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COMP2111 algorithm Assignment

创建日期：2024-05-24 15:43:20

所属分类：计算机computer science

标签： COMP2111

algorithm Assignment

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COMP2111 Assignment

Submission is through give and should consist of:
1. A single pdf file with typeset answers, maximum size 4Mb. Prose should be typed, not handwritten.
Use of LATEX is strongly encouraged, but not required.
2. A file partner.txt with a single line on the format. z . This must be the zID of your
group partner. For individual submissions, write your own zID.
Submit your work using the web interface linked on the course website, or by running the following on a
CSE machine
give cs2111 assn3 assn3.pdf partner.txt
This assignment is to be done in pairs. Individual submissions are discouraged, but will be accepted. Only
one member of the pair should submit. Work out who will submit ahead of time—duplicate submissions
are extremely annoying to sort out.
Late submission is allowed up to 5 days (120 hours) after the deadline. A late penalty of 5% per day will
be deducted from your total mark.
Discussion of assignment material with others is permitted, but you may not exchange or view each
others’ (partial) solutions. The work submitted must be your own, in line with the University’s plagiarism
policy.
Background This assignment is an open-ended research task where you apply Hoare logic to an algo-
rithm of your choice.
This means you should start by choosing an algorithm! It’s your choice, but keep the following in mind:
• The algorithm should contain at least one loop.
• Your need to find an algorithm that makes the tasks below feasible in the given timeframe. If the
algorithm is longer than 10-20 lines or so, you should probably choose something simpler.
• You need an algorithm that you can meaningfully specify with a Hoare triple, and reason about with
Hoare logic.
What does the last bullet point mean exactly? I’ll give some examples. You want to avoid:
• Algorithms that are probabilistic, in the sense that they aren’t guaranteed to always give a correct
answer. An example is the Miller-Rabin primality test.
• Algorithms that give approximate answers only, such as numerical algorithms.
• Concurrent or parallel algorithms.
1
• Algorithms that have fuzzy specifications, where it’s difficult to characterize precisely what it means
for the output to be correct. Examples include most algorithms in machine learning and image
processing.
• Algorithms that require non-compositional control flow operators, such as goto, break or continue.
(But you can probably find an alternative way to formulate the algorithm that doesn’t require these).
A good place to start looking for algorithms is the table of contents of your favourite algorithms text-
book [CLRS09, Knu68, Knu69, Knu73, KT06], or the Wikipedia list of algorithms.1
Once you’ve found an algorithm, do ask your tutor, or Johannes, for input on your choice! We are happy
to offer advice on your choice.
Problem 1 (30 marks)
Present your algorithm, giving at least the following information:
1. The name of the algorithm
2. The origin of the algorithm. Is it known Who invented it? Cite the original paper that introduced the
algorithm, if possible.2 Also cite any other sources you’ve consulted in preparing this assignment.
3. The purpose of the algorithm. What problem does the algorithm solve? What are the requirements
that the algorithm is designed to meet?3
4. Give pseudocode for the algorithm.
5. A brief explanation of the pseudocode, to help the reader understand the algorithm.
Problem 2 (70 marks)
1. Give a vocabulary containing the necessary features to express your algorithm in L, explaining the
intended meaning of each symbol when it’s not obvious.
For algorithms involving numbers, you can use the lectures for inspiration. For algorithms involving
things beyond that, such as arrays, trees, or pointers, there will be design decisions for you to make.
(10 marks)
2. Express your algorithm in L using this vocabulary. (You may have already done so in Problem 1).
Where possible, use the existing primitives in L (possibly including derived operators) to express
the algorithm. This may involve some translation, such as by replacing for loops with while loops.
If your algorithm requires operators that are not expressible in L, you may add them to L provided
you can formulate a reasonable Hoare rule for it. For example, if your algorithm requires generating
random integers in an interval, something like this would do:
{∀I ∈ Z. e ≤ I ≤ e′ → ϕ[I/x]} x := Math.random(e, e′) {ϕ}
(10 marks)
3. Based on the requirements identified in Problem 1, write a formal specification for the algorithm, in
the form of a Hoare triple. Explain any requirements that your Hoare triple does not capture, and
why. (10 marks)
4. Annotate your program with appropriate loop invariants for any loops present in your algorithm.
(10 marks)
2For example, absent a groundbreaking archeological discovery, you may have difficulty finding the primary source for Eratos-
thenes’ sieve.
3Requirements here should be construed broadly. This includes functional requirements of the kind one can fit in Hoare triples
(“the output should be sorted”), as well as non-functional requirements (“the algorithm should run in O(n2) time”, “the algorithm
should run in-place”, “the algorithm should be easier to understand than the competition”, . . . )
3
5. Show that your annotations are correct, in the sense that they follow from the precondition, establish
the postcondition, and are re-established by every loop iteration.
This can take the form of an informal argument (for partial marks), or a complete annotation of the
program (for full marks). (20 marks)
6. If your algorithm is guaranteed to terminate, find a loop variant (aka measure) for every loop in
your program. If it is not, explain why. (10 marks)
The mark distribution and criteria for the above sub-questions is indicative only, and may be adjusted
depending on your choice of algorithm. In particular:
• If you chose a more difficult algorithm, expectations on completeness and formality decrease some-
what. (Note that a shorter algorithm is not necessarily easier to verify than a long one.)
• A failure to solve a problems may still yield marks. For example, a compelling description of the
difficulties you encountered when attempting to formulate an invariant.
Assorted advice
Arrays
Many algorithms require arrays. Adding arrays to L is just a matter of adding array indexing etc. to
the vocabulary. To save you some time though, here is a tempting, but unsound, way to formalise array
updates:
{ϕ[e/x[i]]} x[i] := e {ϕ} UNSOUND RULE DO NOT USE
The issue here is that ϕ can make claims about x[i] without mentioning x[i] directly. For example, the
following is of course not a valid Hoare triple: {x is sorted} x[0] := 17 {x is sorted} — but it is provable
with the incorrect rule above.
The solution is to model array updates as updates to the array as a whole. That is, model x[i] := e as
x := x[i 7→ e], where the RHS reads “the array x, but with index i updated to hold the value e. With that
change, array updates can be handled by the standard assignment rule.
This illustrates the more general point that the Hoare logic assignment rule is specifically designed for
situations where the LHS is a variable, not a compound expression.
Pointers
Pointers are tricky, and this assignment is no exception.
If you’re feeling adventurous enough to do a program with non-trivial use of pointers, you’ll need a
slightly different notion of state than we’ve used in the course so far. We’ve mostly been content to let
states be mappings from variable names to values; for pointer programs this will fail to capture the effects
of aliasing, where updating the value pointed to by x may change the value pointed to by y indirectly.
My recommendation is to let the state have two components: the first is the standard mapping from
variable names to values, but with the minor twist that values can be memory addresses. The second
component is the memory, which is a mapping from memory addresses to values.
4
Scope
If your algorithm turns out to be too complicated to verify, one way to reduce the scope is to axiomatise
auxiliary functions. By this, I mean specifying auxiliary functions with appropriate Hoare triples, and
then using those Hoare triples (without proof) in the correctness proof for the main function.
As an example, if you’re verifying merge sort, you might axiomatise an auxiliary merge function.
This may feel a bit like cheating, but it’s done all the time in real-world verification projects: somewhere,
you need to draw the boundary between what’s verified and what isn’t. Even if some code is left out of
scope, the verification is still valuable: it tells you exactly where you need to beware of bugs.

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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 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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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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