Parallel Algorithm Design & Performance Analysis

Parallelism Generally

Here's a survey of data structures used by scientific simulation software. I prepared this while in Illinois, working on a variety of scientific simulation software.

Some applications are quite easy to parallelize. Others... aren't.

"Naturally Parallel" applications don't need any communication. For example, Mandelbrot set rendering is naturally parallel, because each Mandelbrot pixel is a stand-alone problem you can solve independently of all the other Mandelbrot pixels. GPUs can handle naturally parallel applications in a single pass. The other common term for this is "Trivially Parallel" or "Embarrasingly Parallel", which makes it sound like a bad thing--but parallelism is both natural and good!
"Neighbor" applications communicate with their immediate neighbors, and that's it. For example, heat flow in a plate can be computed based solely on one's immediate neighbors (new value at arr[i] is a function of the old value of arr[i-1], arr[i], and arr[i+1]). Neighbor applications naturally fit into the "ghost exchange" communication pattern, and for large problem sizes usually can be tweaked to get good performance.
"Other" applications have a weirder, often collective communication pattern. Depending on the structure of the problem, such applications can sometimes have relatively good performance, but are often network bound.
"Sequential" applications might have multiple threads or processes, but they don't have any parallelism--only one thread/process can do useful work at a time. Many I/O limited applications are basically sequential, since ten CPUs can wait on one disk no faster than one CPU! (Solution: add more disks!)

Amdahl's Law

Fact: Amdahl's Law limits the speedup of a partially parallelized program to 1/F, where F is the fraction of the program's runtime that is still sequential, even assuming everything else is so parallel it takes no time at all.
    speedup = old time / new time = sequential time / parallel time
                    = 1 / ( F + (1-F)/P ) => 1/F     (as number of processors P approaches infinity)

Question: what fraction of real programs MUST be sequential? I claim it's only a vanishingly small amount--the user interface probably should be sequential in places, since there's only one user (one mouse, one button). So Amdahl's law is in some sense true but useless--if the sequential part of the code is a 2-ns button click, a 2-second program has F=1/billion, so the speedup is limited to one billion. Big deal!

However, it really takes programming effort to parallelize some parts of the program--some algorithms are sequential (you need new algorithms), some hardware like I/O is sequential (you need better hardware), etc.

Mathematically, since
    F proportional to 1/pain
then Ahmdal's law really says
    speedup is proportional to pain

Which is both a true and useful statement.

I/O Performance

One recurring problem in parallel computing is I/O: talking to the disk, network, or output devices. Oddly, most real hardware has wildly different latency (time for one individual operation) and bandwidth (total bytes per second)--you might expect latency = 1/bandwidth, but in practice that first byte is way more expensive than the next bytes. For example, loading a single byte from RAM requires the whole row to be addressed and sensed, so reading adjacent bytes gets much cheaper. Sending one byte over the network requires an entire packet, so sending multiple bytes at once decreases the cost substantially.

Storage Device	Latency (sec/request)	Requests/sec	Bandwidth (bytes/sec)	Source
Floppy Drive	500ms	2	5KB/s - Horrible	Hazy memories
Modem	100ms (round trip)	10	5KB/s - Horrible	"
DVD-ROM	150ms (seek time)	8	5MB/s - Sad	Experiment (laptop drive)
Flash Drive	0.5ms	2000	13MB/s - Sad	Experiment (USB flashdrive)
Hard Disk	10ms	100	50MB/s - Decent	Experiment (laptop hard drive)
Gigabit Ethernet	100us	10,000	100MB/s - Decent	Experiment (powerwall)
RAM	50ns	20 million	2GB/s - Great	Experiment (netrun Pentium 4)
Cache	1ns	1 billion	4GB/s - Great	Experiment (netrun Pentium 4)

Parallel Performance In MPI

Selected outputs, running under MPICH 1.2.7 on our gigabit ethernet powerwall:


Test send/recv_int took 52.468us per iteration (10000 iterations)

Sending even a small, one-int message takes a *long* time (high alpha cost as listed below).

Test sandbag (100) took 104.236us per iteration (1000 iterations)

The "sandbag" is my function that does CPU work.

Test send/recv_int + sandbag(100) took 176.449us per iteration (10000 iterations)

Nonlinear slowdown! I think the CPU and network are interfering with one another here--the bottom line is that communication is expensive, especially when it won't overlap with computation.

Test isend/irecv_int + sandbag(100) took 114.860us per iteration (100 iterations)

For better performance, use Isend and Irecv, which are "nonblocking": the CPU can keep working while the network talks.


Test send/recv_1meg took 10529.570us per iteration (100 iterations)

Communication costs just 10ns/byte for long messages, since the startup overhead amortizes away.

Test isend/irecv_1meg + sandbag(10000) took 20864.780us per iteration (100 iterations)

However, MPI doesn't seem to overlap long messages very well--this is clearly first compute, then communicate. (I might need to add an "MPI_Test" call or something inside my sandbag function.)

Test barrier took 61.669us per iteration (1000 iterations)

A barrier requires only one synchronizing message per CPU.

Test bcast_int + barrier took 111.750us per iteration (1000 iterations)

A broadcast costs about the same as one network message on two processors.

Test reduce_int + barrier took 116.753us per iteration (1000 iterations)

Similarly, a reduction costs basically one message (on two processors).

