cme323_lec5.pdf - CME 323 Distributed Algorithms and Optimization Spring 2016 http\/\/~rezab\/dao Instructor Reza Zadeh Matroid and Stanford

cme323_lec5.pdf - CME 323 Distributed Algorithms and...

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CME 323: Distributed Algorithms and Optimization, Spring 2016 ~ rezab/dao . Instructor: Reza Zadeh, Matroid and Stanford. Lecture 5, 4/17/2017. Scribed by Andreas Santucci. Lecture contents 1. QuickSort 2. Parallel algorithm for minimum spanning trees (Boruvka) 3. Parallel connected components (random mates) 1 QuickSort First, we’ll finish the analysis of QuickSort. The algorithm is as follows. Algorithm 1: QuickSort Input: An array A Output: Sorted A 1 p element of A chosen uniformly at random 2 L [ a | a A s.t. a < p ] // Implicitly: B L { a i < p } n i =1 , prefixSum ( B l ) , 3 R [ a | a A s.t. a > p ] // which requires Θ( n ) work and O (log n ) depth. 4 return [ QuickSort ( L ) , p, QuickSort ( R )] 1.1 Analysis on Memory Management Recall that in Lecture 4, we designed an algorithm to construct L and R in O ( n ) work and O (log n ) depth. Since we know the algorithm used to construct L and R (which is the main work required of QuickSort ), let us take this opportunity to take a closer look at memory management during the algorithm. Selecting a pivot uniformly at random We denote the size our input array A by n . To be precise, we can perform step 1 in Θ(log n ) work and O (1) depth. That is, to generate a number uniformly from the set { 1 , 2 , . . . , n } we can assign log n processors to independently flip a bit “on” with probability 1/2. The resulting bit-sequence can be interpreted as a log 2 representation of a number from { 1 , . . . , n } . 1
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Allocating storage for L and R Start by making a call to the OS to allocate an array of n elements; this requires O (1) work and depth, since we do not require the elements to be initialized. We compare each element in the array with the pivot, p , and write a 1 to the corresponding element if the element belongs in L (i.e. it’s smaller) and a 0 otherwise. This requires Θ( n ) work but can be done in parallel, i.e. O (1) depth. We are left with an array of 1’s and 0’s indicating whether an element belongs in L or not, call it L , L = { a A s.t. a < p } . We then apply PrefixSum on the indicator array L , which requires O ( n ) work and O (log n ) depth. Then, we may examine the value of the last element in the output array from prefixSum to learn the size of L . Looking up the last element in array L requires O (1) work and depth. We can further allocate a new array for L in constant time and depth. Since we know | L | and we know n , we also know | R | = n - | L | ; computing | R | and allocating corresponding storage requires O (1) work and depth. Thus, allocating space for L and R requires O ( n ) work and O (log n ) depth. Filling L and R Now, we use n processors, assigning each to exactly one element in our input array A , and in parallel we perform the following steps. Each processor 1 , 2 , . . . , n is assigned to its corresponding entry in A . Suppose we fix attention to the k th processor, which is responsible for assigning the k th entry in A to its appropriate location in either L or R . We first examine L [k] to determine whether the element belongs in L or R . In addition, examine the corresponding entry in prefixSum output, denote this value by i = prefixSum ( L ) [k] . If the k th entry of A belongs in L
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