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CS330 Midterm

创建日期：2024-04-10 17:25:44

所属分类：计算机computer science

标签： CS330

Midterm

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CS330 Midterm

Instructions
• This exam is open book. You may consult your textbook, notes, slides, or other course
materials, but no external sources.
• You may reference and use anything without having to prove or explain in detail (e.g.,
algorithms, proofs, running time) discussed in lecture, lab or the textbook.
• Absolutely no collaboration is allowed. You cannot discuss the contents of this
exam with anyone except the instructors.
• The exam consists of 3 questions. With question 1 consisting of subproblems a - e.
• The exam has quite a few questions. Don’t get stuck on one, if it takes you a long time,
move on to the next question and you can come back to it later. Also, the deadline
is a hard deadline. It’s better to submit a not fully completed exam, than miss your
deadline.
• You can write your answers by hand on paper, on a tablet, or you can type your
solutions. Make sure to clearly write which problem you are answering and also label
your answers with the problem number and the problem part at the top of each page
when you upload to Gradescope. If we cannot find or read your answers they will not
be graded.
• From the time you download this pdf you have exactly 120 minutes time to upload
your final pdf to Gradescope. The system is set not to accept exams after your time
has expired.
• If you have questions, you may ask them via private post on Piazza or via the Zoom
link. (This link is also posted on Piazza in @309)
1
Problem 1 Short answers (8 points)
(a) (2 points) For each of the below determine whether the complexity is Θ(n2). Write
True or False, no explanation needed.
(a) Smallest complexity of an algorithm which finds the largest and smallest element
in a list.
(b) Complexity of Dijkstra’s algorithm (with the implementation discussed in class).
(c) Smallest complexity of finding a minimum weight spanning tree in a graph that
has n vertices and 3n many edges. (Answer this question with regard to the
algorithms and implementations that we studied in class.)
(d) Let G(U,W,E) be a complete bipartite graph. Suppose that |U | = |W | = √n.
The number of edges in G is Θ(n2).
Here is the definition of a complete bipartite graph: the nodes in G are divided in
to two sets U and W that form a partition, i.e. each node is assigned to exactly
one of the two. There is an edge between every pair of nodes u,w where u ∈ U
and w ∈ W , but there are NO edges between pairs of nodes in the same set.
(b) (3 points) Consider the graph G below. Note, that this graph is undirected. Answer
both parts (a.) and (b.). In both parts there may be more than one correct answer,
and you should list all the answers that are correct. No explanation needed.
a
b
c
d e
f
g
h
(a) Which of the following is NOT a possible order in which vertices of G are dis-
covered when a depth-first search (DFS) on G is done? Assume that the search
starts at vertex a.
(A) abcdefgh (B) acfghedb (C) abceghedb
(D) acefhgbd (E) None of them
(b) Which of the following ARE possible orders in which the vertices of G are dis-
covered when a breadth-first search (BFS) on G is done? Assume that the search
starts at vertex f .
(A) fchgaedb (B) fhgecabd (C) fgchabd
(D) fghcaedb (E) chgebda (F) None of them
2
(c) ( 1 point) Suppose you are given an undirected, connected graph G(V,E). Suppose
you run DFS on this graph started from a randomly chosen node. How can you use the
outcome of DFS to assign a direction to each edge, so that the resulting directed graph
is a DAG? Give a very short explanation (you should be able to do this in one or two
sentences). You don’t need to describe an efficient implementation of your algorithm
or prove its correctness, only state your approach clearly.
(d) (1 point) Given a weighted graph where weights of all edges are unique (no two edge
have same weights), there is always a unique shortest path from a source to destination
in such a graph. Explain why or provide a counter example.
(e) (1 point) Let G = (V,E, c) be an undirected graph with costs c(e) on each edge e
and let T be a minimum weight spanning tree on G. Suppose the cost of each edge is
increased by 1. Is T still an MST? Explain why or provide a counter example.
3
Problem 2 (7 points)
Exam 2 In the Interval Partitioning problem we are given a set of n requests, where the
ith request starts at time si and finishes at time fi, find the minimum number of resources
needed to schedule all requests so that no two requests are assigned to the same resource at
the same time.
We propose the Earliest-Finish-Time-First (EFTF) algorithm to solve this problem. It
works as follows; first, we sort the requests in ascending order of finish time. Then we
consider requests in this order. We assign a request to one of the resources it’s compatible
with at random. If a request is incompatible with all resources being used, then we choose
a new resource to use for this request.
(a) (3 points) Give an example of jobs, such that the EFTF algorithm does not yield an
optimal solution (that is a minimum size list of resources).
Note: For this part of the problem it is sufficient to draw a picture of the intervals
(jobs) that are in your example as was done in our textbook. You should number the
jobs and use this numbering to specify which jobs are scheduled using ESTF as well
as a list of jobs which are optimal for this problem.
(b) (4 points) Prove that when all jobs are of equal length, then the Earliest-Finish-Time-
First algorithm does yield an optimal solution. (You may reference proofs from class
if you think they are relevant.)
4
Problem 3 (7 points)
A graph G(V,E) is called BIPARTITE if V can be decomposed into two disjoint sets U
and W such that every edge in E joins a vertex in U to one in W . (Thus there is no edge
between two nodes in the same set.) You can see an example in the picture on the right.
(a) (3 points) Observe the two graphs depicted in the picture. If you look at it carefully,
you can see that the two graphs are identical, just drawn in a different layout. You
can also clearly see from the drawing on the right that it’s bipartite.
Suppose we have a connected graph G(V,E) that we know to be bipartite, but don’t
know which of the two sets U and W each node belongs to. (e.g. you would see
something like the drawing on the left.) Design a simple greedy algorithm that assigns
the nodes to the two sets. (Hint: start from an arbitrary node v and assign it to set
U . Then consider the neighbors of v. ) Describe your algorithm and give a short
explanation why it is correct in assigning the vertices (we are not expecting a rigorous
formal proof here.)
(b) ( 1 point) Analyze the algorithm’s running time assuming it’s stored in an adjacency
list.
(c) (3 points) A cycle in a graph is even length if it consists of even number of nodes (and
of course the same number of edges). Based on the algorithm you designed in (a.),
prove that any cycle in a bipartite graph has to be of even length. (Hint: think of
which sets U or W each subsequent node along the cycle is in.)
5

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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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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 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