Merge sort implementation
•Download as PPTX, PDF•
0 likes•25 views
this slide describes one of the most popular sorting algorithm- merge sort algorithm and how it works.
Report
Share
Report
Share
Recommended
Multiplying & dividing rational algebraic expressions
This document discusses how to multiply and divide rational algebraic expressions. It explains that to multiply rational expressions, one multiplies the numerators and denominators separately. Rational expressions must first be factored to cancel common factors before multiplying. Several examples of multiplying rational expressions are shown. The document also explains that to divide rational expressions, one multiplies the numerator by the reciprocal of the denominator. Some examples of dividing rational expressions are provided.
Rational algebraic expressions
This document provides information about rational algebraic expressions. It defines a rational algebraic expression as a ratio of two polynomials where the denominator is not equal to zero. It gives examples of rational expressions like 7x/2y and 10x-5/(x+3). It explains that rational expressions are defined for all real numbers except those that would make the denominator equal to zero. The document also discusses how to simplify rational expressions by factoring the numerator and denominator and cancelling common factors. It provides examples of when cancellation is and is not allowed.
Algebra 2 Section 2-2
This document discusses linearity and symmetry in functions. It defines linear equations and functions as those where the highest power of the variable is 1. Nonlinear functions have powers higher than 1. Line symmetry occurs when a graph can be folded over a line of symmetry so the two halves match. Point symmetry is when a graph can be rotated 180 degrees around a point of symmetry to match the original orientation. Examples are provided to demonstrate identifying linear versus nonlinear functions from equations and graphs, as well as determining if graphs have line or point symmetry and the associated lines or points of symmetry.
3.2 notes
This document discusses parallel lines and transversals. It defines corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. It states that if two parallel lines are cut by a transversal, then corresponding angles, alternate interior angles, and alternate exterior angles are congruent. It also states that if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Students are instructed to complete guided practice problems with a partner.
6.3 solve by elimination
This document provides steps for solving systems of equations by elimination. It outlines the process of writing the system so like terms are aligned, adding or subtracting entire equations to eliminate a variable, and then substituting the result into one equation to solve for the remaining variable. It also provides examples of systems of equations to solve using this elimination method and assigns homework problems from the textbook.
Algaebraic forms
This document defines key algebraic concepts:
1. Terms are the individual parts that make up an algebraic expression, such as coefficients and variables.
2. Like terms contain the same variables and exponents that can be combined by addition or subtraction.
3. A polynomial is an algebraic expression with multiple terms.
It provides examples of terms, coefficients, variables, exponents, and like terms in algebraic expressions. Exercises are included for the reader to practice identifying and simplifying algebraic forms.
Joins and unions
This document provides an overview of JOINs and UNIONS in MySQL. It defines different types of JOINs (INNER JOIN, LEFT JOIN, RIGHT JOIN) and provides syntax examples to retrieve data from multiple tables based on relationships between columns. While FULL JOIN is not supported in MySQL, the document demonstrates how to emulate it using a UNION of a LEFT JOIN and RIGHT JOIN. It also covers the UNION and UNION ALL operators to combine result sets from multiple queries.
Engineering maths/Exact Differential equations
This video is helpful for Engineering maths students. Here, we discuss what are exact differential equations and learn various means and methods to solve them. 3 types of methods are well illustrated here.
Consider subscribing to this channel for more videos.
http://lnnk.in/@mathmadeeasy22
You can also visit my page
https://www.mathmadeeasy.co/lessons-1
You can subscribe to my playlist
http://lnnk.in/@engineering-maths-1
Feel free to contact me for any further help
Recommended
Multiplying & dividing rational algebraic expressions
This document discusses how to multiply and divide rational algebraic expressions. It explains that to multiply rational expressions, one multiplies the numerators and denominators separately. Rational expressions must first be factored to cancel common factors before multiplying. Several examples of multiplying rational expressions are shown. The document also explains that to divide rational expressions, one multiplies the numerator by the reciprocal of the denominator. Some examples of dividing rational expressions are provided.
Rational algebraic expressions
This document provides information about rational algebraic expressions. It defines a rational algebraic expression as a ratio of two polynomials where the denominator is not equal to zero. It gives examples of rational expressions like 7x/2y and 10x-5/(x+3). It explains that rational expressions are defined for all real numbers except those that would make the denominator equal to zero. The document also discusses how to simplify rational expressions by factoring the numerator and denominator and cancelling common factors. It provides examples of when cancellation is and is not allowed.
