{"id":15484,"date":"2020-11-24T09:11:42","date_gmt":"2020-11-24T09:11:42","guid":{"rendered":"http:\/\/onlineclassesguru.com\/index.php\/2020\/11\/24\/foundations-of-algorithms\/"},"modified":"2020-11-24T09:11:42","modified_gmt":"2020-11-24T09:11:42","slug":"foundations-of-algorithms","status":"publish","type":"post","link":"https:\/\/onlineclassesguru.com\/index.php\/2020\/11\/24\/foundations-of-algorithms\/","title":{"rendered":"Foundations of Algorithms"},"content":{"rendered":"<style type=\"text\/css\"><\/style><p>&nbsp;<\/p>\n<p>The following are analytical and programming problems to be completed by individual students (i.e., this is NOT<br \/>\na collaborative assignment). Please follow the requirements provided under the link Programming Assignment<br \/>\nRequirements.<br \/>\nProblems for Grading<\/p>\n<ol>\n<li>[50 Points] Closest Pairs<br \/>\nIn this form of algorithm development there are several practical applications, such as but not limited to moving<br \/>\nobjects in a game, classification algorithms, computational biology, computational finance, genetic algorithms,<br \/>\nand N-body simulations.<br \/>\nIn these implementations ensure you output your results to the command window for a two dimensional arrays<br \/>\nA \u2264 30 values, e.g. \u201cdouble[][] A= new double[30][2];\u201d. For a two dimensional array A &gt; 30 please<br \/>\ncreate output files.<br \/>\n(a) [12.5 Points] Construct a brute-force algorithm (pseudocode) for finding the closest pair of points in a<br \/>\nset of n points in a two dimensional plane and show the worst-case big-O estimate for the number of<br \/>\noperations used by the algorithm. Hint: First determine the distance between every pair of points and<br \/>\nthen find the points with the closest distance.<br \/>\n(b) [12.5 Points] Implement the brute-force algorithm from part (a).<br \/>\n(c) [12.5 Points] Given a set of n points (x1, y1), \u2026,(xn, yn) in a two dimensional plan, where the distance between two points (xi<br \/>\n, yi) and (xj , yj ) is measured by using the Euclidian distance p<br \/>\n(xi \u2212 xj )<br \/>\n2 + (yi \u2212 yj )<br \/>\n2,<br \/>\ndesign an algorithm (pseudocode) to find the closest pair of points in a more efficient efficient way than<br \/>\npart a and the worst-case big-O estimate.<br \/>\n(d) [12.5 Points] Implement the the algorithm from part (c) with the following methods:<br \/>\ni. The distance of the closest pair of points.<br \/>\nii. The n pair of closest points.<br \/>\niii. The distance values for the n points.<\/li>\n<li>[50 Points] Deterministic Turing Machine (DTM)<br \/>\nIn 1900, mathematician David Hilbert enumerated 23 mathematical problems he considered challenges for the<br \/>\ncoming century. The tenth problem is his list concerned algorithms: Given a polynomial, find an \u201calgorithm\u201d<br \/>\nfor determining whether the polynomial has an integral root.In the way Hilbert phrased the above problem, he<br \/>\nexplicitly asked that an algorithm be \u201cdevised\u201d.As we now know, no algorithm exists for this task if the polynomial is de-fined on several variables. That is, the above problem is algorithmically unsolvable. However,<br \/>\nmathematicians of that period could not come to this conclusion with their intuitive notion of algorithm.<br \/>\nAlthough our intuitive notion of an algorithm is enough to \u201cdevise\u201d solutions for certain problems (the ones<br \/>\na computer can solve), it is useless for showing that no algorithm exists for a particular task, as our intuitive<br \/>\nnotion is not a formal one.So, proving that an algorithm does not exist required having a clear definition of<br \/>\nalgorithm.Such a definition was simultaneously given in 1936 by Alonzo Church and Alan Turing.Church used<br \/>\na notational system called the lamda calculus to precisely define algorithms.Turing did it with his \u201cmachines\u201d<br \/>\n1<br \/>\nThis connection between the informal notion of algorithm and the precise definition is known as the the<br \/>\nChurch-Turing thesis. More precisely, Turing proposed to adopt the Turing machine that halts on all inputs as<br \/>\nthe precise formal notion corresponding to our intuitive notion of an \u201calgorithm\u201d. Note that the Church-Turing<br \/>\nthesis is not a theorem, but a \u201cthesis\u201d, as it asserts that a certain informal concept (algorithm) corresponds to<br \/>\na certain mathematical object (Turing machine).