06 - 20 Jan - Divide and Conquer
•
1 like•259 views
The document discusses algorithms for finding the closest pair of points from a set of n points given as (x,y) coordinates. It presents three approaches: computing all pairwise distances which is O(n^2); sorting the points and comparing adjacent pairs which is the time to sort plus O(n); and divide and conquer which splits the points, solves subproblems recursively, and merges in O(n log n) time.
Report
Share
Report
Share
Download to read offline
Recommended
The dev project
This document contains 4 math problems with step-by-step solutions: 1) factoring and graphing a polynomial function, finding vertical and horizontal asymptotes, x-intercepts, and holes; 2) finding the maximum area of a garden within a fence of a given length; 3) fully distributing a polynomial; 4) factoring a quadratic equation.
Wati lg
The document discusses indefinite integrals and their relationship to derivatives. It states that the indefinite integral is the inverse of the derivative, so finding the integral of a function means finding its antiderivative. The document provides examples of using formulas to find the antiderivatives of functions like 5x^4 by recognizing it as the derivative of x^5. It also demonstrates how integrals follow properties like scalar multiplication and addition/subtraction.
Understanding Reed-Solomon code
Understanding the math of Reed-Solomon code (based on FEC code RS(544,514) used in IEEE 200/400GBASE-R)
Reed Solomon encoder and decoder \ ريد سلمون
This document provides information about Reed-Solomon encoding and decoding. It introduces Reed-Solomon codes, including their parameters like code length, number of information symbols, and error correcting capability. It describes the generator polynomial and how it is used to encode messages. The document also discusses the Massey FSR Synthesis Algorithm for Reed-Solomon decoding and Forney's Equation for calculating error magnitudes. An example is provided to demonstrate decoding a received codeword using the Massey algorithm.
Ch 2
The document introduces numerical methods for finding the roots or zeros of equations of the form f(x) = 0, where f(x) is an algebraic or transcendental function. It focuses on the bisection method, also called the Bolzano method, which uses interval bisection to bracket the root between two values where f(x) has opposite signs. The method iteratively narrows down the interval to find the root to within a specified tolerance. Several examples demonstrate applying the bisection method to find roots of polynomial, logarithmic, and trigonometric equations.
File ...handouts complete
The document provides examples for calculating squares of numbers nearest to certain values like 50, 100, 200, 300, 400, 1000, 1100 using a basic calculation method. It involves taking the base as the number rounded to the nearest value and comparing it to the given value. The difference is divided by the square of the difference and added to the base number to get the square. It also provides examples for squares ending with 5, examples where the unit digit sum is 10, methods to find square roots and cube roots. The document further discusses number systems, divisibility rules, number of divisors, recurring decimals to fractions conversion, remainder theorem and alligation.
Fascio di parabole
This document is an exercise in Italian that considers a bundle of parabolas with the equation y = kx^2 + 5x + k + 5. It asks the student to:
1) Find the base points and characteristics of the bundle, which has no base points.
2) Find the parabola in the bundle that passes through the origin, which is y = -5x^2 + 5x.
3) Find the parabola with axis x = 15/2, which is y = -1/3x^2 + 5x + 14/3.
4) Find the parabolas with the focus on the x-axis, which are y
Domain & range intro presentation
The document discusses domain and range in functions. [1] The domain is the set of possible values that the independent variable can take on. [2] The range is the set of possible values of the dependent variable. [3] Interval notation uses ( ) for open intervals and [ ] for closed intervals to describe spans of values for the domain.
Recommended
The dev project
This document contains 4 math problems with step-by-step solutions: 1) factoring and graphing a polynomial function, finding vertical and horizontal asymptotes, x-intercepts, and holes; 2) finding the maximum area of a garden within a fence of a given length; 3) fully distributing a polynomial; 4) factoring a quadratic equation.
Wati lg
The document discusses indefinite integrals and their relationship to derivatives. It states that the indefinite integral is the inverse of the derivative, so finding the integral of a function means finding its antiderivative. The document provides examples of using formulas to find the antiderivatives of functions like 5x^4 by recognizing it as the derivative of x^5. It also demonstrates how integrals follow properties like scalar multiplication and addition/subtraction.
Understanding Reed-Solomon code
Understanding the math of Reed-Solomon code (based on FEC code RS(544,514) used in IEEE 200/400GBASE-R)
Reed Solomon encoder and decoder \ ريد سلمون
This document provides information about Reed-Solomon encoding and decoding. It introduces Reed-Solomon codes, including their parameters like code length, number of information symbols, and error correcting capability. It describes the generator polynomial and how it is used to encode messages. The document also discusses the Massey FSR Synthesis Algorithm for Reed-Solomon decoding and Forney's Equation for calculating error magnitudes. An example is provided to demonstrate decoding a received codeword using the Massey algorithm.
Ch 2
The document introduces numerical methods for finding the roots or zeros of equations of the form f(x) = 0, where f(x) is an algebraic or transcendental function. It focuses on the bisection method, also called the Bolzano method, which uses interval bisection to bracket the root between two values where f(x) has opposite signs. The method iteratively narrows down the interval to find the root to within a specified tolerance. Several examples demonstrate applying the bisection method to find roots of polynomial, logarithmic, and trigonometric equations.
File ...handouts complete
The document provides examples for calculating squares of numbers nearest to certain values like 50, 100, 200, 300, 400, 1000, 1100 using a basic calculation method. It involves taking the base as the number rounded to the nearest value and comparing it to the given value. The difference is divided by the square of the difference and added to the base number to get the square. It also provides examples for squares ending with 5, examples where the unit digit sum is 10, methods to find square roots and cube roots. The document further discusses number systems, divisibility rules, number of divisors, recurring decimals to fractions conversion, remainder theorem and alligation.
