Randomized Algorithms How Do We Evaluate This?

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Analyzing Algorithms Goal: “Runs fast on typical real problem instances” Randomized Algorithms How do we evaluate this? Example: Binary search Given a sorted array, determine if the array contains the number 157? 2 3 Complexity " Measuring efficiency T! analysis n! Time ≈ # of instructions executed in a simple Problem size n programming language Best-case complexity: min # steps algorithm only simple operations (+,*,-,=,if,call,…) takes on any input of size n each operation takes one time step Average-case complexity: avg # steps algorithm each memory access takes one time step takes on inputs of size n Worst-case complexity: max # steps algorithm takes on any input of size n 4 5 1 Complexity The complexity of an algorithm associates a number Complexity T(n), the worst-case time the algorithm takes on problems of size n, with each problem size n. T(n)! Mathematically, T: N+ → R+ ! I.e., T is a function that maps positive integers (problem Time sizes) to positive real numbers (number of steps). Problem size ! 6 7 Simple Example Complexity Array of bits. The complexity of an algorithm associates a number I promise you that either they are all 1’s or # 0’s T(n), the worst-case time the algorithm takes on and # 1’s. problems of size n, with each problem size n. Give me a program that will tell me which it is. For randomized algorithms, look at worst-case value of E(T), where the Best case? expectation is taken over randomness in Worst case? algorithm. Neat idea: use randomization to reduce the worst case 8 9 2 Quicksort Analysis of Quicksort Worst case number of comparisons: n Sorting algorithm (assume for now all elements 2 distinct) ✓ ◆ How can we use randomization to improve running Given array of some length n time? If n = 0 or 1, halt Pick random element as a pivot each step Else pick element p of array as “pivot” Split array into subarrays <p, > p => Randomized algorithm Recursively sort elements < p Recursively sort elements > p 10 11 Analysis of Randomized Quicksort Analysis of Randomized Quicksort" Quicksort with random pivots Fix pair i,j. Compute E(Xij) Define A k indicator r.v. that is 1 if elt in [ei, ej] X = # of comparisons. first selected at kth level of tree X = Xij E(Xij)=Pr(Xij = 1) 1 i<j n X = Pr(X =1A )Pr(A ) What is condition for elements ith smallest and jth ij | k k 1 k n smallest to get directly compared? X 2 2 = Pr(Ak) = j i +1 Claim: fate determined first time an elt in [ei, ej] 1 k n j i +1 − X − picked. 2 12 Pr(X =1A )= 13 ij | k j i +1 − 3 Analysis of Randomized Quicksort E(X)= E(Xij) 1 i<j n X 2 = j i +1 1 i<n j>i X X − 1 1 1 2 + + ...+ 2 3 n i +1 1 i<n ✓ ◆ X − 2n ln(n)+O(n) 14 4 .

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The Randomized Quicksort Algorithm
Outline The Randomized Quicksort Algorithm K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University 7 February, 2012 Subramani Sample Analyses Outline Outline 1 The Randomized Quicksort Algorithm Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that A[1] < A[2]... A[n]. Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that A[1] < A[2]... A[n]. Note The assumption of distinctness simpliﬁes the analysis. Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that A[1] < A[2]... A[n]. Note The assumption of distinctness simpliﬁes the analysis. It has no bearing on the running time. Subramani Sample Analyses The Randomized Quicksort Algorithm The Partition subroutine Function PARTITION(A,p,q) 1: {We partition the sub-array A[p,p + 1,...,q] about A[p].} 2: for (i = (p + 1) to q) do 3: if (A[i] < A[p]) then 4: Insert A[i] into bucket L.
