Chapter 4 Partitioning and DivideandConquer Strategies Two fundamental

Chapter 4 Partitioning and Divide-and-Conquer Strategies • Two fundamental techniques in parallel programming - Partitioning - Divide-and-conquer • Typical problems solved with these techniques Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 1

Partitioning/Divide and Conquer Examples Many possibilities. • Operations on sequences of number such as simply adding them together • Several sorting algorithms can often be partitioned or constructed in a recursive fashion • Numerical integration • N-body problem Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 2

Partitioning • Dividing the problem into parts. • Basis for all parallel programming, in one form or another. • Domain decomposition - partitioning applied to the program data. • Dividing the data and operating upon the divided data concurrently. • Functional decomposition - partitioning applied to the functions of a program. • Dividing the program into independent functions and executing the functions concurrently. • Domain decomposition forms the foundation of most parallel algorithms Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 3

Partitioning a sequence of numbers into parts and adding the parts Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 4

Number Summation Revisited • Our previous implementation used broadcast() and reduce() to send the data and combine the result • The following pseudocode uses individual send()s and receive()s: 1. Master: s = n/p; for (i=0, x=0; I<p; I++, x=x+s) send(&numbers[x], s, P[I]); sum = 0; for (i=0; i<p; i++){ recv(&part_sum, P[ANY]); sum = sum + part_sum; } 2. Slave: recv(numbers, s, P[MASTER]); part_sum = 0; for (i=0; i<s; i++) part_sum = part_sum + numbers[i]; send(&part_sum, P[MASTER]); Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 5

Divide-and-Conquer • Dividing a problem into subproblems that are of the same form as the larger problem. • Further divisions into still smaller sub-problems are usually done by recursion. int add(int *s){ if (number(s) == 2) return (s[0] + s[2]); else { divide(s, s 1, s 2); part_sum 1 = add(s 1); part_sum 2 = add(s 2); return (part_sum 1 + part_sum 2); } } Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 6

Tree construction • When each division creates two parts, a recursive divide-and-conquer formulation forms a binary tree. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 7

Parallel Implementation • • • In a sequential implementation, only one node of the tree can be visited at a time. A parallel solution offers the prospect of traversing several parts of the tree simultaneously. Could assign one processor to each node in the tree. • Requires 2 m+1 -1 processors to divide the tasks into 2 m parts A more efficient solution is to reuse processors at each level of the tree At each stage, each processor keeps half of the list and passes on the other half Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 8

Dividing a list into parts • The division stops when the total number of processors is committed. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 9

Partial summation Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 10

Adding a List of Numbers • Suppose we statically create eight processes to add a list of numbers: Process P 0: divide(s 1, s 2); send(s 2, P 4); divide(s 1, s 2); send(s 2, P 2); part_sum = *s 1; recv(&part_sum 1, P 1); part_sum = part_sum + part_sum 1; recv(&part_sum 1, P 2); part_sum = part_sum + part_sum 1; recv(&part_sum 1, P 4); part_sum = part_sum + part_sum 1; Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 11

Adding a List of Numbers (cont’d) Process P 4: recv(s 1, P 0); divide(s 1, s 2); send(s 2, P 6); divide(s 1, s 2); send(s 2, P 5); part_sum = *s 1; recv(&part_sum 1, P 5); part_sum = part_sum + part_sum 1; recv(&part_sum 1, P 6); part_sum = part_sum + part_sum 1; send(&part_sum, P 0); Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 12

Analysis • Assume that n is a power of 2. • Communication • log p steps with p processes (division phase) • Time complexity is O(n) for constant p. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 13

Analysis (cont’d) • Computation • At the end of the divide phase, the n/p numbers are added together. Then one addition occurs at each stage during the combining phase, leading to • Time complexity of O(n) for constant p. For large n and variable p, we get O(n/p). • Total parallel execution time becomes: Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 14

M-ary Divide and Conquer) • Divide and conquer can also be applied where a task is divided into more than two parts at each stage • Sample code for 4 -are divide and conquer: int add(int *s){ if(number(s)<=4) return (n 1+n 2+n 3+n 4); else{ Divide(s, s 1, s 2, 34); p_sum 1 = add(s 1); p_sum 2 = add(s 2); p_sum 3 = add(s 3); p_sum 4 = add(s 4); return (p_sum 1 + p_sum 2+ p_sum 3 + p_sum 4); } } Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 15

Quadtree Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 16

Dividing an image Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 17

Example 1: Sorting Using Bucket Sort • Most sequential sorting algorithms are based on comparing and exchanging pairs of numbers • Bucket sort is not based on compare and exchange, but is naturally a partitioning method • Works well only if the numbers to be sorted are uniformly distributed across a known interval • If all the numbers in the interval 0 to a-1 – Divide numbers into m regions 0 to a/m-1, a/m to 2 a/m-1, … – One “bucket” is assigned to hold numbers that fall a region Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 18

