Difference between revisions of "GPU610 Team Tsubame"

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  Pi: 3.14172063999999995331791069475002586841583251953125000000000000
 
  Pi: 3.14172063999999995331791069475002586841583251953125000000000000
  
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'''Analysis'''
+
'''Summary'''
 +
 
 
Both programs calculate finite results of Pi, however the Leibniz implementation has higher accuracy and calculates more digits per iteration. Since both algorithms have O(n) runtime, the Leibniz implementation is also superior in terms of speed.
 
Both programs calculate finite results of Pi, however the Leibniz implementation has higher accuracy and calculates more digits per iteration. Since both algorithms have O(n) runtime, the Leibniz implementation is also superior in terms of speed.
  

Revision as of 00:05, 10 February 2017

TBD...

Team Members

  1. Mark Anthony Villaflor (Leader)
  2. Huachen Li
  3. Yanhao Lei
eMail All

Progress

PHASE 1

Pi

This is a comparison between two programs that calculate Pi.

  • Test System Specifications
OS: Windows 7 (64-bit)
CPU: Intel Core i3-2350M @ 2.30GHz
GPU: GeForce GT 520MX (48 CUDA cores)

IDE: Microsoft Visual Studio Enterprise 2015; Version 14.0.25431.01 Update 3
  • How To Execute On Linux?

1. Here is the Makefile: 

# Change this to "monte-carlo" if needed
VER = leibniz
# Uncomment and modify the following lines to specify a specific version of GCC
#GCC_VERSION = 5.2.0
#PREFIX = /usr/local/gcc/${GCC_VERSION}/bin/
CC = ${PREFIX}gcc
CPP = ${PREFIX}g++

$(VER): $(VER).o ; \
$(CPP) -g -pg -o$(VER) $(VER).o

$(VER).o: $(VER).cpp ; \
$(CPP) -c -O2 -g -pg -std=c++14 $(VER).cpp

# Remember to ">make clean" after ">make" to cleanup if the cleaning does not happen automatically
clean: ; \
rm *\.o

2. Copy leibniz.cpp and/or monte-carlo.cpp and put them into the same directory as the Makefile.

3. Execute the following command (in the same directory as the Makefile): 

> make

4. Execute the binary with: 

> ./leibniz [ n ]

OR 

> ./monte-carlo [ n ]

Where n is the number of iterations.


Leibniz formula implementation:

leibniz.cpp 

00 #include <iostream>
01 #include <iomanip>
02 #include <string>
03 
04 #include <chrono>
05
06 // Function duplicated from Workshop 2 - BLAS
07 void reportTime(const char* msg, std::chrono::steady_clock::duration span) {
08     auto ms = std::chrono::duration_cast<std::chrono::milliseconds>(span);
09     std::cout << msg << " - took - " 
10     << ms.count() << " millisecs" << std::endl;
11 }
12 
13 // Arctangent of 1; execute for ~n iterations to refine the result
14 long double arctan1( unsigned long long int it ) { 
15     long double r = 0.0; // 1 op. (=) runs 1 time
16 
17     // v0.25; due to performing the operations in a different order, there are rounding issues...
18     for ( long double d = 0.0; d < it; d++ ) { // 1 op. (=) runs 1 time; 2 op.s (<, ++) run n times
19         long double de = d * 4.0 + 1.0; // 3 op.s (=, *, +) run n times
20         r += (1.0 / de) - (1.0 / (de + 2.0)); // 6 op.s (+ & =, /, -, /, +) run n times
21     }   
22 
23     return r; // 1 op. (return) runs 1 time
24 }
25 
26 int main( int argc, char* argv[] ) {
27     unsigned long long int n = std::stoull(argv[1], 0); 
28 
29     std::chrono::steady_clock::time_point ts, te; 
30     ts = std::chrono::steady_clock::now();
31    long double pi = 4.0 * arctan1( n );
32    te = std::chrono::steady_clock::now();
33    reportTime("Arctangent(1) ", te - ts);
34 
35     // Maximum length of a long double is 64 digits; minus "3." gives 62 digits.
36     std::cout.precision(62);
37     std::cout << "Pi: " << std::fixed << pi << std::endl;
40 }
  • STAGE 1 - Big-O:

There is only one for loop in this program (on line 18); it executes d times, where d is the first argument provided to the program on the command line. Summing up the operations in the (predicted) hotspot, T(n) = 11n + 3; therefore O(n) runtime.