Fascio di parabole
This document is an exercise in Italian that considers a bundle of parabolas with the equation y = kx^2 + 5x + k + 5. It asks the student to:
1) Find the base points and characteristics of the bundle, which has no base points.
2) Find the parabola in the bundle that passes through the origin, which is y = -5x^2 + 5x.
3) Find the parabola with axis x = 15/2, which is y = -1/3x^2 + 5x + 14/3.
4) Find the parabolas with the focus on the x-axis, which are y
Domain & range intro presentation
The document discusses domain and range in functions. [1] The domain is the set of possible values that the independent variable can take on. [2] The range is the set of possible values of the dependent variable. [3] Interval notation uses ( ) for open intervals and [ ] for closed intervals to describe spans of values for the domain.
Quadraticapplications.ppt
This document discusses solving problems involving quadratic functions. It provides examples of finding the maximum or minimum value of quadratic functions, including finding the dimensions of a rectangle that will maximize its area given a fixed perimeter. It also discusses key concepts related to quadratic functions like parabolas, vertices, axes of symmetry, and finding the zeros of a function. Examples are provided to illustrate solving problems requiring the application of these quadratic function concepts.
Math presentation on domain and range
This document defines key concepts related to functions including domain, co-domain, and range. It provides examples of determining the domain and range of various functions. The domain of a function is the set of inputs, the co-domain is the set of all possible outputs, and the range is the set of actual outputs. Examples show how to use inequalities to find the domain by determining when a function gives real values and how to manipulate equations to find the range.
Calculus
The document defines domain and range. Domain is the set of all possible input values of a function. Range is the set of all output values of a function as the input variable takes on all possible values. It then verifies the domain and range of several functions, including square root, square, and inverse functions. It also evaluates compositions of functions where f(x)=x^2 and g(x)=x-3. Finally, it evaluates compositions of functions where f(x)=sqrt(x), g(x)=x/2, and h(x)=x-8.
Iterative1
The document describes the iterative method for finding roots of equations. It discusses:
1) How to set up an iterative process to find a root by taking an initial approximation x0 and repeatedly applying the function pi(x) to get closer approximations x1, x2, etc. until the change between successive values is very small.
2) An example of using the method to find the root of the quadratic equation 2x^2 - 4x + 1 = 0. It shows calculating successive approximations that converge to the root.
3) How the iterative method can be implemented in C++ using a function that takes the initial value and returns the updated value at each iteration until the change is negligible.
10.5 more on language of functions x
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
Polynomial functionsandgraphs
This document discusses polynomial functions and their graphs. It defines polynomial functions as functions of the form P(x) = anxn + an-1xn-1 + ... + a1x + a0, where an is the leading coefficient. The degree of the polynomial determines features of its graph like the maximum number of x-intercepts. The leading coefficient test determines the end behavior of the graph. Key features of polynomial graphs are intercepts, extrema, and end behavior.
Gre 403 jeopardy
This document contains a quiz about linear equations and slope with 100 multiple choice questions across 6 categories: With Your Host, Tables And More, Identifying Equations, Old School, Things A Geek Might Know, and Daily Place A Wager. The questions cover topics such as finding the slope of a line given two points, writing equations of lines given slope and y-intercept, evaluating equations for given x-values, and solving multi-step equations.
9 the basic language of functions x
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
2 1 polynomials
The document discusses identifying and evaluating polynomial functions. It defines polynomial functions as functions with whole number exponents and real number coefficients, and no variables in denominators. Polynomials can be written in standard form showing the degree and coefficients. The key points covered are: identifying the degree of a polynomial, finding the zeros of a polynomial function, directly substituting values for x, and using synthetic substitution to evaluate polynomials for any value of x.
7.2 abs value function
This document provides notes for a math chapter on absolute value and reciprocal functions. It covers graphing and expressing absolute value functions as piecewise functions. Key points include: absolute value is defined as the distance from zero on the number line; absolute value functions are continuous and have the same x-intercept as the linear function inside the absolute value; taking the absolute value of a quadratic function preserves the vertex but changes the range. Care must be taken with the domains of absolute value functions depending on if the coefficient of the x-term inside is positive or negative.
Module 3 polynomial functions
This module discusses polynomial functions of degree greater than two. The key points are:
1. The graph of a third-degree polynomial has both a minimum and maximum point, while higher degree polynomials have one less turning point than their degree.
2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
Produccion escrita expresiones algebraicas
The document discusses various algebraic operations including:
1) Summing algebraic expressions by adding like terms.
2) Subtracting algebraic expressions by adding the opposite of like terms.
3) Multiplying algebraic expressions by multiplying each term in one factor by each term in the other factor.
4) Dividing algebraic expressions using long division.
3.3 Zeros of Polynomial Functions
This document discusses key concepts related to finding zeros of polynomial functions including:
1) The factor theorem, which states that a polynomial x - k is a factor of a function f(x) if and only if f(k) = 0.
2) The rational zeros theorem, which gives possible rational zeros based on the factors of the leading coefficient and constant term.
3) The fundamental theorem of algebra, which states that every polynomial of degree n has n complex zeros and examples of finding functions based on given zeros.
4) The conjugate zeros theorem, which states that if z = a + bi is a zero, then z = a - bi is also a zero for polynomials with real coefficients.