Algebra 2 Section 2-2
This document discusses linearity and symmetry in functions. It defines linear equations and functions as those where the highest power of the variable is 1. Nonlinear functions have powers higher than 1. Line symmetry occurs when a graph can be folded over a line of symmetry so the two halves match. Point symmetry is when a graph can be rotated 180 degrees around a point of symmetry to match the original orientation. Examples are provided to demonstrate identifying linear versus nonlinear functions from equations and graphs, as well as determining if graphs have line or point symmetry and the associated lines or points of symmetry.
3.2 notes
This document discusses parallel lines and transversals. It defines corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. It states that if two parallel lines are cut by a transversal, then corresponding angles, alternate interior angles, and alternate exterior angles are congruent. It also states that if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. Students are instructed to complete guided practice problems with a partner.
6.3 solve by elimination
This document provides steps for solving systems of equations by elimination. It outlines the process of writing the system so like terms are aligned, adding or subtracting entire equations to eliminate a variable, and then substituting the result into one equation to solve for the remaining variable. It also provides examples of systems of equations to solve using this elimination method and assigns homework problems from the textbook.
Algaebraic forms
This document defines key algebraic concepts:
1. Terms are the individual parts that make up an algebraic expression, such as coefficients and variables.
2. Like terms contain the same variables and exponents that can be combined by addition or subtraction.
3. A polynomial is an algebraic expression with multiple terms.
It provides examples of terms, coefficients, variables, exponents, and like terms in algebraic expressions. Exercises are included for the reader to practice identifying and simplifying algebraic forms.
Joins and unions
This document provides an overview of JOINs and UNIONS in MySQL. It defines different types of JOINs (INNER JOIN, LEFT JOIN, RIGHT JOIN) and provides syntax examples to retrieve data from multiple tables based on relationships between columns. While FULL JOIN is not supported in MySQL, the document demonstrates how to emulate it using a UNION of a LEFT JOIN and RIGHT JOIN. It also covers the UNION and UNION ALL operators to combine result sets from multiple queries.
Engineering maths/Exact Differential equations
This video is helpful for Engineering maths students. Here, we discuss what are exact differential equations and learn various means and methods to solve them. 3 types of methods are well illustrated here.
Consider subscribing to this channel for more videos.
http://lnnk.in/@mathmadeeasy22
You can also visit my page
https://www.mathmadeeasy.co/lessons-1
You can subscribe to my playlist
http://lnnk.in/@engineering-maths-1
Feel free to contact me for any further help
3.2 use parallel lines and transversals
This document discusses parallel lines and transversals. It defines key terms like corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. It presents theorems about angle relationships that are created when parallel lines are cut by a transversal, such as corresponding angles being congruent and consecutive interior angles being supplementary. Examples are provided to demonstrate applying these concepts and theorems to find missing angle measures. Students are assigned practice problems to reinforce their understanding of using parallel lines and transversals to solve for unknown angle measures.
Lesson 1 student notes
Notes for student NB to save time!! I created these notes for students in CCMath 2, the lesson is a review of CCMath 1 content.
Adding and subtracting positive and negative rational number notes 1
This document discusses adding and subtracting positive and negative rational numbers. It begins by stating the objectives which are for students to understand the differences between adding/subtracting versus multiplying/dividing positive and negative rational numbers. It also aims for students to be able to explain steps for adding/subtracting decimals and perform operations with whole numbers and fractions. The document then reviews steps for adding and subtracting fractions and decimals before introducing the rules for adding, subtracting, multiplying and dividing positive and negative rational numbers.
Rational Expressions
The document discusses rational expressions and operations involving them. It begins with an introduction to rational expressions, noting they are algebraic expressions with both the numerator and denominator being polynomials. It then outlines the lessons that will be covered in the module, including illustrating, simplifying, and performing operations on rational expressions. Several examples are then provided of simplifying rational expressions by factoring the numerator and denominator and cancelling common factors. The document also discusses multiplying rational expressions by using the same process as multiplying fractions, and provides examples of multiplying rational expressions.
Content schedule bt
This document provides a tentative schedule for the content and exams in a Modern Geometry course. It outlines the chapters and topics to be covered over 15 weeks, including two-dimensional and three-dimensional shapes, perimeter, area, volume, logic and proofs of triangle congruence, parallel lines, similarity, right triangles, trigonometry, circles, arcs, chords, and secants. Recommended textbook problems are listed for each section. There are 4 exams scheduled to assess comprehension of the material as it is covered throughout the course.
3.2 use parallel lines and transversals
This document discusses parallel lines and transversals. It provides examples of using theorems about alternate interior angles, corresponding angles, vertical angles, and consecutive interior angles to solve for unknown angle measures. Theorems demonstrated include the alternate interior angles theorem, vertical angles congruence theorem, and consecutive interior angles theorem. Students are guided through practice problems applying these theorems to find missing angle measures given information about parallel lines cut by a transversal.
