FIT1045 Algorithms and programming in Python, S1-2019Assignment 1 (value 12%)Due: Friday 12th Apr 2019, 11:55 pm.ObjectivesThe objectives of this assignment are: To demonstrate the ability to implement algorithms using basic data structures and operations on them. To gain experience in designing an algorithm for a given problem description and implementing thatalgorithm in Python.Submission Procedure
Put you name and student ID on each page of your solution.Save your files into a zip file called yourFirstName yourLastName.zipSubmit your zip file containing your solution to Moodle.Your assignment will not be accepted unless it is a readable zip file.Important Note: Please ensure that you have read and understood the university’s policies on plagiarismand collusion available at http://www.monash.edu.au/students/policies/academic-integrity.html. Youwill be required to agree to these policies when you submit your assignment.Marks: This assignment will be marked both by the correctness of your code and by an interview withyour lab demonstrator, to assess your understanding. This means that although the quality of your code(commenting, decomposition, good variable names etc.) will not be marked directly, it will help to write cleancode so that it is easier for you to understand and explain.This assignment has a total of 30 marks and contributes to 12% of your final mark. Late submission will have10% off the total assignment marks per day (including weekends) deducted from your assignment mark. (In thecase of Assignment 1, this means that a late assignment will lose 3 marks for each day (including weekends)).Assignments submitted 7 days after the due date will normally not be accepted.Detailed marking guides can be found at the end of each task. Marks are subtracted when you are unable toexplain your code via a code walk-through in the assessment interview. Readable code is the basis of aconvincing code walk-through.1Task 1: Race to Pi (12 Marks)BackgroundThe ratio of a circle’s circumference to its diameter, usually denoted by the Greek letter , is an extremelyimportant constant in mathematical computations. Unfortunately, the exact value of cannot be storedexplicitly in a computer, because it is irrational, i.e., it does not correspond to any ratio between two integervalues. To help us out, there are many different ways to approximate , most of which based on computing afinite prefix of an infinite sum or product. For instance:via a solution to the basel problem: derived from the taylor expansion of 1/(1 + x): via the wallis algorithm: via the spigot algorithm: Each of these sums (or the product in case of 3.) converges to the respective value on the left-hand-side of theequals sign. That is for every desired precision > 0, the absolute difference between the left-hand-side and thesum/product on the right-hand-side is guaranteed to be less than if a sufficient number of terms is computed.Thus, they are all valid ways to compute . However, the speed of convergence, i.e., the number of terms thathave to be computed to reach a desired precision is different for each method. (Note that in case of wallis andspigot every expression in parenthesis is considered one term; e.g., the first term of the wallis product equals4/3, the third term of the spigot sum equals 2/15.) In this task, we will determine which of the above methodsis preferable from this perspective.InstructionsCreate a Python module called pi.py. Within that module:a) Write four different functions for approximating corresponding to the four different approaches listedabove (the functions must be called basel, taylor, wallis, and spigot). Each of these functions mustobey the following input/output specification:Input: a float parameter precision which specifies the precision to which should be calculated, i.e.,the absolute difference of the approximation to the Python constant pi.Output: two values (x,n) such that x is the first approximation to with an absolute difference frompi of at most precision, and n is the number of terms of the sum/product that were computed inthe process.For example, if the input was 0.1, then the function should continue to compute terms of the series untilthe approximation x to is less than 0.1 away from the Python constant pi (see more detailed examplesbelow).b) Write another function called race(precision, algorithms) that can be used to compare the convergencespeed of different approximation algorithms via the following input/output specification:Input: a float parameter precision and a list of Python functions algorithms.Output: a list of integer pairs [(i1, n1), (i2, n2),..., (ik,nk)] such that– for each pair (i,n) in the list, n is the number of steps that the i-th algorithm takes to approximate within the given precision– the list is in ascending order with respect to the number of steps (the second number of the pair)with ties resolved arbitrarily.Note that the input to race are functions and not strings. That is, in principle race can be used forevaluating arbitrary functions for approximating , not only the four covered here.c) Write a function print results which takes as input the output of the function race and prints theresults in a human readable format. Each line of the output should be “Algorithm i finished in nsteps” where i is first element of the integer pair, and n is the second.The only two import statements allowed in your module are from math import pi and from math importsqrt. Note that the subtasks can be solved independently and the given order (a, b, c) is only a recommendation.2Examplesa) Calling wallis(0.2) returns (2.972154195011337, 4) as the result of approximating with2.972154195011337 = 2 termsusing 4 terms because pi - 2.972154195011337 < 0.2 and non of the three prior wallis approximationssatisfies this condition.b) Calling basel(0.1) returns (3.04936163598207, 10) as the result of approximating withusing 10 terms because pi - 3.04936163598207 < 0.1 and non of the nine prior basel approximationssatisfies this condition.c) Calling taylor(0.2) returns (3.3396825396825403, 5) as the result of approximating with3.3396825396825403 = 4using 5 terms because pi - 3.3396825396825403 < 0.2 and non of the four prior taylor approximationssatisfies this condition.d) Calling spigot(0.1) returns (3.0476190476190474, 4) as the result of approximating with3.0476190476190474 = 2 using 4 terms because pi - 3.0476190476190474 < 0.1 and non of the prior three spigot approximationssatisfies this condition.e) Calling race(0.01, [taylor, wallis, basel]) returns [(2,78),(3,96),(1,100)] because functions taylor,basel, and wallis need 100, 96, and 78 terms, respectively, to approximate within the required precisionof 0.01.f) Calling print results([(2,78),(3,96),(1,100)]) printsAlgorithm 2 finished in 78 stepsAlgorithm 3 finished in 96 stepsAlgorithm 1 finished in 100 stepsMarking Guide (total 12 marks)Marks are given for the correct behaviour of the different functions:(a) 2 marks each for basel, taylor, wallis, and spigot(b) 3 marks for race(c) 1 mark for print resultsAll functions are assessed independently. For instance, even if function race does not always produce the correctoutput, function print results can still be marked as correct.3Task 2: Reversi (18 Marks)BackgroundReversi is a game for two players (Black and White), which is played on a squared board of 8 rows/columnswith stones that are white on one side and black on the other. In the beginning of the game, stones of Player 1(B) and Player 2 (W) are placed in a cross arrangement at the center of the board as shown in Table ??. Thenthe players take turns placing stones on empty fields on the board with their colour facing up according to thefollowing rules:All opponent’s stones that are enclosed in a straight line (horizontal, vertical, or diagonal) between thenewly placed stone and another stone of the same color are flipped (see Table).A move is only valid if it turns at least one opponent’s stone.If a player has no valid moves, their turn is skipped (see example in Table).The game ends when both players have no legal moves.When the game ends, each player scores points equal to the number of stones which have their colour face up,and the player with a higher score wins (draw if scores equal). In this task you will create a Python programthat allows one or two players to play a game of Reversi.Table 1: Initial board state at the beginning of game.Player 1 (B) has the first move.Table 2: Board state after Player 1 (B) played “e6”as her first move.a b c d e f g hTable 3: Valid moves (indicated by) for Player 2(W) following “e6” from Player 1.Table 4: Board state with no valid move for Player 2(W), who, in case it is her turn, has to pass back toPlayer 1 (B), whose only valid move is “e1”.InstructionsCreate a python module reversi.py. The only import statement allowed in your module is import copy. Yourmodule must at least contain the nine functions described in the subtasks below, but you are allowed, and infact, encouraged, to implement additional functions as you deem appropriate.4a) Create three functions new board(), print board(board), and score(board) that represent the basic elementsof Reversi as follows. The function new board() takes no parameters and returns a fresh boardconfiguration representing the Reversi starting position. The other two functions each take as input a boardconfiguration. The function score(board) returns as output a pair of integers (s1, s2) where s1 representsthe number of stones of Player 1 in the given board configuration and s2 represents the number of stones ofPlayer 2. The function print board(board) has no return value but prints the given board configurationin human-readable form to the console (in particular, all rows of the board must be printed in separate linesof the console and labelled by their names 1, 2, . . . and all columns must be properly aligned and labelled bytheir names a, b, . . . — similar to the illustrations above). For these and all other functions of this task, aboard configuration must be represented as a list of lists in the following way: A board configuration is a list of length 8, the elements of which represent the rows of the board fromtop to bottom: the first element represents the topmost row of the board, row 1, the second representsrow 2, etc.. Each row, i.e., each element of a list representing a board configuration, is also a list of length 8. Theseinner lists represent the rows of the board from left to right: the first element represents the field in theleftmost column, Column A, the second element represents the field in second column, Column B, etc. Each field, i.e., each element of the inner lists representing the rows, is an integer from the set {0, 1, 2}representing the state of that field: 0 represents an empty square, 1 represents a square with a Blackstone, and 2 represents a square with a White stone.See example ?? for an illustration of this representation of a board configuration.b) Create three functions enclosing(board, player, pos, direct), valid moves(board, player), andnext state(board, player, pos) that represent the rules of Reversi according to the following specifi-cations. A board position is represented by a pair (r, c) where r and c are each integers from the rangerange(8) representing a row index and a column index, respectively. For instance, the