5
Polynomials Grade 10
This document provides instructions on how to multiply, divide, and factor polynomials. It discusses:
1) Multiplying polynomials by distributing terms and using FOIL for binomials.
2) Dividing polynomials using long division.
3) Factoring polynomials using grouping, finding two numbers whose product is the constant and sum is the coefficient, and recognizing difference of squares.
Algebraic functions powerpoint
This document defines and provides examples of algebraic functions. It explains that algebraic functions involve algebraic operations like addition, subtraction, multiplication, and division on variables and constants. It also discusses linear algebraic functions, which can be represented by a straight line on the Cartesian plane in the form of f(x)=mx+b. The document provides examples of linear functions using tables, graphs, and solving linear equations for y. It compares linear and quadratic functions, noting that while linear functions can be written as fractions, quadratic functions involve squared variables and are not linear.
Application of derivative
Additional Applications of the Derivative related to engineering maths
this presentation is only for the knowledge...............
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
Christian Jett Morales presents on the quadratic function derived from zeros of the function. The presentation includes:
- An introduction to deriving the quadratic function f(x) = ax^2 + bx + c from the zeros r1 and r2 of the function.
- Examples of deriving quadratic functions when given different zeros, such as 3/2 and -1, or 2+√3 and 2-√3.
- An activity for participants to practice deriving quadratic functions.
- A dedication to God and the presenter's teacher for the opportunity to create the presentation.
Ani agustina (a1 c011007) polynomial
The document describes aspects of polynomials including:
- Definitions of polynomials and their components
- Algebraic operations on polynomials like addition, subtraction, and multiplication
- Dividing polynomials using long division and synthetic (Horner's) division methods
- The remainder theorem stating the remainder of dividing a polynomial by (x - k) is the value of the polynomial at k
- The factorization theorem relating the factors of a polynomial to its roots
- Techniques for factorizing polynomials based on the sums of coefficients
The document contains examples and explanations of key polynomial concepts over multiple pages.
Finding zeros of a quadratic function
The document provides examples of finding the zeros of quadratic functions by factorizing and setting each factor equal to 0. It then lists 5 additional quadratic functions and assigns the reader to find their zeros.
4.2,4.3 graphing
The document discusses how to graph linear equations. It states that the graph of an equation is the set of all solutions, where a solution is an ordered pair that makes the equation true. It provides steps for graphing an equation, which include rewriting the equation to solve for y, making an input/output table using values for x, plotting the ordered pairs, and connecting the points with a straight line. It also discusses that the graph of y=b is a horizontal line and the graph of x=a is a vertical line. Finally, it notes that intercepts are found by plugging values into the equation - the x-intercept uses y=0 and the y-intercept uses x=0.
Jurnal informatika
The document describes the mergesort algorithm for sorting data in linear time complexity of O(n log n). Mergesort uses a divide-and-conquer approach by recursively dividing an array into two halves, sorting each half using mergesort, and then merging the two sorted halves back into a single sorted array. It analyzes the runtime using recurrence relations and proves it is O(n log n).
Java code to find the closest pair using divide and conquer algorith.pdf
Java code to find the closest pair using divide and conquer algorithm.
Solution
The Brute force solution is O(n^2), compute the distance between each pair and return the
smallest. But Devide and conquer is better and completes in O(nLogn) time.
Please find below the code:
import java.util.*;
// compile: javac ClosestPair.java
// run: Java ClosestPair 500
// Here 500 is some randomly generated set of input points.
/*
The algorithm is :
Input: An array of n points P[]
Output: The smallest distance between two points in the given array.
As a pre-processing step, input array is sorted according to x coordinates.
1) Find the middle point in the sorted array, we can take P[n/2] as middle point.
2) Divide the given array in two halves. The first subarray contains points from P[0] to P[n/2].
The second subarray contains points from P[n/2+1] to P[n-1].
3) Recursively find the smallest distances in both subarrays. Let the distances be dl and dr. Find
the minimum of dl and dr. Let the minimum be d.
*/
public class ClosestPair
{
public static class Point
{
public final double x;
public final double y;
public Point(double x, double y)
{
this.x = x;
this.y = y;
}
public String toString()
{ return \"(\" + x + \", \" + y + \")\"; }
}
public static class Pair
{
public Point point1 = null;
public Point point2 = null;
public double distance = 0.0;
public Pair()
{ }
public Pair(Point point1, Point point2)
{
this.point1 = point1;
this.point2 = point2;
calcDistance();
}
public void update(Point point1, Point point2, double distance)
{
this.point1 = point1;
this.point2 = point2;
this.distance = distance;
}
public void calcDistance()
{ this.distance = distance(point1, point2); }
public String toString()
{ return point1 + \"-\" + point2 + \" : \" + distance; }
}
public static double distance(Point p1, Point p2)
{
double xdist = p2.x - p1.x;
double ydist = p2.y - p1.y;
return Math.hypot(xdist, ydist);
}
public static Pair bruteForce(List points)
{
int numPoints = points.size();
if (numPoints < 2)
return null;
Pair pair = new Pair(points.get(0), points.get(1));
if (numPoints > 2)
{
for (int i = 0; i < numPoints - 1; i++)
{
Point point1 = points.get(i);
for (int j = i + 1; j < numPoints; j++)
{
Point point2 = points.get(j);
double distance = distance(point1, point2);
if (distance < pair.distance)
pair.update(point1, point2, distance);
}
}
}
return pair;
}
public static void sortByX(List points)
{
Collections.sort(points, new Comparator() {
public int compare(Point point1, Point point2)
{
if (point1.x < point2.x)
return -1;
if (point1.x > point2.x)
return 1;
return 0;
}
}
);
}
public static void sortByY(List points)
{
Collections.sort(points, new Comparator() {
public int compare(Point point1, Point point2)
{
if (point1.y < point2.y)
return -1;
if (point1.y > point2.y)
return 1;
return 0;
}
}
);
}
public static Pair divideAndConquer(List points)
{
List pointsSortedByX = new ArrayList(points);
sortByX(pointsSortedByX);
List pointsSortedByY = new ArrayList(points);
sortByY.
