i++) { /* n passes thru the array */
/* From start to the end of unsorted part */
for(j=1;j<(n-i);j++) {
/* If adjacent items out of order, swap */
if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]);
}
}
}
7. Bubble Sort - Analysis
/* Bubble sort for integers */
#define SWAP(a,b)
{ int t; t=a; a=b; b=t; }
void bubble( int a[], int n ) {
int i, j;
for(i=0;i<n;i++) { /* n passes thru the array */
/* From start to the end of unsorted part */
for(j=1;j<(n-i);j++) {
/* If adjacent items out of order, swap */
if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]);
}
}
O(1) statement
}
8. Bubble Sort - Analysis
/* Bubble sort for integers */
#define SWAP(a,b)
{ int t; t=a; a=b; b=t; }
void bubble( int a[], int n ) {
int i, j;
for(i=0;i<n;i++) { /* n passes thru the array */
/* From start to the end of unsorted part */
for(j=1;j<(n-i);j++) {
/* If adjacent items out of order, swap */
if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]);
}
}
Inner loop
O(1) statement
}
n-1, n-2, n-3, … , 1 iterations
9. Bubble Sort - Analysis
/* Bubble sort for integers */
#define SWAP(a,b)
{ int t; t=a; a=b; b=t; }
void bubble( int a[], int n ) {
int i, j;
for(i=0;i<n;i++) { /* n passes thru the array */
/* From start to the end of unsorted part */
for(j=1;j<(n-i);j++) {
/* If adjacent items out of order, swap */
if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]);
}
}
}
Outer loop n iterations
10. Bubble Sort - Analysis
/* Bubble sort for integers */
#define SWAP(a,b)
{ int t; t=a; a=b; b=t; }
void bubble( int a[], int n ) {
int i, j;
Overall
for(i=0;i<n;i++) { /* n passes thru the array */
/* From start to the end of unsorted part */
1
n(n+1)
for(j=1;j<(n-i);j++) {
Σ i =
= O(n2)
2
i=n-1
/* If adjacent items out of order, swap */
if( a[j-1]>a[j] ) SWAP(a[j-1],a[j]);
}
}
inner loop iteration count
} outer loop iterations
n
12. Selection sort
Given an array of length n,
0 through n-1 and select the smallest
Swap it with the element in location 0
Search elements 1 through n-1 and select the smallest
Swap it with the element in location 1
Search elements 2 through n-1 and select the smallest
Swap it with the element in location 2
Search elements 3 through n-1 and select the smallest
Swap it with the element in location 3
Search elements
Continue in this fashion until there’s nothing left to
search
13. Example and analysis of
selectionsort
The selection sort might swap an
7 2 8 5 4
2 7 8 5 4
2 4 8 5 7
2 4 5 8 7
array element with itself--this is
harmless, and not worth checking for
Analysis:
The outer loop executes n-1 times
The inner loop executes about n/2
times on average (from n to 2 times)
Work done in the inner loop is
2 4 5 7 8
constant (swap two array elements)
(n-1)*(n/2)
You should recognize this as O(n2)
Time required is roughly
13
14. Selection sort
How does it work:
first find the smallest in the array and exchange it with the element in
the first position, then find the second smallest element and
exchange it with the element in the second position, and continue in
this way until the entire array is sorted.
How does it sort the list in a non increasing order?
Selection sort is:
The simplest sorting techniques.
a good algorithm to sort a small number of elements
an incremental algorithm – induction method
Selection sort is Inefficient for large lists.
Incremental algorithms process the input elements oneby-one and maintain the solution for the elements processed
so far.
15. Selection Sort Algorithm
Input: An array A[1..n] of n elements.
Output: A[1..n] sorted in nondecreasing order.
1. for i ← 1 to n - 1
2. k ← i
3. for j ← i + 1 to n {Find the i th smallest element.}
4. if A[j] < A[k] then k ← j
5. end for
6. if k ≠ i then interchange A[i] and A[k]
7. end for
16. Sorting
Card players all know how to sort …
First card is already sorted
With all the rest,
Scan back from the end until you find the first card larger than
the new one,
Move all the lower ones up one slot
insert it
A
K
10
2
J
2
2
9
Q
9
17. One step of insertion sort
sorted
3
4
next to be inserted
7 12 14 14 20 21 33 38 10 55 9 23 28 16
less than
10
3
4
temp
10
7 10 12 14 14 20 21 33 38 55 9 23 28 16
sorted
18. Algorithm: INSERTIONSORT
Input: An array A[1..n] of n elements.
