1. Parallel Algorithms

2. Computation Models
Goal of computation model is to provide a realistic representation of the costs of programming.
Model provides algorithm designers and programmers a measure of algorithm complexity which helps them decide what is “good” (i.e. performance-efficient)

3. Goal for Modeling
We want to develop computational models which accurately represent the cost and performance of programs
If model is poor, optimum in model may not coincide with optimum observed in practice
Model
Real World x
optimum A
optimum B Y

4. Models of Computation What’s a model good for??
Provides a way to think about computers. Influences design of:
Architectures
Languages
Algorithms
Provides a way of estimating how well a program will perform.
Cost in model should be roughly same as cost of executing program

5. The Random Access Machine Model
RAM model of serial computers:
Memory is a sequence of words, each capable of containing an integer.
Each memory access takes one unit of time
Basic operations (add, multiply, compare) take one unit time.
Instructions are not modifiable
Read-only input tape, write-only output tape

6. Has RAM influenced our thinking?
Language design:
No way to designate registers, cache, DRAM.
Most convenient disk access is as streams.
How do you express atomic read/modify/write?
Machine & system design:
It’s not very easy to modify code.
Systems pretend instructions are executed in-order.
Performance Analysis:
Primary measures are operations/sec (MFlop/sec, MHz, ...)
What’s the difference between Quicksort and Heapsort??

7. What about parallel computers
RAM model is generally considered a very successful “bridging model” between programmer and hardware.
“Since RAM is so successful, let’s generalize it for parallel computers ...”

8. PRAM [Parallel Random Access Machine]
(Introduced by Fortune and Wyllie, 1978)
PRAM composed of:
P processors, each with its own unmodifiable program.
A single shared memory composed of a sequence of words, each capable of containing an arbitrary integer.
a read-only input tape.
a write-only output tape.
PRAM model is a synchronous, MIMD, shared address space parallel computer.

9. PRAM model of computation
Shared memory
p processors, each with local memory
Synchronous operation
Shared memory reads and writes
Each processor has unique id in range 1-p

10. Characteristics
At each unit of time, a processor is either active or idle (depending on id)
All processors execute same program
At each time step, all processors execute same instruction on different data (“data- parallel”)
Focuses on concurrency only

11. Variants of PRAM model

12. More PRAM taxonomy
Different protocols can be used for reading and writing shared memory.
EREW - exclusive read, exclusive write
A program isn’t allowed to have two processors access the same memory location at the same time.
CREW - concurrent read, exclusive write
CRCW - concurrent read, concurrent write
Needs protocol for arbitrating write conflicts
CROW – concurrent read, owner write
Each memory location has an official “owner”
PRAM can emulate a message-passing machine by partitioning memory into private memories.

13. Sub-variants of CRCW Common CRCW Arbitrary CRCW Priority CRCW
CW iff all processors writing same value
Arbitrary CRCW
Arbitrary value of write set stored
Priority CRCW
Value of min-index processor stored
Combining CRCW

14. Why study PRAM algorithms?
Well-developed body of literature on design and analysis of such algorithms
Baseline model of concurrency
Explicit model
Specify operations at each step
Scheduling of operations on processors
Robust design paradigm

15. optimal sequential Algorithm
Work-Time paradigm
Higher-level abstraction for PRAM algorithms
WT algorithm = (finite) sequence of time steps with arbitrary number of operations at each step
Two complexity measures
Step complexity T(n)
Work complexity W(n)
WT algorithm work-efficient if W(n) = Q(TS(n))
optimal sequential Algorithm

16. Designing PRAM algorithms
Balanced trees
Pointer jumping
Euler tours
Divide and conquer
Symmetry breaking
. . .

17. Balanced trees
Key idea: Build balanced binary tree on input data, sweep tree up and down
“Tree” not a data structure, often a control structure (e.g., recursion)

18. Alg : Sum Given: Sequence a of n = 2k elements
Given: Binary associative operator +
Compute: S = a an

19. WT description of sum integer B[1..n] forall i in 1 : n do B[i] := ai
enddo
for h = 1 to k do
forall i in 1 : n/2h do
B[i] := B[2i-1] + B[2i]
S := B[1]

20. Points to note about WT pgm
Global program: no references to processor id
Contains both serial and concurrent operations
Semantics of forall
Order of additions different from sequential order: associativity critical

21. Analysis of scan operation
Algorithm is correct
Q(lg n) steps, Q(n) work
EREW model
Two variants
Inclusive: as discussed
Exclusive: s1 = I, sk = x xk-1
If n not power of 2, pad to next power

22. Complexity measures of Sum
Recall definitions of step complexity T(n) and work complexity W(n)
Concurrent execution reduces number of steps