More Related Content
What's hot
Quadraticapplications.ppt
This document discusses solving problems involving quadratic functions. It provides examples of finding the maximum or minimum value of quadratic functions, including finding the dimensions of a rectangle that will maximize its area given a fixed perimeter. It also discusses key concepts related to quadratic functions like parabolas, vertices, axes of symmetry, and finding the zeros of a function. Examples are provided to illustrate solving problems requiring the application of these quadratic function concepts.
Math presentation on domain and range
This document defines key concepts related to functions including domain, co-domain, and range. It provides examples of determining the domain and range of various functions. The domain of a function is the set of inputs, the co-domain is the set of all possible outputs, and the range is the set of actual outputs. Examples show how to use inequalities to find the domain by determining when a function gives real values and how to manipulate equations to find the range.
Calculus
The document defines domain and range. Domain is the set of all possible input values of a function. Range is the set of all output values of a function as the input variable takes on all possible values. It then verifies the domain and range of several functions, including square root, square, and inverse functions. It also evaluates compositions of functions where f(x)=x^2 and g(x)=x-3. Finally, it evaluates compositions of functions where f(x)=sqrt(x), g(x)=x/2, and h(x)=x-8.
Iterative1
The document describes the iterative method for finding roots of equations. It discusses:
1) How to set up an iterative process to find a root by taking an initial approximation x0 and repeatedly applying the function pi(x) to get closer approximations x1, x2, etc. until the change between successive values is very small.
2) An example of using the method to find the root of the quadratic equation 2x^2 - 4x + 1 = 0. It shows calculating successive approximations that converge to the root.
3) How the iterative method can be implemented in C++ using a function that takes the initial value and returns the updated value at each iteration until the change is negligible.
10.5 more on language of functions x
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
Polynomial functionsandgraphs
This document discusses polynomial functions and their graphs. It defines polynomial functions as functions of the form P(x) = anxn + an-1xn-1 + ... + a1x + a0, where an is the leading coefficient. The degree of the polynomial determines features of its graph like the maximum number of x-intercepts. The leading coefficient test determines the end behavior of the graph. Key features of polynomial graphs are intercepts, extrema, and end behavior.
Gre 403 jeopardy
This document contains a quiz about linear equations and slope with 100 multiple choice questions across 6 categories: With Your Host, Tables And More, Identifying Equations, Old School, Things A Geek Might Know, and Daily Place A Wager. The questions cover topics such as finding the slope of a line given two points, writing equations of lines given slope and y-intercept, evaluating equations for given x-values, and solving multi-step equations.
9 the basic language of functions x
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
2 1 polynomials
The document discusses identifying and evaluating polynomial functions. It defines polynomial functions as functions with whole number exponents and real number coefficients, and no variables in denominators. Polynomials can be written in standard form showing the degree and coefficients. The key points covered are: identifying the degree of a polynomial, finding the zeros of a polynomial function, directly substituting values for x, and using synthetic substitution to evaluate polynomials for any value of x.
7.2 abs value function
This document provides notes for a math chapter on absolute value and reciprocal functions. It covers graphing and expressing absolute value functions as piecewise functions. Key points include: absolute value is defined as the distance from zero on the number line; absolute value functions are continuous and have the same x-intercept as the linear function inside the absolute value; taking the absolute value of a quadratic function preserves the vertex but changes the range. Care must be taken with the domains of absolute value functions depending on if the coefficient of the x-term inside is positive or negative.
Module 3 polynomial functions
This module discusses polynomial functions of degree greater than two. The key points are:
1. The graph of a third-degree polynomial has both a minimum and maximum point, while higher degree polynomials have one less turning point than their degree.
2. Methods like finding upper and lower bounds and Descartes' Rule of Signs can help determine properties of the graph like zeros.
3. Odd degree polynomials increase on the far left and right if the leading term is positive, and decrease if negative. Even degree polynomials increase on the far left and decrease on the far right, or vice versa.
Produccion escrita expresiones algebraicas
The document discusses various algebraic operations including:
1) Summing algebraic expressions by adding like terms.
2) Subtracting algebraic expressions by adding the opposite of like terms.
3) Multiplying algebraic expressions by multiplying each term in one factor by each term in the other factor.
4) Dividing algebraic expressions using long division.
3.3 Zeros of Polynomial Functions
This document discusses key concepts related to finding zeros of polynomial functions including:
1) The factor theorem, which states that a polynomial x - k is a factor of a function f(x) if and only if f(k) = 0.
2) The rational zeros theorem, which gives possible rational zeros based on the factors of the leading coefficient and constant term.
3) The fundamental theorem of algebra, which states that every polynomial of degree n has n complex zeros and examples of finding functions based on given zeros.
4) The conjugate zeros theorem, which states that if z = a + bi is a zero, then z = a - bi is also a zero for polynomials with real coefficients.
5
Polynomials Grade 10
This document provides instructions on how to multiply, divide, and factor polynomials. It discusses:
1) Multiplying polynomials by distributing terms and using FOIL for binomials.
2) Dividing polynomials using long division.
