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Revision as of 11:12, 7 April 2018 by Tsarkarcd (talk | contribs) (→‎Idea 1 - Jacobi Method for 2D Poisson Problem)

GPUSquad

         (           (                         (      
 (       )\ )        )\ )   (            (     )\ )   
 )\ )   (()/(    (  (()/( ( )\      (    )\   (()/(   
(()/(    /(_))   )\  /(_)))((_)     )\((((_)(  /(_))  
 /(_))_ (_))  _ ((_)(_)) ((_)_   _ ((_))\ _ )\(_))_   
(_)) __|| _ \| | | |/ __| / _ \ | | | |(_)_\(_)|   \  
  | (_ ||  _/| |_| |\__ \| (_) || |_| | / _ \  | |) | 
   \___||_|   \___/ |___/ \__\_\ \___/ /_/ \_\ |___/

Team Members

  1. Tanvir Sarkar
  2. Michael Overall
  3. Igor Krasnyanskiy
  4. Email All

Progress

Assignment 1

Idea 1 - Jacobi Method for 2D Poisson Problem

This topic is based on the following pdf: https://math.berkeley.edu/~wilken/228A.F07/chr_lecture.pdf

Background:

Poisson's equation according to Wikipedia:

"In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics."

According to the intro in the pdf above, finite-difference and finite-element methods are the solution techniques of choice for solving elliptic PDE problems. Regardless of which type of technique you choose, you will end up with sets of linear relationships between various variables, and will need to solve for the solution u_k which satisfies all the linear relationships prescribed by the PDE.

This can be written, according to the pdf, as a matrix "Au = b, where we wish to find a solution u, given that A is a matrix capturing the differentiation operator, and b corresponds to any source or boundary terms". The Jacobi method can be used to solve this matrix, and is used in the code sample later.

Jacobi method according to Wikipedia:

"In numerical linear algebra, the Jacobi method (or Jacobi iterative method[1]) is an algorithm for determining the solutions of a diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges."

**************************

CODE - Based on PDF

**************************

The following code is based on the jacobi.cc and common.cc code at the bottom of the source PDF, but there are some changes for these tests. The files use .cpp extensions instead of .cc, the code in common.cc was added to jacobi.cpp so it's a single file, and code for carrying out the Jacobi iterations in the main were placed into a seperate function called dojacobi() for gprof profiling.

The std::cout line inside the output_and_error function was commented out to avoid excessive printing to the terminal. The error_and_output function was rewritten as outputImage to get rid of terminal and file I/O during calculations, and only output the final image at the end of the calculations (this would be necessary for the parallel versions, so the changes were applied to the serial code as well to make things fair). 

Compilation on Linux:
g++ -std=c++0x -O2 -g -pg -o jacobi jacobi.cpp
// Load standard libraries
#include <cstdio>
#include <cstdlib>
#include <iostream>
#include <fstream>
#include <cmath>
#include <chrono>
using namespace std;

// Set grid size and number of iterations
const int save_iters = 20;
const int total_iters = 5000;
const int error_every = 2;
const int m = 32, n = 1024;
const float xmin = -1, xmax = 1;
const float ymin = -1, ymax = 1;

// Compute useful constants
const float pi = 3.1415926535897932384626433832795;
const float omega = 2 / (1 + sin(2 * pi / n));
const float dx = (xmax - xmin) / (m - 1);
const float dy = (ymax - ymin) / (n - 1);
const float dxxinv = 1 / (dx*dx);
const float dyyinv = 1 / (dy*dy);
const float dcent = 1 / (2 * (dxxinv + dyyinv));

// Input function
inline float f(int i, int j) {
	float x = xmin + i*dx, y = ymin + j*dy;
	return abs(x)>0.5 || abs(y)>0.5 ? 0 : 1;
}

// Common output and error routine
void output_and_error(char* filename, float *a, const int sn) {
	// Computes the error if sn%error every==0
	if (sn%error_every == 0) {
		float z, error = 0; int ij;
		for (int j = 1; j<n - 1; j++) {
			for (int i = 1; i<m - 1; i++) {
				ij = i + m*j;
				z = f(i, j) - a[ij] * (2 * dxxinv + 2 * dyyinv)
					+ dxxinv*(a[ij - 1] + a[ij + 1])
					+ dyyinv*(a[ij - m] + a[ij + m]);
				error += z*z;
			}
		}
		//cout << sn << " " << error*dx*dy << endl;
	}

