The document describes various divide and conquer algorithms including binary search, merge sort, quicksort, and finding maximum and minimum elements. It begins by explaining the general divide and conquer approach of dividing a problem into smaller subproblems, solving the subproblems independently, and combining the solutions. Several examples are then provided with pseudocode and analysis of their divide and conquer implementations. Key algorithms covered in the document include binary search (log n time), merge sort (n log n time), and quicksort (n log n time on average).
The document discusses divide and conquer algorithms. It explains that divide and conquer algorithms follow three steps: 1) divide the problem into subproblems, 2) conquer the subproblems by solving them recursively, and 3) combine the results to solve the original problem. Binary search, merge sort, and quicksort are provided as examples of divide and conquer algorithms. Binary search divides a sorted array in half at each step to search for a target value. Merge sort divides the array in half, recursively sorts the halves, and then merges the sorted halves. Quicksort chooses a pivot to partition the array into left and right halves, recursively sorts the halves, and returns the fully sorted array.
1) The document describes the divide-and-conquer algorithm design paradigm. It can be applied to problems where the input can be divided into smaller subproblems, the subproblems can be solved independently, and the solutions combined to solve the original problem.
2) Binary search is provided as an example divide-and-conquer algorithm. It works by recursively dividing the search space in half and only searching the subspace containing the target value.
3) Finding the maximum and minimum elements in an array is also solved using divide-and-conquer. The array is divided into two halves, the max/min found for each subarray, and the overall max/min determined by comparing the subsolutions.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
Quick sort Algorithm Discussion And AnalysisSNJ Chaudhary
Quicksort is a divide-and-conquer algorithm that works by partitioning an array around a pivot element and recursively sorting the subarrays. In the average case, it has an efficiency of Θ(n log n) time as the partitioning typically divides the array into balanced subproblems. However, in the worst case of an already sorted array, it can be Θ(n^2) time due to highly unbalanced partitioning. Randomizing the choice of pivot helps avoid worst-case scenarios and achieve average-case efficiency in practice, making quicksort very efficient and commonly used.
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
The document discusses divide and conquer algorithms. It explains that divide and conquer algorithms follow three steps: 1) divide the problem into subproblems, 2) conquer the subproblems by solving them recursively, and 3) combine the results to solve the original problem. Binary search, merge sort, and quicksort are provided as examples of divide and conquer algorithms. Binary search divides a sorted array in half at each step to search for a target value. Merge sort divides the array in half, recursively sorts the halves, and then merges the sorted halves. Quicksort chooses a pivot to partition the array into left and right halves, recursively sorts the halves, and returns the fully sorted array.
1) The document describes the divide-and-conquer algorithm design paradigm. It can be applied to problems where the input can be divided into smaller subproblems, the subproblems can be solved independently, and the solutions combined to solve the original problem.
2) Binary search is provided as an example divide-and-conquer algorithm. It works by recursively dividing the search space in half and only searching the subspace containing the target value.
3) Finding the maximum and minimum elements in an array is also solved using divide-and-conquer. The array is divided into two halves, the max/min found for each subarray, and the overall max/min determined by comparing the subsolutions.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
Quick sort Algorithm Discussion And AnalysisSNJ Chaudhary
Quicksort is a divide-and-conquer algorithm that works by partitioning an array around a pivot element and recursively sorting the subarrays. In the average case, it has an efficiency of Θ(n log n) time as the partitioning typically divides the array into balanced subproblems. However, in the worst case of an already sorted array, it can be Θ(n^2) time due to highly unbalanced partitioning. Randomizing the choice of pivot helps avoid worst-case scenarios and achieve average-case efficiency in practice, making quicksort very efficient and commonly used.
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly.
The document discusses various sorting algorithms including insertion sort, selection sort, bubble sort, merge sort, and quick sort. It provides detailed explanations of how each algorithm works through examples using arrays or lists of numbers. The key steps of each algorithm are outlined in pseudocode to demonstrate how they sort a set of data in either ascending or descending order.
The document discusses recursion, which is a method for solving problems by breaking them down into smaller subproblems. It provides examples of recursive algorithms like summing a list of numbers, calculating factorials, and the Fibonacci sequence. It also covers recursive algorithm components like the base case and recursive call. Methods for analyzing recursive algorithms' running times are presented, including iteration, recursion trees, and the master theorem.
Here are the key steps:
1. Guess the solution: T(n) = O(n log n)
2. Set the induction goal: T(n) ≤ c n log n for some c > 0 and n ≥ n0
3. Apply the induction hypothesis: T(n/2) ≤ c (n/2) log(n/2)
4. Substitute into the recurrence: T(n) = 2T(n/2) + n ≤ 2c(n/2)log(n/2) + n = cn log n
5. Simplify and show it meets the induction goal.
Therefore, by mathematical induction, the solution T(n) =
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach and defining the sorting problem. It then describes the three steps of merge sort as divide, conquer, and combine. It provides pseudocode for the merge sort and merge algorithms. Finally, it analyzes the running time of merge sort, showing that it runs in O(n log n) time using the recursion tree method.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
The document introduces algorithms for sorting and searching tasks. It discusses sequential search, binary search, selection sort, bubble sort, merge sort, and quick sort algorithms. For each algorithm, it provides pseudocode to describe the steps, an example, and analysis of time complexity in the best, worst and average cases. The time complexities identified are Θ(n) for sequential search average case, Θ(log n) for binary search, Θ(n2) for selection, bubble and quick sort worst cases, and Θ(n log n) for merge and quick sort average cases.
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.
The document discusses two algorithms for matrix multiplication and finding the median of an unsorted list:
1) Strassen's algorithm improves on the traditional O(n^3) matrix multiplication algorithm by using divide and conquer to achieve O(n^lg7) time complexity.
2) Finding the median can be done in expected O(n) time using quickselect, or deterministically in O(n) time by choosing the median of medians as the pivot.
The document describes the quicksort algorithm. Quicksort works by:
1) Partitioning the array around a pivot element into two sub-arrays of less than or equal and greater than elements.