3) Factoring polynomials using grouping, finding two numbers whose product is the constant and sum is the coefficient, and recognizing difference of squares.
Algebraic functions powerpoint
This document defines and provides examples of algebraic functions. It explains that algebraic functions involve algebraic operations like addition, subtraction, multiplication, and division on variables and constants. It also discusses linear algebraic functions, which can be represented by a straight line on the Cartesian plane in the form of f(x)=mx+b. The document provides examples of linear functions using tables, graphs, and solving linear equations for y. It compares linear and quadratic functions, noting that while linear functions can be written as fractions, quadratic functions involve squared variables and are not linear.
Application of derivative
Additional Applications of the Derivative related to engineering maths
this presentation is only for the knowledge...............
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
Christian Jett Morales presents on the quadratic function derived from zeros of the function. The presentation includes:
- An introduction to deriving the quadratic function f(x) = ax^2 + bx + c from the zeros r1 and r2 of the function.
- Examples of deriving quadratic functions when given different zeros, such as 3/2 and -1, or 2+√3 and 2-√3.
- An activity for participants to practice deriving quadratic functions.
- A dedication to God and the presenter's teacher for the opportunity to create the presentation.
Ani agustina (a1 c011007) polynomial
The document describes aspects of polynomials including:
- Definitions of polynomials and their components
- Algebraic operations on polynomials like addition, subtraction, and multiplication
- Dividing polynomials using long division and synthetic (Horner's) division methods
- The remainder theorem stating the remainder of dividing a polynomial by (x - k) is the value of the polynomial at k
- The factorization theorem relating the factors of a polynomial to its roots
- Techniques for factorizing polynomials based on the sums of coefficients
The document contains examples and explanations of key polynomial concepts over multiple pages.
Finding zeros of a quadratic function
The document provides examples of finding the zeros of quadratic functions by factorizing and setting each factor equal to 0. It then lists 5 additional quadratic functions and assigns the reader to find their zeros.
What's hot (19)
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
Similar to 06 - 20 Jan - Divide and Conquer
4.2,4.3 graphing
The document discusses how to graph linear equations. It states that the graph of an equation is the set of all solutions, where a solution is an ordered pair that makes the equation true. It provides steps for graphing an equation, which include rewriting the equation to solve for y, making an input/output table using values for x, plotting the ordered pairs, and connecting the points with a straight line. It also discusses that the graph of y=b is a horizontal line and the graph of x=a is a vertical line. Finally, it notes that intercepts are found by plugging values into the equation - the x-intercept uses y=0 and the y-intercept uses x=0.
Jurnal informatika
The document describes the mergesort algorithm for sorting data in linear time complexity of O(n log n). Mergesort uses a divide-and-conquer approach by recursively dividing an array into two halves, sorting each half using mergesort, and then merging the two sorted halves back into a single sorted array. It analyzes the runtime using recurrence relations and proves it is O(n log n).
Java code to find the closest pair using divide and conquer algorith.pdf
Java code to find the closest pair using divide and conquer algorithm.
Solution
The Brute force solution is O(n^2), compute the distance between each pair and return the
smallest. But Devide and conquer is better and completes in O(nLogn) time.
Please find below the code:
import java.util.*;
// compile: javac ClosestPair.java
// run: Java ClosestPair 500
// Here 500 is some randomly generated set of input points.
/*
The algorithm is :
Input: An array of n points P[]
Output: The smallest distance between two points in the given array.
As a pre-processing step, input array is sorted according to x coordinates.
1) Find the middle point in the sorted array, we can take P[n/2] as middle point.
2) Divide the given array in two halves. The first subarray contains points from P[0] to P[n/2].
The second subarray contains points from P[n/2+1] to P[n-1].
3) Recursively find the smallest distances in both subarrays. Let the distances be dl and dr. Find
the minimum of dl and dr. Let the minimum be d.
*/
public class ClosestPair
{
public static class Point
{
public final double x;
public final double y;
public Point(double x, double y)
{
this.x = x;
this.y = y;
}
public String toString()
{ return \"(\" + x + \", \" + y + \")\"; }
}
public static class Pair
{
public Point point1 = null;
public Point point2 = null;
public double distance = 0.0;
public Pair()
{ }
public Pair(Point point1, Point point2)
{
this.point1 = point1;
this.point2 = point2;
calcDistance();
}
public void update(Point point1, Point point2, double distance)
{
this.point1 = point1;
this.point2 = point2;
this.distance = distance;
}
public void calcDistance()
{ this.distance = distance(point1, point2); }
public String toString()
{ return point1 + \"-\" + point2 + \" : \" + distance; }
}
public static double distance(Point p1, Point p2)
{
double xdist = p2.x - p1.x;
double ydist = p2.y - p1.y;
return Math.hypot(xdist, ydist);
}
public static Pair bruteForce(List points)
{
int numPoints = points.size();
if (numPoints < 2)
return null;
Pair pair = new Pair(points.get(0), points.get(1));
if (numPoints > 2)
{
for (int i = 0; i < numPoints - 1; i++)
{
Point point1 = points.get(i);
for (int j = i + 1; j < numPoints; j++)
{
Point point2 = points.get(j);
double distance = distance(point1, point2);
if (distance < pair.distance)
pair.update(point1, point2, distance);
}
}
}
return pair;
}
public static void sortByX(List points)
{
Collections.sort(points, new Comparator() {
public int compare(Point point1, Point point2)
{
if (point1.x < point2.x)
return -1;
if (point1.x > point2.x)
return 1;
return 0;
}
}
);
}
public static void sortByY(List points)
{
Collections.sort(points, new Comparator() {
public int compare(Point point1, Point point2)
{
if (point1.y < point2.y)
return -1;
if (point1.y > point2.y)
return 1;
return 0;
}
}
);
}
public static Pair divideAndConquer(List points)
{
List pointsSortedByX = new ArrayList(points);
sortByX(pointsSortedByX);
List pointsSortedByY = new ArrayList(points);
sortByY.