	// Saves the matrix if sn<=save iters
	if (sn <= save_iters) {
		int i, j, ij = 0, ds = sizeof(float);
		float x, y, data_float; const char *pfloat;
		pfloat = (const char*)&data_float;

		ofstream outfile;
		static char fname[256];
		sprintf(fname, "%s.%d", filename, sn);
		outfile.open(fname, fstream::out
			| fstream::trunc | fstream::binary);

		data_float = m; outfile.write(pfloat, ds);

		for (i = 0; i<m; i++) {
			x = xmin + i*dx;
			data_float = x; outfile.write(pfloat, ds);
		}

		for (j = 0; j<n; j++) {
			y = ymin + j*dy;
			data_float = y;
			outfile.write(pfloat, ds);

			for (i = 0; i<m; i++) {
				data_float = a[ij++];
				outfile.write(pfloat, ds);
			}
		}

		outfile.close();
	}
}

void outputImage(char* filename, float *a) {
	// Computes the error if sn%error every==0


	// Saves the matrix if sn<=save iters
	int i, j, ij = 0, ds = sizeof(float);
	float x, y, data_float; const char *pfloat;
	pfloat = (const char*)&data_float;

	ofstream outfile;
	static char fname[256];
	sprintf(fname, "%s.%d", filename, 101);
	outfile.open(fname, fstream::out
		| fstream::trunc | fstream::binary);

	data_float = m; outfile.write(pfloat, ds);

	for (i = 0; i < m; i++) {
		x = xmin + i*dx;
		data_float = x; outfile.write(pfloat, ds);//will need to be deferred
	}

	for (j = 0; j < n; j++) {
		y = ymin + j*dy;
		data_float = y;
		outfile.write(pfloat, ds);

		for (i = 0; i < m; i++) {
			data_float = a[ij++];
			outfile.write(pfloat, ds);
		}
	}

	outfile.close();
}

void dojacobi(int i, int j, int ij, int k, float error, float u[],
	float v[], float z, float* a, float* b) {

	// Set initial guess to be identically zero
	for (ij = 0; ij<m*n; ij++) u[ij] = v[ij] = 0;
	//output_and_error("jacobi out", u, 0);

	// Carry out Jacobi iterations
	for (k = 1; k <= total_iters; k++) {
		// Alternately flip input and output matrices
		if (k % 2 == 0) { a = u; b = v; }
		else { a = v; b = u; }

		// Compute Jacobi iteration
		for (j = 1; j<n - 1; j++) {
			for (i = 1; i<m - 1; i++) {
				ij = i + m*j;
				a[ij] = (f(i, j) + dxxinv*(b[ij - 1] + b[ij + 1])
					+ dyyinv*(b[ij - m] + b[ij + m]))*dcent;
			}
		}

		// Save and compute error if necessary
		//output_and_error("jacobi out", a, k);
	}
	outputImage("jacobi out", a);

}

int main() {
	int i, j, ij, k;
	//float error, u[m*n], v[m*n], z;
	float error, z;
	float *a, *b;
	float *u;
	float *v;
	u = new float[m*n];
	v = new float[m*n];

	std::chrono::steady_clock::time_point ts, te;
	ts = std::chrono::steady_clock::now();
	
	dojacobi(i, j, ij, k, error, u, v, z, a, b);
	
	delete[] u;
	delete[] v;

	te = std::chrono::steady_clock::now();
	std::chrono::steady_clock::duration duration = te - ts;
	auto ms = std::chrono::duration_cast<std::chrono::milliseconds>(duration);
	std::cout << "Serial Code Time: " << ms.count() << " ms" << std::endl;

	return 0;
}
Plotting Images with gnuplot

1. cd into the folder that contains the jacobi out.# files
2. run gnuplot
3. enter: set terminal png
4. enter: set output "outfilename.png"
5. enter: plot "jacobi out.#" binary

PNG image should be created in the same folder as the jacobi out.# files

************************

Inputs / Performance

************************

In a 2D PDE such as the 2D Poisson problem, the matrix will have m * n gridpoints.