2) Recursively sorting the two sub-arrays.
3) Combining the now sorted sub-arrays.
In the average case, quicksort runs in O(n log n) time due to balanced partitions at each recursion level. However, in the worst case of an already sorted input, it runs in O(n^2) time due to highly unbalanced partitions. A randomized version of quicksort chooses pivots randomly to avoid worst case behavior.
Quicksort AlgorithmQuicksort is a divide and conquer algorithm. Q.pdfanupamfootwear
Quicksort Algorithm:
Quicksort is a divide and conquer algorithm. Quicksort first divides a large array into two
smaller sub-arrays: the low elements and the high elements. Quicksort can then recursively sort
the sub-arrays.
The steps are:
The base case of the recursion is arrays of size zero or one, which never need to be sorted.
The pivot selection and partitioning steps can be done in several different ways; the choice of
specific implementation schemes greatly affects the algorithm\'s performance.
This algorithm is based on Divide and Conquer paradigm. It is implemented using merge sort. In
this approach the time complexity will be O(n log(n)) . Actually in divide step we divide the
problem in two parts. And then two parts are solved recursively. The key concept is two count
the number of inversion in merge procedure. In merge procedure we pass two sub-list. The
element is sorted and inversion is found as follows
a)Divide : Divide the array in two parts a[0] to a[n/2] and a[n/2+1] to a[n].
b)Conquer : Conquer the sub-problem by solving them recursively.
1) Set count=0,0,i=left,j=mid. C is the sorted list.
2) Traverse list1 and list2 until mid element or right element is encountered .
3) Compare list1[i] and list[j].
i) If list1[i]<=list2[j]
c[k++]=list1[i++]
else
c[k++]=list2[j++]
count = count + mid-i;
4) add rest elements of list1 and list2 in c.
5) copy sorted list c back to original list.
6) return count.
void quickSort(int arr[], int left, int right) {
int i = left, j = right;
int tmp;
int pivot = arr[(left + right) / 2];
/* partition */
while (i <= j) {
while (arr[i] < pivot)
i++;
while (arr[j] > pivot)
j--;
if (i <= j) {
tmp = arr[i];
arr[i] = arr[j];
arr[j] = tmp;
i++;
j--;
}
};
/* recursion */
if (left < j)
quickSort(arr, left, j);
if (i < right)
quickSort(arr, i, right);
}
Solution
Quicksort Algorithm:
Quicksort is a divide and conquer algorithm. Quicksort first divides a large array into two
smaller sub-arrays: the low elements and the high elements. Quicksort can then recursively sort
the sub-arrays.
The steps are:
The base case of the recursion is arrays of size zero or one, which never need to be sorted.
The pivot selection and partitioning steps can be done in several different ways; the choice of
specific implementation schemes greatly affects the algorithm\'s performance.
This algorithm is based on Divide and Conquer paradigm. It is implemented using merge sort. In
this approach the time complexity will be O(n log(n)) . Actually in divide step we divide the
problem in two parts. And then two parts are solved recursively. The key concept is two count
the number of inversion in merge procedure. In merge procedure we pass two sub-list. The
element is sorted and inversion is found as follows
a)Divide : Divide the array in two parts a[0] to a[n/2] and a[n/2+1] to a[n].
b)Conquer : Conquer the sub-problem by solving them recursively.
1) Set count=0,0,i=left,j=mid. C is the sorted list.
2) Traverse list1 and list2 until mid elem.
Quicksort is a divide and conquer sorting algorithm that works by partitioning an array around a pivot value and recursively sorting the subarrays. In the best case when the array is partitioned evenly, quicksort runs in O(n log n) time as the array is cut in half at each recursive call. However, in the worst case when the array is already sorted, each partition only cuts off one element, resulting in O(n^2) time as the recursion reaches a depth of n. Choosing a better pivot value can improve quicksort's performance on poorly sorted arrays.
The document discusses algorithms and their analysis. It begins by defining an algorithm and key aspects like correctness, input, and output. It then discusses two aspects of algorithm performance - time and space. Examples are provided to illustrate how to analyze the time complexity of different structures like if/else statements, simple loops, and nested loops. Big O notation is introduced to describe an algorithm's growth rate. Common time complexities like constant, linear, quadratic, and cubic functions are defined. Specific sorting algorithms like insertion sort, selection sort, bubble sort, merge sort, and quicksort are then covered in detail with examples of how they work and their time complexities.
Divide and conquer is a general algorithm design paradigm where a problem is divided into subproblems, the subproblems are solved independently, and the results are combined to solve the original problem. Binary search is a divide and conquer algorithm that searches for a target value in a sorted array by repeatedly dividing the search interval in half. It compares the target to the middle element of the array, and then searches either the upper or lower half depending on whether the target is greater or less than the middle element. Finding the maximum and minimum elements in an array can also be solved using divide and conquer by recursively finding the max and min of halves of the array and combining the results.
The document discusses the divide and conquer algorithm design strategy. It begins by explaining the general concept of divide and conquer, which involves splitting a problem into subproblems, solving the subproblems, and combining the solutions. It then provides pseudocode for a generic divide and conquer algorithm. Finally, it gives examples of divide and conquer algorithms like quicksort, binary search, and matrix multiplication.
Analysis and design of algorithms part2Deepak John
Analysis of searching and sorting. Insertion sort, Quick sort, Merge sort and Heap sort. Binomial Heaps and Fibonacci Heaps, Lower bounds for sorting by comparison of keys. Comparison of sorting algorithms. Amortized Time Analysis. Red-Black Trees – Insertion & Deletion.
This document discusses the divide and conquer algorithm design strategy and provides an analysis of the merge sort algorithm as an example. It begins by explaining the divide and conquer strategy of dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions. It then provides pseudocode and explanations for the merge sort algorithm, which divides an array in half, recursively sorts the halves, and then merges the sorted halves back together. It analyzes the time complexity of merge sort as Θ(n log n), proving it is more efficient than insertion sort.