Python week4-lecture1-handout
The document discusses the merge sort algorithm. It explains that merge sort uses a divide and conquer strategy to sort an array. It divides the array into halves, recursively sorts the halves, and then merges the sorted halves back together into a fully sorted array. The key steps are dividing, conquering by recursively sorting the halves, and merging the sorted halves back together.
Parallel Lines Angles
This document discusses properties of parallel lines and how to determine congruent and specific angles. It states that angles 1, 4, 5, and 8 are congruent at 47 degrees each since they are vertical, corresponding, or alternate exterior angles of parallel lines. It also notes that angle 2 is 133 degrees since it is supplementary to angle 1.
Math chapter 3 multiplying exponential expressions
When multiplying exponential expressions with the same base but different exponents, such as 34 * 35, the base is kept the same and the exponents are added. So, 34 * 35 = 39, because it is equivalent to 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3, which is 3 to the power of 9. More generally, when multiplying expressions of the form xm * xn, the result is x(m+n).
Math chapter 3 dividing exponential expressions
This document discusses how to simplify exponential expressions when dividing numbers with the same base but different exponents. It provides examples of simplifying 37/33 and 53/58. The key rule explained is that when dividing numbers with the same base but different exponents, you copy the base and subtract the denominator's exponent from the numerator's exponent.
Understanding exponents
The document explains exponents and how to write expressions using exponents. It provides examples of expressions written with multiplication and shows how to rewrite them using exponents. Some key examples include rewriting 3 x 3 x 3 x 3 as 34 and showing that 2 x 2 x 2 x 2 x 2 x 2 x 2 can be written as 27. It also gives practice problems for rewriting expressions using exponents and substituting values.
3rd Test - Slope and parallel lines
The document discusses using slope and perpendicular lines (⊥) on a coordinate plane. It explains that two non-vertical lines are parallel if they have the same slope. Two non-vertical lines are perpendicular if the product of their slopes is -1. The document poses three examples of line pairs and asks to determine if they are parallel, perpendicular, or neither based on their slopes.
L 50
To solve trigonometric equations involving inverse trigonometric functions like arcsin, restrictions must be placed on the possible solutions. For arcsin(1/√2), the restriction is that the answer must be between -π/2 and π/2 since the range of arcsin is [-π/2, π/2]. Solving trig equations generally requires using inverse trig identities to isolate the inverse trig function and determine its possible values based on its restricted range.
Rationalizing the Denominator of a Radical Expression
The document discusses rationalizing the denominator of a radical expression. Rationalizing means finding an equivalent expression where the denominator is a perfect square by multiplying the numerator and denominator by the radical in the denominator. This removes the radical from the denominator and makes the expression rational. For example, to rationalize 3/√3, we multiply the numerator and denominator by √3, giving (3√3)/(3) = 1.
Unit 6 review
This document provides a review for a unit test on fractions. It lists lessons covered on multiplying fractions, multiplying mixed numbers, and dividing fractions and mixed numbers. It then provides 25 true/false questions and 3 word problems about operations with fractions, mixed numbers, and fraction equations.
Unit 1 BP801T a introduction mean median mode
This document provides information and examples for calculating various statistical measures - mean, median, mode, range - from data sets. It explains that the mean is the average found by adding all values and dividing by the number of values, the median is the middle number of a data set arranged in order, and the mode is the most frequently occurring value. Examples are given to demonstrate calculating each measure from sets of test score data. Frequency and cumulative frequency are also defined.
Roots and radical expressions
This document discusses operations involving radicals such as square roots. Some key points covered include:
1) Simplifying radicals by pulling out perfect squares and simplifying inside and outside the radical.
2) Multiplying radicals by multiplying only the numbers outside the radicals while keeping numbers inside the same.
3) Rationalizing denominators by multiplying the denominator and numerator by the radical in the denominator to remove radicals from the denominator.
4) Adding and subtracting radicals by combining like terms where only the numbers outside the radicals are added/subtracted and numbers inside are kept the same.
Mean, Median And Mode
The document defines and provides examples for calculating the mean, median, and mode of data sets. It explains that the mode is the most common number, the median is the middle number once the data is ordered, and the mean is the average found by adding all numbers and dividing by the count. It then provides 6 practice data sets and asks the reader to calculate the mean, median, and mode of each, showing their work.
Adding similar fractions
Fraction represents equal parts of a whole. A fraction consists of a numerator and denominator. To add similar fractions, you must make sure the denominators are the same. Then you add the numerators and put the sum over the original denominator. Finally, simplify the sum if possible. Examples are provided where the fractions 1/3 + 1/3 = 2/3, 2/5 + 1/5 = 3/5, and 3/5 + 2/5 = 1.