  • STAGE 2 - Potential Speedup

Using Amdahl's Law: 

P = 1708ms / 1725ms (using the third test data from below...)
P = 0.99014

n = 48 (processors reported by deviceQuery.exe)

Sn = 1 / ( 1 - P + P / n )
S48 = 1 / ( 1 - 0.99014 + 0.99014 / 48 )
S48 = 32.79988

The maximum speedup on the test system is approximately 33 times.

  • STAGE 3 - Test Runs

The following tests were done on the Matrix server:

4 digits are correct at 10K iterations: 

> time ./leibniz 10000
Arctangent(1)  - took - 0 millisecs
Pi: 3.14154265358982449354505184224706226814305409789085388183593750

real 	0m0.016s
user	0m0.004s
sys	0m0.008s

7 digits are correct at 10M iterations: 

> time ./leibniz 10000000
Arctangent(1)  - took - 171 millisecs
Pi: 3.14159260358979321929411010483335076060029678046703338623046875

real	0m0.187s
user	0m0.172s
sys	0m0.012s

No difference at 100M iterations: 

> time ./leibniz 100000000
Arctangent(1)  - took - 1708 millisecs
Pi: 3.14159264858979533105269588144636827564681880176067352294921875

real	0m1.725s
user	0m1.704s
sys	0m0.008s

The following tests were done on the test system; these results will be used for the comparison between the serial and parallel versions.

n = 10,000 

Arctangent(1)  - took - 0 millisecs
Pi: 3.14154265358982032196877298702020198106765747070312500000000000

n = 10,000,000 

Arctangent(1)  - took - 199 millisecs
Pi: 3.14159260358809833135751432564575225114822387695312500000000000

n = 100,000,000 

Arctangent(1)  - took - 2087 millisecs
Pi: 3.14159264457621567601108836242929100990295410156250000000000000


Monte-Carlo algorithm implementation:

monte-carlo.cpp 

00 #include <iostream>
01 #include <random>
02 #include <string>
03 
04 #include <chrono>
05 
06 // Duplicated from https://scs.senecac.on.ca/~gpu610/pages/workshops/w2.html
07 void reportTime(const char* msg, std::chrono::steady_clock::duration span) {
08     auto ms = std::chrono::duration_cast<std::chrono::milliseconds>(span);
09     std::cout << msg << " - took - " <<
10     ms.count() << " millisecs" << std::endl;
11 }
12 
13 int main(int argc, char* argv[]) {
14     std::chrono::steady_clock::time_point ts, te;
15 
16     ts = std::chrono::steady_clock::now();
17     unsigned long long int n = std::stoull(argv[1], 0),
18                            totalCircle = 0;
19 
20     int stride = 1000,
21         circleSize = n / stride;
22 
23     unsigned int* circle = new unsigned int[circleSize];
24 
25     for (int i = 0; i < circleSize; i++)
26         circle[i] = 0;
27 
28     std::random_device rd;
29     std::mt19937 mt(rd());
30     std::uniform_real_distribution<long double> dist(0.0, 1.0);
31     te = std::chrono::steady_clock::now();
32     reportTime("Init. ", te - ts);
33 
34     ts = std::chrono::steady_clock::now();
35     for (unsigned long long int i = 0; i < circleSize; i++) {
36         for (int j = 0; j < stride; j++) {
37             long double x = dist(mt),
38                         y = dist(mt);
39             // if position is inside the circle...
40             if (x * x + y * y < 1.0) {
41                 circle[i]++;
42             }
43         }
44     }
45 
46     for (int i = 0; i < circleSize; i++)
47         totalCircle += circle[i];
48 
49     long double pi = 4.0 * ((long double) totalCircle) / ((long double) n);
50     te = std::chrono::steady_clock::now();
51     reportTime("Drop points ", te - ts);
52 
53     std::cout.precision(62);
54     std::cout << "Pi: " << std::fixed << pi << std::endl;
55 
56     delete [] circle;
57 }
  • STAGE 1 - Big-O:

The (predicted) hotspot begins from line 35 and ends at line 44. Although there are two for loops, the outer for loop executes n / stride times while the inner for loop executes stride times; the actual iteration is just n ( O(n) runtime ).

  • STAGE 2 - Potential Speedup:

Using Amdahl's Law: 

P = 10883ms / 10903ms (using the last sample of the third test data from below...)
P = 0.99817

n = 48 (processors reported by deviceQuery.exe)

Sn = 1 / ( 1 - P + P / n )
S48 = 1 / ( 1 - 0.99817 + 0.99817 / 48 )
S48 = 44.19015

The maximum speedup on the test system is approximately 44 times.

