monte carlo method - west virginia university
TRANSCRIPT
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Outline
Monte Carlo Method
Vamshi Krishna Vudepu1
1Lane Department of Computer Science and Electrical EngineeringWest Virginia University
05 April, 2012
Vamshi Randomized Algorithms 1of 26
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Outline
Outline
1 IntroductionMonte Carlo DefinitionExamplesDefinitions
2 ApplicationsThe DNF Counting ProblemDNF Couting Algorithms
Vamshi Randomized Algorithms 2of 26
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Outline
Outline
1 IntroductionMonte Carlo DefinitionExamplesDefinitions
2 ApplicationsThe DNF Counting ProblemDNF Couting Algorithms
Vamshi Randomized Algorithms 2of 26
![Page 4: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/4.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Outline
1 IntroductionMonte Carlo DefinitionExamplesDefinitions
2 ApplicationsThe DNF Counting ProblemDNF Couting Algorithms
Vamshi Randomized Algorithms 3of 26
![Page 5: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/5.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
What is a Monte Carlo Method?
The Monte Carlo Method refers to a collection of tools for evaluating values throughsampling and simulation.
Why Monte Carlo Method?
(i) Used when inputs are uncertain and are specified as probability distribution.
(ii) Predicts output values based on input samples.
Vamshi Randomized Algorithms 4of 26
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
What is a Monte Carlo Method?
The Monte Carlo Method refers to a collection of tools for evaluating values throughsampling and simulation.
Why Monte Carlo Method?
(i) Used when inputs are uncertain and are specified as probability distribution.
(ii) Predicts output values based on input samples.
Vamshi Randomized Algorithms 4of 26
![Page 7: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/7.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
What is a Monte Carlo Method?
The Monte Carlo Method refers to a collection of tools for evaluating values throughsampling and simulation.
Why Monte Carlo Method?
(i) Used when inputs are uncertain and are specified as probability distribution.
(ii) Predicts output values based on input samples.
Vamshi Randomized Algorithms 4of 26
![Page 8: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/8.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
What is a Monte Carlo Method?
The Monte Carlo Method refers to a collection of tools for evaluating values throughsampling and simulation.
Why Monte Carlo Method?
(i) Used when inputs are uncertain and are specified as probability distribution.
(ii) Predicts output values based on input samples.
Vamshi Randomized Algorithms 4of 26
![Page 9: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/9.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
What is a Monte Carlo Method?
The Monte Carlo Method refers to a collection of tools for evaluating values throughsampling and simulation.
Why Monte Carlo Method?
(i) Used when inputs are uncertain and are specified as probability distribution.
(ii) Predicts output values based on input samples.
Vamshi Randomized Algorithms 4of 26
![Page 10: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/10.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
General steps involved in Monte Carlo Methods
(i) Define a domain of possible inputs.
(ii) Choose inputs at random from the probability distribution over the domain.
(iii) Perform computation on inputs and aggregate the results.
Vamshi Randomized Algorithms 5of 26
![Page 11: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/11.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
General steps involved in Monte Carlo Methods
(i) Define a domain of possible inputs.
(ii) Choose inputs at random from the probability distribution over the domain.
(iii) Perform computation on inputs and aggregate the results.
Vamshi Randomized Algorithms 5of 26
![Page 12: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/12.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
General steps involved in Monte Carlo Methods
(i) Define a domain of possible inputs.
(ii) Choose inputs at random from the probability distribution over the domain.
(iii) Perform computation on inputs and aggregate the results.
Vamshi Randomized Algorithms 5of 26
![Page 13: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/13.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
General steps involved in Monte Carlo Methods
(i) Define a domain of possible inputs.
(ii) Choose inputs at random from the probability distribution over the domain.
(iii) Perform computation on inputs and aggregate the results.
Vamshi Randomized Algorithms 5of 26
![Page 14: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/14.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators. Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 15: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/15.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations?
Using Random number generators. Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 16: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/16.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators.
Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 17: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/17.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators. Consider 3black balls and 7 red balls in a bin.
The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 18: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/18.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators. Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively.
A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 19: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/19.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators. Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball.
We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 20: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/20.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators. Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input.
If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 21: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/21.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators. Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked.