ClosestPairClosestPairClosestPairClosestPair
ClosestPairClosestPairClosestPairClosestPairClosestPairClosestPairClosestPairClosestPairClosestPairClosestPairClosestPair
Cost allocation in vertex weighted Steiner tree games
The document summarizes a research paper on vertex weighted Steiner tree games. It introduces:
1) A variant of the minimum cost spanning tree game where players receive rewards for connecting to a central supplier and must share the total profit of rewards minus connection costs.
2) An allocation rule that divides the total profit among players, which yields a core element if and only if the outcome is efficient.
3) Conditions under which the allocation rule results in an efficient outcome are investigated.
Polynomials
This document defines polynomials and discusses their key properties. It begins by defining a polynomial as an algebraic expression with two or more terms where the power of each variable is a positive integer. The degree of a polynomial is defined as the highest power of the variable. Polynomials are then classified based on their degree as constant, linear, quadratic, cubic, etc. The document also discusses the zeros or roots of a polynomial, which are the values that make the polynomial equal to zero. It shows how the zeros relate to the coefficients of the polynomial and can be found using factoring or solving techniques. Examples are provided to illustrate dividing polynomials using the division algorithm.
Approximate Nearest Neighbour in Higher Dimensions
This document discusses approximate nearest neighbor (ANN) search in high dimensional spaces. It begins by introducing the ANN problem and noting the "curse of dimensionality" that makes exact searches inefficient in high dimensions. It then discusses constructing a (1+ε)-approximate NN data structure for the Hamming cube using locality sensitive hashing (LSH). The data structure uses O(dn + n1+ρ) space and O(nρ) hash probes per query, where ρ depends on sensitivity properties of the hash family. The document also discusses using LSH for ANN search in Euclidean spaces by projecting points to random lines, using multiple projections to amplify probabilities of nearby points hashing to the same value.
An algorithm for computing resultant polytopes
Presentation of the paper "An output-sensitive algorithm for computing (projections of) resultant polytopes" in the Annual Symposium on Computational Geometry (SoCG 2012)
Nbhm m. a. and m.sc. scholarship test 2005
This document contains instructions for a scholarship test consisting of 60 multiple choice questions across three sections: Algebra, Analysis, and Geometry. The instructions specify that candidates are to write their answers in the spaces provided at the end of each question in the form of words, numbers, or simple mathematical expressions. Certain questions require identifying true statements from multiple choices, for which candidates should write the label(s) of the true statement(s).
factoring
The security of the RSA algorithm depends on the difficulty of factoring large numbers. The best known factoring algorithms are trial division, Dixon's algorithm, the quadratic sieve, and the number field sieve. The quadratic sieve and number field sieve are parallelizable algorithms that improve on Dixon's algorithm by using a "sieving" technique to more efficiently find relations between factors. While factoring performance improves incrementally over time, a large key size (over 300 bits) is still considered secure against the best known factoring methods.
Applied Math 40S June 2 AM, 2008
The document discusses arithmetic sequences, including defining them recursively or through an implicit linear equation, finding the common difference and nth term, and examples of determining terms and sums of arithmetic sequences given initial terms or the values of a and d. It also provides homework problems involving identifying arithmetic sequences, writing terms, and calculating sums of sequences.
Cs6402 design and analysis of algorithms may june 2016 answer key
The document discusses algorithms and complexity analysis. It provides Euclid's algorithm for computing greatest common divisor, compares the orders of growth of n(n-1)/2 and n^2, and describes the general strategy of divide and conquer methods. It also defines problems like the closest pair problem, single source shortest path problem, and assignment problem. Finally, it discusses topics like state space trees, the extreme point theorem, and lower bounds.
Algorithms
1) The lower bound theory is used to determine if an algorithm is the most efficient possible by comparing its time complexity f(n) to a lower bound complexity g(n).
2) For comparison-based sorting algorithms, the decision tree model shows that any sorting algorithm requires at least O(nlogn) comparisons in the worst case.
3) This lower bound of O(nlogn) time complexity for comparison-based sorting algorithms means that no algorithm can sort faster than nlogn time using only element comparisons.
NUMERICAL METHODS
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
Share with it ur friends & Follow me for more updates.!
January 13, 2014
The document provides information and practice questions about linear functions and slope:
- It instructs the reader to prepare for an upcoming final exam and coordinate test.
- Several practice questions are provided about slope, equations of lines, graphs, and using linear functions to model real-world scenarios.
- The questions cover topics like determining the slope between two points, writing equations in point-slope form, interpreting slope in context, and using linear models to solve word problems.
A Tutorial on Computational Geometry
My talk about computational geometry in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
Polynomial Function and Synthetic Division
This file is about Polynomial Function and Synthetic Division. A project passed to Mrs. Marissa De Ocampo. Submitted by Group 6 of Grade 10-Galilei of Caloocan National Science and Technology High School '15-'16
Disjoint sets
The document summarizes the disjoint-set data structure and algorithms for implementing union-find operations on disjoint sets. Key points:
- Disjoint sets allow representing partitions of elements into disjoint groups and supporting operations to merge groups and find the group a element belongs to.