At m = 33, n = 33, iterations = 5000: 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  Ts/call  Ts/call  name    
100.00      0.05     0.05                             dojacobi(int, int, int, int, double, double*, double*, double, double*, double*)
  0.00      0.05     0.00     5001     0.00     0.00  output_and_error(char*, double*, int)
  0.00      0.05     0.00        1     0.00     0.00  _GLOBAL__sub_I__Z16output_and_errorPcPdi

At m = 165, n = 165, iterations = 5000: 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  us/call  us/call  name    
 99.15      1.16     1.16                             dojacobi(int, int, int, int, double, double*, double*, double, double*, double*)
  0.85      1.17     0.01     5001     2.00     2.00  output_and_error(char*, double*, int)
  0.00      1.17     0.00        1     0.00     0.00  _GLOBAL__sub_I__Z16output_and_errorPcPdi

At m = 330, n = 330, iterations = 5000: 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total           
 time   seconds   seconds    calls  us/call  us/call  name    
 99.43      5.26     5.26                             dojacobi(int, int, int, int, double, double*, double*, double, double*, double*)
  0.57      5.29     0.03     5001     6.00     6.00  output_and_error(char*, double*, int)
  0.00      5.29     0.00        1     0.00     0.00  _GLOBAL__sub_I__Z16output_and_errorPcPdi

The hotspot seems to clearly be the triple for-loop in the Jacobi iterations code of the dojacobi() function. I believe these matrix calculations could be parallelized for improved performance.

Idea 2 - LZW Compression

BACKGROUND:(Paraphrased from "LZW compression" at http://whatis.techtarget.com/definition/LZW-compression) LZW compression is a compression algorithm that creates a dictionary of tokens from collections of characters. These tokens correspond to different patterns of bit values. Unique tokens are generated from the longest possible series of characters that recur in sequence any time later in a body of text. These patterns of bits that correspond to string tokens are written to an output file. Since commonly recurring sequences of words are registered to the dictionary as smaller (perhaps 12 bit) tokens, the corresponding dictionary bit code comes to represent a series of characters that would otherwise be longer than 12 bits. This is what facilitates the compression.

************************************************

CODE:

The following code is an example program that performs Compression and Decompression of text files using a LZW algorithm that encodes to 12 bit values.

This is algorithm was provided with a link as a potential project through the group projects page, but here it is again: https://codereview.stackexchange.com/questions/86543/simple-lzw-compression-algorithm

The body of code for the algorithm is as follows:


// Compile with gcc 4.7.2 or later, using the following command line: // // g++ -std=c++0x lzw.c -o lzw // //LZW algorithm implemented using fixed 12 bit codes.

  1. include <iostream>
  2. include <sstream>
  3. include <fstream>
  1. include <bitset>
  2. include <string>
  3. include <unordered_map>
  1. define MAX_DEF 4096

using namespace std;

string convert_int_to_bin(int number) { 

   string result = bitset<12>(number).to_string();
   return result;

}

void compress(string input, int size, string filename) { 

   unordered_map<string, int> compress_dictionary(MAX_DEF);
   //Dictionary initializing with ASCII
   for ( int unsigned i = 0 ; i < 256 ; i++ ){
   compress_dictionary[string(1,i)] = i;
   }
   string current_string;
   unsigned int code;
   unsigned int next_code = 256;
   //Output file for compressed data
   ofstream outputFile;
   outputFile.open(filename + ".lzw");

   for(char& c: input){
   current_string = current_string + c;
   if ( compress_dictionary.find(current_string) ==compress_dictionary.end() ){
           if (next_code <= MAX_DEF)
               compress_dictionary.insert(make_pair(current_string, next_code++));
           current_string.erase(current_string.size()-1);
           outputFile << convert_int_to_bin(compress_dictionary[current_string]);
           current_string = c;
       }   
   }   
   if (current_string.size())
           outputFile << convert_int_to_bin(compress_dictionary[current_string]);
   outputFile.close();

}


void decompress(string input, int size, string filename) { 

   unordered_map<unsigned int, string> dictionary(MAX_DEF);
   //Dictionary initializing with ASCII
   for ( int unsigned i = 0 ; i < 256 ; i++ ){
   dictionary[i] = string(1,i);
   }
   string previous_string;
   unsigned int code;
   unsigned int next_code = 256;
   //Output file for decompressed data
   ofstream outputFile;
   outputFile.open(filename + "_uncompressed.txt");

   int i =0;
   while (i<size){
       //Extracting 12 bits and converting binary to decimal
       string subinput = input.substr(i,12);
       bitset<12> binary(subinput);
       code = binary.to_ullong();
       i+=12;

       if ( dictionary.find(code) ==dictionary.end() ) 
           dictionary.insert(make_pair(code,(previous_string + previous_string.substr(0,1))));
       outputFile<<dictionary[code];
       if ( previous_string.size())
           dictionary.insert(make_pair(next_code++,previous_string + dictionary[code][0])); 
       previous_string = dictionary[code];
       }
   outputFile.close();