For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 22: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/22.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definition
Sampling
How can you ensure that an input is selected uniformly at random from the given inputdomain in Computer Simulations? Using Random number generators. Consider 3black balls and 7 red balls in a bin. The probability of choosing a black ball and a redball is is 0.3 and 0.7 respectively. A random number generator which generates anynumber between 0 and 1 (equally likely) can be used to choose a ball. We canassociate each possible random number generated to the input. If the numbergenerated is less than or equal to 0.3, then a black ball will be picked. For the numbersbetween 0.3 and 1, red ball will be selected.
Vamshi Randomized Algorithms 6of 26
![Page 23: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/23.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Outline
1 IntroductionMonte Carlo DefinitionExamplesDefinitions
2 ApplicationsThe DNF Counting ProblemDNF Couting Algorithms
Vamshi Randomized Algorithms 7of 26
![Page 24: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/24.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice. For example, the probability that the sum shows up 6 is 0.138. Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice. Ifthe sum six showed up 15 times, the probability could be concluded as 0.15. How canyou increase accuracy? Accuracy increases with increase in number of times we rollthe dice.Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 25: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/25.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice.
For example, the probability that the sum shows up 6 is 0.138. Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice. Ifthe sum six showed up 15 times, the probability could be concluded as 0.15. How canyou increase accuracy? Accuracy increases with increase in number of times we rollthe dice.Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 26: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/26.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice. For example, the probability that the sum shows up 6 is 0.138.
Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice. Ifthe sum six showed up 15 times, the probability could be concluded as 0.15. How canyou increase accuracy? Accuracy increases with increase in number of times we rollthe dice.Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 27: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/27.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice. For example, the probability that the sum shows up 6 is 0.138. Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice.
Ifthe sum six showed up 15 times, the probability could be concluded as 0.15. How canyou increase accuracy? Accuracy increases with increase in number of times we rollthe dice.Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 28: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/28.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice. For example, the probability that the sum shows up 6 is 0.138. Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice. Ifthe sum six showed up 15 times, the probability could be concluded as 0.15.
How canyou increase accuracy? Accuracy increases with increase in number of times we rollthe dice.Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 29: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/29.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice. For example, the probability that the sum shows up 6 is 0.138. Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice. Ifthe sum six showed up 15 times, the probability could be concluded as 0.15. How canyou increase accuracy?
Accuracy increases with increase in number of times we rollthe dice.Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 30: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/30.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice. For example, the probability that the sum shows up 6 is 0.138. Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice. Ifthe sum six showed up 15 times, the probability could be concluded as 0.15. How canyou increase accuracy? Accuracy increases with increase in number of times we rollthe dice.
Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 31: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/31.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Consider two fair dice.Calculate the probability for particular sum of outcomes of thetwo dice. For example, the probability that the sum shows up 6 is 0.138. Roll the dicemanually 100 times and note down how many times the sum six turns up on the dice. Ifthe sum six showed up 15 times, the probability could be concluded as 0.15. How canyou increase accuracy? Accuracy increases with increase in number of times we rollthe dice.Monte Carlo method simulates rolling of dice for given number of times (say forinstance 1000 times) and gives the probability.
Vamshi Randomized Algorithms 8of 26
![Page 32: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/32.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 33: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/33.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of π
Draw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 34: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/34.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it.
Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 35: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/35.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random.
What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 36: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/36.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle?
Ratio of area of circle to the area of square which is π4 . Ratio of number of
grains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 37: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/37.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square
which is π4 . Ratio of number of
grains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 38: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/38.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is
π4 . Ratio of number of
grains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 39: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/39.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 .
Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 40: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/40.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value.
Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 41: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/41.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 43: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/43.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 44: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/44.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 45: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/45.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimate the value of πDraw a square of 2× 2 units on floor and inscribe a circle in it. Scatter some objects ofsame size (such as grains) at random. What is the probability that the object falls insidethe circle? Ratio of area of circle to the area of square which is π
4 . Ratio of number ofgrains inside the circle to the total number of grains inside the square givesapproximately the same value. Multiplying with 4 gives the approximate value of π.
Note
To get the accurate value
(i) The number of objects should be large enough.