position “b3” isrepresented by (2, 1). The row and column order of the “b3” style and the specification of positionis reversed. This is because it is natural to constuct lists of lists in such a way that the outer listscorrespond to rows, and the inner list positions corerspond to column positions, so we reference themwith row first, then column. This row, column convention is quite common across maths and computerscience. A direction is represented by a pair (dr, dc) where dr and dc are each integers from the set {-1, 0,1}. For instance, “horizontal to the left” is described by (0, -1), “diagonal to the right and down” isdescribed by (1, 1), and so on). The function enclosing(board, player, pos, dir) represents whether putting a player’s stone ona given position would enclose a straight line of opponent’s stones in a given direction:Input: A board configuration board, an integer player from the set {1, 2}, a board position posand a direction dir.Output: True if board contains a stone of player in direction dir from position pos and all positionson the straight line between that stone and pos contain stones of the other player, and there is atleast one stone belonging to the other player on the straight line; False otherwise. The function valid moves(board, player) has:Input: A board configuration board as well as an integer player from the set {1, 2}.Output: A list of board positions (represented as pairs of integers as described above) [(r1, c1),(r2, c2), ..., (rk, ck)] such that the elements of that list correspond to all valid positions thatPlayer player is allowed to drop a stone on in the given input configuration board (in particular,the list is empty if the player has not valid moves in the given board). The function next state(board, player, pos) has:Input: A board configuration board as well as an integer player from the set {1, 2} and a board positionrepresenting the move of Player player to place one of their stones on the board configurationboard on the field described by pos.Output: A pair (next board, next player) such that next board is the board configuration resultingfrom Player player placing a stone on position pos on the input board configuration board,and next player is an integer from the set {0, 1, 2} corresponding to the player who gets tomove next or 0 in case the game ends after the input move is carried out.5c) Create functions position(string), run two players() and run single player() that put all of theabove together and that allow one or two players to play a game of Reversi through the console as follows. The function position(string) accepts as input any string string (e.g., "b5" or "Genghis Khan")and has as output the board position (r, c) described by the string or None in case the string doesnot correspond to a valid board position. Running run two players() should repeatedly do the following until the user selects to stop:1: Print the current board state2: Ask the player whose turn it is to input a position to drop a stone (in the format "d4"). The playercan also choose to quit by entering "q"3: Your program should then respond in one of the following ways:– If the move is not valid, print "invalid move"– If the move is valid, place the move on the board.4: If the game ended (no valid move remaining for either player), print the game result and exit5: If the game did not end, the current player’s turn is over. It is now the other player’s turn Running run single player() should behave identical to run two players() except that all movesof Player 2 are carried out automatically such that the resulting board configuration has the maximalscore for Player 2 (among all their valid moves).Examplesa) Calling score(new board()) returns (2, 2) because the score in the starting configuration is 2 for Player(B) and 2 for player 2 (W).b) Calling enclosing(new board(), 1, (4, 5), (0, -1)) returns True because Player 1 can enclose oneopponent’s stone to the left when placing a stone on field “f5”. In contrast, there is no straight line ofopponent’s stones to enclose on the diagonal line to the bottom right. Therefore, enclosing(new board(),1, (4, 5), (1, 1)) returns False.c) Calling next state(new board(), 1, (4, 5)) will return (next board, 2) where next board is the boardconfiguration[[0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0],[0,0,0,2,1,0,0,0],[0,0,0,1,1,1,0,0],[0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0]]because that is the configuration resulting from Player 1 putting a stone on field “f5” (encoded by integerpair (4, 5)) in the starting configuration, and Player 2 has the next move afterwards.d) Calling valid moves(next state(new board(), 1, (4, 5))[0], 2) returns the list [(3, 5), (5, 3),(5, 5)] or any of its permutations (list with same elements but different order) because the valid moves ofPlayer 2 after Player 1 played “f5” in Turn 1 are “d6”, “f4”, and “f6”.e) Calling position("e3") returns the pair (2, 4) because that is the internal representation of board position“e3”. Calling position("l1"), position("a0"), or position("Genghis Khan") all return None becauseall these input do not correspond to valid board positions.Marking Guide (total 18 marks)Marks are given for the correct behaviour of the different functions:a) 1 mark each for new board, print board, and scoreb) 4 marks for enclosing, 3 marks for valid moves, and 2 marks next statec) 1 mark for position, 2 marks for run two players and 3 marks for run single playerAll functions are assessed independently to the degree possible. For instance, even if function valid moves doesnot always produce the correct output, function run two players can still be marked as correct. WX：codehelp