  • STAGE 3 - Test Runs

The following tests were done on the Matrix server:

At around 10K iterations, the first decimal is stable. 

> time ./monte-carlo 10000
Init.  - took - 0 millisecs
Drop points  - took - 1 millisecs
Pi: 3.11679999999999999995940747066214271399076096713542938232421875

real	0m0.019s
user	0m0.004s
sys	0m0.008s
> time ./monte-carlo 10000
Init.  - took - 0 millisecs
Drop points  - took - 1 millisecs
Pi: 3.16480000000000000009124645483638005316606722772121429443359375

real	0m0.018s
user	0m0.008s
sys	0m0.004s
> time ./monte-carlo 10000
Init.  - took - 0 millisecs
Drop points  - took - 1 millisecs
Pi: 3.16639999999999999995108079797745403993758372962474822998046875

real	0m0.018s
user	0m0.004s
sys	0m0.008s

The next digit is stable at around 10M iterations 

> time ./monte-carlo 10000000
Init.  - took - 0 millisecs
Drop points  - took - 1096 millisecs
Pi: 3.14150879999999999990685506379151092914980836212635040283203125

real	0m1.114s
user	0m1.092s
sys	0m0.008s
> time ./monte-carlo 10000000
Init.  - took - 0 millisecs
Drop points  - took - 1097 millisecs
Pi: 3.14219679999999999993332000514101309818215668201446533203125000

real	0m1.114s
user	0m1.092s
sys	0m0.016s
> time ./monte-carlo 10000000
Init.  - took - 0 millisecs
Drop points  - took - 1097 millisecs
Pi: 3.14158840000000000010696443730751070688711479306221008300781250

real	0m1.115s
user	0m1.088s
sys	0m0.012s

By 100M, the third digit appears to be stable. 

> time ./monte-carlo 100000000
Init.  - took - 1 millisecs
Drop points  - took - 10910 millisecs
Pi: 3.14138611999999999989559296142971334120375104248523712158203125

real	0m10.930s
user	0m10.881s
sys	0m0.012s
> time ./monte-carlo 100000000
Init.  - took - 1 millisecs
Drop points  - took - 10847 millisecs
Pi: 3.14185203999999999998835042980260823242133483290672302246093750

real	0m10.868s
user	0m10.833s
sys	0m0.016s
> time ./monte-carlo 100000000
Init.  - took - 1 millisecs
Drop points  - took - 10883 millisecs
Pi: 3.14160056000000000009896028441147564080893062055110931396484375

real	0m10.903s
user	0m10.865s
sys	0m0.016s

The following tests were done on the test system; these results will be used for the comparison between the serial and parallel versions.

n = 10,000 

Init.  - took - 2 millisecs
Drop points  - took - 1 millisecs
Pi: 3.12440000000000006608047442568931728601455688476562500000000000

Init.  - took - 1 millisecs
Drop points  - took - 2 millisecs
Pi: 3.14880000000000004334310688136611133813858032226562500000000000

Init.  - took - 2 millisecs
Drop points  - took - 2 millisecs
Pi: 3.12760000000000015774048733874224126338958740234375000000000000

n = 10,000,000 

Init.  - took - 2 millisecs
Drop points  - took - 2046 millisecs
Pi: 3.14098960000000015924115359666757285594940185546875000000000000

Init.  - took - 2 millisecs
Drop points  - took - 2059 millisecs
Pi: 3.14231840000000017809611563279759138822555541992187500000000000

Init.  - took - 2 millisecs
Drop points  - took - 2052 millisecs
Pi: 3.14155439999999996913970790046732872724533081054687500000000000

n = 100,000,000 

Init.  - took - 2 millisecs
Drop points  - took - 21263 millisecs
Pi: 3.14143352000000009027758096635807305574417114257812500000000000

Init.  - took - 2 millisecs
Drop points  - took - 21101 millisecs
Pi: 3.14146268000000006281879905145615339279174804687500000000000000

Init.  - took - 3 millisecs
Drop points  - took - 20884 millisecs
Pi: 3.14172063999999995331791069475002586841583251953125000000000000

Summary

Both programs calculate finite results of Pi, however the Leibniz implementation has higher accuracy and calculates more digits per iteration. Since both algorithms have O(n) runtime, the Leibniz implementation is also superior in terms of speed.