(ii) Each object has to be scattered uniformly at random. Object should not bedropped purposefully at a particular location inside the square.
Vamshi Randomized Algorithms 9of 26
![Page 46: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/46.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo method
Let (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 48: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/48.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0).
Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 49: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/49.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.
Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 50: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/50.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{
1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 51: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/51.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,
0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 52: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/52.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 53: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/53.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 54: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/54.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 55: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/55.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Estimating π value using Monte Carlo methodLet (X ,Y ) be a point chosen uniformly at random in a 2× 2 square centered at origin(0, 0). Consider a circle of radius 1 unit centered at origin inscribed in the square.Consider a random variable Z such that
Z =
{1, if (X ,Y ) lies inside the circle,0, Otherwise.
Pr(Z = 1) =π
4
Assuming that the experiment is run m times with X and Y chosen independentlyamong the runs, let
W =m∑
i=1
Zi
E[W ] = E
[ m∑i=1
Zi
]=
m∑i=1
E[Zi ] =m · π
4
Vamshi Randomized Algorithms 10of 26
![Page 56: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/56.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
![Page 58: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/58.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
![Page 59: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/59.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
![Page 60: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/60.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )
= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
![Page 61: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/61.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
![Page 62: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/62.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
![Page 63: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/63.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Examples
Example
Hence, W ′ = 4m ·W is an estimate of π.
Applying the following Chernoff bound,
Pr( |X − µ | ≥ δ · µ) ≤ 2 · e−µ·δ2/3
Pr( |W ′ − Π | ≥ ε · π) = Pr( |W − m·π|4 | ≥ ε·m·π
4 )= Pr( |W − E[W ] | ≥ ε · E[W ])
≤ 2 · e−m·Π·ε2/12
Therefore, m should be sufficiently large to obtain a tight approximation of π w.h.p.
Vamshi Randomized Algorithms 11of 26
![Page 64: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/64.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Outline
1 IntroductionMonte Carlo DefinitionExamplesDefinitions
2 ApplicationsThe DNF Counting ProblemDNF Couting Algorithms
Vamshi Randomized Algorithms 12of 26
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?As long as ε ≤ 1, m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 66: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/66.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?As long as ε ≤ 1, m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 67: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/67.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?As long as ε ≤ 1, m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 68: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/68.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?As long as ε ≤ 1, m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 69: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/69.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?
As long as ε ≤ 1, m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 70: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/70.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?As long as ε ≤ 1,
m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 71: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/71.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?As long as ε ≤ 1, m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 72: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/72.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A randomized algorithm gives an (ε, δ)- approximation for the value V if the output X ofthe algorithm satisfies
Pr( |X − V | ≤ εV ) ≥ 1− δ
Question
In the π example, what should be the value of m for the algorithm to give an(ε, δ)-appromixation?As long as ε ≤ 1, m should be large enough to make
2 · e−m·π·ε2
12 ≤ δ
Solving the above equation, we get
m ≥12 · ln(2/δ)
π · ε2
Vamshi Randomized Algorithms 13of 26
![Page 73: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/73.jpg)
IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Theorem
Let X1, ...,Xm be independent and identically distributed indicator random variables,with µ = E[Xi ]. If m ≥ 3·ln(2/δ)
ε2·µ , then
Pr
( ∣∣∣∣∣ 1m
m∑i=1
Xi − µ
∣∣∣∣∣ ≥ ε · µ)≤ δ
i.e, m samples provides an (ε, δ) approximation for µ.
Vamshi Randomized Algorithms 14of 26
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Theorem
Let X1, ...,Xm be independent and identically distributed indicator random variables,
with µ = E[Xi ]. If m ≥ 3·ln(2/δ)
ε2·µ , then
Pr
( ∣∣∣∣∣ 1m
m∑i=1
Xi − µ
∣∣∣∣∣ ≥ ε · µ)≤ δ
i.e, m samples provides an (ε, δ) approximation for µ.
Vamshi Randomized Algorithms 14of 26
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Theorem
Let X1, ...,Xm be independent and identically distributed indicator random variables,with µ = E[Xi ].
If m ≥ 3·ln(2/δ)
ε2·µ , then
Pr
( ∣∣∣∣∣ 1m
m∑i=1
Xi − µ
∣∣∣∣∣ ≥ ε · µ)≤ δ
i.e, m samples provides an (ε, δ) approximation for µ.
