Parallel
vs
Sequential Algorithms
Design of efficient algorithms
A parallel computer is of little use unless
efficient parallel algorithms are available.
–
The issue in designing parallel algorithms are very
different from those in designing their sequential
counterparts.
–
A significant amount of work is being done to
develop efficient parallel algorithms for a variety
of parallel architectures.
Processor Trends
•
Moore’s Law
–
performance doubles every 18 months
•
Parallelization within processors
–
pipelining
–
multiple pipelines
Why Parallel Computing
•
Practical:
–
Moore’s Law cannot hold forever
–
Problems must be solved immediately
–
Cost

effectiveness
–
Scalability
•
Theoretical:
–
challenging problems
Efficient and optimal parallel
algorithms
•
A parallel algorithm is efficient
iff
–
it is fast (e.g. polynomial time) and
–
the product of the parallel time and number of processors is
close to the time of at the best know sequential algorithm
T
sequential
吠
parallel
丠
processors
•
A parallel algorithms
is optimal iff this product is of the
same order as the best known sequential time
•
The basic parallel complexity class is
NC
.
•
NC
is a class of problems computable
in poly

logarithmic
time
(log
c
n, for a constant c)
using a polynomial number of
processors
.
•
P
is a class of problems computable sequentially in a
polynomial time
The main open question in parallel computations is
NC = P
?
The main open question
PRAM
•
PRAM

Parallel Random Access Machine
•
Shared

memory multiprocessor
•
unlimited number of processors, each
–
has unlimited local memory
–
knows its ID
–
able to access the shared
memory in constant time
–
unlimited shared memory
A very reasonable question: Why do we need a PRAM model?
•
to make it easy to reason about algorithms
•
to achieve complexity bounds
•
to analyze the maximum parallelism
.
.
.
P
1
P
2
P
n
.
.
1
2
3
m
P
i
PRAM
MODEL
.
.
.
P
1
P
2
P
n
.
.
?
1
2
3
m
Common Memory
P
i
PRAM
n
RAM processors connected to a common memory of
m
cells
ASSUMPTION:
at each time unit each
P
i
can read a memory cell, make an internal
computation and write another memory cell.
CONSEQUENCE:
any pair of processor
P
i
P
j
can communicate in
constant time!
P
i
writes the message in cell
x
at time
t
P
i
reads the message in cell
x
at time
t+1
Summary of assumptions for PRAM
PRAM
•
Inputs/Outputs are placed in the shared memory (designated
address)
•
Memory cell stores an arbitrarily large integer
•
Each instruction takes unit time
•
Instructions are synchronized across the processors
PRAM Instruction Set
•
accumulator architecture
–
memory cell
R
0
accumulates results
•
multiply/divide instructions take only constant operands
–
prevents generating exponentially large numbers in polynomial time
PRAM Complexity Measures
•
for each individual processor
–
time
: number of instructions executed
–
space
: number of memory cells accessed
•
PRAM machine
–
time
: time taken by the longest running processor
–
hardware
: maximum number of active processors
Two Technical Issues for PRAM
•
How processors are activated
•
How shared memory is accessed
Processor Activation
•
P
0
places the number of processors (
p
) in
the designated shared

memory cell
–
each active
P
i
, where
i
<
p
, starts executing
–
O
(1) time to activate
–
all processors halt when
P
0
halts
•
Active processors explicitly activate
additional processors via FORK instructions
–
tree

like activation
–
O
(log
p
) time to activate
1
0
0
0
0
0
0
i
processor will activate a processor
2i
and a processor
2i+1
...
p
PRAM
•
Too many interconnections gives problems with synchronization
•
However it is the best conceptual model
for designing efficient
parallel algorithms
–
due to simplicity and possibility of simulating efficiently PRAM algorithms
on more realistic parallel architectures
Basic parallel statement
for all x in X do in parallel
instruction (x)
For each x PRAM will assign a
processor which will execute
instruction(x)
Shared

Memory Access
Concurrent
(C) means, many processors can do the operation
simultaneously in the same memory
Exclusive
(E) not concurent
•
EREW (
Exclusive
Read Exclusive Write)
•
CREW (Concurrent Read Exclusive Write)
–
Many processors can read simultaneously the same location, but only
one can attempt to write to a given location
•
ERCW (
Exclusive
Read Concurrent Write)
•
CRCW (
Concurrent
Read Concurrent Write)
–
Many processors can write
/read
at
/from
the same memory location
Concurrent Write (CW)
•
What value gets written finally?
•
Priority CW
–
processors have priority based on which write
value is decided
•
Common CW
–
multiple processors can simultaneously
write only if values are the same
•
Arbitrary/Random CW
–
any one of the values are
randomly chosen
Example CRCW

