It is a puzzle game where players insert number on a grid consisting of squares with equal number of smaller squares inside. Most games consist of 9x9 grid with higher difficulty and run-time at bigger sizesand more missing numbers. The rules of the game are to insert numbers in such a way that every number appears once in each horizontal and vertical line and the inner square. ==Profiling and analysis== The following program was run on Matrix command line without any modification to the original code. ===Easy Problem===<code> <nowiki>command: ./Sudoku sample-puzzle-1 Input: 0 6 0 0 0 0 9 7 20 5 0 0 0 2 0 0 30 7 0 3 9 0 5 0 02 0 0 0 0 5 4 0 80 0 0 0 0 0 0 0 03 0 1 8 0 0 0 0 60 0 4 0 2 3 0 8 07 0 0 9 0 0 0 2 09 2 5 0 0 0 0 4 0 Solution: 1 6 3 4 5 8 9 7 24 5 9 7 1 2 8 6 38 7 2 3 9 6 5 1 42 9 7 1 6 5 4 3 85 8 6 2 3 4 1 9 73 4 1 8 7 9 2 5 66 1 4 5 2 3 7 8 97 3 8 9 4 1 6 2 59 2 5 6 8 7 3 4 1 </nowiki></code>=====Flat Profile - Easy=====  From this flat profile we can see that most of the functions being called were for checks of rows and columns with the third most called function being placeNum(int,int). As the problem complexity was low, the program completed quickly with far less calls compared to the hard problem. <code> <nowiki>Flat profile: Each sample counts as 0.01 seconds. no time accumulated  % cumulative self self total time seconds seconds calls Ts/call Ts/call name 0.00 0.00 0.00 4539 0.00 0.00 checkRow(int, int) 0.00 0.00 0.00 1620 0.00 0.00 checkColumn(int, int) 0.00 0.00 0.00 1120 0.00 0.00 placeNum(int, int) 0.00 0.00 0.00 698 0.00 0.00 checkSquare(int, int, int) 0.00 0.00 0.00 476 0.00 0.00 goBack(int&, int&) 0.00 0.00 0.00 2 0.00 0.00 print(int (*) [9]) 0.00 0.00 0.00 1 0.00 0.00 _GLOBAL__sub_I_sudoku 0.00 0.00 0.00 1 0.00 0.00 _GLOBAL__sub_I_temp 0.00 0.00 0.00 1 0.00 0.00 solveSudoku() 0.00 0.00 0.00 1 0.00 0.00 storePositions() 0.00 0.00 0.00 1 0.00 0.00 __static_initialization_and_destruction_0(int, int) 0.00 0.00 0.00 1 0.00 0.00 __static_initialization_and_destruction_0(int, int) </nowiki></code> =====Granularity - Easy===== The call graph gives us a further breakdown of the program and the number of calls that were made to the functions in order to solve the easy puzzle. <code> <nowiki>Call graph granularity: each sample hit covers 4 byte(s) no time propagated index % time self children called name 0.00 0.00 4539/4539 placeNum(int, int) [7][5] 0.0 0.00 0.00 4539 checkRow(int, int) [5]----------------------------------------------- 0.00 0.00 1620/1620 placeNum(int, int) [7][6] 0.0 0.00 0.00 1620 checkColumn(int, int) [6]----------------------------------------------- 0.00 0.00 1120/1120 solveSudoku() [13][7] 0.0 0.00 0.00 1120 placeNum(int, int) [7] 0.00 0.00 4539/4539 checkRow(int, int) [5] 0.00 0.00 1620/1620 checkColumn(int, int) [6] 0.00 0.00 698/698 checkSquare(int, int, int) [8]----------------------------------------------- 0.00 0.00 698/698 placeNum(int, int) [7][8] 0.0 0.00 0.00 698 checkSquare(int, int, int) [8]----------------------------------------------- 0.00 0.00 476/476 solveSudoku() [13][9] 0.0 0.00 0.00 476 goBack(int&, int&) [9]----------------------------------------------- 0.00 0.00 2/2 main [4][10] 0.0 0.00 0.00 2 print(int (*) [9]) [10]----------------------------------------------- 0.00 0.00 1/1 __do_global_ctors_aux [22][11] 0.0 0.00 0.00 1 _GLOBAL__sub_I_sudoku [11] 0.00 0.00 1/1 __static_initialization_and_destruction_0(int, int) [15]----------------------------------------------- 