<br \/>\nSince Turing machines can only deal with decision problems, you may find very strange the fact that a Turing<br \/>\nmachine is a formal counterpart of our intuitive notion of an algorithm, as many computational problems are<br \/>\nnot decision problems.Well, keep in mind that every computational problem is either equivalent to a decision<br \/>\nproblem or is at least as \u201chard\u201d as some decision problem. So, the restriction to decision problems is not a<br \/>\nproblem.Besides helping us understand what we can or cannot do with computers, Turing machines can also<br \/>\nbe used to formally define the complexity of an algorithm.<br \/>\nIn this problem you will implement two variants of a DTM. The definition and example are provided below<br \/>\nfollowed by the two required implementations. A deterministic Turing machine consists of the following:<br \/>\n\u2022 A finite state control,<br \/>\n\u2022 A tape consisting of a two-way sequence of tape squares, and<br \/>\n\u2022 A read-write head.<br \/>\nA given DTM is manipulated through the execution of a program utilizing the following components:<br \/>\n\u2022 A finite set \u0393 of symbols,<br \/>\n\u2022 A finite set Q of states (q0 is designated as the start state and qY and qN are designated as halting<br \/>\nstates), where {q0, qH} \u2208 Q,<br \/>\n\u2022 The direction. s = (left, right, halt), where left = -1 and right = +1, in which to move the tape head s \u2208 Q<br \/>\n\u2022 A transition function \u03b4 : (Q \u2212 {qY , qN }) \u00d7 \u0393 \u2192 Q \u00d7 {+1, 0, \u22121},<br \/>\n\u2022 \u03a3 \u2282 \u0393 as a set of input symbols, and<br \/>\n\u2022 b \u2208 (\u0393 \u2212 \u03a3) corresponds to the \u201dblank symbol\u201d (#)<br \/>\nThis allows a DTM to be represented as a finite set of program lines with the form of a 6-tuple TM M =<br \/>\n(\u0393, Q, s, \u03b4, \u03a3, b).<br \/>\nIn\u00a0<a href=\"https:\/\/homeworkhandlers.com\/other\/\">other<\/a>\u00a0<a href=\"https:\/\/homeworkhandlers.com\/words\/\">words<\/a>, the DTM will scan the tape, and the transition function will read the symbol from \u0393 currently<br \/>\nwritten on the table and determine what character to write in its place as well as which direction s to move<br \/>\nthe read-write head. Presumably, if the program indicates +1, the head moves to the right, if the program<br \/>\nindicates \u22121, the head moves to the left and if the program indicates 0, the head stays still.<br \/>\nTo see how a DTM\u00a0<a href=\"https:\/\/homeworkhandlers.com\/works\/\">works<\/a>, suppose we have \u03a3 \u2282 \u0393 as a set of input symbols and b \u2208 (\u0393 \u2212 \u03a3) a \u201dblank symbol\u201d<br \/>\n(#). Next suppose we have an input string x \u2208 \u03a3<br \/>\n\u2217 where we place the symbols of x in consecutive cells of the<br \/>\ntape in positions 1\u2026|x|. All other cells on the tape are assumed to have b. The program will begin in state q0<br \/>\nwith the read-write head scanning square 1 and proceed according to the transition function.<br \/>\nIf the current state of the DTM is ever qY or qN , then the program terminates. On the other hand, if the current<br \/>\nstate is in Q \u2212 {qY , qN }, then the program continues. When transitioning, the read-write head will first erase<br \/>\nthe symbol in the square it is examining and then write the new symbol as specified by the transition function.<br \/>\nThe read-write head then either moves one position to the right or one position to the left, and the Finite State<br \/>\nControl updates the state to some successor q<br \/>\n0<br \/>\n.<br \/>\nBecause we have limited the set of halting (or terminal) states to qY , qN , we note that a DTM is only able<br \/>\nto solve problems that return a Boolean result. In particular, we say that a DMT is used to solve decision<br \/>\nproblems where qY indicates the DMT has decided yes and qN indicates the DTM has decided no.<br \/>\nThe set of states is defined to be Q = q0, q1, q2, q3, qY , qN , and the transition function \u03b4 is defined by the<br \/>\nfollowing table:<br \/>\n2<br \/>\nstart q0 q1 q2<br \/>\nq3<br \/>\nqY<br \/>\nqN<br \/>\n0, 1, [R]<br \/>\nb, [L]<br \/>\n0, !b, [L]<br \/>\n1, !b, [L]<br \/>\nb, [L]<br \/>\n0, !b, [L]<br \/>\n1, !b, [L]<br \/>\nb, [L]<br \/>\n0, 1, b, [L]<br \/>\nFigure 1: DTM State Diagram<br \/>\nTable 1: DTM model illustrated in Figure 1.<br \/>\nQ \u2212 {qY , qN } 0 1 b<br \/>\nq0 (q0, 0, +1) (q0, 1, +1) (q1, b, \u22121)<br \/>\nq1 (q2, b, \u22121) (q3, b, \u22121) (qN , b, \u22121)<br \/>\nq2 (qY , b, \u22121) (qN , b, \u22121) (qN , b, \u22121)<br \/>\nq3 (qN , b, \u22121) (qN , b, \u22121) (qN , b, \u22121)<br \/>\nTo reiterate how the DTM works, Table 1 represents the states q = Q \u2212 {qY , qN }, symbols in x, and the<br \/>\ndirection the read-write head moves (s). Starting with the initial state represented with q0 the finite controller<br \/>\nreads the initial symbol represented by x. At each step the controller reads the symbol x on the tape at state<br \/>\nq, enters the state q<br \/>\n0 = Q = q0, q1, q2, q3, qY , qN , writes the symbol x<br \/>\n0<br \/>\nin the current cell, erasing x, and moves<br \/>\nin the direction identified by s. This is represented as the five-tuple (q, x, q0<br \/>\n, x0<br \/>\n, s) TM. For Table 1 the DTM is<br \/>\nrepresented by the following finite-state machine in Figure 1. The edges in Figure 1 are labeled with the symbol(s) on the tape, a symbol that overwrites the current symbol (if any), and the direction of tape movement.