23. How to do prefix sum ?
Input: Sequence x of n = 2k elements, binary associative operator +
Output: Sequence s of n = 2k elements, with sk = x xk
Example:
x = [1, 4, 3, 5, 6, 7, 0, 1]
s = [1, 5, 8, 13, 19, 26, 26, 27]

24. { List Ranking List ranking problem If d denotes the distance
Given a singly linked list L with n objects, for each node, compute the distance to the end of the list
If d denotes the distance
node.d = if node.next = nil
node.next.d otherwise
Serial algorithm: O(n)
Parallel algorithm
Assign one processor for each node
Assume there are as many processors as list objects
For each node i, perform
i.d = i.d + i.next.d
i.next = i.next.next // pointer jumping {

25. List Ranking - Pointer Jumping
9/21/00
List Ranking - Pointer Jumping
List_ranking(L)
for each node i, in parallel do
if i.next = nil then i.d = 0
else i.d = 1
while exists a node i, such that i.next != nil do
if i.next != nil then
i.d = i.d + i.next.d // i updates i itself
i.next = i.next.next
Analysis
After a pointer jumping, a list is transformed into two (interleaved) lists
After that, four (interleaved) lists
Each pointer jumping doubles the number of lists and halves their length
After log n, all lists contain only one node
Total time: O(log n)

26. 9/21/00
List Ranking - Example
Fig. 30.2

27. List Ranking - Discussion
Synchronization is important
In step 8 (i.next = i.next.next), all processors must read right hand side before any processor write left hand side
The list ranking algorithm is EREW
If we assume in step 7 (i.d = i.d + i.next.d) all processors read i.d and then read i.next.d
If j.next = i, i and j do not read i.d concurrently
Work performance
performs O(n log n) work since n processors in O(log n) time
Work efficient
A PRAM algorithm is work efficient w.r.t another algorithm if two algorithms are within a constant factor
Is the link ranking algorithm work-efficient w.r.t the serial algorithm?
No, because O(n log n) versus O(n)
Speedup
S = n / log n

28. Parallel Prefix on a List
Prefix computation
Input <x1, x2, .., xn>, a binary, associative operator 
Output <y1, y2, .., yn>
Prefix computation: yk = x1  x2 ..  xk
Example
if xk = 1 for k=1..n and  = +
Then yk = k, for k = 1..n
Serial algorithm: O(n)
Notation
[i, j] = xi  xi+1  ..  xj
[k, k] = xk
[i, k]  [k+1, j] = [i, j]
Idea: perform prefix computation on a linked list so that
each node k contains [k, k] = xk initially
finally each node k contains [1, k] = yk

29. Parallel Prefix on a List (2)
List_prefix(L, X)
// L: list, X: <x1, x2, .., xn>
for each node i, in parallel
i.y = xi
While exists a node i such that i.next != nil do
for each node i, in parallel do
if i.next != nil then
i.next.y = i.y  i.next.y // i updates its successor
i.next = i.next.next
Analysis
Initially k-th node has [k,k] as y-value, points to (k+1)-th node
At the first iteration,
k-th node fetches [k+1,k+1] from its successor and
perform [k,k]  [k+1,k+1] resulting in [k,k+1] and
update its successor
At the second iteration
k-th node fetches [k+1,k+2] from its successor and
perform [k-1,k]  [k+1,k+2] resulting in [k-1,k+2] and

30. Parallel Prefix on a List (3)
9/21/00
Parallel Prefix on a List (3)
Fig. 30.3

31. Parallel Prefix on a List (4)
Running time: O(log n)
After log n, all lists contain only one node
Work performed: O(n log n)
Speedup
S = n / log n

32. Pointer jumping
Fast parallel processing of linked data structures (lists, trees)
Convention: Draw trees with edges directed from children to parents
Example: Finding the roots of forest represented as parent array P
P[i] = j if and only if (i, j) is a forest edge
P[i] = i if and only if i is a root

33. Algorithm (Roots of forest)
forall i in 1:n do
S[i] := P[i]
while S[i] != S[S[i]] do
S[i] := S[S[i]]
endwhile
enddo

34. Initial state of forest

35. After one iteration

36. After another iteration

37. Concurrent Read – Finding Roots
9/21/00
Concurrent Read – Finding Roots
Fig. 30.5

38. Analysis of pointer jumping
Termination detection?
At each step, tree distance between i and S[i] doubles unless S[i] is a root
CREW model
Correctness by induction on h
O(lg h) steps, O(n lg h) work
TS(n) = O(n)
Not work-efficient unless h constant

39. Concurrent Read – Finding Roots
This is a CREW algorithm
Suppose Exclusive-Read is used, what will be the running time?
Initially only one node i has root information
First iteration: Another node reads from the node i
Totally two nodes are filled up
Second iteration: Another two nodes can reads from the two nodes
Totally four nodes are filled up
k-th iteration: 2k-1 nodes are filled up
If there are n nodes, k=log n
So Find_root with Exclusive-Read takes O(log n).
O(log log n) vs. O(log n)

40. Euler tours Technique for fast optimal processing of tree data
Euler circuit of directed graph: directed cycle that traverses each edge exactly once
Represent (rooted) tree by Euler circuit of its directed version

41. Trees = balanced parentheses
( ( ( ) ( ) ) ( ) ( ( ) ( ) ( ) ) )
Key property: The parenthesis subsequence
corresponding to a subtree is balanced.