Exponents Rules
This document provides rules and examples for operations involving exponents. It explains that when bases are the same, exponents can be added or subtracted. It also discusses the power of a power rule where the outer exponent is multiplied by the inner one. There are warnings that these rules only apply when bases are the same or there is a single base inside brackets. Negative exponents are explained as reciprocals with the base moving above or below the fraction line.
Sorting in data structures and algorithms , it has all the necessary points t...
1) The document discusses various sorting algorithms like bubble sort, selection sort, insertion sort, merge sort, quick sort and heap sort.
2) It provides detailed explanations of how merge sort and quick sort algorithms work, including examples with diagrams showing the sorting process step-by-step.
3) Merge sort has a time complexity of O(n log n) as it recursively divides the array into halves and then merges the sorted halves, while quick sort selects a pivot element and partitions the array into elements less than and greater than the pivot in O(n) time on average.
Sorting-algorithmbhddcbjkmbgjkuygbjkkius.pdf
The document discusses various sorting algorithms in Java including bubble sort, insertion sort, selection sort, merge sort, heapsort, and quicksort. It provides explanations of how each algorithm works and comparisons of the time performance of each algorithm based on testing multiple runs. Quicksort and heapsort generally had the best performance while bubble sort consistently had the worst performance.
More Related Content
What's hot
3.2 use parallel lines and transversals
This document discusses parallel lines and transversals. It defines key terms like corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. It presents theorems about angle relationships that are created when parallel lines are cut by a transversal, such as corresponding angles being congruent and consecutive interior angles being supplementary. Examples are provided to demonstrate applying these concepts and theorems to find missing angle measures. Students are assigned practice problems to reinforce their understanding of using parallel lines and transversals to solve for unknown angle measures.
Lesson 1 student notes
Notes for student NB to save time!! I created these notes for students in CCMath 2, the lesson is a review of CCMath 1 content.
Adding and subtracting positive and negative rational number notes 1
This document discusses adding and subtracting positive and negative rational numbers. It begins by stating the objectives which are for students to understand the differences between adding/subtracting versus multiplying/dividing positive and negative rational numbers. It also aims for students to be able to explain steps for adding/subtracting decimals and perform operations with whole numbers and fractions. The document then reviews steps for adding and subtracting fractions and decimals before introducing the rules for adding, subtracting, multiplying and dividing positive and negative rational numbers.
Rational Expressions
The document discusses rational expressions and operations involving them. It begins with an introduction to rational expressions, noting they are algebraic expressions with both the numerator and denominator being polynomials. It then outlines the lessons that will be covered in the module, including illustrating, simplifying, and performing operations on rational expressions. Several examples are then provided of simplifying rational expressions by factoring the numerator and denominator and cancelling common factors. The document also discusses multiplying rational expressions by using the same process as multiplying fractions, and provides examples of multiplying rational expressions.
Content schedule bt
This document provides a tentative schedule for the content and exams in a Modern Geometry course. It outlines the chapters and topics to be covered over 15 weeks, including two-dimensional and three-dimensional shapes, perimeter, area, volume, logic and proofs of triangle congruence, parallel lines, similarity, right triangles, trigonometry, circles, arcs, chords, and secants. Recommended textbook problems are listed for each section. There are 4 exams scheduled to assess comprehension of the material as it is covered throughout the course.
3.2 use parallel lines and transversals
This document discusses parallel lines and transversals. It provides examples of using theorems about alternate interior angles, corresponding angles, vertical angles, and consecutive interior angles to solve for unknown angle measures. Theorems demonstrated include the alternate interior angles theorem, vertical angles congruence theorem, and consecutive interior angles theorem. Students are guided through practice problems applying these theorems to find missing angle measures given information about parallel lines cut by a transversal.
Python week4-lecture1-handout
The document discusses the merge sort algorithm. It explains that merge sort uses a divide and conquer strategy to sort an array. It divides the array into halves, recursively sorts the halves, and then merges the sorted halves back together into a fully sorted array. The key steps are dividing, conquering by recursively sorting the halves, and merging the sorted halves back together.
Parallel Lines Angles
This document discusses properties of parallel lines and how to determine congruent and specific angles. It states that angles 1, 4, 5, and 8 are congruent at 47 degrees each since they are vertical, corresponding, or alternate exterior angles of parallel lines. It also notes that angle 2 is 133 degrees since it is supplementary to angle 1.
Math chapter 3 multiplying exponential expressions
When multiplying exponential expressions with the same base but different exponents, such as 34 * 35, the base is kept the same and the exponents are added. So, 34 * 35 = 39, because it is equivalent to 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3, which is 3 to the power of 9. More generally, when multiplying expressions of the form xm * xn, the result is x(m+n).