The document discusses various sorting algorithms and their time complexities:
1. Comparison sorts like merge sort and quicksort have a best case time complexity of O(n log n).
2. Counting sort runs in O(n+k) time where k is the range of input values, and is not a comparison sort.
3. Radix sort treats input as d-digit numbers in some base k and uses counting sort to sort on each digit, achieving O(dn+dk) time which is O(n) when d and k are constants.
4. A randomized selection algorithm finds the ith order statistic in expected O(n) time using randomized partition.
This document discusses the divide and conquer algorithm called merge sort. It begins by explaining the general divide and conquer approach of dividing a problem into subproblems, solving the subproblems recursively, and then combining the solutions. It then provides an example of how merge sort uses this approach to sort a sequence. It walks through the recursive merge sort algorithm on a sample input. The document explains the merge procedure used to combine the sorted subproblems and proves its correctness. It analyzes the running time of merge sort using recursion trees and determines it is O(n log n). Finally, it introduces recurrence relations and methods like substitution, recursion trees, and the master theorem for solving recurrences.
The document discusses different sorting algorithms including merge sort and quicksort. Merge sort has a divide and conquer approach where an array is divided into halves and the halves are merged back together in sorted order. This results in a runtime of O(n log n). Quicksort uses a partitioning approach, choosing a pivot element and partitioning the array into subarrays of elements less than or greater than the pivot. In the best case, this partitions the array in half at each step, resulting in a runtime of O(n log n). In the average case, the runtime is also O(n log n). In the worst case, the array is already sorted, resulting in unbalanced partitions and a quadratic runtime of O(n^2
The document provides an introduction to unsupervised learning and reinforcement learning. It then discusses eigen values and eigen vectors, showing how to calculate them from a matrix. It provides examples of covariance matrices and using Gaussian elimination to solve for eigen vectors. Finally, it discusses principal component analysis and different clustering algorithms like K-means clustering.
Cross validation is a technique for evaluating machine learning models by splitting the dataset into training and validation sets and training the model multiple times on different splits, to reduce variance. K-fold cross validation splits the data into k equally sized folds, where each fold is used once for validation while the remaining k-1 folds are used for training. Leave-one-out cross validation uses a single observation from the dataset as the validation set. Stratified k-fold cross validation ensures each fold has the same class proportions as the full dataset. Grid search evaluates all combinations of hyperparameters specified as a grid, while randomized search samples hyperparameters randomly within specified ranges. Learning curves show training and validation performance as a function of training set size and can diagnose underfitting
In computer science, divide and conquer is an algorithm design paradigm based on multi-branched recursion. A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly.
The document discusses various sorting algorithms including insertion sort, selection sort, bubble sort, merge sort, and quick sort. It provides detailed explanations of how each algorithm works through examples using arrays or lists of numbers. The key steps of each algorithm are outlined in pseudocode to demonstrate how they sort a set of data in either ascending or descending order.
The document discusses recursion, which is a method for solving problems by breaking them down into smaller subproblems. It provides examples of recursive algorithms like summing a list of numbers, calculating factorials, and the Fibonacci sequence. It also covers recursive algorithm components like the base case and recursive call. Methods for analyzing recursive algorithms' running times are presented, including iteration, recursion trees, and the master theorem.
Here are the key steps:
1. Guess the solution: T(n) = O(n log n)
2. Set the induction goal: T(n) ≤ c n log n for some c > 0 and n ≥ n0
3. Apply the induction hypothesis: T(n/2) ≤ c (n/2) log(n/2)
4. Substitute into the recurrence: T(n) = 2T(n/2) + n ≤ 2c(n/2)log(n/2) + n = cn log n
5. Simplify and show it meets the induction goal.
Therefore, by mathematical induction, the solution T(n) =
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach and defining the sorting problem. It then describes the three steps of merge sort as divide, conquer, and combine. It provides pseudocode for the merge sort and merge algorithms. Finally, it analyzes the running time of merge sort, showing that it runs in O(n log n) time using the recursion tree method.
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
The document introduces algorithms for sorting and searching tasks. It discusses sequential search, binary search, selection sort, bubble sort, merge sort, and quick sort algorithms. For each algorithm, it provides pseudocode to describe the steps, an example, and analysis of time complexity in the best, worst and average cases. The time complexities identified are Θ(n) for sequential search average case, Θ(log n) for binary search, Θ(n2) for selection, bubble and quick sort worst cases, and Θ(n log n) for merge and quick sort average cases.
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.
The document discusses two algorithms for matrix multiplication and finding the median of an unsorted list:
1) Strassen's algorithm improves on the traditional O(n^3) matrix multiplication algorithm by using divide and conquer to achieve O(n^lg7) time complexity.
2) Finding the median can be done in expected O(n) time using quickselect, or deterministically in O(n) time by choosing the median of medians as the pivot.
The document describes the quicksort algorithm. Quicksort works by:
1) Partitioning the array around a pivot element into two sub-arrays of less than or equal and greater than elements.
2) Recursively sorting the two sub-arrays.
3) Combining the now sorted sub-arrays.
In the average case, quicksort runs in O(n log n) time due to balanced partitions at each recursion level. However, in the worst case of an already sorted input, it runs in O(n^2) time due to highly unbalanced partitions. A randomized version of quicksort chooses pivots randomly to avoid worst case behavior.
Quicksort AlgorithmQuicksort is a divide and conquer algorithm. Q.pdfanupamfootwear
Quicksort Algorithm:
Quicksort is a divide and conquer algorithm. Quicksort first divides a large array into two
smaller sub-arrays: the low elements and the high elements. Quicksort can then recursively sort
the sub-arrays.
The steps are:
The base case of the recursion is arrays of size zero or one, which never need to be sorted.
The pivot selection and partitioning steps can be done in several different ways; the choice of
specific implementation schemes greatly affects the algorithm\'s performance.