Theoretical Message-Passing Performance

Most network cards require some (fairly large) fixed amount of time per message, plus some (smaller) amount of time per byte of the message:
tnet = a + b * L;

Where

tnet: network time. The total time, in seconds, spent on the network.
a: network latency. The time, in seconds, to send a zero-length message. For MPICH running on gigabit Ethernet, this is something like 50us/message, which is absurd (it's the time from the CPU to the northbridge to the PCI controller to the network card to the network to the switch to the other network card to the CPU interrupt controller through the OS to MPI and finally to the application). Fancier, more expensive networks such as Myrinet or Infiniband have latencies as low as 5us/message (in 2005; Wikipedia now claims 1.3us/message). (Network latency is often written as alpha or α).
b: 1/(network bandwidth). The time, in seconds, to send each byte of the message. For 100MB/s gigabit ethernet (1000Mb/s is megabits), this is 10ns/byte. For 4x Infiniband, which sends 1000MB/s, this is 1ns/byte. (Network 1/bandwidth is often written as beta or β).
L: number of bytes in message.

The bottom line is that shorter messages are always faster. *But* due to the per-message overhead, they may not be *appreciably* faster. For example, for Ethernet a zero-length message takes 50us. A 100-byte message takes 51us (50us/message+100*10ns/byte). So you might as well send 100 bytes if you're going to bother sending anything!

In general, you'll hit 50% of the network's achievable bandwidth when sending messages so
a = b * L
or
L = a / b
For Ethernet, this breakeven point is at L = 5000 bytes/message, which is amazingly long! Shorter messages are "alpha dominated", which means you're waiting for the network latency (speed of light, acknowledgements, OS overhead). Longer messages are "beta dominated", which means you're waiting for the network bandwidth (signalling rate). Smaller messages don't help much if you're alpha dominated!

The large per-message cost of most networks means many applications cannot get good performance with an "obvious" parallel strategy, like sending individual floats when they're needed. Instead, you have to figure out beforehand what data needs to be sent, and send everything at once.

Parallel CPU Performance

OK, that's the network's performance. What about the CPU? A typical model is:
tcpu = k * n / p

Where

tcpu is the total time spent computing, in seconds.
k is the (average) time spent computing each element. For example, Mandelbrot set rendering takes about 300ns/pixel. Some folks break k down into the (average) number of floating-point operations per element (flops/element), multiplied by the time taken per floating-point operation (seconds/flop, typically around 1ns (CPU) to 0.1ns (GPU)).
n is the number of elements in the computation. For example, a 1000x1000 image has 1m pixels.
p is the number of processors. Dividing the total work by p implicitly assumes the problem's load is (more or less) evenly balanced!

Parallel Problem Performance

Say we're doing an n=1mpixel Mandelbrot rendering in parallel on p=10 processors. Step one is to render all the pixels:
    tcpu = k * n / p
    tcpu = 300 ns/pixel * 1m pixels / 10 processors = 30ms / processor

After rendering the pixels, we've got to send them to one place for output. So each processor sends a message of size L = (3 bytes/pixel * n / p) = 300Kbytes/processor, and thus:
    tnet = a + b * L
    tnet = a + b * (3 bytes/pixel * 1m bytes / 10 processors) = 3.05 ms/processor
(Careful: this assumes all the messages sent can be received at one place in parallel, which often isn't the case!)

Hence in this case we spend almost 10 times longer computing the data as we do sending it, which is pretty good.

Unlike the network, the CPU spends its time actually solving the problem. So CPU time is usually seen as "good" useful work, while network time is "bad" parallel overhead. So more generally, you'd like to make sure the "computation to communication ratio", tcpu/tnet is as high as possible. For the mandelbrot problem:

    tcpu / tnet = k * n / p / ( a + b * n / p * 3)

Or, simplifying:
    tcpu / tnet = k / ( a * p / n + b * 3)

Remember, a higher CPU/net ratio means more time spent computing, and hence higher efficiency. So we can immediately see that:

As the amount of work k per element grows, the efficiency goes up. (This makes sense: more time spent computing is "better", at least from the point of view of CPU/net ratio!)
As the network's per-message cost a grows, efficiency goes down. (Slower network is worse)
As you add processors to a fixed-size problem, the per-message cost goes up, and efficiency goes down. (More processors means more messages.)
As the problem size n grows, the per-message cost gets less important, and efficiency goes up. (Messages get longer with bigger problems, which uses the network more efficiently.)
As the network's time per byte b goes down, efficiency goes up. (Faster network is better)

Or, to spend more time computing *relative* to the time spent communicating:

Get a faster network.
Find a bigger problem.
Do *more* work per element (or get a *slower* CPU!).
Use *fewer* processors.

Of course, what users really care about is the *total* time spent, tcpu + tnet:
tcpu + tnet = k * n / p + a + b * n / p * 3

To spend less time overall:

Get a faster network.
Find a *smaller* problem.
Do *less* work per element (or get a *faster* CPU).
Use more processors.

Note that even if you do all these things, you can *never* solve the mandelbrot problem in parallel in less than the per-message time a. For Ethernet, this means parallel problems are always going to take 50us. In particular, if the problem can be solved serially in less than 50us, going parallel can't possibly help! Luckily, it's rare you care about a problem that takes less than 50us total. And if that sub-50us computation is just a part of a bigger problem, you can parallelize the *bigger* problem.

In general, the per-message cost means you want to parallelize the *coarsest* pieces of the computation you can!