Math chapter 3 dividing exponential expressions
This document discusses how to simplify exponential expressions when dividing numbers with the same base but different exponents. It provides examples of simplifying 37/33 and 53/58. The key rule explained is that when dividing numbers with the same base but different exponents, you copy the base and subtract the denominator's exponent from the numerator's exponent.
Understanding exponents
The document explains exponents and how to write expressions using exponents. It provides examples of expressions written with multiplication and shows how to rewrite them using exponents. Some key examples include rewriting 3 x 3 x 3 x 3 as 34 and showing that 2 x 2 x 2 x 2 x 2 x 2 x 2 can be written as 27. It also gives practice problems for rewriting expressions using exponents and substituting values.
3rd Test - Slope and parallel lines
The document discusses using slope and perpendicular lines (⊥) on a coordinate plane. It explains that two non-vertical lines are parallel if they have the same slope. Two non-vertical lines are perpendicular if the product of their slopes is -1. The document poses three examples of line pairs and asks to determine if they are parallel, perpendicular, or neither based on their slopes.
L 50
To solve trigonometric equations involving inverse trigonometric functions like arcsin, restrictions must be placed on the possible solutions. For arcsin(1/√2), the restriction is that the answer must be between -π/2 and π/2 since the range of arcsin is [-π/2, π/2]. Solving trig equations generally requires using inverse trig identities to isolate the inverse trig function and determine its possible values based on its restricted range.
Rationalizing the Denominator of a Radical Expression
The document discusses rationalizing the denominator of a radical expression. Rationalizing means finding an equivalent expression where the denominator is a perfect square by multiplying the numerator and denominator by the radical in the denominator. This removes the radical from the denominator and makes the expression rational. For example, to rationalize 3/√3, we multiply the numerator and denominator by √3, giving (3√3)/(3) = 1.
Unit 6 review
This document provides a review for a unit test on fractions. It lists lessons covered on multiplying fractions, multiplying mixed numbers, and dividing fractions and mixed numbers. It then provides 25 true/false questions and 3 word problems about operations with fractions, mixed numbers, and fraction equations.
Unit 1 BP801T a introduction mean median mode
This document provides information and examples for calculating various statistical measures - mean, median, mode, range - from data sets. It explains that the mean is the average found by adding all values and dividing by the number of values, the median is the middle number of a data set arranged in order, and the mode is the most frequently occurring value. Examples are given to demonstrate calculating each measure from sets of test score data. Frequency and cumulative frequency are also defined.
Roots and radical expressions
This document discusses operations involving radicals such as square roots. Some key points covered include:
1) Simplifying radicals by pulling out perfect squares and simplifying inside and outside the radical.
2) Multiplying radicals by multiplying only the numbers outside the radicals while keeping numbers inside the same.
3) Rationalizing denominators by multiplying the denominator and numerator by the radical in the denominator to remove radicals from the denominator.
4) Adding and subtracting radicals by combining like terms where only the numbers outside the radicals are added/subtracted and numbers inside are kept the same.
Mean, Median And Mode
The document defines and provides examples for calculating the mean, median, and mode of data sets. It explains that the mode is the most common number, the median is the middle number once the data is ordered, and the mean is the average found by adding all numbers and dividing by the count. It then provides 6 practice data sets and asks the reader to calculate the mean, median, and mode of each, showing their work.
Adding similar fractions
Fraction represents equal parts of a whole. A fraction consists of a numerator and denominator. To add similar fractions, you must make sure the denominators are the same. Then you add the numerators and put the sum over the original denominator. Finally, simplify the sum if possible. Examples are provided where the fractions 1/3 + 1/3 = 2/3, 2/5 + 1/5 = 3/5, and 3/5 + 2/5 = 1.
Exponents Rules
This document provides rules and examples for operations involving exponents. It explains that when bases are the same, exponents can be added or subtracted. It also discusses the power of a power rule where the outer exponent is multiplied by the inner one. There are warnings that these rules only apply when bases are the same or there is a single base inside brackets. Negative exponents are explained as reciprocals with the base moving above or below the fraction line.
What's hot (20)
Adding and subtracting positive and negative rational number notes 1
Adding and subtracting positive and negative rational number notes 1
Math chapter 3 multiplying exponential expressions
Math chapter 3 multiplying exponential expressions
Rationalizing the Denominator of a Radical Expression
Rationalizing the Denominator of a Radical Expression
Similar to Merge sort implementation
Sorting in data structures and algorithms , it has all the necessary points t...
1) The document discusses various sorting algorithms like bubble sort, selection sort, insertion sort, merge sort, quick sort and heap sort.
2) It provides detailed explanations of how merge sort and quick sort algorithms work, including examples with diagrams showing the sorting process step-by-step.