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Theorem
Let X1, ...,Xm be independent and identically distributed indicator random variables,with µ = E[Xi ]. If m ≥ 3·ln(2/δ)
ε2·µ , then
Pr
( ∣∣∣∣∣ 1m
m∑i=1
Xi − µ
∣∣∣∣∣ ≥ ε · µ)≤ δ
i.e, m samples provides an (ε, δ) approximation for µ.
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Theorem
Let X1, ...,Xm be independent and identically distributed indicator random variables,with µ = E[Xi ]. If m ≥ 3·ln(2/δ)
ε2·µ , then
Pr
( ∣∣∣∣∣ 1m
m∑i=1
Xi − µ
∣∣∣∣∣ ≥ ε · µ)≤ δ
i.e, m samples provides an (ε, δ) approximation for µ.
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Theorem
Let X1, ...,Xm be independent and identically distributed indicator random variables,with µ = E[Xi ]. If m ≥ 3·ln(2/δ)
ε2·µ , then
Pr
( ∣∣∣∣∣ 1m
m∑i=1
Xi − µ
∣∣∣∣∣ ≥ ε · µ)≤ δ
i.e, m samples provides an (ε, δ) approximation for µ.
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A fully polynomial randomized approximation scheme (FPRAS) for a problem is arandomized algorithm for which, given an input x and any parameters ε and δ with0 < ε, δ < 1, the algorithm outputs an (ε, δ)-approximation to V (x) in time that ispolynomial in 1/ε, ln δ−1, and the size of the input x .
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A fully polynomial randomized approximation scheme (FPRAS) for a problem is arandomized algorithm for which,
given an input x and any parameters ε and δ with0 < ε, δ < 1, the algorithm outputs an (ε, δ)-approximation to V (x) in time that ispolynomial in 1/ε, ln δ−1, and the size of the input x .
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A fully polynomial randomized approximation scheme (FPRAS) for a problem is arandomized algorithm for which, given an input x and any parameters ε and δ with0 < ε, δ < 1,
the algorithm outputs an (ε, δ)-approximation to V (x) in time that ispolynomial in 1/ε, ln δ−1, and the size of the input x .
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A fully polynomial randomized approximation scheme (FPRAS) for a problem is arandomized algorithm for which, given an input x and any parameters ε and δ with0 < ε, δ < 1, the algorithm outputs an (ε, δ)-approximation to V (x) in time that ispolynomial in
1/ε, ln δ−1, and the size of the input x .
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A fully polynomial randomized approximation scheme (FPRAS) for a problem is arandomized algorithm for which, given an input x and any parameters ε and δ with0 < ε, δ < 1, the algorithm outputs an (ε, δ)-approximation to V (x) in time that ispolynomial in 1/ε,
ln δ−1, and the size of the input x .
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A fully polynomial randomized approximation scheme (FPRAS) for a problem is arandomized algorithm for which, given an input x and any parameters ε and δ with0 < ε, δ < 1, the algorithm outputs an (ε, δ)-approximation to V (x) in time that ispolynomial in 1/ε, ln δ−1,
and the size of the input x .
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IntroductionApplications
Monte Carlo DefinitionExamplesDefinitions
Definitions
Definition
A fully polynomial randomized approximation scheme (FPRAS) for a problem is arandomized algorithm for which, given an input x and any parameters ε and δ with0 < ε, δ < 1, the algorithm outputs an (ε, δ)-approximation to V (x) in time that ispolynomial in 1/ε, ln δ−1, and the size of the input x .
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
Outline
1 IntroductionMonte Carlo DefinitionExamplesDefinitions
2 ApplicationsThe DNF Counting ProblemDNF Couting Algorithms
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Note
Sampling in Monte Carlo methods is not as simple as shown for estimating Π value.Sampling is often nontrivial.
What is a DNF?
A Boolean formula which comprises of disjuntion (OR) of clauses where each clause isa conjunction (AND) of literals.
Example
(x1 ∧ x̄2) ∨ (x3 ∧ x4) ∨ (x̄4 ∧ x1)
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Note
Sampling in Monte Carlo methods is not as simple as shown for estimating Π value.