PRAM
•
Initially
–
table
A
contains values 0 and 1
–
output
contains value 0
•
The program computes the
“Boolean OR”
of
A[1], A[2], A[3], A[4], A[5]
Example CREW

PRAM
•
Assume initially table
A
contains [0,0,0,0,0,1] and we
have the parallel program
Pascal triangle
PRAM CREW
Membership problem
•
p processors PRAM with n numbers (p
≤ n)
•
Does x exist within the n numbers?
•
P0 contains x and finally P0 has to know
Algorithm
step1: Inform everyone what x is
step2: Every processor checks [n/p] numbers and sets a flag
step3: Check if any of the flags are set to 1
One more time about
PRAM model
•
N synchronized processors
•
Shared memory
–
EREW, ERCW,
–
CREW, CRCW
•
Constant time
–
access to the memory
–
standard multiplication/addition
–
Communication
(implemented via access to shared memory)
Two problems for PRAM
Problem 1.
Min of n numbers
Problem 2.
Computing a position of the first
one in the sequence of 0’s and 1’s.
How
fast we can compute with many processor and how to reduce
the number of processors?
Min of n numbers
•
Input: Given an array A with n numbers
•
Output: the minimal number in an array A
Sequential algorithm
…
At least n comparisons should
be performed!!!
COST = (num. of processors)
(time)
Cost = 1
n
?
Sequential vs. Parallel
Optimal
Par.Cost = O(n)
Mission: Impossible
…
computing in a constant time
•
Archimedes:
Give me a lever long
enough and a place to stand and I
will move the earth
•
NOWDAYS….
Give me a parallel machine with
enough processors and I will find
the smallest number in any giant
set in a constant time!
Parallel solution 1
Min of n numbers
•
Comparisons between numbers can be done independently
•
The second part is to find the result using concurrent write mode
•
For n numbers


> we have ~ n
2
pairs
[a
1
,a
2
,a
3
,a
4
]
(a
1
,a
2
)
(a
2
, a
3
)
(a
3
, a
4
)
(a
2
, a
4
)
(a
1
, a
3
)
(a
1
, a
4
)
000000000000000000000000000000000000000000000000
1
0
(a
i
,a
j
)
If a
i
> a
j
then a
i
cannot be the minimal number
i
j
1
n
M[1..n]
The following program computes MIN of n numbers stored in
the array C[1..n] in O(1) time with n
2
processors.
Algorithm A1
for each 1
i
n do in parallel
M[i]:=0
for each 1
i,j
n do in parallel
if i
j C[i]
C[j] then M[j]:=1
for each 1
i
n do in parallel
if M[i]=0 then output:=i
From n
2
processors to n
1+1/2
Step 1: Partition into disjoint blocks of size
Step 2: Apply A1 to each block
Step 3: Apply A1 to the results from the step 2
A1
A1
A1
A1
A1
A1
A1
A1
A1
A1
A1
n
n
n
n
From n
1+1/2
processors to n
1+1/4
Step 1: Partition into disjoint blocks of size
Step 2: Apply A2 to each block
Step 3: Apply A2 to the results from the step 2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
n
n
2

> n
1+1/2

> n
1+1/4

> n
1+1/8

> n
1+1/16

>…

> n
1+1/k
~ n
1
•
Assume that we have an algorithm A
k
working in O(1)
time with processors
Algorithm A
k+1
1.Let
=1/2
2. Partition the input array C of size n into disjoint
blocks of size n
each
3. Apply in parallel algorithm A
k
to each of these blocks
4. Apply algorithm A
k
to the array C’ consisting of n/ n
minima in the blocks.
k
n
1
Complexity
•
We can compute minimum of n numbers
using CRCW PRAM model in O(log log n) with
n processors by applying a strategy of
partitioning the input
ParCost = n
log log n
Mission: Impossible
(Part 2)
Computing a position of the first one in the sequence of 0’s and 1’s
in a constant time.
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1
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Problem 2.
Computing a position of the first one in the sequence of 0’s
and 1’s.
FIRST

ONE

POSITION(C)=4 for
the input array
C=[0,0,0,1,0,0,0,1,1,1,0,0,0,1]
Algorithm A
(2 parallel steps and n
2
processors)
for each 1
i
<j
n do in parallel
if
C[
i
] =1 and C[j]=1 then C[j]:=0
for each 1
i
n do in parallel
if
C[
i
] =1 then
FIRST