Output: A[1..n] sorted in nondecreasing order.
1. for
i ← 2 to n
2. x ← A[i]
3. j ← i - 1
4. while (j >0) and (A[j] > x)
5. A[j + 1] ← A[j]
6.
j← j-1
7. end while
8. A[j + 1] ← x
9. end for
Example sort : 34 8 64 51 32 21
19. Analysis of insertion sort
We run once through the outer loop, inserting
each of n elements; this is a factor of n
On average, there are n/2 elements already sorted
The inner loop looks at (and moves) half of these
This gives a second factor of n/4
Hence, the time required for an insertion sort of
an array of n elements is proportional to n2/4
Discarding constants, we find that insertion sort is
O(n2)
20. Summary
Bubble sort, selection sort, and insertion sort are all
O(n2)
As we will see later, we can do much better than this
with somewhat more complicated sorting algorithms
Within O(n2),
Bubble sort is very slow, and should probably never be used
for anything
Selection sort is intermediate in speed
Insertion sort is usually the fastest of the three--in fact, for
small arrays (say, 10 or 15 elements), insertion sort is faster
than more complicated sorting algorithms
Selection sort and insertion sort are “good enough” for
small arrays
21. Radix Sort
Consider the following 9 numbers:
493 812 715 710 195 437 582 340 385
We should start sorting by comparing and ordering the one's digits:
Digit
Sublist
0
340 710
1
2
812 582
3
4
493
715 195
385
437
5
6
7
8
9
Notice that the numbers were added onto
the list in the order that they were found,
which is why the numbers appear to be
unsorted in each of the sublists above.
Now, we gather the sublists (in order from
the 0 sublist to the 9 sublist) into the main
list again:
340 710 812 582 493 715 195
385 437
22. Now, the sublists are created again, this time based on the ten's digit:
Now the sublists are gathered in order from 0 to 9:
710 812 715 437 340 582 385 493 195
Digit
Sublist
0
1
710 812 715
2
3
4
5
6
7
437
340
8
582 385
9
493 195
Now the sublists are gathered in
order from 0 to 9:
710 812 715 437 340
582 385 493 195
23. Finally, the sublists are created according to the hundred's digit:
At last, the list is gathered up again:
195 340 385 437 493 582 710 715 812
Digit
Sublist
0
1
2
195
3
340 385
4
437 493
5
6
582
7
710 715
8
9
812
At last, the list is gathered up
again:
195 340 385 437 493
582 710 715 812
24. Disadvantages
Still, there are some tradeoffs for Radix Sort that can make it less
preferable than other sorts.
The speed of Radix Sort largely depends on the inner basic
operations, and if the operations are not efficient enough, Radix
Sort can be slower than some other algorithms such as Quick
Sort and Merge Sort. These operations include the insert and
delete functions of the sublists and the process of isolating the
digit you want.
In the example above, the numbers were all of equal length, but
many times, this is not the case. If the numbers are not of the
same length, then a test is needed to check for additional digits
that need sorting. This can be one of the slowest parts of Radix
Sort, and it is one of the hardest to make efficient.
25. Radix Sort can also take up more space than other sorting
algorithms, since in addition to the array that will be sorted, you
need to have a sublist for each of the possible digits or letters. If
you are sorting pure English words, you will need at least 26
different sublists, and if you are sorting alphanumeric words or
sentences, you will probably need more than 40 sublists in all!
Since Radix Sort depends on the digits or letters, Radix Sort is
also much less flexible than other sorts. For every different type
of data, Radix Sort needs to be rewritten, and if the sorting order
changes, the sort needs to be rewritten again. In short, Radix Sort
takes more time to write, and it is very difficult to write a general
purpose Radix Sort that can handle all kinds of data.