This algorithm is based on Divide and Conquer paradigm. It is implemented using merge sort. In
this approach the time complexity will be O(n log(n)) . Actually in divide step we divide the
problem in two parts. And then two parts are solved recursively. The key concept is two count
the number of inversion in merge procedure. In merge procedure we pass two sub-list. The
element is sorted and inversion is found as follows
a)Divide : Divide the array in two parts a[0] to a[n/2] and a[n/2+1] to a[n].
b)Conquer : Conquer the sub-problem by solving them recursively.
1) Set count=0,0,i=left,j=mid. C is the sorted list.
2) Traverse list1 and list2 until mid element or right element is encountered .
3) Compare list1[i] and list[j].
i) If list1[i]<=list2[j]
c[k++]=list1[i++]
else
c[k++]=list2[j++]
count = count + mid-i;
4) add rest elements of list1 and list2 in c.
5) copy sorted list c back to original list.
6) return count.
void quickSort(int arr[], int left, int right) {
int i = left, j = right;
int tmp;
int pivot = arr[(left + right) / 2];
/* partition */
while (i <= j) {
while (arr[i] < pivot)
i++;
while (arr[j] > pivot)
j--;
if (i <= j) {
tmp = arr[i];
arr[i] = arr[j];
arr[j] = tmp;
i++;
j--;
}
};
/* recursion */
if (left < j)
quickSort(arr, left, j);
if (i < right)
quickSort(arr, i, right);
}
Solution
Quicksort Algorithm:
Quicksort is a divide and conquer algorithm. Quicksort first divides a large array into two
smaller sub-arrays: the low elements and the high elements. Quicksort can then recursively sort
the sub-arrays.
The steps are:
The base case of the recursion is arrays of size zero or one, which never need to be sorted.
The pivot selection and partitioning steps can be done in several different ways; the choice of
specific implementation schemes greatly affects the algorithm\'s performance.
This algorithm is based on Divide and Conquer paradigm. It is implemented using merge sort. In
this approach the time complexity will be O(n log(n)) . Actually in divide step we divide the
problem in two parts. And then two parts are solved recursively. The key concept is two count
the number of inversion in merge procedure. In merge procedure we pass two sub-list. The
element is sorted and inversion is found as follows
a)Divide : Divide the array in two parts a[0] to a[n/2] and a[n/2+1] to a[n].
b)Conquer : Conquer the sub-problem by solving them recursively.
1) Set count=0,0,i=left,j=mid. C is the sorted list.
2) Traverse list1 and list2 until mid elem.
Quicksort is a divide and conquer sorting algorithm that works by partitioning an array around a pivot value and recursively sorting the subarrays. In the best case when the array is partitioned evenly, quicksort runs in O(n log n) time as the array is cut in half at each recursive call. However, in the worst case when the array is already sorted, each partition only cuts off one element, resulting in O(n^2) time as the recursion reaches a depth of n. Choosing a better pivot value can improve quicksort's performance on poorly sorted arrays.
The document discusses algorithms and their analysis. It begins by defining an algorithm and key aspects like correctness, input, and output. It then discusses two aspects of algorithm performance - time and space. Examples are provided to illustrate how to analyze the time complexity of different structures like if/else statements, simple loops, and nested loops. Big O notation is introduced to describe an algorithm's growth rate. Common time complexities like constant, linear, quadratic, and cubic functions are defined. Specific sorting algorithms like insertion sort, selection sort, bubble sort, merge sort, and quicksort are then covered in detail with examples of how they work and their time complexities.
Divide and conquer is a general algorithm design paradigm where a problem is divided into subproblems, the subproblems are solved independently, and the results are combined to solve the original problem. Binary search is a divide and conquer algorithm that searches for a target value in a sorted array by repeatedly dividing the search interval in half. It compares the target to the middle element of the array, and then searches either the upper or lower half depending on whether the target is greater or less than the middle element. Finding the maximum and minimum elements in an array can also be solved using divide and conquer by recursively finding the max and min of halves of the array and combining the results.
The document discusses the divide and conquer algorithm design strategy. It begins by explaining the general concept of divide and conquer, which involves splitting a problem into subproblems, solving the subproblems, and combining the solutions. It then provides pseudocode for a generic divide and conquer algorithm. Finally, it gives examples of divide and conquer algorithms like quicksort, binary search, and matrix multiplication.
Analysis and design of algorithms part2Deepak John
Analysis of searching and sorting. Insertion sort, Quick sort, Merge sort and Heap sort. Binomial Heaps and Fibonacci Heaps, Lower bounds for sorting by comparison of keys. Comparison of sorting algorithms. Amortized Time Analysis. Red-Black Trees – Insertion & Deletion.
This document discusses the divide and conquer algorithm design strategy and provides an analysis of the merge sort algorithm as an example. It begins by explaining the divide and conquer strategy of dividing a problem into smaller subproblems, solving those subproblems recursively, and combining the solutions. It then provides pseudocode and explanations for the merge sort algorithm, which divides an array in half, recursively sorts the halves, and then merges the sorted halves back together. It analyzes the time complexity of merge sort as Θ(n log n), proving it is more efficient than insertion sort.
The document discusses various sorting algorithms and their time complexities:
1. Comparison sorts like merge sort and quicksort have a best case time complexity of O(n log n).
2. Counting sort runs in O(n+k) time where k is the range of input values, and is not a comparison sort.
3. Radix sort treats input as d-digit numbers in some base k and uses counting sort to sort on each digit, achieving O(dn+dk) time which is O(n) when d and k are constants.
4. A randomized selection algorithm finds the ith order statistic in expected O(n) time using randomized partition.
This document discusses the divide and conquer algorithm called merge sort. It begins by explaining the general divide and conquer approach of dividing a problem into subproblems, solving the subproblems recursively, and then combining the solutions. It then provides an example of how merge sort uses this approach to sort a sequence. It walks through the recursive merge sort algorithm on a sample input. The document explains the merge procedure used to combine the sorted subproblems and proves its correctness. It analyzes the running time of merge sort using recursion trees and determines it is O(n log n). Finally, it introduces recurrence relations and methods like substitution, recursion trees, and the master theorem for solving recurrences.