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Randomized Algorithms, Quicksort and Randomized Selection Carola Wenk Slides Courtesy of Charles Leiserson with Additions by Carola Wenk
CMPS 2200 – Fall 2014 Randomized Algorithms, Quicksort and Randomized Selection Carola Wenk Slides courtesy of Charles Leiserson with additions by Carola Wenk CMPS 2200 Intro. to Algorithms 1 Deterministic Algorithms Runtime for deterministic algorithms with input size n: • Best-case runtime Attained by one input of size n • Worst-case runtime Attained by one input of size n • Average runtime Averaged over all possible inputs of size n CMPS 2200 Intro. to Algorithms 2 Deterministic Algorithms: Insertion Sort for j=2 to n { key = A[j] // insert A[j] into sorted sequence A[1..j-1] i=j-1 while(i>0 && A[i]>key){ A[i+1]=A[i] i-- } A[i+1]=key } • Best case runtime? • Worst case runtime? CMPS 2200 Intro. to Algorithms 3 Deterministic Algorithms: Insertion Sort Best-case runtime: O(n), input [1,2,3,…,n] Attained by one input of size n • Worst-case runtime: O(n2), input [n, n-1, …,2,1] Attained by one input of size n • Average runtime : O(n2) Averaged over all possible inputs of size n •What kind of inputs are there? • How many inputs are there? CMPS 2200 Intro. to Algorithms 4 Average Runtime • What kind of inputs are there? • Do [1,2,…,n] and [5,6,…,n+5] cause different behavior of Insertion Sort? • No. Therefore it suffices to only consider all permutations of [1,2,…,n] . • How many inputs are there? • There are n! different permutations of [1,2,…,n] CMPS 2200 Intro. to Algorithms 5 Average Runtime Insertion Sort: n=4 • Inputs: 4!=24 [1,2,3,4]0346 [4,1,2,3] [4,1,3,2] [4,3,2,1] [2,1,3,4]1235 [1,4,2,3] [1,4,3,2] [3,4,2,1] [1,3,2,4]1124 [1,2,4,3] [1,3,4,2] [3,2,4,1] [3,1,2,4]2455 [4,2,1,3] [4,3,1,2] [4,2,3,1] [3,2,1,4]3244 [2,1,4,3] [3,4,1,2] [2,4,3,1] [2,3,1,4]2333 [2,4,1,3] [3,1,4,2] [2,3,4,1] • Runtime is proportional to: 3 + #times in while loop • Best: 3+0, Worst: 3+6=9, Average: 3+72/24 = 6 CMPS 2200 Intro.
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IBM Research Report Derandomizing Arthur-Merlin Games And
H-0292 (H1010-004) October 5, 2010 Computer Science IBM Research Report Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds Dan Gutfreund, Akinori Kawachi IBM Research Division Haifa Research Laboratory Mt. Carmel 31905 Haifa, Israel Research Division Almaden - Austin - Beijing - Cambridge - Haifa - India - T. J. Watson - Tokyo - Zurich LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , payment of royalties). Copies may be requested from IBM T. J. Watson Research Center , P. O. Box 218, Yorktown Heights, NY 10598 USA (email: [email protected]). Some reports are available on the internet at http://domino.watson.ibm.com/library/CyberDig.nsf/home . Derandomization Implies Exponential-Size Lower Bounds 1 DERANDOMIZING ARTHUR-MERLIN GAMES AND APPROXIMATE COUNTING IMPLIES EXPONENTIAL-SIZE LOWER BOUNDS Dan Gutfreund and Akinori Kawachi Abstract. We show that if Arthur-Merlin protocols can be deran- domized, then there is a Boolean function computable in deterministic exponential-time with access to an NP oracle, that cannot be computed by Boolean circuits of exponential size. More formally, if prAM ⊆ PNP then there is a Boolean function in ENP that requires circuits of size 2Ω(n).
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January 30 4.1 a Deterministic Algorithm for Primality Testing
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CMSC 420 Dave Mount CMSC 420: Lecture 7 Randomized Search Structures: Treaps and Skip Lists Randomized Data Structures: A common design techlque in the field of algorithm design in- volves the notion of using randomization. A randomized algorithm employs a pseudo-random number generator to inform some of its decisions. Randomization has proved to be a re- markably useful technique, and randomized algorithms are often the fastest and simplest algorithms for a given application. This may seem perplexing at first. Shouldn't an intelligent, clever algorithm designer be able to make better decisions than a simple random number generator? The issue is that a deterministic decision-making process may be susceptible to systematic biases, which in turn can result in unbalanced data structures. Randomness creates a layer of \independence," which can alleviate these systematic biases. In this lecture, we will consider two famous randomized data structures, which were invented at nearly the same time. The first is a randomized version of a binary tree, called a treap. This data structure's name is a portmanteau (combination) of \tree" and \heap." It was developed by Raimund Seidel and Cecilia Aragon in 1989. (Remarkably, this 1-dimensional data structure is closely related to two 2-dimensional data structures, the Cartesian tree by Jean Vuillemin and the priority search tree of Edward McCreight, both discovered in 1980.) The other data structure is the skip list, which is a randomized version of a linked list where links can point to entries that are separated by a significant distance. This was invented by Bill Pugh (a professor at UMD!).