27. Merge Sort
Merge sort is based on
the divide-and-conquer
paradigm. It consists of
three steps:
Divide: partition input
sequence S into two
sequences S1 and S2 of
about n/2 elements each
Recur: recursively sort S1
and S2
Conquer: merge S1 and S2
into a unique sorted
sequence
Algorithm mergeSort(S, C)
Input sequence S, comparator C
Output sequence S sorted
according to C
if S.size() > 1 {
(S1, S2) := partition(S, S.size()/2)
S1 := mergeSort(S1, C)
S2 := mergeSort(S2, C)
S := merge(S1, S2)
}
return(S)
28. Merge Sort Execution Tree (recursive calls)
An execution of merge-sort is depicted by a binary
tree
each node represents a recursive call of merge-sort and stores
unsorted sequence before the execution and its partition
sorted sequence at the end of the execution
the root is the initial call
the leaves are calls on subsequences of size 0 or 1
7 2 9 4 → 2 4 7 9
7 2 → 2 7
7→ 7
2→ 2
Divide-and-Conquer
9 4 → 4 9
9→ 9
4→ 4
39. Another Analysis of Merge-Sort
The height h of the merge-sort tree is O(log n)
at each recursive call we divide in half the sequence,
The work done at each level is O(n)
At level i, we partition and merge 2i sequences of size n/2i
Thus, the total running time of merge-sort is O(n log n)
depth
#seqs
size
0
1
n
Cost for
level
n
1
2
n/2
n
i
2i
n/2i
n
…
…
…
…
logn
2logn = n n/2logn = 1
n
Divide-and-Conquer
40. Summary of Sorting Algorithms
(so far)
Vectors
Algorithm
Time
Notes
Selection Sort
O(n2)
Slow, in-place
For small data sets
Insertion Sort
O(n2) WC, AC
O(n) BC
Slow, in-place
For small data sets
Heap Sort
O(nlog n)
Fast, in-place
For large data sets
Merge Sort
O(nlogn)
Fast, sequential data access
For huge data sets
42. Quick-Sortrandomized
Quick-sort is a
sorting algorithm based on
the divide-and-conquer
paradigm:
x
Divide: pick a random
element x (called pivot) and
partition S into
L elements less than x
E elements equal x
G elements greater than x
x
L
E
Recur: sort L and G
Conquer: join L, E and G
Divide-and-Conquer
x
G
43. Analysis of Quick Sort using
Recurrence Relations
• Assumption: random pivot
expected to give equal
sized sublists
• The running time of Quick
Sort can be expressed as:
T(n) = 2T(n/2) + P(n)
• T(n) - time to run quicksort()
on an input of size n
• P(n) - time to run partition() on
input of size n
Divide-and-Conquer
Algorithm QuickSort(S, l, r)
Input sequence S, ranks l and r
Output sequence S with the
elements of rank between l and r
rearranged in increasing order
if l ≥ r
return
i ← a random integer between l and r
x ← S.elemAtRank(i)
(h, k) ← Partition(x)
QuickSort(S, l, h − 1)
QuickSort(S, k + 1, r)
44. Quicksort
Efficient sorting algorithm
Discovered by C.A.R. Hoare
Example of Divide and Conquer algorithm
Two phases
Partition phase
Divides the work into half
Sort phase
Conquers the halves!
45. Quicksort
Partition
Choose a pivot
Find the position for the pivot so that
all elements to the left are less
all elements to the right are greater
< pivot
pivot
> pivot
46. Quicksort algorithm to each half
Apply the same
Conquer
< pivot
< p’
p’
> pivot
> p’
pivot
< p”
p”
> p”
47. Quicksort
Implementation
quicksort( void *a, int low, int high )
{
int pivot;
/* Termination condition! */
if ( high > low )
{
pivot = partition( a, low, high );
quicksort( a, low, pivot-1 );
quicksort( a, pivot+1, high );
}
}
Divide
Conquer
48. int partition( int *a, int low, int high ) {
int left, right;
int pivot_item; Partition
pivot_item = a[low];
pivot = left = low;
right = high;
while ( left < right ) {
/* Move left while item < pivot */
while( a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
}
/* right is final position for the pivot */
a[low] = a[right];
a[right] = pivot_item;
return right;
}
Quicksort -
49. This example
uses int’s
to keep things
simple!
int partition( int *a, int low, int high ) {
int left, right;
int pivot_item; Partition
pivot_item = a[low];
pivot = left = low;
right = high;
Any item will do as the pivot,
while ( left < right ) { choose the leftmost one!