}

string convert_char_to_string(const char *pCh, int arraySize){ 

   string str;
   if (pCh[arraySize-1] == '\0') str.append(pCh);
   else for(int i=0; i<arraySize; i++) str.append(1,pCh[i]);
   return str;

}

static void show_usage() { 

       cerr << "Usage: \n"
             << "Specify the file that needs to be compressed or decompressed\n"
             <<"lzw -c input    #compress file input\n"
             <<"lzw -d input    #decompress file input\n"
             <<"Compressed data will be found in a file with the same name but with a .lzw extension\n"
             <<"Decompressed data can be found in a file with the same name and a _uncompressed.txt extension\n"
             << endl;

}


int main (int argc, char* argv[]) { 

   streampos size;
   char * memblock;

   if (argc <2)
   {
       show_usage();   
       return(1);
   }
   ifstream file (argv[2], ios::in|ios::binary|ios::ate);
   if (file.is_open())
   {
       size = file.tellg();
       memblock = new char[size];
       file.seekg (0, ios::beg);
       file.read (memblock, size);
       file.close();
       string input = convert_char_to_string(memblock,size);
       if (string( "-c" ) == argv[1] )
           compress(input,size, argv[2]);
       else if (string( "-d" ) == argv[1] )
           decompress(input,size, argv[2]);
       else
           show_usage();
   }
   else {
   cout << "Unable to open file."<<endl;
   show_usage();
   }
   return 0;

}


PROFILING:

The above program needs an input file to compress and decompress text in. For the purposes of testing, the Gutenberg press' "Complete Works of Shakespeare" was used as an input text file (http://www.gutenberg.org/files/100/100-0.txt) because it represents a large enough body of text to actually have perceptible run times for compression. Increases in the size of the data used are created through copying one full version of the text in the last iteration of testing and appending it to the end of the text file (so one Shakespeare's complete works becomes two back to back, two becomes four, etc).

PROFILING WITH THE ORIGINAL TEXT:

[[File:Flat profile:

Each sample counts as 0.01 seconds. 

 %   cumulative   self              self     total           
time   seconds   seconds    calls  ns/call  ns/call  name    
46.15      0.24     0.24                             compress(std::string, int, std::string)
25.00      0.37     0.13  7954538    16.34    16.34  show_usage()
21.15      0.48     0.11  2091647    52.59    52.59  convert_int_to_bin(int)
 7.69      0.52     0.04  2091647    19.12    35.47  std::__detail::_Map_base<std::string, std::pair<std::string const, int>, std::_Select1st<std::pair<std::string const, int> >, true, std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true> >::operator[](std::string const&)
 0.00      0.52     0.00     3841     0.00     0.00  std::__detail::_Hashtable_iterator<std::pair<std::string const, int>, false, false> std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true>::_M_insert_bucket<std::pair<std::string, unsigned int> >(std::pair<std::string, unsigned int>&&, unsigned int, unsigned int)
 0.00      0.52     0.00      256     0.00     0.00  std::__detail::_Hashtable_iterator<std::pair<std::string const, int>, false, false> std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true>::_M_insert_bucket<std::pair<std::string, int> >(std::pair<std::string, int>&&, unsigned int, unsigned int)
 0.00      0.52     0.00      256     0.00    16.34  std::__detail::_Map_base<std::string, std::pair<std::string const, int>, std::_Select1st<std::pair<std::string const, int> >, true, std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true> >::operator[](std::string&&)
 0.00      0.52     0.00        1     0.00     0.00  _GLOBAL__sub_I__Z18convert_int_to_bini]]


PROFILING WITH TWICE THE TEXT:

Flat profile:

Each sample counts as 0.01 seconds. 