With regards to the hotspots, both programs contain one area that holds over 99% of their total execution time; these areas will be the focus for parallelization. The programs were revised to remove all data dependencies, thus there should be no conflicts between threads when the parallel codes are applied.


Maze

Testing Environment:

  • Operating system: Linux
  • Compiler: g++ 5.2.0
  • Processor: Dual-Core AMD Opteron(tm) Processor 8220
  • System memory: 3098088 kB

How to setup:

1. Download source file: https://github.com/corzani/maze

2. Create “Makefile” file: 

# Makefile for GPU610/assigment1/maze
#

GCC_VERSION = 5.2.0
PREFIX = /usr/local/gcc/${GCC_VERSION}/bin/
CC = ${PREFIX}gcc
CPP = ${PREFIX}g++
OBJS = Maze.o MazeDebug.o MazePng.o main.o
SOURCE = Maze.cpp MazeDebug.cpp MazePng.cpp main.cpp
LIBS = -lpng
TARGET = maze

$(TARGET): $(OBJS)
        $(CPP) -pg -o$(TARGET) $(OBJS) $(LIBS)

$(OBJS): $(SOURCE)
        $(CPP) -c -O2 -g -pg -std=c++14 $(SOURCE)
#all: $(TARGET)

clean:
        rm *.o

3. Complile and run: 

> make
> maze <maze x axes> <maze y axes>

Example: 

> maze 5000 5000

Analysis:

The program function named toPng() takes up an average of 42% (min: 35%; max: 50%) of the execution time. The reason for toPng() to take so much runtime is because of the for loop that building the walls of the maze. 

for (int i = 0; i < height; ++i) {
    row_pointers[i] = new png_byte[width * 3];
    for (int j = 0; j < width * 3; j += 3) {
        row_pointers[i][j] = WALL;
        row_pointers[i][j + 1] = WALL;
        row_pointers[i][j + 2] = WALL;
    }
}

The function also has a runtime of O(n^2).

GPU610 Maze-Graph.png

For n = 1000 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  ms/call  ms/call  name    
 50.00      0.05     0.05                             MazePng::toPng(unsigned int)
 30.00      0.08     0.03  1999978     0.00     0.00  AbstractMaze::getNext(unsigned int, short*, unsigned int*)
 10.00      0.09     0.01        1    10.00    10.00  MazePng::createImage(unsigned char**, unsigned int)
 10.00      0.10     0.01                             AbstractMaze::generate(int)
  0.00      0.10     0.00   999999     0.00     0.00  AbstractMaze::knockWall(unsigned int, unsigned int, AbstractMaze::directions)
  0.00      0.10     0.00     7798     0.00     0.00  _ZNSt5dequeIjSaIjEE16_M_push_back_auxIJjEEEvDpOT_
  0.00      0.10     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::_M_initialize_map(unsigned int)
  0.00      0.10     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::~_Deque_base()
  0.00      0.10     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN12AbstractMaze3mapE
  0.00      0.10     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN7MazePngC2Ejj
  0.00      0.10     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN9MazeDebugC2Ejj
  0.00      0.10     0.00        1     0.00     0.00  _GLOBAL__sub_I_main
  0.00      0.10     0.00        1     0.00     0.00  AbstractMaze::AbstractMaze(unsigned int, unsigned int)

For n = 2000 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  ms/call  ms/call  name    
 36.21      0.21     0.21  7999992     0.00     0.00  AbstractMaze::getNext(unsigned int, short*, unsigned int*)
 24.14      0.35     0.14                             MazePng::toPng(unsigned int)
 12.07      0.42     0.07  3999999     0.00     0.00  AbstractMaze::knockWall(unsigned int, unsigned int, AbstractMaze::directions)
 12.07      0.49     0.07        1    70.00    70.00  MazePng::createImage(unsigned char**, unsigned int)
 10.34      0.55     0.06                             AbstractMaze::generate(int)
  5.17      0.58     0.03        1    30.00    30.00  AbstractMaze::AbstractMaze(unsigned int, unsigned int)
  0.00      0.58     0.00    31113     0.00     0.00  _ZNSt5dequeIjSaIjEE16_M_push_back_auxIJjEEEvDpOT_
  0.00      0.58     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::_M_initialize_map(unsigned int)
  0.00      0.58     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::~_Deque_base()
  0.00      0.58     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN12AbstractMaze3mapE
  0.00      0.58     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN7MazePngC2Ejj
  0.00      0.58     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN9MazeDebugC2Ejj
  0.00      0.58     0.00        1     0.00     0.00  _GLOBAL__sub_I_main