Bucket Sort Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 19

Sequential Bucket Sort: Partitioning into Buckets • Need to determine placement of each number into appropriate bucket • Given a number r, compare it with the start of regions a/m, 2 a/m, 3 a/m, … – Can take m-1 comparisons to determine right bucket • Better alternative: multiply r by m/a to determine its bucket – Requires one computational step, except that division can be expensive • If m is a power of 2, bucket identification even faster – If m = 2 k, higher k bits of r determine r’s bucket • Assume that placing a number into a bucket requires one step – placing all the numbers requires n steps. • If the numbers are uniformly distributed each bucket will contain n/m numbers Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 20

Sequential Bucket Sort: Sorting the Buckets • After placing the numbers, each bucket needs to be sorted • n numbers can be sorted sequentially in O(n log n) time • Best time for sequential compare and exchange sorting • Thus, each bucket can be sorted in O(n/m log n/m) time, sequentially • Assuming no concatenation overhead, the sequential time is Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 21

Parallel Bucket Sort • Can be parallelized by assigning one processor per bucket • The computation time will now be (with m=p): • This equation means that each processor examines each of the numbers – inefficient • Can be improved by removing each examined number from the list at it is bucketed Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 22

Parallel version of bucket sort Simple approach Assign one processor for each bucket. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 23

Further Parallelization • Partition sequence into m regions, one region for each processor. • Each processor maintains p “small” buckets and separates the numbers in its region into its own small buckets. • Small buckets then emptied into p final buckets for sorting – Requires each processor to send one small bucket to each of the other processors (bucket i to processor i). • The algorithms has four phases: – – Partitioning numbers Placement into p small buckets Sending small buckets to large buckets Sorting large buckets Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 24

Another parallel version of bucket sort Introduces new message-passing operation - all-to-all broadcast. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 25

Analysis • Phase 1 - Computation and Communication • the p partitions, containing n/p numbers each, are sent to the processes. Using a broadcast (or scatter) routine • tcomm 1 = tstartup + ntdata • Phase 2 – Computation • To separate each partition of n/p numbers into p small buckets requires the time • tcomp 2 = n/p Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 26

Analysis • Phase 3 – Communication • Next, the small buckets are distributed (no computation). • Each small bucket will have about n/p 2 numbers. • Each process must send the contents of p-1 small buckets to other processes • Since each process of the p processes must make this communication, we have tcomm 3 = p(p-1)(tstartup+(n/p 2)tdata • The above assumes the communications do not overlap in time and individual send()s are used • If the communication could overlap (MPI_Alltoall()), we have: tcomm 3 = (p-1)(tstartup+(n/p 2)tdata Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 27

Analysis • Phase 4 – computation • Sort the largest buckets simultaneously tcomp 4 = (n/p)log(n/p) • Overall execution time: Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 28

“all-to-all” broadcast routine Sends data from each process to every other process Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 29

“all-to-all” routine actually transfers rows of an array to columns: Transposes a matrix. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 30

Example 2: Numerical integration • In many areas of mathematics and science it is necessary to calculate the area bounded by y = f(x), the axis, and the lines x = a, and x = b. • The area can be calculated using definite integrals • Can be difficult or impossible sometimes • Typically, approximations methods are used to calculate such area • Trapezoidal rule • Simpson rule • Adaptive quadrature methods • Area approximated by partitioning the interval [a, b] into n equal pieces and inscribing a rectangle above each subinterval thus obtained Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 31

Numerical integration using rectangles. Each region calculated using an approximation given by rectangles: Aligning the rectangles: Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 32

Trapezoidal Rule: Static Assignment • One process is assigned statically to compute each region, before the computation begins • As before an SPMD programming model is used • Each calculation has same form • With p processors to compute the area between x=a and x=b, each process will handle a region of size • (b-a)/p • Each process will compute its area in the manner shown by the code overleaf Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 33

Numerical integration using trapezoidal method May not be better! Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 34

Trapezoidal Rule (cont’d) if(i==master){ printf(“Enter # of intervals “); scanf(“%d”, &n); } bcast(&n, p[group]); region=(b-1)/p; start = a + region*i; end = start+region; d =(b-a)/n; area=0. 0; for(x=start; x<end; x=x+d) area = area + 0. 5*(f(x)+f(x+d))*d; } reduce_add(&integral, &area, p[group]) Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 35

Trapezoidal Rule: Optimization • An implementation of this formula: area = 0. 5*(f(start)+f(end))*d; for(x=start+d; x<end; x=x+d) area = area + f(x); area = area * d; Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 36

Adaptive Quadrature Solution adapts to shape of curve. Use three areas, A, B, and C. Computation terminated when largest of A and B sufficiently close to sum of remain two areas. Slides for Parallel Programming Techniques & Applications Using Networked Workstations & Parallel Computers 2 nd ed. , by B. Wilkinson & M. Allen, @ 2004 Pearson Education Inc. All rights reserved. 37