Definitions• A Synchronous application is one where all

processes must reach certain points before execution continues.

• Local synchronization is a requirement that a subset of processes (usually neighbors) reach a synchronous point before execution continues.

• A barrier is the basic message passing mechanism for synchronizing processes.

• Deadlock occurs when groups of processors are permanently waiting for messages that cannot be satisfied because the sending processes are also permanently waiting for messages.

Barrier Illustration

P0

P0

P0

P0

P0

Barrie

r

Waiting

Executing

C: MPI_Barrier(MPI_COMM_WORLD);mpiJava: MPI.COMM_WORLD.Barrier();

Counter (linear) Barrier

Master ProcessorFor (i=0; i<P; i++) // Arrival Phase

Receive null message from any processorFor (i=0; i<P; i++) // Departure Phase

Send null message to release slaves

Slave ProcessorsSend null message to enter barrierReceive null message for barrier release

Note: This logic avoids processors arriving before prior release

Tree (non-linear) BarrierP

0P

1P

2P

3P

4P

5P

6P

7

P0

P1

P2

P3

P4

P5

P6

P7Entry Phase Release Phase

Note: Implementation logic is similar to divide and conquer

Barrier Barrier

– Stage 1: P0p1; p2p3; p4p5; p6p7

– Stage 2: p0 p2; p1p3; p4p6; p5p7

– Stage 3: p0p4; p1p5; p2p6; p3p7

P0 P1 P2 P3 P4 P5 P6 P7

P0 P1 P2 P3 P4 P5 P6 P7

P0 P1 P2 P3 P4 P5 P6 P7

P0 P1 P2 P3 P4 P5 P6 P7

Local Synchronization

• Even ProcessorsSend null message to processor i-1Receive null message from processor i-1Send null message to processor i+1Receive null message from processor i+1

• Odd Numbered ProcessorsReceive null message from processor i+1Send null message to processor i+1Receive null message from processor i-1Send null message to processor i-1

• Notes: – Local Synchronization is an incomplete barrier

• processors exit after receiving messages from their neighbors– Deadlock can occur if the message passing order is incorrect.

• MPI_Sendrecv() and MPI_Sendrecv_replace() are deadlock free

Synchronize with neighbors before proceeding

Local Synchronization Example

• Heat Distribution Problem– Goal

• Determine final temperature at each n x n grid point

– Initial boundary condition• Know initial temperatures at the end

points

– Cannot proceed to next iteration until local synchronization completes

DOAverage each grid point with its neighbors

UNTIL temperature changes are small enough

New Value =(∑neighbors)/4

Sequential Heat Distribution Code

Initialize rows 0,n and columns 0,n of g and h

Iteration = 0

DO

FOR (i=1; i<n; i++)

FOR (j=1; j<n; j++)

IF (iteration %2) hi,j = (gi-1,j+gi+1,j+gi,j-1+gi,j+1)/4

ELSE gi,j = (hi-1,j+hi+1,j+hi,j-1+hi,j+1)/4

iteration++

UNTIL max (|gi – hi|)<tolerance or iteration>MAX

Block or Strip Partitioning

p0 p1 p2 p3

p4 p5 p6 p7

p8 p9 p10 p11

p12 p13 p14 p15

p0 p1 p2 p3 p4 p5 p6 p7

Blocks

Column Strips

• Block Partitioning– Eight messages exchanged

at each iteration– Data exchanged per

message is n/P1/2

• Strip Partitioning– Four messages exchanged

at each iteration– Data exchanged per

message is n • Question: Which is better?

Assign portions of the grid to processors in the topology

Parallel Implementation

• Algorithm Modifications– Declare “ghost” rows to hold adjacent data (10 x 10 for 8 x 8 block)

– Exchange data with neighbor processors

– Perform the calculation for the local grid cells

PiCells to east

Cells to south

Cells to west

Cells to north

Heat Distribution PartitioningSendRcv(row,col) if row,col is not local if myrank even Send(point,prow,col) Recv(point,prow,col)Else Recv(point,prow,col) Send(point,prow,col)

Main logicFor each iteration

For each point compute new temperature

SendRcv(row-1,col,point)SendRcv(row+1,col,point)SendRcv(row,col-1,point)SendRcv(row,col+1,point)

Fully Synchronized Example

• Data Parallel Computations– Simultaneously apply the same operation to different data

• Sequential Codefor (i=1; i<n; i++) a[i] = someFunction(a[i])

• Shared Memory CodeForall (i=0; i<n; i++) {bodyOfInstructions}

– In these cases, the for loop is a natural barrier

• Distributed processingFor local a[i]; {someFunction(a[i])}

barrier();

