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CSCI3104: Algorithms

创建日期：2024-07-23 18:06:39

所属分类：计算机computer science

标签： CSCI3104

Algorithms

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CSCI3104: Algorithms

Final Exam Questions

Notes:

Due date: 2pm on December 8,2018
Submit a pdf file of your written answers and one py file with your Python codes. Your written answers must be typed to receive credit. All Python solutions should be clearly commented. Your codes need to run to get credit for your answers.
You should not ask about the exam during the office hours or post any question about it on Piazza. However, classes on Wednesday, November 28 and Wednesday, December 5 will be reserved for questions about the exam.
All work on this exam needs to be independent. You may consult the textbook, the lecture notes and homework sets, but you should not use any other resources. If we suspect that you collaborated with anyone in the class or on the Internet, we will enforce the honor code policy strictly. If there is a reasonable doubt about an honor code violation, you will fail this course.

Problems:

1.(15 pts) Doctor Jean needs your help!

She’s been relabeling her collection of chro- mosomesequences and has found three different sequences that were displaced from their original locations in the lab. She knows that two of the sequences represent two chromosomes from a human (Homo Sapiens) and the other sequence represents a chromosome from a soybean (Glycine Max).

Help Doctor Jean by parsing in the following sequences into memory (be careful of new- lines!) and applying the Longest Common Subsequence algorithm learned in recitation to each unique pair of sequences.

Unzip the ”sequence data.zip” file located on Canvas to access the data. When back- tracing through the computed two dimensional matrix to find the longest common

subsequence, break ties uniformly at random. Compare the results from your imple- mentation to determine which species the sequences pertain to (try to come up with a sensible metric that will compare the results).

(a)Showa table that maps the sequence to the species.

(b)Submit your separate python (.py) file along with your PDF

Note: the data provided is minified data (first 50 lines of each sequence) from the National Center for Biotechnology Information. The data was originally in FASTA format but it was modified when removing the descriptions of sequences from the files. You can treat the file as a text file. Below are useful links for information relating to this question.

2.(15 pts) Draco Malfoy is struggling

with the problem of making change for n cents usingthe smallest number of Malfoy has coin values of v1 < v2 < · · · < vr for r coins types, where each coin’s value vi is a positive integer. His goal is to obtain a set

of counts {di}, one for each coin type, such that Σr di = k and where k is minimized.

(a)A greedy algorithm for making change is the wizard’s algorithm, which all youngwizards Malfoy writes the following pseudocode on the whiteboard to illustrate it,Algorithms代写

where n is the amount of money to make change for and v is a vector of the coin denominations:

wizardChange(n,v,r) : d[1 .. r] = 0 // initial histogram of coin types in solution while n > 0 { k = 1 while ( k < r and v[k] > n ) { k++ } if k==r { return ’no solution’ } else { n = n - v[k] } } return d

Hermione snorts and says Malfoy’s code has bugs. Identify the bugs and explain why each would cause the algorithm to fail.Algorithms代写

(b)Sometimesthegoblins at Gringotts Wizarding Bank run out of coins,1 and make change using whatever is left on Identify a set of U.S. coin denominations

1It’s a little known secret, but goblins like to eat the coins. It isn’t pretty for the coins, in the end.

for which the greedy algorithm does not yield an optimal solution. Justify your answer in terms of optimal substructure and the greedy-choice property. (The set should include a penny so that there is a solution for every value of n.)

(c)Onthe advice of computer scientists, Gringotts has announced that they will be changing all wizard coin denominations into a new set of coins denominated in powers of c, e., denominations of c0, c1, . . . , cA for some integers c > 1 and A 1. (This will be done by a spell that will magically transmute old coins into new coins, before your very eyes.) Prove that the wizard’s algorithm will always yield an optimal solution in this case.

Hint: first consider the special case of c = 2.

3.Inthe two-player game “Pandas Peril”

an even number of cards are laid out in a row, face up. On each card, is written a positive integer. Players take turns removing a card from either end of the row and placing the card in their pile. The player whose cards add up to the highest number wins the game. One strategy is to use a greedy approach and simply pick the card at the end that is the However, this is not always optimal, as the following example shows: (The first player would win if she would first pick the 4 instead of the 5.)

4 2 10 5

(a)(10 pts) Write a dynamic programming algorithm for a strategy to playPandas Peril. Player 1 will use this strategy and Player 2 will use a greedy strategy of choosing the largest

(b)(10 pts) Prove that your strategy will do no worse than the greedy strategy for maximizing the sum of each

(c)(10pts) Implement your strategy and the greedy strategy in Python and simulate a game, or two, of Pandas Peril. Your simulation should include a randomly generated collection of cards and show the sum of cards in each hand at the end of the game.

4.Acommon problem in computer graphics

is to approximate a complex shape with a bounding For a set, S, of n points in 2-dimensional space, the idea is to find the smallest rectangle, R. with sides parallel to the coordinate axes that contains all the points in S. Once S is approximated by such a bounding box, computations involving S can be sped up. But, the savings is wasted if considerable time is spent constructing R, therefore, having an efficient algorithm for constructing R is crucial.

(a)(10pts) Design a divide and conquer algorithm for constructing R in O( 3n) com-parisons.Algorithms代写

(b)(10pts) Implement your algorithm in Generate 50 points randomly and show that your bounding box algorithm correctly bounds all points generated. Your code should output the list of points, as well as the coordinates of R. You should include an explanation of the results in your pdf file with your algorithm.

(10 pts) Professor Voldemort has designed an algorithm that can take any graph G with n vertices and determine in O(nk) time whether G contains a clique of size k. Has the Professor just shown that P = NP ? Why or whynot?
(10points) Consider the special case of TSP where the vertices correspond to points in the plane, with the cost defined for an edge for every pair (p, q) being the usual Euclidean distance between p and Prove that an optimal tour will not have any pair of crossing edges.

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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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