The document discusses different sorting algorithms including merge sort and quicksort. Merge sort has a divide and conquer approach where an array is divided into halves and the halves are merged back together in sorted order. This results in a runtime of O(n log n). Quicksort uses a partitioning approach, choosing a pivot element and partitioning the array into subarrays of elements less than or greater than the pivot. In the best case, this partitions the array in half at each step, resulting in a runtime of O(n log n). In the average case, the runtime is also O(n log n). In the worst case, the array is already sorted, resulting in unbalanced partitions and a quadratic runtime of O(n^2
The document provides an introduction to unsupervised learning and reinforcement learning. It then discusses eigen values and eigen vectors, showing how to calculate them from a matrix. It provides examples of covariance matrices and using Gaussian elimination to solve for eigen vectors. Finally, it discusses principal component analysis and different clustering algorithms like K-means clustering.
Cross validation is a technique for evaluating machine learning models by splitting the dataset into training and validation sets and training the model multiple times on different splits, to reduce variance. K-fold cross validation splits the data into k equally sized folds, where each fold is used once for validation while the remaining k-1 folds are used for training. Leave-one-out cross validation uses a single observation from the dataset as the validation set. Stratified k-fold cross validation ensures each fold has the same class proportions as the full dataset. Grid search evaluates all combinations of hyperparameters specified as a grid, while randomized search samples hyperparameters randomly within specified ranges. Learning curves show training and validation performance as a function of training set size and can diagnose underfitting
This document provides an overview of supervised machine learning algorithms for classification, including logistic regression, k-nearest neighbors (KNN), support vector machines (SVM), and decision trees. It discusses key concepts like evaluation metrics, performance measures, and use cases. For logistic regression, it covers the mathematics behind maximum likelihood estimation and gradient descent. For KNN, it explains the algorithm and discusses distance metrics and a numerical example. For SVM, it outlines the concept of finding the optimal hyperplane that maximizes the margin between classes.
The document provides information on solving the sum of subsets problem using backtracking. It discusses two formulations - one where solutions are represented by tuples indicating which numbers are included, and another where each position indicates if the corresponding number is included or not. It shows the state space tree that represents all possible solutions for each formulation. The tree is traversed depth-first to find all solutions where the sum of the included numbers equals the target sum. Pruning techniques are used to avoid exploring non-promising paths.
The document discusses the greedy method and its applications. It begins by defining the greedy approach for optimization problems, noting that greedy algorithms make locally optimal choices at each step in hopes of finding a global optimum. Some applications of the greedy method include the knapsack problem, minimum spanning trees using Kruskal's and Prim's algorithms, job sequencing with deadlines, and finding the shortest path using Dijkstra's algorithm. The document then focuses on explaining the fractional knapsack problem and providing a step-by-step example of solving it using a greedy approach. It also provides examples and explanations of Kruskal's algorithm for finding minimum spanning trees.
What is an Algorithm
Time Complexity
Space Complexity
Asymptotic Notations
Recursive Analysis
Selection Sort
Insertion Sort
Recurrences
Substitution Method
Master Tree Method
Recursion Tree Method
This document provides an outline for a machine learning syllabus. It includes 14 modules covering topics like machine learning terminology, supervised and unsupervised learning algorithms, optimization techniques, and projects. It lists software and hardware requirements for the course. It also discusses machine learning applications, issues, and the steps to build a machine learning model.
The document discusses problem-solving agents and their approach to solving problems. Problem-solving agents (1) formulate a goal based on the current situation, (2) formulate the problem by defining relevant states and actions, and (3) search for a solution by exploring sequences of actions that lead to the goal state. Several examples of problems are provided, including the 8-puzzle, robotic assembly, the 8 queens problem, and the missionaries and cannibals problem. For each problem, the relevant states, actions, goal tests, and path costs are defined.
The simplex method is a linear programming algorithm that can solve problems with more than two decision variables. It works by generating a series of solutions, called tableaus, where each tableau corresponds to a corner point of the feasible solution space. The algorithm starts at the initial tableau, which corresponds to the origin. It then shifts to adjacent corner points, moving in the direction that optimizes the objective function. This process of generating new tableaus continues until an optimal solution is found.
The document discusses functions and the pigeonhole principle. It defines what a function is, how functions can be represented graphically and with tables and ordered pairs. It covers one-to-one, onto, and bijective functions. It also discusses function composition, inverse functions, and the identity function. The pigeonhole principle states that if n objects are put into m containers where n > m, then at least one container must hold more than one object. Examples are given to illustrate how to apply the principle to problems involving months, socks, and selecting numbers.
The document discusses relations and their representations. It defines a binary relation as a subset of A×B where A and B are nonempty sets. Relations can be represented using arrow diagrams, directed graphs, and zero-one matrices. A directed graph represents the elements of A as vertices and draws an edge from vertex a to b if aRb. The zero-one matrix representation assigns 1 to the entry in row a and column b if (a,b) is in the relation, and 0 otherwise. The document also discusses indegrees, outdegrees, composite relations, and properties of relations like reflexivity.
This document discusses logic and propositional logic. It covers the following topics:
- The history and applications of logic.
- Different types of statements and their grammar.
- Propositional logic including symbols, connectives, truth tables, and semantics.
- Quantifiers, universal and existential quantification, and properties of quantifiers.
- Normal forms such as disjunctive normal form and conjunctive normal form.
- Inference rules and the principle of mathematical induction, illustrated with examples.
1. Set theory is an important mathematical concept and tool that is used in many areas including programming, real-world applications, and computer science problems.
2. The document introduces some basic concepts of set theory including sets, members, operations on sets like union and intersection, and relationships between sets like subsets and complements.
3. Infinite sets are discussed as well as different types of infinite sets including countably infinite and uncountably infinite sets. Special sets like the empty set and power sets are also covered.
The document discusses uncertainty and probabilistic reasoning. It describes sources of uncertainty like partial information, unreliable information, and conflicting information from multiple sources. It then discusses representing and reasoning with uncertainty using techniques like default logic, rules with probabilities, and probability theory. The key approaches covered are conditional probability, independence, conditional independence, and using Bayes' rule to update probabilities based on new evidence.