Data Parallel Example

A[0] += k A[1] += k A[n-1] += k

A[] += k

•All processors execute instructions in “lock step”

•Forall (i=0; i<n; i++) a[i] += k

Note: Multicomputer configurations partition numbers in blocks

p0 p1 pn

Prefix Sum Problem

• Definition: Given numbers a[i]; i=0; i<n, the prefix sum of a[i] is: a[i] += a[0] + a[1] + … + a[i-1]

• Application: Radix Sort

• Sequential codefor (j=0;j<lg(n);j++) for (i=2j; i<n; i++) a[i] += a[i-2j];

• Parallel shared memory codefor (j=0; j<lg(n); j++) forall (i=2j; i<n; i++) a[i] += a[i-2j];

• Parallel distributed memory code for (j=1; j<= log(n); j++) if (myrank>=2j-1 receive(sum, myrank – 2j-1) a[myrank] = a[myrank] += sum

else send(a[myrank], myrank + 2j)

Note: Prefix Sum algorithm works for any associative operation

Prefix Sum Illustration

Synchronous Iteration

• Processes synchronize at each iteration step• Example: Simulation of Natural Processes• Shared memory code for (j=0; j<n; j++) forall (i=0; i<N; i++) body(i);

• Distributed memory code for (j=0; j<n; j++) body(myRank); barrier();

Example: n equations of n unknowns

an-1,0x0 + an,1x1 … + an,n-1xn-1 = bk

∙∙∙

ak,0x0 + ak,1x1 … + ak,n-1xn-1 = bk

∙∙∙

a1,0x0 + a1,1x1 … + a1,n-1xn-1 = b1

a0,0x0 + a0,1x1 … + a0,n-1xn-1 = b0

• Or rewrite equations as follows:xk=(bk–ak,0x0-…-ak,j-1xj-1-ak,j+1xj+1-…-ak,n-1xn-1)/ak,k

= (bk - ∑j≠kai,j xj)/ai,i

Jacobi Iteration

• Jacobi Iterationxnewi = initial guess

DO

xi = xnewi

xnewi = Calculated next guess

UNTIL ∑i|xnewi – xi|<tolerance

• Jacobi iteration always converges if:

ak,k > ∑i≠k ai,0

i i+1

Error

Iteration

xi

Parallel Jacobi Code

xi = bi

DO for each i

sum = -ai,i * xi

FOR (j=0; j<n; j++) sum += ai,i * xj

xnewi = (bi – sum)/ai,i

allgather(xnewi)

barrier()

Until iterations>MAX or ∑i|xnewi – xi|<tolerance

xnew0 xnew1 xnewn-1

xi

Allgather() xnewi into xi

Additional Jacobi Notes

• What if P (processor count) < n?– Answer: Allocate blocks of variables

to processors

• Block Allocation– Allocate consecutive xi to processors

• Cyclic Allocation– Allocate x0, xP, … to p0– Allocate x1, xp+1, … to p1 … etc.

• Question: Which allocation scheme is better?

Time

Processors4 8 12 16 20 24

Computation

Communication

Jacobi Performance

Cellular Automata

• The System has a finite grid of cells

• Each cell can assume a finite number of states

• Neighbor cells affect a cell according to rule set

• All cell changes of state occur simultaneously

• The system iterates through a number of generations

Serious Applications:•Fluid and gas dynamics•Biological growth•Airplane wing airflow•Erosion modeling•Groundwater pollution

Definition

Conway’s Game of Life• The grid is a two dimension array of cells

– The grid ends can optionally wrap around (like a torus)• Each cell

– Can hold one “organism” – There are eight neighbor cells

o North, Northeast, East, Southeast, South, Southwest, West, Northwest

• Rules (run the simulation over many generations)1. Organism dies from loneliness if 0 or 1 organisms live in

neighbor cells2. Organism survives if 2 organisms live in adjacent cells3. An empty cell with 3 living neighbors gives birth to

organisms in every empty adjacent cell 4. Organism dies from overpopulation >= 4 organisms live in

neighbor cells

Sharks and Fishes• The grid (ocean) is modeled by a three dimension array

– The grid ends can optionally wrap around (like a torus)• Each cell

– Can hold either a fish or a shark, but not both– There are twenty six neighbor cells

• Rules for fish1. Fish move randomly to empty adjacent cells2. If there are no empty adjacent cells, fish stay put3. Fish of breeding age leave a baby fish in the vacating cell4. Fish die after x generations

• Rules for sharks1. Sharks randomly move to adjacent cells with fish, eating the fish2. If no adjacent cells have fish, the shark moves randomly to empty

cells. It stays put if there are no empty cells3. Sharks of breeding age leave a baby shark in the vacating cell4. Sharks that die if they don’t eat a fish for y generations