42. Computing the Depth Problem definition
Given a binary tree with n nodes, compute the depth of each node
Serial algorithm takes O(n) time
A simple parallel algorithm
Starting from root, compute the depths level by level
Still O(n) because the height of the tree could be as high as n
Euler tour algorithm
Uses parallel prefix computation

43. { Computing the Depth (2) { {
Euler tour: A cycle that traverses each edge exactly once in a graph
It is a directed version of a tree
Regard an undirected edge into two directed edges
Any directed version of a tree has an Euler tour by traversing the tree
in a DFS way forming a linked list.
Employ 3*n processors
Each node i has fields i.parent, i.left, i.right
Each node i has three processors, i.A, i.B, and i.C.
Three processors in each node of the tree are linked as follows
i.A = i.left.A if i.left != nil
i.B if i.left = nil
i.B = i.right.A if i.right != nil
i.C if i.right = nil
i.C = i.parent.B if i is the left child
i.parent.C if i is the right child
nil if i.parent = nil { { {

44. Computing the Depth (3) Algorithm O(log n)
Construct the Euler tour for the tree – O(1) time
Assign 1 to all A processors, 0 to B processors, -1 to C processors
Perform a parallel prefix computation
The depth of each node resides in its C processor
O(log n)
Actually log 3n
EREW because no concurrent read or write
Speedup
S = n/log n

45. 9/21/00
Computing the Depth (4)
Fig. 30.4

46. Broadcasting on a PRAM
“Broadcast” can be done on CREW PRAM in O(1) steps:
Broadcaster sends value to shared memory
Processors read from shared memory
Requires lg(P) steps on EREW PRAM. M P B

47. Concurrent Write – Finding Max
Finding max problem
Given an array of n elements, find the maximum(s)
sequential algorithm is O(n)
Data structure for parallel algorithm
Array A[1..n]
Array m[1..n]. m[i] is true if A[i] is the maximum
Use n2 processors
Fast_max(A, n)
for i = 1 to n do, in parallel
m[i] = true // A[i] is potentially maximum
for i = 1 to n, j = 1 to n do, in parallel
if A[i] < A[j] then
m[i] = false
if m[i] = true then max = A[i]
return max
Time complexity: O(1)

48. Concurrent Write – Finding Max
In step 4 and 5, processors with A[i] < A[j] write the same value ‘false’ into the same location m[i]
This actually implements m[i] = (A[i]  A[1])  …  (A[i]  A[n])
Is this work efficient?
No, n2 processors in O(1)
O(n2) work vs. sequential algorithm is O(n)
What is the time complexity for the Exclusive-write?
Initially elements “think” that they might be the maximum
First iteration: For n/2 pairs, compare.
n/2 elements might be the maximum.
Second iteration: n/4 elements might be the maximum.
log n th iteration: one element is the maximum.
So Fast_max with Exclusive-write takes O(log n).
O(1) (CRCW) vs. O(log n) (EREW)

49. Simulating CRCW with EREW
CRCW algorithms are faster than EREW algorithms
How much fast?
Theorem
A p-processor CRCW algorithm can be no more than O(log p) times faster than the best p-processor EREW algorithm
Proof by simulating CRCW steps with EREW steps
Assumption: A parallel sorting takes O(log n) time with n processors
When CRCW processor pi write a datum xi into a location li, EREW pi writes the pair (li, xi) into a separate location A[i]
Note EREW write is exclusive, while CRCW may be concurrent
Sort A by li
O(log p) time by assumption
Compare adjacent elements in A
For each group of the same elements, only one processor, say first, write xi into the global memory li.
Note this is also exclusive.
Total time complexity: O(log p)

50. Simulating CRCW with EREW (2)
9/21/00
Simulating CRCW with EREW (2)
Fig. 30.7

51. CRCW versus EREW - Discussion
Hardware implementations are expensive
Used infrequently
Easier to program, runs faster, more powerful.
Implemented hardware is slower than that of EREW
In reality one cannot find maximum in O(1) time
EREW
Programming model is too restrictive
Cannot implement powerful algorithms