The document outlines the objectives, outcomes, and learning outcomes of a course on artificial intelligence. The objectives include conceptualizing ideas and techniques for intelligent systems, understanding mechanisms of intelligent thought and action, and understanding advanced representation and search techniques. Outcomes include developing an understanding of AI building blocks, choosing appropriate problem solving methods, analyzing strengths and weaknesses of AI approaches, and designing models for reasoning with uncertainty. Learning outcomes include knowledge, intellectual skills, practical skills, and transferable skills in artificial intelligence.
Planning involves representing an initial state, possible actions, and a goal state. A planning agent uses a knowledge base to select action sequences that transform the initial state into a goal state. STRIPS is a common planning representation that uses predicates to describe states and logical operators to represent actions and their effects. A STRIPS planning problem specifies the initial state, goal conditions, and set of operators. A solution is a sequence of ground operator instances that produces the goal state from the initial state.
Online train ticket booking system project.pdfKamal Acharya
Rail transport is one of the important modes of transport in India. Now a days we
see that there are railways that are present for the long as well as short distance
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of the railway database also plays a major role in the smooth running of this
system. The Online Train Ticket Management System will help in reserving the
tickets of the railways to travel from a particular source to the destination.
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Cricket management system ptoject report.pdfKamal Acharya
The aim of this project is to provide the complete information of the National and
International statistics. The information is available country wise and player wise. By
entering the data of eachmatch, we can get all type of reports instantly, which will be
useful to call back history of each player. Also the team performance in each match can
be obtained. We can get a report on number of matches, wins and lost.
Data Communication and Computer Networks Management System Project Report.pdfKamal Acharya
Networking is a telecommunications network that allows computers to exchange data. In
computer networks, networked computing devices pass data to each other along data
connections. Data is transferred in the form of packets. The connections between nodes are
established using either cable media or wireless media.
Sachpazis_Consolidation Settlement Calculation Program-The Python Code and th...Dr.Costas Sachpazis
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This program calculates the consolidation settlement for a foundation based on soil layer properties and foundation data. It allows users to input multiple soil layers and foundation characteristics to determine the total settlement.
Sri Guru Hargobind Ji - Bandi Chor Guru.pdfBalvir Singh
Sri Guru Hargobind Ji (19 June 1595 - 3 March 1644) is revered as the Sixth Nanak.
• On 25 May 1606 Guru Arjan nominated his son Sri Hargobind Ji as his successor. Shortly
afterwards, Guru Arjan was arrested, tortured and killed by order of the Mogul Emperor
Jahangir.
• Guru Hargobind's succession ceremony took place on 24 June 1606. He was barely
eleven years old when he became 6th Guru.
• As ordered by Guru Arjan Dev Ji, he put on two swords, one indicated his spiritual
authority (PIRI) and the other, his temporal authority (MIRI). He thus for the first time
initiated military tradition in the Sikh faith to resist religious persecution, protect
people’s freedom and independence to practice religion by choice. He transformed
Sikhs to be Saints and Soldier.
• He had a long tenure as Guru, lasting 37 years, 9 months and 3 days
2. The General Method
1. DIVIDE: Reduce problem instance to smaller instance of the same
problem
2. CONQUER: Solve smaller instance recursively, independently
3. COMBINE: Extend solution of smaller instances to obtain solution
to original instance
Shiwani Gupta 2
3. Applications of D & C Appoach
• Binary Search
• Max Min Problem
• Merge Sort
• Quick Sort
• Strassen’s Matrix Multiplication
• Problem of multiplying long integers
• Constructing tennis tournament
Shiwani Gupta 3
4. Sequential Search
Algorithm SequentialSearch(A[0…n-1], K)
//Problem Description: Searches for a given value in a given array by
// sequential search
//Input: An Array A[0…n-1] and a search key K
//Output: Returns the index of the first element of A that matches K
// or -1 if there are no matching elements
i 0
while i < n and A[i] K do
i i+1
if i < n return i
else return -1
1. Worst case: O(n)
2. Best case: O(1)
3. Average case: O(n/2)
Thus, we say sequential search is O(n)
Shiwani Gupta 4
5. Binary Search
Search requires the following steps:
1. Inspect the middle item of an array of size N.
2. Inspect the middle of an array of size N/2.
3. Inspect the middle item of an array of size N/power(2,2) and so on
until N/power(2,k) = 1.
– This implies k = log2N
– k is the number of partitions.
• Requires that the array be sorted.
• Rather than start at either end, binary search splits the array in
half and works only with the half that may contain the value.
• This action of dividing continues until the desired value is found
or the remaining values are either smaller or larger than the
search value.
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6. Binary Search: pseudo-code (ITERATIVE)
binary_search(a, x)
left ← 0
right ← N-1
while(left <= right)
mid= (left+right)/2
if (a[mid] > x)
right ← mid – 1
else if (x > a[mid])
low ← mid + 1
else
return mid //found
return -1 //not found
Shiwani Gupta 6
7. Binary Search: pseudo-code (RECURSIVE)
binary_search(a, x, left, right)
if (right < left)
return -1 // not found
mid= floor((left+right)/2) //floor(.)=(int)(.)
if (a[mid] == x)
return mid
else if (x < a[mid])
binary_search(a, x, left, mid-1)
else
binary_search(a, x, mid+1, right)
Binary Search: Time Complexity (for successful search)
Number of comparisons:
T(n) = T(n/2) + 1, if n>1;
= 1, if n=1
which solves to T(n) = Ѳ(logn) with Case 2 of Master Method. 7
8. Search algorithm efficiency
• Sequential search: (n)
• Binary search: (log2n)
– no more than 10 iterations, to find an element in a sorted
list of 1000 elements
– no more than 20 iterations, for a sorted list of one
million!
Shiwani Gupta 8
9. Finding Max Min
StraightMaxMin(a, n, max, min)
max = min = a[1]
for i = 2 to n do
if (a[i]>max) then max = a[i] //n-1 comparisons
if (a[i]<min) then min = a[i] //n-1 comparisons
Total = 2*(n-1) comparisons
1. Worst case: O(n)
2. Best case: O(n)
3. Average case: O(n)
Thus, we say Straight Max Min is O(n).