Sampling is often nontrivial.
What is a DNF?
A Boolean formula which comprises of disjuntion (OR) of clauses where each clause isa conjunction (AND) of literals.
Example
(x1 ∧ x̄2) ∨ (x3 ∧ x4) ∨ (x̄4 ∧ x1)
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Note
Sampling in Monte Carlo methods is not as simple as shown for estimating Π value.Sampling is often nontrivial.
What is a DNF?
A Boolean formula which comprises of disjuntion (OR) of clauses where each clause isa conjunction (AND) of literals.
Example
(x1 ∧ x̄2) ∨ (x3 ∧ x4) ∨ (x̄4 ∧ x1)
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Note
Sampling in Monte Carlo methods is not as simple as shown for estimating Π value.Sampling is often nontrivial.
What is a DNF?
A Boolean formula which comprises of disjuntion (OR) of clauses where each clause isa conjunction (AND) of literals.
Example
(x1 ∧ x̄2) ∨ (x3 ∧ x4) ∨ (x̄4 ∧ x1)
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Note
Sampling in Monte Carlo methods is not as simple as shown for estimating Π value.Sampling is often nontrivial.
What is a DNF?
A Boolean formula which comprises of disjuntion (OR) of clauses where each clause isa conjunction (AND) of literals.
Example
(x1 ∧ x̄2) ∨ (x3 ∧ x4) ∨ (x̄4 ∧ x1)
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Note
Sampling in Monte Carlo methods is not as simple as shown for estimating Π value.Sampling is often nontrivial.
What is a DNF?
A Boolean formula which comprises of disjuntion (OR) of clauses where each clause isa conjunction (AND) of literals.
Example
(x1 ∧ x̄2) ∨ (x3 ∧ x4) ∨ (x̄4 ∧ x1)
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Note
Sampling in Monte Carlo methods is not as simple as shown for estimating Π value.Sampling is often nontrivial.
What is a DNF?
A Boolean formula which comprises of disjuntion (OR) of clauses where each clause isa conjunction (AND) of literals.
Example
(x1 ∧ x̄2) ∨ (x3 ∧ x4) ∨ (x̄4 ∧ x1)
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment?
An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?
False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False.
The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF?
Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using
De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment?
Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF
Questions
(i) What is a satisfying assignment? An assignment which when given to a Booleanformula returns TRUE.
(ii) Determining the satisfiability of CNF is simpler compared to DNF. True/False?False. The input assignment needs to satisfy just one clause in a DNF.
(iii) How to check the satisfiability of a CNF? Convert it into DNF using De Morgan’slaw.
(iv) If H is a CNF of n Boolean variables and H̄ is its DNF, what is the maximumnumber of satisfying assignments can H̄ have for H to have a satisfyingassignment? Strictly less than 2n.
Note
Counting the number of satisfying assignments for a DNF comes under#P − Complete Problems.
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
Outline
1 IntroductionMonte Carlo DefinitionExamplesDefinitions
2 ApplicationsThe DNF Counting ProblemDNF Couting Algorithms
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF Counting Algorithm
Input: A DNF formula F with n variables.Output: Y = an approximation of c(F ).
1: X ← 02: for (k = 1 to m) do3: Generate a random assignment for n variables, chosen uniformly at random from
all 2n possible assignments.4: if (the random assignment satisfies F ) then5: X ← X + 16: end if7: end for8: return Y ← ( X
m ) · 2n
Algorithm 3.1: DNF Counting Algorithm I: Naive approach
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .
Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{
1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.
0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1?
c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
![Page 116: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/116.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
![Page 117: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/117.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
![Page 118: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/118.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
![Page 119: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/119.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
Notes
c(F )→ No. of satisfying assignments of DNF Formaula F .Let Xk be random variable such that
Xk =
{1, if k th iteration generates a satisfying assignment.0, Otherwise.
X =m∑
k=1
Xk .