0.00 0.00 1/1 __do_global_ctors_aux [22][12] 0.0 0.00 0.00 1 _GLOBAL__sub_I_temp [12] 0.00 0.00 1/1 __static_initialization_and_destruction_0(int, int) [16]----------------------------------------------- 0.00 0.00 1/1 main [4][13] 0.0 0.00 0.00 1 solveSudoku() [13] 0.00 0.00 1120/1120 placeNum(int, int) [7] 0.00 0.00 476/476 goBack(int&, int&) [9]----------------------------------------------- 0.00 0.00 1/1 main [4][14] 0.0 0.00 0.00 1 storePositions() [14]----------------------------------------------- 0.00 0.00 1/1 _GLOBAL__sub_I_sudoku [11][15] 0.0 0.00 0.00 1 __static_initialization_and_destruction_0(int, int) [15]----------------------------------------------- 0.00 0.00 1/1 _GLOBAL__sub_I_temp [12][16] 0.0 0.00 0.00 1 __static_initialization_and_destruction_0(int, int) [16]----------------------------------------------- Index by function name  [11] _GLOBAL__sub_I_sudoku [13] solveSudoku() [10] print(int (*) [9]) [12] _GLOBAL__sub_I_temp [14] storePositions() [9] goBack(int&, int&) [6] checkColumn(int, int) [15] __static_initialization_and_destruction_0(int, int) [5] checkRow(int, int) [8] checkSquare(int, int, int) [16] __static_initialization_and_destruction_0(int, int) [7] placeNum(int, int) </nowiki></code> ===Hard Problem=== <code> <nowiki>command: ./Sudoku sample-puzzle-2-hard Input: 0 0 0 0 0 0 0 0 00 0 0 0 0 3 0 8 50 0 1 0 2 0 0 0 00 0 0 5 0 7 0 0 00 0 4 0 0 0 1 0 00 9 0 0 0 0 0 0 05 0 0 0 0 0 0 7 30 0 2 0 1 0 0 0 00 0 0 0 4 0 0 0 9 Output: 9 8 7 6 5 4 3 2 12 4 6 1 7 3 9 8 53 5 1 9 2 8 7 4 61 2 8 5 3 7 6 9 46 3 4 8 9 2 1 5 77 9 5 4 6 1 8 3 25 1 9 2 8 6 4 7 34 7 2 3 1 9 5 6 88 6 3 7 4 5 2 1 9 </nowiki></code> =====Flat Profile - Hard===== From this flat profile we can see how the number of calls exploded as the complexity of the puzzle was increased with more missing numbers without changing the size of the grid. If the grid size was increased with an equal amount of missing numbers then the program would take far longer to complete with exponentially higher number of calls. As the problem became more complex, the previously low called functions in the easy version, checkSquare and goBack had a sudden spike in usage. <code> <nowiki>Flat profile: Each sample counts as 0.01 seconds. % cumulative self self total time seconds seconds calls s/call s/call name 46.41 21.66 21.66 622577597 0.00 0.00 checkRow(int, int) 19.43 30.73 9.07 223365661 0.00 0.00 checkColumn(int, int) 15.31 37.88 7.14 157353814 0.00 0.00 placeNum(int, int) 13.11 43.99 6.12 100608583 0.00 0.00 checkSquare(int, int, int) 3.10 45.44 1.45 69175252 0.00 0.00 goBack(int&, int&) 2.64 46.67 1.23 1 1.23 46.67 solveSudoku() 0.00 46.67 0.00 2 0.00 0.00 print(int (*) [9]) 0.00 46.67 0.00 1 0.00 0.00 _GLOBAL__sub_I_sudoku 0.00 46.67 0.00 1 0.00 0.00 _GLOBAL__sub_I_temp 0.00 46.67 0.00 1 0.00 0.00 storePositions() 0.00 46.67 0.00 1 0.00 0.00 __static_initialization_and_destruction_0(int, int) 0.00 46.67 0.00 1 0.00 0.00 __static_initialization_and_destruction_0(int, int)  </nowiki></code> =====Granularity - Hard===== The call graph gives us a more detailed look on where exactly most of the time was spent in solving the hard puzzle problem and which areas can be possible candidates for optimization. <code> <nowiki>Call graph granularity: each sample hit covers 4 byte(s) for 0.02% of 46.67 seconds index % time self children called name <spontaneous>[1] 100.0 0.00 46.67 main [1] 1.23 45.44 1/1 solveSudoku() [2] 0.00 0.00 2/2 print(int (*) [9]) [11] 0.00 0.00 1/1 