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EECS 376: Foundations of Computer Science

创建日期：2024-04-23 17:11:50

所属分类：计算机computer science

标签： EECS 376

Foundations of Computer Science

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EECS 376: Foundations of Computer Science

EECS 376 Practice Final Exam
Instructions:
This exam is closed book, closed notebook. You may not provide your own scratch paper, but the last
four pages are intentionally left blank for scratch work. No electronic devices are allowed (this includes
calculators, cellphones, etc). You may use one 8.5 × 11-inch study sheet written on both sides, as long as
you prepared it yourself. Check your seat assignment. Make sure you are taking the exam at the time slot
and the classroom you were assigned by the staff.
Any deviation from these rules will constitute an Honor Code violation. In addition, the staff reserves
the right not to grade an exam taken in a violation of this policy.
The exam consists of 15 multiple choice questions, 5 short-answer questions, and 3 questions that require
proofs. You only need to answer two of the proof questions. No extra credit will be given if you answer all
three proof questions. For the multiple choice questions, please mark your answers neatly on the Scantron.
For the short-answer and proof sections, please write your answers clearly in the spaces provided. The exam
has 12 pages, including this page and the 3 blank pages at the end.
PLEASE DO NOT DETACH PAGES FROM THE EXAM. Failing to comply with this
requirement may result in grade deduction!
Write out and sign the full honor pledge below.
Honor pledge:
I have neither given nor received aid on this exam, nor have I concealed any violations of the Honor Code.
I will not discuss the exam with anyone before exam grades are released.
Signature:
Name:
Uniqname:
Section Points
1-15 (Multiple Choice) / 30
16-20 (Short Answer) / 40
21-23 Pick two (Proof) / 30
Total / 100

Recall the following languages:
• LHALT = {〈M,x〉 : M halts on x}
• L = {x : x 6∈ L}
• SAT = {φ : φ is a satisfiable Boolean formula},
• Clique = {(G, k) : graph G has a clique of size k},
• Vertex-Cover = {(G, k) : graph G has a vertex cover of size k},
• Independent-Set = {(G, k) : graph G has a independent set of size k},
• Ham-Cycle = {G : graph G has a Hamiltonian cycle},
• k-Coloring = {G : graph G can be k-colored},
• Maze = {(G, s, t) : graph G has a path from vertex s to vertex t},
• Graph-Isomorphism = {(G1, G2) : graphs G1 and G2 are isomorphic},
• Graph-NonIsomorphism = {(G1, G2) : graphs G1 and G2 are not isomorphic},
• TSP = {(G, k) : weighted graph G has tour of weight at most k},
• Max-Cut = {(G, k) : graph G has a cut of size at least k}.
Remark: Unless otherwise specified, all the graphs are undirected.
Multiple choice (2 points each).
1. Which of the following statements is/are true?
A. Vertex-Cover ≤p Independent-Set
B. Clique ≤p SAT
C. SAT ≤p Vertex-Cover
D. A and B
E. A, B and C
2. In the process of generating shares for Shamir’s secret sharing scheme, the following polynomial is
created:
f(x) = 7 + 3x+ 20x2 + 2x3 + 3x4 + 4x5 (mod 23).
What is the minimum number of shares required to recover the secret?
A. 5
B. 6
C. 7
D. 23
E. Impossible to determine.
3. Use the above setup again. What is the secret?
A. 5
B. 6
C. 7
D. 23
E. Impossible to determine.
Page 2

True/False/Unknown (2 points each). True = A, False = B, Unknown = C
4. Clique is NP-Intermediate.
5. If SAT ∈ P, then P = coNP.
6. The Shortest Path problem is defined as follows: Given a weighted directed graph G = (V,E) with
weight function w : E 7→ R and a pair of vertices s, t ∈ V , determine the (weighted) length of the
shortest path between s and t. Then the Shortest Path problem has an efficient 376-approximation
algorithm.
7. Let A,B,C, and D be the outcomes of rolling 280-sided, 281-sided, 370-sided, and 376-sided fair dice,
where a k-sided not necessarily fair die gives the outcome between 1 and k with the uniform probability.
The following experiment is conducted: The 370-sided die is rolled twice, and the other dice are rolled
once. Let X be the sum of the outcome. Then E[X] = E[A] + 2E[C] + E[D] + E[B].
8. NP is countable.
9. LHALT ≤p SAT.
10. Let S376 be defined as S376 = {x : x is the name of an EECS 376 student in Fall 2018}.
Then SAT ≤p S376.
11. 281-Coloring ≤T 281-Coloring.
12. You are a student of Euclid in Ancient Greece. Your wise teacher has tasked you to travel between
each of the Greek islands and sell his prime numbers. He instructs you to visit each island only once
and then purchase the cheapest boat ride possible to an island you have not visited. Once you have
visited every island you will return home. Euclid asserts that this strategy will minimize the total cost
of your travels. His statement is (true/false/unknown).
Always/Sometimes/Never (2 points each) Always = A, Sometimes = B, Never = C.
13. (All / Some / No) NP-hard languages are undecidable.
14. (All / Some / No) unrecognizable languages are decidable.
15. (All / Some / No) languages in P are polynomial-time mapping reducible to
Graph-NonIsomorphism.
Page 3