Shiwani Gupta 9
10. MaxMin(i,j,max,min) //1<=i<=j<=n
if ( i == j) then max = min = a[i] //single element
else if (i == j -1) then //two elements
if (a[i] < a[j]) then
max = a[j]
min = a[i]
else
max = a[i]
min = a[j]
else //divide P into subproblems
mid = (i+j)/2 //find split
MaxMin(i, mid, max, min) //solve subproblems
MaxMin(mid+1, j, max1, min1) //solve subproblems
if (max<max1) then max = max1 //combine solutions
if (min>min1) then min = min1 //combine solutions
The recurrence for this algorithm is
T(n) = 2T(n/2) + 2 n>2 when
max and min are considered as different procedures
T(n) = 1 n=2
T(n) = 0 n=1
This solves to T(n) = 3n/2-2.
Finding Max Min
10
12. Merge Sort
Sorting Problem: Sort a sequence of n elements into non-
decreasing order.
• Divide: Divide the n-element sequence to be sorted into
two subsequences of n/2 elements each
• Conquer: Sort the two subsequences recursively using
merge sort.
• Combine: Merge the two sorted subsequences to produce
the sorted answer.
Shiwani Gupta 12
14. Merge-Sort (A, p, r)
INPUT: a sequence of n numbers stored in array A
OUTPUT: an ordered sequence of n numbers
MergeSort (A, p, r) // p as beginning pointer of array and r as end pointer
1 if p < r
2 then q (p+r)/2 //split the list at mid
3 MergeSort (A, p, q) //first sublist
4 MergeSort (A, q+1, r) //second sublist
5 Merge (A, p, q, r) // merges A[p..q] with A[q+1..r]
Initial Call: MergeSort(A, 1, n)
Shiwani Gupta 15
15. Procedure Merge
Merge(A, p, q, r)
1 n1 q – p + 1
2 n2 r – q
3 for i 1 to n1
4 do L[i] A[p + i – 1]
5 for j 1 to n2
6 do R[j] A[q + j]
7 L[n1+1]
8 R[n2+1]
9 i 1
10 j 1
11 for k p to r
12 do if L[i] R[j]
13 then A[k] L[i]
14 i i + 1
15 else A[k] R[j]
16 j j + 1
Sentinels, to avoid having to
check if either subarray is
fully copied at each step.
Input: Array containing sorted
subarrays A[p..q] and
A[q+1..r].
Output: Merged sorted
subarray in A[p..r].
Shiwani Gupta 16
16. j
Merge – Example
6 8 26 32 1 9 42 43
… …
A
k
6 8 26 32 1 9 42 43
k k k k k k k
i i i i
i j j j j
6 8 26 32 1 9 42 43
1 6 8 9 26 32 42 43
k
L R
Shiwani Gupta 17
17. Analysis of Merge Sort – Master’s Method
• Running time T(n) of Merge Sort:
• Divide: computing the middle takes (1)
• Conquer: solving 2 subproblems takes 2T(n/2)
• Combine: merging n elements takes (n)
• Total:
T(n) = (1) if n = 1
T(n) = 2T(n/2) + (n) if n > 1
T(n) = (n lg n) case 2
Shiwani Gupta 18
18. • Running time of Merge Sort:
T(n) = (1) if n = 1
T(n) = 2T(n/2) + (n) if n > 1
• Rewrite the recurrence as
T(n) = c if n = 1
T(n) = 2T(n/2) + cn if n > 1
c > 0: Running time for the base case and time per array
element for the divide and combine steps.
Analysis of Merge Sort – Recursion Tree
Method
Shiwani Gupta 19
19. Recursion Tree for Merge Sort
For the original problem,
we have a cost of cn,
plus two subproblems
each of size (n/2) and
running time T(n/2).
cn
T(n/2) T(n/2)
Each of the size n/2 problems
has a cost of cn/2 plus two
subproblems, each costing
T(n/4).
cn
cn/2 cn/2
T(n/4) T(n/4) T(n/4) T(n/4)
Cost of divide and
merge.
Cost of sorting
subproblems.
Shiwani Gupta 20
20. Recursion Tree for Merge Sort
Continue expanding until the problem size reduces to 1.
cn
cn/2 cn/2
cn/4 cn/4 cn/4 cn/4
c c c c
c c
lg n + 1
cn
cn
cn
cn
Total : cnlgn+cn
Shiwani Gupta 21
21. Recursion Tree for Merge Sort
Continue expanding until the problem size reduces to 1.
cn
cn/2 cn/2
cn/4 cn/4 cn/4 cn/4
c c c c
c c
•Each level has total cost cn.
•Each time we go down one level, the
number of subproblems doubles, but the
cost per subproblem halves cost per
level remains the same.
•There are lg n + 1 levels, height is lg n.
(Assuming n is a power of 2.)
• Can be proved by induction.
•Total cost = sum of costs at each level
= (lg n + 1)cn = cnlgn + cn = (n lgn).
Shiwani Gupta 22
22. Quicksort
• Divide: Partition array A[l..r] into 2 subarrays, A[l..s-1] and
A[s+1..r] such that each element of the first array is ≤ A[s]
and each element of the second array is ≥ A[s]. (computing
the index of s is part of partition.)
– Implication: A[s] will be in its final position in the sorted
array.
• Conquer: Sort the two subarrays A[l..s-1] and A[s+1..r] by
recursive calls to Quicksort
• Combine: No work is needed, because A[s] is already in its
correct place after the partition is done, and the two
subarrays have been sorted.