- Several implementations are described, including linked lists and rooted trees (forests).
- The weighted-union heuristic improves the naive linked list implementation from O(n^2) to O(m log n) time by merging shorter lists into longer ones.
- In forests, union-by-rank and path compression heuristics combine sets in nearly linear time by balancing tree heights and flattening paths during finds.
Imc2020 day1&2 problems&solutions
This document contains solutions to 4 problems from an IMC 2020 Online event. Solution 1 computes the number of words of length n over an alphabet that satisfy certain properties, finding the number is 4n-1+2n-1. Solution 2 uses a recurrence relation approach to derive the same formula. Solution 3 uses generating functions to show the number is also 4n-1+2n-1. Solution 4 shows that if a polynomial p satisfies p(x+1)-p(x)=x100 for all x, then p(1-t)≤p(t) for 0≤t≤1/2.
Ch07 6
The document discusses complex eigenvalues and eigenvectors for systems of linear differential equations. It shows that if the matrix A has complex conjugate eigenvalue pairs r1 and r2, then the corresponding eigenvectors and solutions will also be complex conjugates. This leads to real-valued fundamental solutions that can express the general solution. An example demonstrates these concepts, finding the complex eigenvalues and eigenvectors and expressing the general solution in terms of real-valued functions. Spiral points, centers, eigenvalues, and trajectory behaviors are also summarized.
Similar to 06 - 20 Jan - Divide and Conquer (20)
Java code to find the closest pair using divide and conquer algorith.pdf
Java code to find the closest pair using divide and conquer algorith.pdf
Cost allocation in vertex weighted Steiner tree games
Cost allocation in vertex weighted Steiner tree games
Approximate Nearest Neighbour in Higher Dimensions
Approximate Nearest Neighbour in Higher Dimensions
Cs6402 design and analysis of algorithms may june 2016 answer key
Cs6402 design and analysis of algorithms may june 2016 answer key
More from Neeldhara Misra
Erdos Posa Theorem
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
15 - 12 Feb - Stable Matchings
This document summarizes an algorithm lecture about stable matchings. It introduces the stable matching problem and Hall's marriage theorem. It then describes Gale-Shapley's stable matching algorithm, where men and women rank their preferences and men iteratively propose to the highest ranked remaining women. The algorithm is proven to always terminate with a stable matching in polynomial time.
07- 22 Jan - Divide and Conquer
The document summarizes the Master's Theorem for analyzing divide and conquer recurrences. It introduces three cases for the recurrence T(n) ≤ aT(n/b) + O(nd): (1) if a = bd, the solution is T(n) = O(ndlogn); (2) if a < bd, the solution is T(n) = O(nd); (3) if a > bd, the solution is T(n) = O(alogbn). Examples like binary search and merge sort are shown to illustrate the three cases.
10 - 29 Jan - Recursion Part 2
The document discusses algorithms for solving 3CNF formulas using recursion. It describes a recursive algorithm that picks a variable from a clause, and recursively checks if the formula is satisfiable with that variable set to 1 and then 0. This results in a recurrence relation of T(n) = T(n-1) + T(n-2) + T(n-3). The document also discusses using "clever base cases" like eliminating degree 0 and 1 vertices to improve the running time of algorithms on graphs.
05 - 18 Jan - The Sorting Wrap-Up
This document discusses sorting algorithms and finding medians using heaps. It describes how heapify takes linear time overall to turn an array into a heap. To sort an array using a heap, repeatedly extract the maximum element and place it at the end of the array, requiring O(n log n) time. Finding medians can be done by storing the first half of elements in a max-heap and the second half in a min-heap.
02 - 04 Jan - Sorting (Continued)
The document discusses the CS 321 Algorithm Analysis and Design course. It covers announcements about homework, grading, and clarifications. Algorithm analysis is introduced, covering running time and correctness. Insertion sort is presented, including a loop invariant and examples. Best-case, average-case, and worst-case running times are defined. Asymptotics and big O notation are introduced. Merge sort is mentioned along with induction.
04 - 15 Jan - Heap Sort
This document discusses heap sort and how to implement a min heap. It shows the steps of heap sort by organizing elements into a min heap where the first element is always the minimum. It explains that a min heap can support fast insertion and deletion of minimum elements in O(log n) time by maintaining the heap property that a node is larger than its children. The key aspects are using a complete binary tree structure represented as an array and algorithms for heapify, insert, and delete min that utilize the tree's structure.
08 - 25 Jan - Divide and Conquer
This document discusses divide and conquer algorithms for multiplying integers and finding the kth smallest element. It describes how to multiply integers by dividing them into left and right halves, combining results. Finding the kth smallest element involves choosing a pivot, partitioning around it, and recursively searching halves until the kth element is found. Both algorithms take O(n log n) time by dividing the problem size by a constant factor at each step.
13 - 06 Feb - Dynamic Programming
The document summarizes content from a lecture on dynamic programming techniques, including the knapsack problem, shortest paths, and edit distance. It provides examples of using dynamic programming to solve the 0/1 knapsack problem by building up a 2D array from the bottom up where each entry stores the maximum value for items fitting in a knapsack of given size. It also explains calculating edit distance by building a 2D array where each entry stores the minimum edit operations to change one string to another.
00 - 30 Dec - Introduction
This document outlines the course details for CS 321 Algorithm Analysis and Design. It includes information on logistics like office hours and TAs, graded work like assignments and exams, starred problems, discussion policy, and an overview of design techniques, complexity issues, and key tools covered in the course. The document also provides pre-requisites and motivates the importance of algorithms through quotes about using both theory and practice to improve each other.