Short Answer (8 points each)
16. Describe an algorithm that approximates the value of ln(2). You do not need to prove your algorithm
is correct. How many samples n would we need to approximate ln(2) to within absolute error ±0.03
with probability at least 90%? You do not need to simplify your expression for n.
Hint: See picture below.
Figure 1: A curve of y = 1(x+1) . Note that the area in the shaded region is
∫ 1
0
1
x+1dx = ln 2.
17. Prof. Volkovich is sending exam questions to the EECS 376 course staff. To ensure that the staff can
verify that the questions have not been altered, he uses a RSA-based signature scheme with n = 55
and public key e = 27. What would the signed message (m, s) be, if m = 52?
Page 4

18. Given an input string s and a pattern string p, a Wildcard Pattern Matching with support for ‘?’ and
‘*’ will return true if s and p are a match where,
• ‘?’ Matches any single character.
• ‘*’ Matches any sequence of characters (including the empty sequence).
There is only a Wildcard Pattern Matching if the matching covers the entire input string (not partial).
Note: s contains only lowercase letters a-z, and p contains only lowercase letters a-z, and characters
like ? or *. That is ? or * can only appear in p if at all. Here are some of correct input/output pairs
examples:
Inputs (s, p) Outputs Comments
(“aa”, “a”) false “a” does not match the entire string “aa”
(“aa”, “*”) true ‘*’ matches any sequence.
(“acdcb”, “a*c?b”) false
(“acdcb”, “a*cb”) true
Define WPM(i, j) as true if and only if there is a Wildcard Pattern Matching between s[1 · · · i] and
p[1 · · · j] and write the recurrence relation in this situation.
Hint: Recall that the Longest Common Subsequence algorithm relies on the following recurrence
relation: Let S1 and S2 be two strings that we want to find the longest common substring of, and
define LCS(i, j) as the length of the longest common substring between S1[1 · · · i] and S2[1 · · · j]. Then
LCS(i, j) =
{
LCS(i− 1, j − 1) + 1 if (S1[i] = S2[j])
max{LCS(i, j − 1), LCS(i− 1, j)} if (S1[i] 6= S2[j])
where we have ignored the base cases.
Adapt this recurrence relation to fit in an algorithm that would solve the following related problem.
Note: We do not need the full algorithm, just the analogous recurrence relation. You may also ignore
the base case.
Page 5

19. Given two languages A and B, prove that if A ≤p B and B ∈ NP, then A ∈ NP.
20. The Alan Company provides two different products: Cook’s coke and Karp’s soda. Alan claims that
he can tell them apart, since they taste different. However, Betty thinks these two drinks are entirely
identical. How can Betty test Alan’s claim?
Page 6

Proofs (15 pts each)
Please answer two of the following three questions. If you answer all of them we will pick two ar-
bitrarily to grade. Feel free to use the blank pages for your responses, but please clearly mark the
locations/continuations of your answers.
21. Suppose that you are a consultant for the Port Authority of a coastal city. Its profit is determined
almost entirely by how efficiently it can take away the contents of the ships that arrive in port. Here
is a basic model of the problem it faces.
A ship arrives with n containers having positive integer weights w1, . . . , wn. There are many trucks
near the port, each of which can hold some positive integer K units of weight (which is no less than
the container weights). A truck can hold any number of containers, as long as their total weight does
not exceed K. The goal is to minimize the number of trucks that are needed in order to carry away
all the containers. Unfortunately, it is well known that this problem is NP-hard.
Here is a candidate approximation algorithm:
1. for i = 1, 2, ..., n
2. if item i fits in the current truck, put it there
3. else
4. send the current truck away
5. start using the next available truck, putting item i in it
Example: Consider w1 = 3, w2 = 2, w3 = 2,K = 4. Observe that two trucks is optimal: we can put
item 1 on the first truck, and items 2 and 3 on the second truck. Note also that the above algorithm
yields this optimal allocation.
(a) Prove that in the above algorithm, every pair of adjacent trucks carries a combined weight of at
least K.
(b) Show that for any set of weights wi and any truck capacity K, the number of trucks used by
the above algorithm is at most twice the optimal number of trucks, i.e., the algorithm yields a
2-approximation.
Hint: for any weights wi and truck capacity K, we have OPT ≥ (
∑n
i=1 wi)/K.
Page 7

22. Consider the following language:
DOUBLE-SAT = {φ : φ is a Boolean formula with at least two satisfying assignments.}
Prove that DOUBLE-SAT is NP-complete. You may use any of the languages on the list at the top of
page 2.
Page 8

23. A dominating set for a graph G = (V,E) is a subset D ⊆ V of vertices such that every vertex not in
D is adjacent to at least one member of D. See example below.
Figure 2: An example of dominating set (shaded vertices).
The decision version of the problem is defined as follows:
DOM-SET = {(G, k) : graph G has a dominating set of size k}.
Suppose you have an efficient, black-box decider for DOM-SET. Design and analyze an efficient
algorithm that finds a dominating set of minimal size, given a graph G as input.
Page 9

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CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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