3) Merge sort has a time complexity of O(n log n) as it recursively divides the array into halves and then merges the sorted halves, while quick sort selects a pivot element and partitions the array into elements less than and greater than the pivot in O(n) time on average.
Sorting-algorithmbhddcbjkmbgjkuygbjkkius.pdf
The document discusses various sorting algorithms in Java including bubble sort, insertion sort, selection sort, merge sort, heapsort, and quicksort. It provides explanations of how each algorithm works and comparisons of the time performance of each algorithm based on testing multiple runs. Quicksort and heapsort generally had the best performance while bubble sort consistently had the worst performance.
Sorting
The document discusses several common sorting algorithms: bubble sort, selection sort, insertion sort, merge sort, and quick sort. Bubble sort works by scanning an array from left to right, swapping adjacent elements that are out of order. Selection sort finds the smallest element, swaps it into the first position, then repeats to find the next smallest. Insertion sort inserts new elements into the correct position in a sorted array. Merge sort divides an array in half, sorts the halves, and merges them back together. Quick sort chooses a pivot element and partitions the array into sub-arrays based on element values relative to the pivot.
Insertion Sort, Quick Sort And Their complexity
This document provides information on insertion sort, quicksort, and their time complexities. It describes how insertion sort works by dividing an array into sorted and unsorted parts, iterating through elements to insert them in the correct position. Quicksort chooses a pivot element and partitions the array into sub-arrays of smaller size, recursively sorting them. For worst-case complexities, insertion sort is O(n^2) while quicksort can also be O(n^2) if the array is already sorted. On average, insertion sort is still O(n^2) whereas quicksort has a lower complexity of O(nlogn).
Quick sort-170316064200
Quicksort is a divide and conquer sorting algorithm that works by partitioning an array around a pivot value. It then recursively sorts the sub-arrays on each side. The key steps are: 1) Choose a pivot element to split the array into left and right halves, with all elements on the left being less than the pivot and all on the right being greater; 2) Recursively quicksort the left and right halves; 3) Combine the now-sorted left and right halves into a fully sorted array. The example demonstrates quicksorting an array of 6 elements by repeatedly partitioning around a pivot until the entire array is sorted.
Quick sort-Data Structure
Quicksort is a divide and conquer sorting algorithm that works by partitioning an array around a pivot value. It then recursively sorts the sub-arrays on each side. The key steps are: 1) Choose a pivot element to split the array into left and right halves, with all elements on the left being less than the pivot and all on the right being greater; 2) Recursively quicksort the left and right halves; 3) Combine the now-sorted left and right halves into a fully sorted array. The example demonstrates quicksorting an array of 6 elements by repeatedly partitioning around a pivot until the entire array is sorted.
Analysis of Algorithm (Bubblesort and Quicksort)
Quicksort is a recursive divide-and-conquer algorithm that works by selecting a pivot element and partitioning the array into two subarrays of elements less than and greater than the pivot. It recursively sorts the subarrays. The divide step does all the work by partitioning, while the combine step does nothing. It has average case performance of O(n log n) but worst case of O(n^2). Bubble sort repeatedly swaps adjacent elements that are out of order until the array is fully sorted. It has a simple implementation but poor performance of O(n^2).
K-12 Mathematics lesson Presentation.pptx
This document provides instruction on fraction number sequences. It includes:
1) Examples of increasing and decreasing fraction sequences by unit fractions like 1/4 or 1/5.
2) Examples of sequences that increase by non-unit fractions like 2/5 or 3/7.
3) Practice problems for students to complete fraction sequences and identify patterns.
Chapter 8 advanced sorting and hashing for print
Shell sort improves on insertion sort by first sorting elements that are already partially sorted. It does this by using a sequence of increment values to sort sublists within the main list. The time complexity of shell sort is O(n^3/2).
Quicksort uses a divide and conquer approach. It chooses a pivot element and partitions the list into two sublists based on element values relative to the pivot. The sublists are then recursively sorted. The average time complexity of quicksort is O(nlogn) but it can be O(n^2) in the worst case.
Mergesort follows the same divide and conquer strategy as quicksort. It recursively divides the list into halves until single elements
Sorting
data structure notes for bca. only use for education purpose. it help u to understand data structure and its methods.
Lattice multiplication
The document presents the steps of lattice multiplication to multiply whole numbers, decimals, and polynomials. It begins by explaining that lattice multiplication breaks down the traditional long multiplication method into smaller steps by using a grid. For whole number multiplication, the steps are to draw a grid with rows and columns equal to the number of digits in the factors, write the factors in the grid, multiply pairs of digits and record partial products in the grid, and sum the products along the diagonals. The same process is used for decimal multiplication. For polynomial multiplication, the coefficients of the factors determine the grid size, each term is multiplied out, and the results are combined into a polynomial. Examples are provided to demonstrate the lattice multiplication process.