01 - 01 January - Sorting
This document discusses sorting algorithms that are covered in a CS 321 algorithms course. It introduces several sorting algorithms like counting sort, bubble sort, merge sort, quicksort, and insertion sort. For each algorithm, it provides the time complexity. It also mentions that merge sort runs in O(n log n) time due to having log n layers of recursion where each layer takes O(n) work. Quicksort's recursion tree can potentially be skewed, leading to O(n^2) performance in the worst case. References provided direct the reader to a video and website for learning more about sorting algorithms.
14 - 08 Feb - Dynamic Programming
This document discusses dynamic programming techniques for three problems: edit distance, longest increasing subsequence, and matrix chain multiplication. It provides examples of how to calculate the edit distance between two strings by finding the minimum cost of insertions, deletions, and replacements needed to convert one string to the other. It then explains how to find the longest increasing subsequence of a given array using dynamic programming by building up the solution bottom-up. Finally, it briefly mentions finding the most efficient way to multiply a chain of matrices using dynamic programming.
11 - 03 Feb - From Recursion to Dynamic Programming
This document discusses recursion and memoization. It introduces the Fibonacci numbers and the problem of tiling a corridor of length n. It shows the recursive definition of the number of tilings as C(n) = C(n-1) + C(n-2), with base cases C(1)=1 and C(2)=2. It notes that directly implementing this recursion leads to redundant computations. Memoization is proposed as a technique to store computed values in a table to avoid recomputing them, making the algorithm more efficient.
09 - 27 Jan - Recursion Part 1
The document discusses algorithms for solving satisfiability problems on conjunctive normal form (CNF) formulas. It first describes a trivial algorithm that tries all possible assignments, with a runtime of O(2^n*n^3). It then discusses using a recursive approach, but shows this results in an even worse runtime of O(3^n*n^3). Finally, it proposes a recursive algorithm that runs in O(1.8^n) by recursively trying assignments on subsets of variables.
04 - 15 Jan - Heap Sort
This document discusses heap sort and how to implement a min heap. It shows the steps of heap sort by organizing elements into a min heap where the first element is always the minimum. It explains that a min heap can support fast insertion and deletion of minimum elements in O(log n) time by maintaining the heap property that a node is larger than its children. The key aspects are using a complete binary tree structure represented in an array and algorithms for heapify, insert, and delete min that maintain the min heap ordering.
12 - 05 Feb - Dynamic Programming
The document discusses dynamic programming and its application to finding the independent set with the largest total weight on a path. It describes how dynamic programming can be used to solve this problem by breaking it down into subproblems and storing the solution to each subproblem in an array A, such that A[i] stores the maximum weight independent set for the subpath on the first i vertices. The value of each entry A[i] is computed recursively as the maximum of A[i-1] and A[i-2] + the weight of vertex i.
More from Neeldhara Misra (16)
11 - 03 Feb - From Recursion to Dynamic Programming
11 - 03 Feb - From Recursion to Dynamic Programming
Recently uploaded
How to Add Chatter in the odoo 17 ERP Module
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Azure Interview Questions and Answers PDF By ScholarHat
Azure Interview Questions and Answers PDF By ScholarHat
How to Manage Your Lost Opportunities in Odoo 17 CRM
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptxMohd Adib Abd Muin, Senior Lecturer at Universiti Utara Malaysia
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...National Information Standards Organization (NISO)
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Advanced Java[Extra Concepts, Not Difficult].docx
This is part 2 of my Java Learning Journey. This contains Hashing, ArrayList, LinkedList, Date and Time Classes, Calendar Class and more.
The History of Stoke Newington Street Names
Presented at the Stoke Newington Literary Festival on 9th June 2024
www.StokeNewingtonHistory.com
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Aberdeen
How to Fix the Import Error in the Odoo 17
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Recently uploaded (20)
Azure Interview Questions and Answers PDF By ScholarHat
Azure Interview Questions and Answers PDF By ScholarHat
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Executive Directors Chat Leveraging AI for Diversity, Equity, and Inclusion
Pride Month Slides 2024 David Douglas School District
Pride Month Slides 2024 David Douglas School District
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
The simplified electron and muon model, Oscillating Spacetime: The Foundation...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
06 - 20 Jan - Divide and Conquer
- 1. CS 321. Algorithm Analysis & Design Lecture 6 Divide and Conquer
- 2. Closest Pair
- 4. Input: A set of n points given in terms of (x,y) coordinates.
- 5. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 6. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
- 7. Compute distances between all pairs. Approach #0 O(n2)
- 8. Sort the points. Sweep through adjacent pairs. Approach 1 Time to sort + O(n)
- 9. Divide and Conquer. Split the point set, use recursive solutions. Approach 2 Cost of Recursion + Time to merge
- 10. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
- 11. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair.
- 12. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair. Closest Pair on Left - d1 Closest Pair on Right - d2
- 13. Input: A set of n points given in terms of (x) coordinates. Output: The closest pair. Closest Pair on Left - d1 Closest Pair on Right - d2
- 14. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 15. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. 1 2 3 4 5 6 7 8 9 10 11 12 13
- 16. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 17. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. 13 1 7 5 3 9 8 11 10 4 12 2 6
- 18. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 19. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 20. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 21. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 22. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair.
- 23. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. d/2 d/2
- 24. Input: A set of n points given in terms of (x,y) coordinates. Output: The closest pair. d/2 d/2
- 32. T(n) ≤ 2T(n/2) + O(n) T(n) = O(n log n)