Randomized Algorithms Randomized Algorithms Two Types of Randomized - PowerPoint PPT Presentation

Randomized Algorithms Randomized Algorithms Two Types of Randomized Algorithms Two Types of Randomized Algorithms and and Some Complexity Classes Some Complexity Classes Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University 2006/9/20 2006/9/20

References References � Professor Hsueh-I Lu’s slides. � Randomized Algorithms , Rajeev Motwani and Prabhakar Raghavan. � Probability and Computing - Randomized Algorithms and Probabilistic Analysis , Michael Mitzenmacher and Eli Upfal. 2006/9/20 2 2006/9/20 2 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Outline Outline � Las Vegas algorithms and Monte Carlo algorithms � RAMs and Turing machines � Complexity classes – P , NP , RP , ZPP , BPP and their complementary classes – Open problems 2006/9/20 3 2006/9/20 3 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Las Vegas vs. Monte Carlo Las Vegas vs. Monte Carlo � Las Vegas algorithms � Monte Carlo algorithms – Always produces a – Allow a small (correct/optimal) probability for solution. outputting an incorrect/non-optimal – Like RandQS. solution. – Like RandMC. – The name is by von Neumann. 2006/9/20 4 2006/9/20 4 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Las Vegas Algorithms Las Vegas Algorithms � For example, RandQS is a Las Vegas algorithm. � A Las Vegas always gives the correct solution � The only variation from one run to another is its running time, whose distribution we study. 2006/9/20 5 2006/9/20 5 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Randomized quicksort quicksort Randomized algorithm RandQS( X ) { if X is empty then randomization return; select x uniformly at random from X ; let Y = { y ∈ X | y < x } ; let Z = { z ∈ X | z > x } ; call RandQS( Y ); print x ; call RandQS( Z ); } 2006/9/20 6 2006/9/20 6 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

An illustration An illustration 4 2 7 8 1 9 3 6 5 4 2 1 5 7 8 9 6 3 1 2 4 3 6 7 8 9 1 3 4 6 8 9 9 3 2006/9/20 7 2006/9/20 7 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

2 Questions for RandQS RandQS 2 Questions for Is RandQS correct? � – That is, does RandQS “always” output a sorted list of X ? What is the time complexity of RandQS? � – Due to the randomization for selecting x , the running time for RandQS becomes a random variable. – We are interested in the expected time complexity for RandQS. 2006/9/20 8 2006/9/20 8 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Monte Carlo algorithms Monte Carlo algorithms � For example, RandEC (the randomized minimum-cut algorithm we have discussed) is a Monte Carlo algorithm. � A Monte Carlo algorithm may sometimes produce a solution that is incorrect. � For decision problems, there are two kinds of Monte Carlo algorithms: – those with one-sided error – those with two-sided error 2006/9/20 9 2006/9/20 9 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Which is better? Which is better? � The answer depends on the application. � A Las Vegas algorithm is by definition a Monte Carlo algorithm with error probability 0. � Actually, we can derive a Las Vegas algorithm A from a Monte Carlo algorithm B by repeated running B until we get a correct answer. 2006/9/20 10 2006/9/20 10 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Computation model Computation model � Throughout this talk, we use the Turing machine model to discuss complexity theory issues. � As is common, we switch to the RAM (random access machine) as the model of computation when describing and analyzing algorithms. 2006/9/20 11 2006/9/20 11 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Computation model (cont’ ’d) d) Computation model (cont � For simplicity, we will work with the general unit-cost RAM model. � In unit-cost RAM model, each instruction can be performed in one time step. 2006/9/20 12 2006/9/20 12 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

You can refer to any computation theory textbook Deterministic TM Deterministic TM to for more details here. � A deterministic Turing machine is a quadruple M = ( S , ∑ , δ , s ). – Here S is a finite set of states, of which s ∈ S is the machine’s initial state. – ∑ : a finite set of symbols (this set includes special symbols BLANK and FIRST). – δ : the transition function of the Turing machine, mapping S × ∑ to (S ∪ {HALT, YES, NO}) × ∑ × { ← , → , STAY}. � The machine has three states: HALT (the halting state), YES (the accepting state), and NO (the rejecting state) (these are states, but formally not in S .) 2006/9/20 13 2006/9/20 13 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

