1. Introduction to parallel algorithms
CIS – Spring 2005
Introduction to parallel algorithms
Ashok Srinivasan
Florida State University

2. Outline
Background
Primitives
Algorithms
Important points

3. Background Terminology Communication cost model Time complexity
Speedup
Efficiency
Scalability
Communication cost model

4. Time complexity Parallel computation
A group of processors work together to solve a problem
Time required for the computation is the period from when the first processor starts working until when the last processor stops
Sequential
Parallel - bad
Parallel - ideal
Parallel - realistic

5. Other terminology Speedup: S = T1/TP Efficiency: E = S/P
Work: W = P TP
Scalability
How does TP decrease as we increase P to solve the same problem?
How should the problem size increase with P, to keep E constant?
Notation
P = Number of processors
T1 = Time on one processor
TP = Time on P processors

6. Communication cost model
Processes spend some time doing useful work, and some time communicating
Model communication cost as
TC = ts + L tb
L = message size
Independent of location of processes
Any process can communicate with any other process
A process can simultaneously send and receive one message

7. I/O model We will ignore I/O issues, for the most part
We will assume that input and output are distributed across the processors in a manner of our choosing
Example: Sorting
Input: x1, x2, ..., xn
Initially, xi is on processor i
Output xp1, xp2, ..., xpn
xpi on processor i
xpi < xpi+1

8. Primitives
Reduction
Broadcast
Gather/Scatter
All gather
Prefix

9. Reduction -- 1 Tn = n-1 + (n-1)(ts+tb) Sn = 1/(1 + ts + tb)x1
Compute x1 + x xn x2 xn x3 x4
Tn = n-1 + (n-1)(ts+tb)
Sn = 1/(1 + ts + tb)

10. Reduction -- 2 Tn = n/2-1 + (n/2-1)(ts+ tb) + (ts+ tb) + 1x1
Reduction-1
for {x1, ... xn/2}
xn/2+1
Reduction-1
for {xn/2+1, ... xn}
Tn = n/2-1 + (n/2-1)(ts+ tb) + (ts+ tb) + 1
= n/2 + n/2 (ts+ tb)
Sn ~ 2/(1 + ts+ tb)

11. Reduction -- 3 Apply reduction-2 recursively
xn/2+1 x1
Reduction-1
for {x1, ... xn/2}
Reduction-1
for {xn/2+1, ... xn}
xn/4+1
xn/2+1
x3n/4+1 x1
Reduction-1
for {x1, ... xn/4}
Reduction-1
for {xn/4+1, ... xn/2}
Reduction-1
for {xn/2+1, ... x3n/4}
Reduction-1
for {x3n/4+1, ... xn}
Apply reduction-2 recursively
* Divide and conquer
Tn ~ log2n + (ts+ tb) log2n
Sn ~ (n/ log2n) x 1/(1 + ts+ tb)
Note that any associative operator can be used in place of +

12. Parallel addition features
If n >> P
* Each processor adds n/P distinct numbers
Perform parallel reduction on P numbers
TP ~ n/P + (1 + ts+ tb) log P
Optimal P obtained by differentiating wrt P
Popt ~ n/(1 + ts+ tb)
If communication cost is high, then fewer processors ought to be used
E = [1 + (1+ ts+ tb) P log P/n]-1
* As problem size increases, efficiency increases
* As number of processors increases, efficiency decreases

13. Some common collective operations
Broadcast A B C D
A, B, C, D
Gather A
A, B, C, D B C D
Scatter A B C D
A, B, C, D
All Gather

14. Broadcast T ~ (ts+ Ltb) log P L: Length of data x7 x8 x1 x1 x2 x3 x4

15. Gather/Scatter Gather: Data move towards the root
Note: Si=0log P–1 2i
= (2 log P – 1)/(2–1) = P-1
~ P
x18 4L
x14
x58 2L 2L
x12
x34
x56
x78 L L L L x1 x2 x3 x4 x5 x6 x7 x8
Gather: Data move towards the root
Scatter: Review question
T ~ ts log P + PLtb