 %   cumulative   self              self     total           
time   seconds   seconds    calls  ns/call  ns/call  name    
38.71      0.24     0.24                             compress(std::string, int, std::string)
25.81      0.40     0.16  2091647    76.49    76.49  convert_int_to_bin(int)
22.58      0.54     0.14  7954538    17.60    17.60  show_usage()
 9.68      0.60     0.06  2091647    28.69    46.29  std::__detail::_Map_base<std::string, std::pair<std::string const, int>, std::_Select1st<std::pair<std::string const, int> >, true, std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true> >::operator[](std::string const&)
 3.23      0.62     0.02                             convert_char_to_string(char const*, int)
 0.00      0.62     0.00     3841     0.00     0.00  std::__detail::_Hashtable_iterator<std::pair<std::string const, int>, false, false> std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true>::_M_insert_bucket<std::pair<std::string, unsigned int> >(std::pair<std::string, unsigned int>&&, unsigned int, unsigned int)
 0.00      0.62     0.00      256     0.00     0.00  std::__detail::_Hashtable_iterator<std::pair<std::string const, int>, false, false> std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true>::_M_insert_bucket<std::pair<std::string, int> >(std::pair<std::string, int>&&, unsigned int, unsigned int)
 0.00      0.62     0.00      256     0.00    17.60  std::__detail::_Map_base<std::string, std::pair<std::string const, int>, std::_Select1st<std::pair<std::string const, int> >, true, std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true> >::operator[](std::string&&)
 0.00      0.62     0.00        1     0.00     0.00  _GLOBAL__sub_I__Z18convert_int_to_bini


PROFILING WITH 4X THE TEXT:

Flat profile:

Each sample counts as 0.01 seconds. 

 %   cumulative   self              self     total           
time   seconds   seconds    calls  us/call  us/call  name    
40.48      1.02     1.02                             compress(std::string, int, std::string)
30.16      1.78     0.76 31802927     0.02     0.02  show_usage()
22.62      2.35     0.57  8363660     0.07     0.07  convert_int_to_bin(int)
 5.56      2.49     0.14  8363660     0.02     0.04  std::__detail::_Map_base<std::string, std::pair<std::string const, int>, std::_Select1st<std::pair<std::string const, int> >, true, std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true> >::operator[](std::string const&)
 0.79      2.51     0.02                             convert_char_to_string(char const*, int)
 0.40      2.52     0.01      256    39.06    39.09  std::__detail::_Map_base<std::string, std::pair<std::string const, int>, std::_Select1st<std::pair<std::string const, int> >, true, std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true> >::operator[](std::string&&)
 0.00      2.52     0.00     3841     0.00     0.00  std::__detail::_Hashtable_iterator<std::pair<std::string const, int>, false, false> std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true>::_M_insert_bucket<std::pair<std::string, unsigned int> >(std::pair<std::string, unsigned int>&&, unsigned int, unsigned int)
 0.00      2.52     0.00      256     0.00     0.00  std::__detail::_Hashtable_iterator<std::pair<std::string const, int>, false, false> std::_Hashtable<std::string, std::pair<std::string const, int>, std::allocator<std::pair<std::string const, int> >, std::_Select1st<std::pair<std::string const, int> >, std::equal_to<std::string>, std::hash<std::string>, std::__detail::_Mod_range_hashing, std::__detail::_Default_ranged_hash, std::__detail::_Prime_rehash_policy, false, false, true>::_M_insert_bucket<std::pair<std::string, int> >(std::pair<std::string, int>&&, unsigned int, unsigned int)
 0.00      2.52     0.00        1     0.00     0.00  _GLOBAL__sub_I__Z18convert_int_to_bini


A useful hotspot for parallelization is not immediately obvious through profiling, since the main compress() function contains the bulk of the program logic, and its show_usage(, and convert_int_to_bin are simple functions, that are called frequently in one for loop.

What really affects the runtime of the program based on data is the extent to which long matching strings can be tokenized. When this happens, larger, and larger chunks of text can be processed and compressed in an iteration. This is reflected by the fact that pasting the same blocks of text over again to increase data size does not proportionally increase run time because the same tokens work for subsequent pasted blocks.


POTENTIAL FOR PARALLELIZATION:

The compress() function performs similar operations on a collection of text, however it relies on a dictionary and an expanding string to be tokenized in a dictrionary. This could potentially be paralellized through a divide and conquer strategy where gpu blocks with shared caches share their own dictionary and iterate over their own block of text.

Idea 3 - MergeSort

Based on assignment topic suggestions.

Code for logic is based on BTP500 course note by Catherine Leung - https://cathyatseneca.gitbooks.io/data-structures-and-algorithms/content/sorting/merge_sort_code.html

It has been adjust to work only with int arrays and perform operation on worst case scenario for merge sort, which is something similar to this array - [0, 2, 4, 6, 1, 3, 5, 7].