A Turing machine with one tape A Turing machine with one tape B state \ symbol 0 1 t ( s, B , → ) s ( s, 0 , → ) ( s, 1 , → ) ( q, t , ← ) ( h, B , → ) q ( q 0 , t , → ) ( q 1 , t , → ) ( q, t , − ) ( h, B , → ) q 0 ( s, 0 , ← ) ( s, 0 , ← ) ( s, 0 , ← ) ( h, B , → ) q 1 ( s, 1 , ← ) ( s, 1 , ← ) ( s, 1 , ← ) × × × × h Q: What does this Turing B machine do? 0 1 0 0 2006/9/20 14 2006/9/20 14 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

A Turing machine with one tape A Turing machine with one tape B state \ symbol 0 1 t ( s, B , → ) s ( s, 0 , → ) ( s, 1 , → ) ( q, t , ← ) ( h, B , → ) q ( q 0 , t , → ) ( q 1 , t , → ) ( q, t , − ) ( h, B , → ) q 0 ( s, 0 , ← ) ( s, 0 , ← ) ( s, 0 , ← ) ( h, B , → ) q 1 ( s, 1 , ← ) ( s, 1 , ← ) ( s, 1 , ← ) × × × × h Q: What does this Turing B machine do? 0 1 0 2006/9/20 15 2006/9/20 15 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

A probabilistic TM A probabilistic TM � A probabilistic Turing machine is a (nondeterministic) Turing machine augmented with the ability to generate an unbiased coin flip in one step. � It corresponds to a randomized algorithm. 2006/9/20 21 2006/9/20 21 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

A probabilistic TM (cont’ ’d) d) A probabilistic TM (cont � On any input x , a probabilistic Turing machine accepts x with some probability, and we study this probability. 2006/9/20 22 2006/9/20 22 Computation Theory Lab, CSIE, CCU, Taiwan Computation Theory Lab, CSIE, CCU, Taiwan

Randomized Algorithms Randomized Algorithms Two Types of Randomized - PowerPoint PPT Presentation

Randomized Algorithms Randomized Algorithms Two Types of Randomized Algorithms Two Types of Randomized Algorithms and and Some Complexity Classes Some Complexity Classes Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National

Randomized algorithms Quick-sort Closest pair of points Inge Li Grtz 1 2

Randomized Algorithms Randomized Algorithms Markov Chains and Random Walks Markov Chains and

Probability and Delaunay triangulations 1 Randomized algorithms for Delaunay triangulations

Introduction to Randomized Algorithms: QuickSort Lecture 2 August 27, 2020 Chandra (UIUC)

Randomized algorithms Inge Li Grtz Thank you to Kevin Wayne for inspiration to slides

Optimal Randomized Algorithms for Integration Integration on Function Spaces with underlying

Randomized Algorithms, Quicksort and Randomized Selection Carola Wenk Slides courtesy of Charles

ADVANCED ALGORITHMS Lecture 14: randomized algorithms 1 ANNOUNCEMENTS HW 3 out; due

Randomized algorithms Guessing cards Three examples: Median/Select. Inge Li Grtz

ECS231 Low-rank approximation revisited (Introduction to Randomized Algorithms) May 23, 2019

Randomized Algorithms Lecture 3: Occupancy, Moments and deviations, Randomized selection

8. Average-Case Analysis of Algorithms + Randomized Algorithms 1 outline and goals 1)

Chapter 11 Randomized Algorithms III Min Cut CS 573: Algorithms, Fall 2013 October 1, 2013

Approximation and Randomized Algorithms Lecturer: Shi Li Department of Computer Science and

Approximation and Randomized Algorithms Lecturer: Shi Li Department of Computer Science and

Randomized Network Algorithms: An Overview and Recent Results Balaji Prabhakar Departments of EE

P and NP Lecture 22 1 Today Computational Complexity P , NP , PSPACE , EXP NP -completeness

Advanced Topics in Theoretical Computer Science Part 5: Complexity (Part 2) 5.07.2016 Viorica

randomized computation Sometimes randomness helps in computation. randomized computation Augment

INF2080 Context-Sensitive Langugaes Daniel Lupp Universitetet i Oslo 15th February 2017

Machine Models and Lower Bounds for Query Processing Nicole Schweikardt Humboldt-University

CESNET storage activities overview Ji Hork (jiri.horky@cesnet.cz) 6th of March, 2013

SCSI support on Xen MATSUMOTO Hitoshi matsumotohitosh@jp.fujitsu.com Fujitsu Ltd. Why SCSI?

Comparing Tape and Cloud Storage for Long-Term Data Preservation Matt Ninesling Matt Starr Eric