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Randomized Algorithms
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CS265/CME309: Randomized Algorithms and Probabilistic Analysis Lecture #1:Computational Models, and the Schwartz-Zippel Randomized Polynomial Identity Test
CS265/CME309: Randomized Algorithms and Probabilistic Analysis Lecture #1:Computational Models, and the Schwartz-Zippel Randomized Polynomial Identity Test Gregory Valiant,∗ updated by Mary Wootters September 7, 2020 1 Introduction Welcome to CS265/CME309!! This course will revolve around two intertwined themes: • Analyzing random structures, including a variety of models of natural randomized processes, including random walks, and random graphs/networks. • Designing random structures and algorithms that solve problems we care about. By the end of this course, you should be able to think precisely about randomness, and also appreciate that it can be a powerful tool. As we will see, there are a number of basic problems for which extremely simple randomized algorithms outperform (both in theory and in practice) the best deterministic algorithms that we currently know. 2 Computational Model During this course, we will discuss algorithms at a high level of abstraction. Nonetheless, it's helpful to begin with a (somewhat) formal model of randomized computation just to make sure we're all on the same page. Definition 2.1 A randomized algorithm is an algorithm that can be computed by a Turing machine (or random access machine), which has access to an infinite string of uniformly random bits. Equivalently, it is an algorithm that can be performed by a Turing machine that has a special instruction “flip-a-coin”, which returns the outcome of an independent flip of a fair coin. ∗©2019, Gregory Valiant. Not to be sold, published, or distributed without the authors' consent. 1 We will never describe algorithms at the level of Turing machine instructions, though if you are ever uncertain whether or not an algorithm you have in mind is \allowed", you can return to this definition.
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Lecture 3: Randomness in Computation
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Randomized Algorithms: Quicksort and Selection
Randomized Algorithms: Quicksort and Selection Version of September 6, 2016 Randomized Algorithms: Quicksort and Selection Version1 / 30 of September 6, 2016 Outline Outline: Quicksort Average-Case Analysis of QuickSort Randomized quicksort Selection The selection problem First solution: Selection by sorting Randomized Selection Randomized Algorithms: Quicksort and Selection Version2 / 30 of September 6, 2016 Quicksort: Review Quicksort(A; p; r) begin if p < r then q = Partition(A; p; r); Quicksort(A; p; q − 1); Quicksort(A; q + 1; r); end end Partition(A; p; r) reorders items in A[p ::: r]; items < A[r] are to its left; items > A[r] to its right. Showed that if input is a random input (permutation) of n items, then average running time is O(n log n) Randomized Algorithms: Quicksort and Selection Version3 / 30 of September 6, 2016 Average Case Analysis of Quicksort Formally, the average running time can be deﬁned as follows: In is the set of all n! inputs of size n I 2 In is any particular size-n input R(I ) is the running time of the algorithm on input I Then, the average running time over the random inputs is X 1 X Pr(I )R(I ) = R(I ) = O(n log n) n! I 2In I 2In Only fact that was used was that A[r] was a random item in A[p ::: r], i.e., the partition item is equally likely to be any item in the subset. Randomized Algorithms: Quicksort and Selection Version4 / 30 of September 6, 2016 Outline Outline: Quicksort Average-Case Analysis of QuickSort Randomized Quicksort Selection The selection problem First solution: Selection by sorting Randomized Selection Randomized Algorithms: Quicksort and Selection Version5 / 30 of September 6, 2016 Randomized-Partition(A; p; r) Idea: In the algorithm Partition(A; p; r), A[r] is always used as the pivot x to partition the array A[p::r] In the algorithm Randomized-Partition(A; p; r), we randomly choose j, p ≤ j ≤ r, and use A[j] as pivot Idea is that if we choose randomly, then the chance that we get unlucky every time is extremely low.
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Lecture 01: Randomized Algorithm for Reachability Problem
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Linear Advice for Randomized Logarithmic Space
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Probabilistic Verification and Approximation Schemes
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