For n = 3000 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  ms/call  ms/call  name    
 33.08      0.44     0.44                             MazePng::toPng(unsigned int)
 24.06      0.76     0.32 17999970     0.00     0.00  AbstractMaze::getNext(unsigned int, short*, unsigned int*)
 14.29      0.95     0.19  8999999     0.00     0.00  AbstractMaze::knockWall(unsigned int, unsigned int, AbstractMaze::directions)
 12.78      1.12     0.17        1   170.00   170.00  MazePng::createImage(unsigned char**, unsigned int)
 11.28      1.27     0.15                             AbstractMaze::generate(int)
  3.01      1.31     0.04        1    40.00    40.00  AbstractMaze::AbstractMaze(unsigned int, unsigned int)
  1.50      1.33     0.02    70076     0.00     0.00  _ZNSt5dequeIjSaIjEE16_M_push_back_auxIJjEEEvDpOT_
  0.00      1.33     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::_M_initialize_map(unsigned int)
  0.00      1.33     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::~_Deque_base()
  0.00      1.33     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN12AbstractMaze3mapE
  0.00      1.33     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN7MazePngC2Ejj
  0.00      1.33     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN9MazeDebugC2Ejj
  0.00      1.33     0.00        1     0.00     0.00  _GLOBAL__sub_I_main

For n = 4000 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  ms/call  ms/call  name    
 34.51      0.78     0.78                             MazePng::toPng(unsigned int)
 28.32      1.42     0.64 31999957     0.00     0.00  AbstractMaze::getNext(unsigned int, short*, unsigned int*)
 13.72      1.73     0.31        1   310.00   310.00  MazePng::createImage(unsigned char**, unsigned int)
 10.18      1.96     0.23 15999999     0.00     0.00  AbstractMaze::knockWall(unsigned int, unsigned int, AbstractMaze::directions)
 10.18      2.19     0.23                             AbstractMaze::generate(int)
  3.10      2.26     0.07        1    70.00    70.00  AbstractMaze::AbstractMaze(unsigned int, unsigned int)
  0.00      2.26     0.00   124808     0.00     0.00  _ZNSt5dequeIjSaIjEE16_M_push_back_auxIJjEEEvDpOT_
  0.00      2.26     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::_M_initialize_map(unsigned int)
  0.00      2.26     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::~_Deque_base()
  0.00      2.26     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN12AbstractMaze3mapE
  0.00      2.26     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN7MazePngC2Ejj
  0.00      2.26     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN9MazeDebugC2Ejj
  0.00      2.26     0.00        1     0.00     0.00  _GLOBAL__sub_I_main

For n = 5000 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  ms/call  ms/call  name    
 35.16      1.35     1.35                             MazePng::toPng(unsigned int)
 27.08      2.39     1.04 49999831     0.00     0.00  AbstractMaze::getNext(unsigned int, short*, unsigned int*)
 12.50      2.87     0.48        1   480.00   480.00  MazePng::createImage(unsigned char**, unsigned int)
 11.72      3.32     0.45                             AbstractMaze::generate(int)
 10.68      3.73     0.41 24999999     0.00     0.00  AbstractMaze::knockWall(unsigned int, unsigned int, AbstractMaze::directions)
  2.86      3.84     0.11        1   110.00   110.00  AbstractMaze::AbstractMaze(unsigned int, unsigned int)
  0.00      3.84     0.00   195749     0.00     0.00  _ZNSt5dequeIjSaIjEE16_M_push_back_auxIJjEEEvDpOT_
  0.00      3.84     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::_M_initialize_map(unsigned int)
  0.00      3.84     0.00        2     0.00     0.00  std::_Deque_base<unsigned int, std::allocator<unsigned int> >::~_Deque_base()
  0.00      3.84     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN12AbstractMaze3mapE
  0.00      3.84     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN7MazePngC2Ejj
  0.00      3.84     0.00        1     0.00     0.00  _GLOBAL__sub_I__ZN9MazeDebugC2Ejj
  0.00      3.84     0.00        1     0.00     0.00  _GLOBAL__sub_I_main

Summary:

Function toPng() has an average of 42% execution time. It also has O(n^2) runtime.

PHASE 2

PHASE 3

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