<br \/>\nFor example, (1, !b, [R]) on the edge between q1 and q3 denotes when the read\/write head is over a symbol 1<br \/>\nwhile in state q1 overwrite the 1 with b and move the tape to the right.<br \/>\nPlease note that you can define other symbols in \u0393 and you should. Consider representing two unary numbers<br \/>\non the tape and performing the operations in your assignment. If you include S,E in your set \u0393 you could use<br \/>\nthose symbols to recognize the start of data on the tape (S) and the end of data on the tape (E) making those<br \/>\nsymbols available to you in your state modeling. Remember that each state must fully define the operation of<br \/>\nthe DTM in that state for each symbol in \u0393.<br \/>\n(a) [25 Points] Implement the DTM example provided in the Course Content under Module 3. Ensure that<br \/>\nyour components as listed above are clearly identified in your variable list. The finite-state machine<br \/>\nwith the individual states, state table and five-tuple TM are shown in the above example. You should<br \/>\nimplement the following methods:<br \/>\ni. States method, this method should have all of the operations of your TM.<br \/>\nii. Program line that executes the operations for each identified state, this should follow the n-tuple TM<br \/>\nas described above for M.<br \/>\niii. Print method that outputs the change in tape (x \u2264 30) after each transition function is executed.<br \/>\niv. Write method, for tape larger than x &gt; 30 write the outputs to a file<br \/>\n(b) [25 Points] Expand the implementation of the DTM in part (a) to perform additional binary operations<br \/>\n(addition, subtraction, multiplication) at each state. You should implement the following methods that will<br \/>\nallow for other operations:<br \/>\n3<br \/>\ni. Addition \u2013 The tape will need to handle variable length binary inputs, e.g., tape values of 01100101<br \/>\nadd 101 or add 101101101.<br \/>\nTable 2: Binary Addition Rules<br \/>\nOperation<br \/>\nx + x<br \/>\n0<br \/>\nSum Carry<br \/>\n0 + 0 0 0<br \/>\n0 + 1 1 0<br \/>\n1 + 0 1 0<br \/>\n1 + 1 0 1<br \/>\nii. Subtraction \u2013 The tape will need to handle variable length binary inputs, e.g., tape values of 01100101<br \/>\nsubtract 101 or subtract 101101101.<br \/>\nTable 3: Binary Subtraction Rules<br \/>\nOperation<br \/>\nx \u2212 x<br \/>\n0<br \/>\nSum Carry<br \/>\n0 \u2212 0 0 0<br \/>\n0 \u2212 1 0 1<br \/>\n1 \u2212 0 1 0<br \/>\n1 \u2212 1 0 0<br \/>\niii. Multiplication \u2013 The tape will need to handle variable length binary inputs, e.g., tape values of<br \/>\n01100101 multiply 101 or multiply 101101101.<br \/>\nTable 4: Binary Multiplication Rules<br \/>\nOperation<br \/>\nx \u00d7 x<br \/>\n0<br \/>\nMultiplication<br \/>\n0 \u00d7 0 0<br \/>\n0 \u00d7 1 0<br \/>\n1 \u00d7 0 0<br \/>\n1 \u00d7 1 1<br \/>\n4<\/li>\n<\/ol>\n<p><center><a href=\"http:\/\/onlineclassesguru.com\/orders\/ordernow\"><img decoding=\"async\" src=\"https:\/\/encrypted-tbn0.gstatic.com\/images?q=tbn:ANd9GcTyj99p60XCLyLk1htB7-1neRt8-2QdnenNlQ&usqp=CAU\"target=\"_http:\/\/onlineclassesguru.com\/orders\/ordernow\"\/><\/center><p>","protected":false},"excerpt":{"rendered":"<p>&nbsp; The following are analytical and programming problems to be completed by individual students (i.e., this is NOT a collaborative assignment). Please follow the requirements provided under the link Programming Assignment Requirements. Problems for Grading [50 Points] Closest Pairs In this form of algorithm development there are several practical applications, such as but not limited&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-15484","post","type-post","status-publish","format-standard","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Foundations of Algorithms - onlineclassesguru<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/onlineclassesguru.com\/index.php\/2020\/11\/24\/foundations-of-algorithms\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Foundations of Algorithms - onlineclassesguru\" \/>\n<meta property=\"og:description\" content=\"&nbsp; The following are analytical and programming problems to be completed by individual students (i.e., this is NOT a collaborative assignment). 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