CODE

To compile on matrix - g++ -O2 -g -pg -oa1 a1.cpp

It performs merge sort on array with 100000000 which has worst case scenario positioning of elements for sorting. 

#include<iostream>

int SIZE = 100000000;
/*This function merges the two halves of the array arr into tmp and then copies it back into arr*/
void merge(int arr[], int tmp[], int startA, int startB, int end) {
	int aptr = startA;
	int bptr = startB;
	int i = startA;
	while (aptr<startB && bptr <= end) {
		if (arr[aptr]<arr[bptr])
			tmp[i++] = arr[aptr++];
		else
			tmp[i++] = arr[bptr++];
	}
	while (aptr<startB) {
		tmp[i++] = arr[aptr++];
	}
	while (bptr <= end) {
		tmp[i++] = arr[bptr++];
	}
	for (i = startA; i <= end; i++) {
		arr[i] = tmp[i];
	}
}

//this function sorts arr from index start to end
void mSort(int* arr, int* tmp, int start, int end) {
	if (start<end) {
		int mid = (start + end) / 2;
		mSort(arr, tmp, start, mid);
		mSort(arr, tmp, mid + 1, end);
		merge(arr, tmp, start, mid + 1, end);
	}
}

void mergeSort(int* arr, int size) {
	int* tmp = new int[size];
	mSort(arr, tmp, 0, size - 1);
	delete[] tmp;
}

void printArr(int* arr) {
	for (int i = 0; i < SIZE; i++) {
		std::cout << arr[i] << std::endl;
	}
}
int main(){
	int* sampleArr = new int[SIZE];
	bool worstCaseFlag = false;
	int j = 1;
	for (int i = 0; i < SIZE; i++) {
		if (worstCaseFlag == false) {
			sampleArr[i] = i;
			worstCaseFlag = true;
		}
		else {
			sampleArr[i] = j;
			j += 2;
			worstCaseFlag = false;
		}
	}
	mergeSort(sampleArr, SIZE);
	//printArr(sampleArr);
	std::cin.get();
	return 0;
}


Profiling

Profiling with 1000000 elements 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total
 time   seconds   seconds    calls  ms/call  ms/call  name
 69.23      0.09     0.09   999999     0.00     0.00  merge(int*, int*, int, int, int)
 30.77      0.13     0.04        1    40.00   130.00  mSort(int*, int*, int, int)
  0.00      0.13     0.00        1     0.00     0.00  _GLOBAL__sub_I_SIZE


Profiling with 10000000 elements 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total
 time   seconds   seconds    calls   s/call   s/call  name
 88.89      1.04     1.04  9999999     0.00     0.00  merge(int*, int*, int, int, int)
 11.11      1.17     0.13        1     0.13     1.17  mSort(int*, int*, int, int)
  0.00      1.17     0.00        1     0.00     0.00  _GLOBAL__sub_I_SIZE


Profiling with 100000000 elements 

Flat profile:

Each sample counts as 0.01 seconds.
  %   cumulative   self              self     total
 time   seconds   seconds    calls   s/call   s/call  name
 90.40     13.85    13.85 99999999     0.00     0.00  merge(int*, int*, int, int, int)
  9.60     15.32     1.47        1     1.47    15.32  mSort(int*, int*, int, int)
  0.00     15.32     0.00        1     0.00     0.00  _GLOBAL__sub_I_SIZE

Analysing


The most time consuming part is merging, which can be at least partially paralleled. The Big(O) of this case is O(n).

Assignment 2

#include "cuda_runtime.h"
#include "device_launch_parameters.h"

#include <stdio.h>

// Load standard libraries
#include <cstdio>
#include <cstdlib>
#include <iostream>
#include <fstream>
#include <cmath>
#include <chrono>
using namespace std;

__global__ void matrixKernel(float* a, float* b, float dcent, int n, int m, int xmin, int xmax,
	int ymin, int ymax, float dx, float dy, float dxxinv, float dyyinv) {//MODIFY to suit algorithm

	int j = threadIdx.x + 1;
	//above: we are using block and thread indexes to replace some of the iteration logic
	if (j < n - 1) {
		for (int i = 1; i < m - 1; i++) {
			int  ij = i + m*j;

			float x = xmin + i*dx, y = ymin + j*dy;

			float input = abs(x) > 0.5 || abs(y) > 0.5 ? 0 : 1;

			a[ij] = (input + dxxinv*(b[ij - 1] + b[ij + 1])
				+ dyyinv*(b[ij - m] + b[ij + m]))*dcent;
		}
	}