/* Move left while item < pivot */
while( a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
}
23 12 15 38 42 18 36 29 27
/* right is final position for the pivot */
a[low] = a[right];
a[right] = pivot_item;
return right;
low
high
}
Quicksort -
50. int partition( int *a, int low, int high ) {
int left, right;
int pivot_item; Partition
pivot_item = a[low];
pivot = left = low;
Set left and right markers
right = high;
while ( left < right ) {
/* Move left while item < pivot */
while( a[left] <= pivot_item ) left++; right
left
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if (23 12 right 38 SWAP(a,left,right);27
left < 15
) 42 18 36 29
}
/* right is final position for the pivot */
a[low]low a[right];
=
high
pivot: 23
a[right] = pivot_item;
return right;
}
Quicksort -
51. int partition( int *a, int low, int high ) {
Quicksort - Partition
int left, right;
int pivot_item;
pivot_item = a[low];
pivot = left = low;
right = high;
Move the markers
until they cross over
while ( left < right ) {
/* Move left while item < pivot */
while( a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
left
}
/* right is final position for the pivot */
a[low] = a[right]; 15 38 42 18 36 29
23 12
a[right] = pivot_item;
return right;
low
pivot: 23
}
right
27
high
52. int partition( int *a, int low, int high ) {
Quicksort - Partition
int left, right;
int pivot_item;
pivot_item = a[low];
pivot = left = low;
right = high;
Move the left pointer while
it points to items <= pivot
while ( left < right ) {
/* Move left while item < pivot */
while( a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
left
right
}
Move right
/* right is final position for the pivot */ similarly
a[low] = a[right];
23 12 15 38 42
a[right] = pivot_item; 18 36 29 27
return right;
}
low
high
pivot: 23
53. int partition( int *a, int low, int high ) {
Quicksort - Partition
int left, right;
int pivot_item;
pivot_item = a[low];
pivot = left = low;
right = high;
Swap the two items
on the wrong side of the pivot
while ( left < right ) {
/* Move left while item < pivot */
while( a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
}
left
right
/* right is final position for the pivot */
a[low] = a[right];
a[right] = pivot_item; 18 36 29 27
23 12 15 38 42
return right;
pivot: 23
}
low
high
54. int partition( int *a, int low, int high ) {
Quicksort - Partition
int left, right;
int pivot_item;
pivot_item = a[low];
pivot = left = low;
right = high;
left and right
have swapped over,
so stop
while ( left < right ) {
/* Move left while item < pivot */
while( a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
}
left
/* right isright
final position for the pivot */
a[low] = a[right];
a[right] = pivot_item; 38 36 29 27
23 12 15 18 42
return right;
}
low
high
pivot: 23
55. int partition( int *a, int low, int high ) {
Quicksort - Partition
int left, right;
int pivot_item;
pivot_item = a[low];
pivot = left = low;
right = high;
right
left
while ( left < right ) {
/* Move left while item < pivot */
23
while( 18 42 38 36 29 27
12 15 a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
low
high
pivot: 23
}
/* right is final position for the pivot */
a[low] = a[right];
Finally, swap the pivot
a[right] = pivot_item;
and right
return right;
}
56. int partition( int *a, int low, int high ) {
Quicksort - Partition
int left, right;
int pivot_item;
pivot_item = a[low];
pivot = left = low;
right = high;
right
while ( left < right ) {
/* Move left while item < pivot */
18
pivot: 23
while( 23 42 38 36 29 27
12 15 a[left] <= pivot_item ) left++;
/* Move right while item > pivot */
while( a[right] >= pivot_item ) right--;
if ( left < right ) SWAP(a,left,right);
low
high
}
/* right is final position for the pivot */
a[low] = a[right];
Return the position
a[right] = pivot_item; of the pivot
return right;
}