Algo PPT.pdf
The document compares the performance of heap sort and insertion sort algorithms using different sized data sets. It implements both algorithms and analyzes their time complexities in best, average, and worst cases. The results show that insertion sort performs better on small and average sized data, while heap sort scales better to large data sets and has more consistent performance across cases. Heap sort is generally more suitable than insertion sort when sorting large amounts of data.
Section 8: Symmetric Groups
The document discusses symmetric groups and permutations. Some key points:
- A symmetric group SX is the group of all permutations of a set X under function composition.
- Sn, the symmetric group of degree n, represents permutations of the set {1,2,...,n}.
- Permutations can be written using cycle notation, such as (1 2 3) or (1 4).
- Disjoint cycles commute; the product of two permutations is their composition as functions.
- Any permutation can be written as a product of disjoint cycles of length ≥2. Cycles of even length decompose into an odd number of transpositions, and vice versa for odd cycles.
Merge sort
An in-depth walk through of merge sort behavior and javascript implementation of both top-down and bottom-up approaches.
Sortings .pptx
This document discusses various sorting algorithms and their time complexities. It covers common sorting algorithms like bubble sort, selection sort, insertion sort, which have O(N^2) time complexity and are slow for large data sets. More efficient algorithms like merge sort, quicksort, heapsort with O(N log N) time complexity are also discussed. Implementation details and examples are provided for selection sort, insertion sort, merge sort and quicksort algorithms.
CS253: Divide & Conquer Sort (2019)
The document discusses two sorting algorithms: Quicksort and Mergesort. Quicksort works by picking a pivot element and partitioning the array around that pivot, recursively sorting the subarrays. It has average time complexity of O(n log n) but worst case of O(n^2). Mergesort works by dividing the array into halves, recursively sorting the halves, and then merging the sorted halves together. It has time complexity of O(n log n) in all cases. The document also includes Java code for implementing MergeSort and discusses how it works.
Coordinate Geometry
The document defines key terms related to coordinate geometry and coordinate grids. It explains that a coordinate grid uses horizontal and vertical lines to locate points using their distance from two intersecting lines. It defines the x-axis and y-axis, and describes how to plot and find the ordered pair coordinates of points on the grid, including in the four quadrants when the number lines are extended into negatives.
Aizaz comb sort
Comb sort is a sorting algorithm that improves on bubble sort by allowing larger gaps between elements to be compared. It starts with a large gap that shrinks on each iteration. This eliminates more swaps than bubble sort and moves high and low values towards their final positions more quickly. Rabbits refer to large values at the beginning, and turtles to small values at the end, which comb sort handles more efficiently than bubble sort.
Quick_sort1.pptx
Quick sort is a divide and conquer sorting algorithm that works by partitioning an array around a pivot value. It then recursively sorts the sub-arrays on each side of the pivot. The key steps are: 1) Choose a pivot element to divide the array; 2) Partition the array by swapping elements such that all elements less than the pivot come before it and all greater elements come after; 3) Recursively apply the same process to the sub-arrays on each side of the pivot until the entire array is sorted.
Coordinate Geometry
The document defines key concepts in coordinate geometry including coordinate grids, the x-axis, y-axis, ordered pairs, plotting points, and the four quadrants. It provides examples of plotting various ordered pairs in a coordinate grid including those with positive and negative x and y values. Instructions are given on how to find the ordered pair of a point, plot ordered pairs, and extend the grid to include the four quadrants.
Similar to Merge sort implementation (20)
Sorting in data structures and algorithms , it has all the necessary points t...
Sorting in data structures and algorithms , it has all the necessary points t...
Recently uploaded
New techniques for characterising damage in rock slopes.pdf
rock mass characterization and New techniques for characterising damage in rock slopes
Embedded machine learning-based road conditions and driving behavior monitoring
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Modelagem de um CSTR com reação endotermica.pdf
Modelagem em função de transferencia. CSTR não-linear.
ACEP Magazine edition 4th launched on 05.06.2024
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
22CYT12-Unit-V-E Waste and its Management.ppt
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
DfMAy 2024 - key insights and contributions
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Technical Drawings introduction to drawing of prisms
Method of technical Drawing of prisms,and cylinders.