storePositions() [14]----------------------------------------------- 1.23 45.44 1/1 main [1][2] 100.0 1.23 45.44 1 solveSudoku() [2] 7.14 36.85 157353814/157353814 placeNum(int, int) [3] 1.45 0.00 69175252/69175252 goBack(int&, int&) [7]----------------------------------------------- 7.14 36.85 157353814/157353814 solveSudoku() [2][3] 94.3 7.14 36.85 157353814 placeNum(int, int) [3] 21.66 0.00 622577597/622577597 checkRow(int, int) [4] 9.07 0.00 223365661/223365661 checkColumn(int, int) [5] 6.12 0.00 100608583/100608583 checkSquare(int, int, int) [6]----------------------------------------------- 21.66 0.00 622577597/622577597 placeNum(int, int) [3][4] 46.4 21.66 0.00 622577597 checkRow(int, int) [4]----------------------------------------------- 9.07 0.00 223365661/223365661 placeNum(int, int) [3][5] 19.4 9.07 0.00 223365661 checkColumn(int, int) [5]----------------------------------------------- 6.12 0.00 100608583/100608583 placeNum(int, int) [3][6] 13.1 6.12 0.00 100608583 checkSquare(int, int, int) [6]----------------------------------------------- 1.45 0.00 69175252/69175252 solveSudoku() [2][7] 3.1 1.45 0.00 69175252 goBack(int&, int&) [7]----------------------------------------------- 0.00 0.00 2/2 main [1][11] 0.0 0.00 0.00 2 print(int (*) [9]) [11]----------------------------------------------- 0.00 0.00 1/1 __do_global_ctors_aux [22][12] 0.0 0.00 0.00 1 _GLOBAL__sub_I_sudoku [12] 0.00 0.00 1/1 __static_initialization_and_destruction_0(int, int) [15]----------------------------------------------- 0.00 0.00 1/1 __do_global_ctors_aux [22][13] 0.0 0.00 0.00 1 _GLOBAL__sub_I_temp [13] 0.00 0.00 1/1 __static_initialization_and_destruction_0(int, int) [16]----------------------------------------------- 0.00 0.00 1/1 main [1][14] 0.0 0.00 0.00 1 storePositions() [14]----------------------------------------------- 0.00 0.00 1/1 _GLOBAL__sub_I_sudoku [12][15] 0.0 0.00 0.00 1 __static_initialization_and_destruction_0(int, int) [15]----------------------------------------------- 0.00 0.00 1/1 _GLOBAL__sub_I_temp [13][16] 0.0 0.00 0.00 1 __static_initialization_and_destruction_0(int, int) [16]----------------------------------------------- Index by function name  [12] _GLOBAL__sub_I_sudoku [2] solveSudoku() [11] print(int (*) [9]) [13] _GLOBAL__sub_I_temp [14] storePositions() [7] goBack(int&, int&) [5] checkColumn(int, int) [15] __static_initialization_and_destruction_0(int, int) [4] checkRow(int, in t) [6] checkSquare(int, int, int) [16] __static_initialization_and_destruction_0(int, int) [3] placeNum(int , int)  </nowiki></code> ====Analysis====The program was quite fast and had a really short run time for solving the given problem on a standard 9x9 block. This project was found on [https://github.com/shafeeq/Sudoku GitHub]. It is GNU C++ compiler compatible and both the flat profile and call graph were generated on matrix successfully. For the easy problem, the run time was quick as the inputted grid had more filled cells compared to the hard problem where most of the cells were set to 0. This increased the complexity scope and which is why the program took 46.67 seconds to complete.  From the hard puzzle call graph we can see that the function which could be the prime candidates for parallelization and optimization are the checkRow and checkColumn functions in which the program spends most of its time. Because of the type of mathematical problem set, this Sudoku solver can be an excellent application for a parallelization project.