16. All gather x7 x8 x3 x4 x5 x6 L x1 x2
Equivalent to each processor broadcasting to all the processors

17. All gather
x78
x78
x34
x34 2L
x56
x56 L
x12
x12

18. All gather
x58
x58
x14
x14 2L
x58
x58 L 4L
x14
x14

19. All gather Tn ~ ts log P + PLtb 2L L 4L x18 x18 x18 x18 x18 x18 x18

20. Review question: Pipelining
* Useful when repeatedly and regularly performing a large number of primitive operations
Optimal time for a broadcast = log P
But doing this n times takes n log P time
Pipelining the broadcasts takes n + P time
Almost constant amortized time per broadcast
if n >> P
n + P << n log P when n >> P
Review question: How can you accomplish this time complexity?

21. Sequential prefix Input Output Algorithm Values xi , 1 < i < n
Xi = x1 * x2 * ... * xi, 1 < i < n
* is an associative operator
Algorithm
X1 = x1
for i = 2 to n
Xi = Xi-1 * xi

22. Parallel prefix Input Output Define f(a,b) as follows
Processor i has xi
Output
Processor i has
x1 * x2 * ... * xi
Define f(a,b) as follows
if a == b
Xi = xi, on Proc Pi
else
compute in parallel
f(a,(a+b)/2)
f((a+b)/2+1,b)
Pi and Pj send Xi and Xj to each other, respectively
a < i < (a+b)/2
j = i + (a+b)/2
Xi = Xi*Xj on Pi
Xj = Xi*Xj on Pj
Divide and conquer
f(a,b) yields the following
Xi = xa *... * xi, Proc Pi
Xi = xa *... * xb, Proc Pi
a < i < b
f(1,n) solves the problem
T(n) = t(n/2) (ts+tw) => T(n) = O(log n)
An iterative implementation improves the constant

23. Iterative parallel prefix example

24. Algorithms
Linear recurrence
Matrix vector multiplication

25. Determine each xi, 2 < i < n
Linear recurrence
Determine each xi, 2 < i < n
xi = ai xi-1 + bi xi-2
x0 = x0, x1 = x1
Sequential solution
for i = 2 to n
Follows directly from the recurrence
This approach is not easily parallelized

26. Linear recurrence in parallel
Given xi = ai xi-1 + bi xi-2
x2i = a2i x2i-1 + b2i x2i-2
x2i+1 = a2i+1 x2i + b2i+1 x2i-1
Rewrite this in matrix form
x2i
x2i+1
x2i-2
x2i-1
b2i a2i
a2i+1 b2i b2i+1 + a2i+1 a2i Ai
Xi-1 Xi
Xi = Ai A i A1X0
This is a parallel prefix computation, since matrix multiplication is associative
Solved in O(log n) time

27. Matrix-vector multiplication
c = A b
Often performed repeatedly
bi = A bi-1
We need same data distribution for c and b
One dimensional decomposition
Example: row-wise block striped for A
b and c replicated
Each process computes its components of c independently
Then all-gather the components of c

28. 1-D matrix-vector multiplication
c: Replicated
A: Row-wise
b: Replicated
Each process computes its components of c independently
Time = Q(n2/P)
Then all-gather the components of c
Time = ts log P + tb n
Note: P < n

29. 2-D matrix-vector multiplicationB0 C1
A10
A11
A12
A13 B1 C2
A20
A21
A22
A23 B2 C3
A30
A31
A32
A33 B3
Processes Pi0 sends Bi to P0i
Time: ts + tbn/P0.5
Processes P0j broadcast Bj to all Pij
Time = ts log P0.5 + tb n log P0.5 / P0.5
Processes Pij compute Cij = AijBj
Time = Q(n2/P)
Processes Pij reduce Cij on to Pi0, 0 < i < P0.5
Total time = Q(n2/P + ts log P + tb n log P / P0.5 )
P < n2
* More scalable than 1-dimensional decomposition

30. Important points Efficiency
Increases with increase in problem size
Decreases with increase in number of processors
Aggregation of tasks to increase granularity
Reduces communication overhead
Data distribution
2-dimensional may be more scalable than 1-dimensional
Has an effect on load balance too
General techniques
Divide and conquer
Pipelining