ONE

POSITION:=
i
1
1
1
0
After the first parallel step
C will contain a single
element 1
Reducing number of processors
Algorithm B
–
it reports if there is any one in the
table.
There

is

one:=0
for each 1
i
n do in parallel
if
C[i] =1 then
There

is

one:=1
000000000000000000
1
1
1
Now we can merge two algorithms A and B
1.
Partition table C into segments of size
2.
In each segment apply the algorithm B
3.
Find position of the first one in these sequence by applying
algorithm A
4.
Apply algorithm A to this single segment and compute the final
value
n
B
B
B
B
B
B
B
B
B
B
A
A
Complexity
•
We apply an algorithm A twice and each time
to the array of length
which need only ( )
2
= n processors
•
The time is O(1) and number of processors is n.
n
n
Tractable and intractable problems
for parallel computers
P (complexity)
•
In computational complexity theory, P is the complexity class
containing decision problems which can be solved by a
deterministic Turing machine using a polynomial amount of
computation time, or polynomial time.
•
P is known to contain many natural problems, including linear
programming, calculating the greatest common divisor, and
finding a maximum matching.
•
In 2002, it was shown that the problem of determining if a
number is prime is in P.
P

complete class
•
In complexity theory, the complexity class P

complete is a set
of decision problems and is useful in the analysis of which
problems can be efficiently solved on parallel computers.
•
A decision problem is in P

complete if it is complete for P,
meaning that it is in P, and that every problem in P can be
reduced to it in polylogarithmic time on a parallel computer
with a polynomial number of processors.
•
In other words, a problem A is in P

complete if, for each
problem B in P, there are constants c and k such that B can be
reduced to A in time O((log n)
c
) using O(n
k
) parallel
processors.
Motivation
•
The class P, typically taken to consist of all the "tractable"
problems for a sequential computer, contains the class NC,
which consists of those problems which can be efficiently
solved on a parallel computer. This is because parallel
computers can be simulated on a sequential machine.
•
It is not known whether NC=P. In other words, it is not known
whether there are any tractable problems that are inherently
sequential.
•
Just as it is widely suspected that P does not equal NP, so it is
widely suspected that NC does not equal P.
P

complete problems
•
The most basic P

complete problem is this:
Given a
Turing machine, an input for that machine, and a number T
(written in unary), does that machine halt on that input within the first
T steps?
•
It is clear that this problem is P

complete: if we can parallelize
a general simulation of a sequential computer, then we will be
able to parallelize any program that runs on that computer.
•
If this problem is in NC, then so is every other problem in P.
•
This problem illustrates a common trick in the theory of P

completeness. We aren't really interested in whether a
problem can be solved quickly on a parallel machine.
•
We're just interested in whether a parallel machine solves it
much more quickly than a sequential machine. Therefore, we
have to reword the problem so that the sequential version is
in P. That is why this problem required T to be written in
unary.
•
If a number T is written as a binary number (a string of n ones
and zeros, where n=log(T)), then the obvious sequential
algorithm can take time 2
n
. On the other hand, if T is written
as a unary number (a string of n ones, where n=T), then it only
takes time n. By writing T in unary rather than binary, we have
reduced the obvious sequential algorithm from exponential
time to linear time. That puts the sequential problem in P.
Then, it will be in NC if and only if it is parallelizable.
P

complete problems
•
Many other problems have been proved to be P

complete, and therefore are widely believed to be
inherently sequential. These include the following
problems, either as given, or in a decision

problem
form:
•
In order to prove that a given problem is P

complete,
one typically tries to reduce a known P

complete
problem to the given one, using an efficient parallel
algorithm.
Examples of P

complete problems
•
Circuit Value Problem (CVP)

Given a circuit, the
inputs to the circuit, and one gate in the circuit,
calculate the output of that gate
•
Game of Life

Given an initial configuration of
Conway's Game of Life
, a particular cell, and a time
T
(in unary), is that cell alive after
T
steps?
•
Depth First Search Ordering

Given a graph with
fixed ordered adjacency lists, and nodes u and v, is
vertex u visited before vertex v in a depth

first
search?
Problems not known to be P

complete
•
Some problems are not known to be either NP

complete or P.
These problems (e.g. factoring) are suspected to be difficult.
•
Similarly there are problems that are not known to be either
P

complete or NC, but are thought to be difficult to
parallelize.
•
Examples include the decision problem forms of finding the
greatest common divisor of two binary numbers, and
determining what answer the extended Euclidean algorithm
would return when given two binary numbers.
Conclusion
•
Just as the class P can be thought of as the tractable
problems, so NC can be thought of as the problems that can
be efficiently solved on a parallel computer.
•
NC is a subset of P because parallel computers can be
simulated by sequential ones.
•
It is unknown whether NC = P, but most researchers suspect
this to be false, meaning that there are some tractable
problems which are probably "inherently sequential" and
cannot significantly be sped up by using parallelism
•
The class P

Complete can be thought of as "probably not
parallelizable" or "probably inherently sequential".
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