EE/CSCI 451 Spring 2018 Homework 8 Total Points: [10 points] Explain the following terms: EREW PRAM CRCW PRAM. Brent s Theorem.

Size: px

Start display at page:

Download "EE/CSCI 451 Spring 2018 Homework 8 Total Points: [10 points] Explain the following terms: EREW PRAM CRCW PRAM. Brent s Theorem."

Silvia Rich
5 years ago
Views:

1 EE/CSCI 451 Spring 2018 Homework 8 Total Points: [10 points] Explain the following terms: EREW PRAM CRCW PRAM Brent s Theorem BSP model 1

2 2 [15 points] Assume two sorted sequences of size n can be merged in M(n) time using P (n) processors on a PRAM (M(2) = 1, P (2) = 1). Write a divide and conquer parallel code to sort n numbers A[0,, n 1] on PRAM. Write a recurrence for parallel time. Obtain the parallel time as a function of M(). What is the total number of processors used? Assume serial merge is used, what is the total parallel time and total number of processors needed? You may use order notation. 2

3 3 [15 points] Based on the Bellman-Ford algorithm shown below, write a CRCW PRAM algorithm to perform single source shortest path (SSSP) using p processors, where p = e = # of edges. Assume the input graph G(V, E) is directed and each edge has a positive weight. Note, your algorithm should terminate if there is no update to shortest path during an iteration. Clearly define the variables and their meaning. What is the total execution time? Bellman-Ford Algorithm Let edge(i, j) denote the edge from vertex i to vertex j Let w(i, j) denote the weight of edge(i, j) Let s(i) denote the weight of shortest path from source to vertex i Bellman-Ford (G(V, E)) 1. For each vertex x in V // Initialization // If x is source then s(x) = 0 At_least_one_vertex_has_update = true Else s(x) = End if End for 2. For k = 1 to #_of_vertices do If At_least_one_vertex_has_update = true then At_least_one_vertex_has_update = false 3. lfor each edge(i, j) E do If s(i) + w(i, j) < s(j) then s(j) = s(i) + w(i, j) At_least_one_vertex_has_update = true End if End for Else Algorithm terminates End if End for 3

4 4 [20 points] Given an array of n elements A[0,, n 1] Show a O(1) time CRCW PRAM algorithm using n processors to determine whether the array is sorted in ascending order. Show a O(1) time CRCW PRAM algorithm using n 2 processors to determine whether the elements are distinct. 4

5 5 [20 points] The objective of this question is to perform triangle detection in a directed graph G(V, E) on a PRAM in O(1) time. Let n be the number of vertices. Vertices {u, v, w} form a triangle if edges < u, v >, < v, w >, < u, w > exist in G. Develop a CREW PRAM algorithm which takes an adjacency matrix A, where A(i, j) = 1 iff there is an edge from vertex i to vertex j, as input and outputs a V V V matrix B with element B(i, j, k) = 1 if i, j, k form a triangle. The algorithm should run in O(1) time. Determine the number of processors required and the total work done (use order notation for work). State any assumptions you may make. Now assume that the edges are weighted. Develop a CRCW algorithm which takes an adjacency matrix A, where A(i, j) is the weight of the edge < i, j >, as input and outputs the minimum weighted triangle in O(1) time. Determine the number of processors required and the total work done (use order notation for work). State any assumptions you may make. 5

6 6 [20 points] Consider the parallel prefix sum (or scan) problem that we discussed in the lecture. An alternative divide and conquer approach is as follows: The array of n elements a 0, a 1,..., a n 1 can be partitioned into two sub-arrays a 0, a 1,..., a n 2 1 and a n,..., a n 1 of size n 2 2 each and the prefix sum of both the partitions can be calculated recursively. The prefix sums of the right partition can be updated by using the last prefix sum of the left partition. Apply the technique described above on an array A with elements a i = i + 1 for 0 i 7 and show the intermediate results corresponding to each recursive invocation of the prefix sum algorithm. Function Invocation Returned Value Prefix-sum(1,2,3,4,5,6,7,8) {1,3,6,10,15,21,28,36} Prefix-sum(1,2,3,4) Prefix-sum(5,6,7,8) Prefix-sum(1,2) Prefix-sum(3,4) Prefix-sum(5,6) Prefix-sum(7,8) Write a recursive algorithm for the technique described above using CREW PRAM. What is the total parallel time (use order notation)? Determine the total work done (use order notation). Write a recursive algorithm for the technique described above using EREW PRAM. Use recursive doubling technique for updating the second partition. Determine the total parallel time (use order notation). 6

Parallel Algorithms for (PRAM) Computers & Some Parallel Algorithms. Reference : Horowitz, Sahni and Rajasekaran, Computer Algorithms

Parallel Algorithms for (PRAM) Computers & Some Parallel Algorithms Reference : Horowitz, Sahni and Rajasekaran, Computer Algorithms Part 2 1 3 Maximum Selection Problem : Given n numbers, x 1, x 2,, x