}
// Set grid size and number of iterations
const int save_iters = 20;
const int total_iters = 5000;
const int error_every = 2;
const int m = 500, n = 500;
const float xmin = -1, xmax = 1;
const float ymin = -1, ymax = 1;

// Compute useful constants
const float pi = 3.1415926535897932384626433832795;
const float omega = 2 / (1 + sin(2 * pi / n));
const float dx = (xmax - xmin) / (m - 1);
const float dy = (ymax - ymin) / (n - 1);
const float dxxinv = 1 / (dx*dx);
const float dyyinv = 1 / (dy*dy);
const float dcent = 1 / (2 * (dxxinv + dyyinv));

// Input function
inline float f(int i, int j) {
	float x = xmin + i*dx, y = ymin + j*dy;
	return abs(x) > 0.5 || abs(y) > 0.5 ? 0 : 1;
}

// Common output and error routine
void outputImage(char* filename, float *a) {
	// Computes the error if sn%error every==0


	// Saves the matrix if sn<=save iters
	int i, j, ij = 0, ds = sizeof(float);
	float x, y, data_float; const char *pfloat;
	pfloat = (const char*)&data_float;

	ofstream outfile;
	static char fname[256];
	sprintf(fname, "%s.%d", filename, 101);
	outfile.open(fname, fstream::out
		| fstream::trunc | fstream::binary);

	data_float = m; outfile.write(pfloat, ds);

	for (i = 0; i < m; i++) {
		x = xmin + i*dx;
		data_float = x; outfile.write(pfloat, ds);
	}

	for (j = 0; j < n; j++) {
		y = ymin + j*dy;
		data_float = y;
		outfile.write(pfloat, ds);

		for (i = 0; i < m; i++) {
			data_float = a[ij++];
			outfile.write(pfloat, ds);
		}
	}

	outfile.close();
}


void dojacobi() {
	int i, j, ij, k;
	float error, z;
	float *a, *b;
	float *u;
	float *v;
	u = new float[m*n];
	v = new float[m*n];

	// Set initial guess to be identically zero
	for (ij = 0; ij < m*n; ij++) u[ij] = v[ij] = 0;

	a = v; b = u;
	float* d_a;
	float* d_b;

	//malloc
	cudaMalloc((void**)&d_a, m*n * sizeof(float));
	cudaMalloc((void**)&d_b, m*n * sizeof(float));

	cudaMemcpy(d_a, a, n* m * sizeof(float), cudaMemcpyHostToDevice);
	cudaMemcpy(d_b, b, n* m * sizeof(float), cudaMemcpyHostToDevice);

	dim3 dGrid(1);
	dim3 dBlock(m);

	// Carry out Jacobi iterations
	for (k = 1; k <= total_iters; k++) {
		if (k % 2 == 0) {
			cudaError_t error = cudaGetLastError();

			matrixKernel << <dGrid, dBlock >> > (d_a, d_b, dcent, n, m, xmin, xmax, ymin, ymax, dx, dy, dxxinv, dyyinv);

			cudaDeviceSynchronize();
			if (cudaGetLastError()) {
				std::cout << "error";
			}
		}

		else {
			cudaError_t error = cudaGetLastError();

			matrixKernel << <dGrid, dBlock >> > (d_b, d_a, dcent, n, m, xmin, xmax, ymin, ymax, dx, dy, dxxinv, dyyinv);

			cudaDeviceSynchronize();
			if (cudaGetLastError()) {
				std::cout << "error";
			}
		}
	}
	cudaMemcpy(a, d_a, n* m * sizeof(float), cudaMemcpyDeviceToHost);
	outputImage("jacobi out", a);
	cudaFree(d_a);
	cudaFree(d_b);
	delete[] u;
	delete[] v;
}

int main() {
	std::chrono::steady_clock::time_point ts, te;
	ts = std::chrono::steady_clock::now();
	dojacobi();
	te = std::chrono::steady_clock::now();
	std::chrono::steady_clock::duration duration = te - ts;
	auto ms = std::chrono::duration_cast<std::chrono::milliseconds>(duration);
	std::cout << "Parallel Code Time: " << ms.count() << " ms" << std::endl;
	cudaDeviceReset();

	return 0;
}

Assignment 3

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