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''The following tests were done on the test system; these results will be used for the comparison between the serial and parallel versions.''
 
''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
 
''n'' = 10,000
 
  Init.  - took - 2 millisecs
 
  Init.  - took - 2 millisecs

Revision as of 23:12, 9 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)
  • 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

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


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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