What is the probability that Xk is 1? c(F )2n
E[Y ] =E[X ] · 2n
m= c(F )
Applying our previous theorem, Y gives an (ε, δ) -approximation of c(F ), when
m ≥3 · 2n · ln(2/δ)
ε2 · c(F )
Vamshi Randomized Algorithms 21of 26
![Page 120: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/120.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
Problems in Naive Approach
If c(F ) ≥ 2n/α(n) for some polynomial α, then we need number of samples mpolynomial in n, 1/ε, ln(1/δ).
However, c(F ) can be much less than 2n. In such case, it takes exponential number ofassignments before finding the first satisfying assignment.
The number of satisfying assignments might not be sufficiently dense enough in set ofall assignments.
To obtain FPRAS for this problem, we need to construct a better sample space thatincludes all the satisfying assignments of F and also, the satisfying assignments mustbe sufficiently dense in the sample space.
Vamshi Randomized Algorithms 22of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
Problems in Naive Approach
If c(F ) ≥ 2n/α(n) for some polynomial α, then we need number of samples mpolynomial in n, 1/ε, ln(1/δ).
However, c(F ) can be much less than 2n. In such case, it takes exponential number ofassignments before finding the first satisfying assignment.
The number of satisfying assignments might not be sufficiently dense enough in set ofall assignments.
To obtain FPRAS for this problem, we need to construct a better sample space thatincludes all the satisfying assignments of F and also, the satisfying assignments mustbe sufficiently dense in the sample space.
Vamshi Randomized Algorithms 22of 26
![Page 122: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/122.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
Problems in Naive Approach
If c(F ) ≥ 2n/α(n) for some polynomial α, then we need number of samples mpolynomial in n, 1/ε, ln(1/δ).
However, c(F ) can be much less than 2n.
In such case, it takes exponential number ofassignments before finding the first satisfying assignment.
The number of satisfying assignments might not be sufficiently dense enough in set ofall assignments.
To obtain FPRAS for this problem, we need to construct a better sample space thatincludes all the satisfying assignments of F and also, the satisfying assignments mustbe sufficiently dense in the sample space.
Vamshi Randomized Algorithms 22of 26
![Page 123: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/123.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
Problems in Naive Approach
If c(F ) ≥ 2n/α(n) for some polynomial α, then we need number of samples mpolynomial in n, 1/ε, ln(1/δ).
However, c(F ) can be much less than 2n. In such case, it takes exponential number ofassignments before finding the first satisfying assignment.
The number of satisfying assignments might not be sufficiently dense enough in set ofall assignments.
To obtain FPRAS for this problem, we need to construct a better sample space thatincludes all the satisfying assignments of F and also, the satisfying assignments mustbe sufficiently dense in the sample space.
Vamshi Randomized Algorithms 22of 26
![Page 124: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/124.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
Problems in Naive Approach
If c(F ) ≥ 2n/α(n) for some polynomial α, then we need number of samples mpolynomial in n, 1/ε, ln(1/δ).
However, c(F ) can be much less than 2n. In such case, it takes exponential number ofassignments before finding the first satisfying assignment.
The number of satisfying assignments might not be sufficiently dense enough in set ofall assignments.
To obtain FPRAS for this problem, we need to construct a better sample space thatincludes all the satisfying assignments of F and also, the satisfying assignments mustbe sufficiently dense in the sample space.
Vamshi Randomized Algorithms 22of 26
![Page 125: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/125.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
Problems in Naive Approach
If c(F ) ≥ 2n/α(n) for some polynomial α, then we need number of samples mpolynomial in n, 1/ε, ln(1/δ).
However, c(F ) can be much less than 2n. In such case, it takes exponential number ofassignments before finding the first satisfying assignment.
The number of satisfying assignments might not be sufficiently dense enough in set ofall assignments.
To obtain FPRAS for this problem, we need to construct a better sample space thatincludes all the satisfying assignments of F and also, the satisfying assignments mustbe sufficiently dense in the sample space.