Understanding Inductive Bias in Machine Learning
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
bank management system in java and mysql report1.pdf
truth is high but higher still is truth full living
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Low power architecture of logic gates using adiabatic techniques
The growing significance of portable systems to limit power consumption in ultra-large-scale-integration chips of very high density, has recently led to rapid and inventive progresses in low-power design. The most effective technique is adiabatic logic circuit design in energy-efficient hardware. This paper presents two adiabatic approaches for the design of low power circuits, modified positive feedback adiabatic logic (modified PFAL) and the other is direct current diode based positive feedback adiabatic logic (DC-DB PFAL). Logic gates are the preliminary components in any digital circuit design. By improving the performance of basic gates, one can improvise the whole system performance. In this paper proposed circuit design of the low power architecture of OR/NOR, AND/NAND, and XOR/XNOR gates are presented using the said approaches and their results are analyzed for powerdissipation, delay, power-delay-product and rise time and compared with the other adiabatic techniques along with the conventional complementary metal oxide semiconductor (CMOS) designs reported in the literature. It has been found that the designs with DC-DB PFAL technique outperform with the percentage improvement of 65% for NOR gate and 7% for NAND gate and 34% for XNOR gate over the modified PFAL techniques at 10 MHz respectively.
Recently uploaded (20)
ACRP 4-09 Risk Assessment Method to Support Modification of Airfield Separat...
ACRP 4-09 Risk Assessment Method to Support Modification of Airfield Separat...
New techniques for characterising damage in rock slopes.pdf
New techniques for characterising damage in rock slopes.pdf
Embedded machine learning-based road conditions and driving behavior monitoring
Embedded machine learning-based road conditions and driving behavior monitoring
Technical Drawings introduction to drawing of prisms
Technical Drawings introduction to drawing of prisms
bank management system in java and mysql report1.pdf
bank management system in java and mysql report1.pdf
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Low power architecture of logic gates using adiabatic techniques
Low power architecture of logic gates using adiabatic techniques
basic-wireline-operations-course-mahmoud-f-radwan.pdf
basic-wireline-operations-course-mahmoud-f-radwan.pdf
Merge sort implementation
- 2. Introduction to Group members Nazmul Hasan Rupu bsse 1034 Sadikul Haque Sadi bsse 1003
- 3. Merge Sort In merge sort algorithm , we use the concept of divide and conquer. We divide the array into two parts ,then sort them and merge them to get sorted array. This sorting method is an implementation of recursive function.
- 4. Merge Sort We take an array and keep dividing it from middle till we get only one element in each part(sub-array) . Then we sort the sub-arrays and merge them to get the final sorted array.
- 5. Example Let’s consider an array of 6 elements. 5 3 2 1 4 6 Arrange them in ascending order using merge sort algorithm.
- 6. Array index Array elements 0 1 2 3 4 5 5 3 2 1 4 6 Begin Starting index End Last index Is Size >1 ? yes Divide the array into 2 part Mid = Size/2 Mid = 6/2 = 3 Size= 6
- 7. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 We will consider the left sub-array
- 8. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 Is Size > 1 ? Size=3 yes Divide the array into 2 part Mid = Size/2 Mid = 3/2 = 1
- 9. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 0 5 Is Size > 1 ? no Size=1 As there is only one element in the array we now go to the right sub array
- 10. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 0 5 0 1 3 2 Size =2 Is Size > 1 ? yes Divide the array into 2 part Mid = Size/2 Mid = 2/2 = 1
- 11. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 0 5 0 1 3 2 0 3 1 2 As there is only one element in the array we now go to the right sub array
- 12. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 0 5 0 1 3 2 0 3 1 2 Both the right sub- array and left sub- array have one element So we can now sort them and merge them
- 13. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 0 5 0 1 3 2 0 3 1 2 0 1 2 2 0 1 2 3
- 14. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 5 3 2 0 5 0 1 2 3 0 1 2 2 3 2 0 1 2 2 3 2 0 1 2 2 3 5
- 15. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 2 3 5 The left sub-array is sorted now we will go to right sub-array 0 1 2 1 4 6 Now we will follow the same way as we followed so far
- 16. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6 0 1 0 1 4 6 0 4 0 6
- 17. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6 0 1 0 1 4 6 0 4 0 6
- 18. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6 0 1 0 1 4 6
- 19. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6 The left and right array are sortedWe can now merge them to get sorted array in ascending order
- 20. 0 1 2 3 4 5 5 3 2 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6
- 21. 0 1 2 3 4 5 1 3 2 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6
- 22. 0 1 2 3 4 5 1 2 2 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6
- 23. 0 1 2 3 4 5 1 2 3 1 4 6 0 1 2 2 3 5 0 1 2 1 4 6
- 24. 0 1 2 3 4 5 1 2 3 4 4 6 0 1 2 2 3 5 0 1 2 1 4 6
- 25. 0 1 2 3 4 5 1 2 3 4 5 6 0 1 2 2 3 5 0 1 2 1 4 6 0 1 2 3 4 5 1 2 3 4 5 6 The array is now sorted ! ! !