Shiwani Gupta 23
23. The Quicksort Algorithm
ALGORITHM Quicksort(A[l..r])
//Problem Description: Sorts a subarray by quicksort
//Input: A subarray A[l..r] of A[0..n-1],defined by its left and right
indices l and r
//Output: The subarray A[l..r] sorted in nondecreasing order
if l < r
s Partition (A[l..r]) // s is a split position
Quicksort(A[l..s-1])
Quicksort(A[s+1..r]
Shiwani Gupta 24
24. Partitioning Algorithm
Algorithm Partition(A[l, r])
p A[l ]
i l; j r+1
repeat
repeat i←i+1 until A[i]>p
repeat j←j-1 until A[i]<=p
swap(A[i],A[j])
until i>j
swap(p,A[j])
Shiwani Gupta 25
25. Example
We are given array of n integers to sort:
40 20 10 80 60 50 7 30 100
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26. Pick Pivot Element
There are a number of ways to pick the pivot element. In this example,
we will use the first element in the array:
40 20 10 80 60 50 7 30 100
Shiwani Gupta 27
31. 40 20 10 80 60 50 7 30 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
p = 0
Shiwani Gupta 32
32. 40 20 10 80 60 50 7 30 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
p = 0
Shiwani Gupta 33
33. 40 20 10 80 60 50 7 30 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
p = 0
Shiwani Gupta 34
34. 40 20 10 30 60 50 7 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
p = 0
Shiwani Gupta 35
35. 40 20 10 30 60 50 7 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
p = 0
Shiwani Gupta 36
36. 40 20 10 30 60 50 7 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
p = 0
Shiwani Gupta 37
37. 40 20 10 30 60 50 7 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
p = 0
Shiwani Gupta 38
38. 40 20 10 30 60 50 7 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
p = 0
Shiwani Gupta 39
39. 40 20 10 30 60 50 7 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
p = 0
Shiwani Gupta 40
40. 40 20 10 30 60 50 7 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
p = 0
Shiwani Gupta 41
41. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
p = 0
Shiwani Gupta 42
42. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
p = 0
Shiwani Gupta 43
43. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
p = 0
Shiwani Gupta 44
44. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
p = 0
Shiwani Gupta 45
45. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
p = 0
Shiwani Gupta 46
46. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
p = 0
Shiwani Gupta 47
47. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
p = 0
Shiwani Gupta 48
48. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 49
49. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
40 20 10 30 7 50 60 80 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 50
50. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
5. swap data[j] and data[p]
40 20 10 30 7 50 60 80 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 51
51. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
5. swap data[j] and data[p]
7 20 10 30 40 50 60 80 100
p = 4
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 52
52. 7 20 10 30 40 50 60 80 100
[0] [1] [2] [3] [4] [5] [6] [7] [8]
<= data[p] > data[p]
Partition Result
Best Case
Shiwani Gupta 53
53. Quicksort: Worst Case
• Assume first element is chosen as p.
• Assume we get array that is already in order:
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 54
54. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
5. swap data[j] and data[p]
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 55
55. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
5. swap data[j] and data[p]
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 56
56. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
5. swap data[j] and data[p]
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 57
57. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
5. swap data[j] and data[p]
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 58
58. 1. while data[i] <= data[p]
++i
2. while data[j] > data[p]
--j
3. if i < j
swap data[i] and data[j]
4. while j > i, go to 1.
5. swap data[j] and data[p]
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 59
59. 1. While data[i] <= data[p]
++i
2. While data[j] > data[p]
--j
3. If i < j
swap data[i] and data[j]
4. While j > i, go to 1.
5. Swap data[j] and data[p]
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
i j
Shiwani Gupta 60
60. 1. While data[i] <= data[p]
++i
2. While data[j] > data[p]
--j
3. If i < j
swap data[i] and data[j]
4. While j > i, go to 1.
5. Swap data[j] and data[p]
2 4 10 12 13 50 57 63 100
p = 0
[0] [1] [2] [3] [4] [5] [6] [7] [8]
> data[p]
<= data[p]
Shiwani Gupta 61
61. Efficiency of Quicksort
Based on whether the partitioning is balanced.
• Best case: split in the middle — Θ( n log n)
– T(n) = 2T(n/2) + Θ(n) , n>1 //2 subproblems of size n/2 each
– T(1)=0
• Worst case: sorted array — Θ( n2)
– T(n) = T(n-1) + T(0) + Θ(n) //2 subproblems of size 0 and n-1
– T(0)= 0
• Average case: random arrays — Θ( n log n) //of size 9:1 or 99:1
62
64. Solving Average case recurrence
log4/3 n levels = log2 n / log2 4/3 = log n
T(n) = nlogn
65. Improve Efficiency of Quick Sort
• Use mid element as pivot
• Take the median as pivot
• Take mean of first, last and middle element as pivot
• Use random element as pivot
– Running time is independent of input ordering
– Worst case determined only by output of random-number generator
Shiwani Gupta 66
66. What is divide and conquer method?
Show that algorithm for finding
Minimum and Maximum using Divide
and Conquer does not use more than 3n/2
comparisons
Implement recursive Binary Search algorithm. Write
complete code along with recursive function call
Implement the Binary Search, prove that the
complexity of binary search is O(log2n)
Explain all sorting techniques based on divide and conquer strategy
Sort the given elements using Merge Sort
technique. 90, 20 80, 89, 70, 65, 85, 74.
Show passes both for divide and combine
Solve Merge sort recurrence using
recursion tree method
Give the analysis of Merge Sort using
divide and conquer?
Write Merge sort algorithm and analyze it
Sort following data using Merge Sort 9,
8, 7, 6, 5, 4, 3, 2, 1, 0
Give the analysis of Merge Sort using
Divide and Conquer?
Sort the given elements using Merge Sort
technique. 90, 20 80, 89, 70, 65, 85, 74
Show how Quick sort can be made to run in
O(nlogn) time in worst case
Analyze worst and best case complexity for Quick
sort
Explain analysis of Quick Sort algorithm using
recursion
Sort following data using Quick Sort 9, 8, 7, 6, 5, 4,
3, 2, 1, 0
Sort the given elements using Quick Sort technique.
90, 20 80, 89, 70,65, 85, 74
Prove that worst case complexity of Quick Sort is
O(n2)
Analyze worst and best case complexity for Quick
Sort
Task