Vamshi Randomized Algorithms 22of 26
![Page 126: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/126.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
FPRAS for DNF Couting
Let F = C1 ∨ C2 ∨ · · · ∨ Ct
A satisfying assignment needs to satisfy at least one clause.Let SCi be set of assignments that satisfy clause i
Let U = {(i, a) |1 ≤ i ≤ t & a ∈ SCi}
U denotes set of satisfying assignments for all clauses.
t∑i=1
|SCi | = |U |
Vamshi Randomized Algorithms 23of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
FPRAS for DNF Couting
Let F = C1 ∨ C2 ∨ · · · ∨ Ct
A satisfying assignment needs to satisfy at least one clause.Let SCi be set of assignments that satisfy clause i
Let U = {(i, a) |1 ≤ i ≤ t & a ∈ SCi}
U denotes set of satisfying assignments for all clauses.
t∑i=1
|SCi | = |U |
Vamshi Randomized Algorithms 23of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
FPRAS for DNF Couting
Let F = C1 ∨ C2 ∨ · · · ∨ Ct
A satisfying assignment needs to satisfy at least one clause.
Let SCi be set of assignments that satisfy clause i
Let U = {(i, a) |1 ≤ i ≤ t & a ∈ SCi}
U denotes set of satisfying assignments for all clauses.
t∑i=1
|SCi | = |U |
Vamshi Randomized Algorithms 23of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
FPRAS for DNF Couting
Let F = C1 ∨ C2 ∨ · · · ∨ Ct
A satisfying assignment needs to satisfy at least one clause.Let SCi be set of assignments that satisfy clause i
Let U = {(i, a) |1 ≤ i ≤ t & a ∈ SCi}
U denotes set of satisfying assignments for all clauses.
t∑i=1
|SCi | = |U |
Vamshi Randomized Algorithms 23of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
FPRAS for DNF Couting
Let F = C1 ∨ C2 ∨ · · · ∨ Ct
A satisfying assignment needs to satisfy at least one clause.Let SCi be set of assignments that satisfy clause i
Let U = {(i, a) |1 ≤ i ≤ t & a ∈ SCi}
U denotes set of satisfying assignments for all clauses.
t∑i=1
|SCi | = |U |
Vamshi Randomized Algorithms 23of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF- Counting Algorithm
FPRAS for DNF Couting
Let F = C1 ∨ C2 ∨ · · · ∨ Ct
A satisfying assignment needs to satisfy at least one clause.Let SCi be set of assignments that satisfy clause i
Let U = {(i, a) |1 ≤ i ≤ t & a ∈ SCi}
U denotes set of satisfying assignments for all clauses.
t∑i=1
|SCi | = |U |
Vamshi Randomized Algorithms 23of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U. Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U? First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣
c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U. Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U? First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U.
Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U? First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
![Page 135: Monte Carlo Method - West Virginia University](https://reader036.vdocuments.site/reader036/viewer/2022081622/613d5169736caf36b75be739/html5/thumbnails/135.jpg)
IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U. Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U? First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U. Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U? First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U. Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U?
First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U. Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U? First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
FPRAS for DNF Couting cont...
The count can be estimated by
c(F ) =
∣∣∣∣∣t⋃
i=1
SCi
∣∣∣∣∣c(F ) ≤ |U | , since an assignment can satisfy more than one clause and thus canappear in more than one pair in U. Let S be subset of U with size c(F ).
S = {(i, a) |1 ≤ i ≤ t , a ∈ SCi , a 6∈ SCj for j < i}
How to sample uniformly from U? First choose i from the pair (i, a) in U. Since i thclause has |SCi | satisfying assignments, we should choose i with probability
|SCi |∑ti=1 |SCi |
=|SCi ||U |
Vamshi Randomized Algorithms 24of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF-Counting Algorithm
Input: A DNF formula F with n variables.Output: Y = an approximation of c(F ).
1: X ← 02: for (k = 1 to m) do3: With probability |SCi |∑t
i=1 |SCi |choose, uniformly at random, an assignment a ∈ SCi .
4: if (a is not in any SCj , j < i) then5: X ← X + 16: end if7: end for8: return Y ← ( X
m ) ·∑t
i=1 |SCi |
Algorithm 3.2: DNF Counting Algorithm II: FPRAS
Vamshi Randomized Algorithms 25of 26
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IntroductionApplications
The DNF Counting ProblemDNF Couting Algorithms
DNF Counting Algorithm
Theorem
DNF counting algorithm II is a fully polynomial randomized approximation scheme(FPRAS) for the DNF counting problem when
m = d(3 · t2/ε2) · ln(2/δ)e
Vamshi Randomized Algorithms 26of 26