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Algorithms 3027/3927
创建日期：2024-04-29 14:54:56
标签： COMP3027
Algorithms

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Algorithms 3027/3927 Assignment 

This assignment is for COMP3027 only.
Grattis! You’ve been hired to assist the packaging operations at Ikea. Your first task is to evaluate a
bid from a box manufacturer. Roughly speaking, the ideal box manufacturer is one whose boxes can fit as
many Ikea products as possible. In particular, you are given a list P of n products p1, . . . , pn where product
pi has length length(pi) and width width(pi), and a list B of m boxes b1, . . . , bm where box bj has length
length(bj) and width width(bj). We say that product pi fits into box bj if length(pi) ≤ length(bj)
and width(pi) ≤ width(bj). (Unfortunately, the packing machine is unable to rotate products due to
damage from the Great Flood of 2022, so no rotations are allowed.) The total fit is the total number
of product-box pairs (pi, bj) such that pi fits in bj . Note that this is different from the total number of
products that can fit in at least one box. The goal is to compute the total fit.
Figure 1: p1 and p2 fit in box b1; p2 fits in box b2 so the total fit is 3.
Note that there may be multiple products and/or boxes that share the same length and/or width.
Task 1: Keep It Simple, Single-Dimensional [5 points]
We start with the simple case where products and boxes are one-dimensional, i.e. they only have a
length. (Believe it or not, thinking about this is actually useful for the other parts of the assignment.)
Here, we are given a list P of n products p1, . . . , pn where product pi has length length(pi), and a list
B of m boxes b1, . . . , bm where box bj has length length(bj). We say that product pi fits into box bj if
length(pi) ≤ length(bj). The total fit is the total number of product-box pairs (pi, bj) such that pi fits
in bj .
Your task is to design an algorithm that computes the total fit in O(N logN) time where N = m+n
and implement your algorithm on Ed. (You are not required to submit a description of the algorithm
and proof of correctness and time complexity, but you should also practice describing your algorithm in
plain English and proving correctness and time complexity as it will help with Task 2.)
Task 2: Two-Dimensional [95 points]
We can think of the products and boxes as points in two-dimensional space with length as the x-coordinate
and width being the y-coordinate. Thus, it is natural to divide the input using a horizontal or vertical
line, i.e. divide the products and boxes according to their width or length, respectively. Before deciding
on exactly what the line should be, it is useful to design the combine step.
(a) Combine step In this subtask, we will design the combine step subroutine assuming that the
divide step divides products and boxes into those whose length are at most D and those that are
more than D, for some D. Formally, the inputs to the combine step are:
1
• 4 lists PL, PR, BL, BR. The list PL consist of products pi with length(pi) ≤ D and PR consist
of products pi with length(pi) > D. Ditto for BL and BR. Moreover, each list is sorted in
ascending order of width,
• FL, the total fit of products PL with boxes BL, i.e. the number of pairs (pi, bj) where pi is in
PL and bj is in BL and pi fits in bj .
• FR, the total fit of products PR with boxes BR, i.e. the number of pairs (pi, bj) where pi is in
PR and bj is in BR and pi fits in bj .
The output of the combine step is the total fit of products in both PL and PR with all boxes in
BL and BR, i.e. the number of pairs (pi, bj) where pi is in PL or PR, bj is in BL or BR, and pi fits
in bj . Your task is to design an O(N) time algorithm for the combine step, where N is the total
number of products and boxes, i.e. N = |PL|+ |PR|+ |BL|+ |BR|. (Note: You may choose to solve
a different version of the combine step. If you choose to do so, make sure you clearly define what
the inputs and outputs of your combine step are.)
(i) Description of how your algorithm works in plain English.
(ii) Prove that your algorithm is correct.
(iii) Prove an upper bound on the time complexity of your algorithm.
(iv) Implement your algorithm on Ed
(b) Use the combine step algorithm above to construct a divide-and-conquer algorithm for the problem.
For full marks, your algorithm should take time O(N logN) where N = m + n, the total number
of products and boxes.
(i) Description of how your algorithm works in plain English. Make sure you describe
• your pre-processing step (if needed)
• the problem your recursive algorithm is solving, i.e. its input and output.
• your divide step (subproblems, base cases)
• your delegate step
• how you use the combine step subroutine
(ii) Prove that your algorithm is correct
(iii) Prove an upper bound on the time complexity of your algorithm
(iv) Implement your algorithm on Ed
Guidelines
To make it easier for you to write and for us to mark, you can simply assume the following without
further explanation/proof when describing/analyzing your algorithm.
• Computing the min or max of an unsorted list of n values takes O(n) time. You can simply write
“Let A be the max of the input list”. Do not tell us the for loop to do this.
• Sorting a list of n values takes O(n log n) time. You can simply write “Sort list L in ascending order
of blah” and assume it takes O(n log n) time without specifying what sort you are using. Please do
not re-write or re-prove mergesort or any other sorting algorithm.
• Check whether a product pi fits in a box bj in O(1) time. You can simply write “If pi fits in bj then
blah” instead of writing “if length(pi) ≤ length(bj) and width(pi) ≤ width(bj) then blah”.
Submission details
• Submission deadline is Wednesday 23 March, at 23:59. Late submissions without special
consideration will be subject to the penalties specified in the first lecture (5% per day). Submis-
sions later than Friday 25 March, 23:59 will not be accepted.
• Submit your answers as a single document to Gradescope. Your work must be typed (no images of
text, although you can use diagrams if you think it helps.) Please try to be reasonably concise (1-2
pages of a4 for 3027 and 2-3 for 3927).
2
• The implementations should be done in Ed, and submitted via Ed.
• Your report will be subject to automatic and manual plagiarism detection systems. Remember, it’s
acceptable to discuss high level ideas with your peers, but you should not share the detail of your
work, such as parts of the precise algorithms, examples, proofs, writing, or code.
• To facilitate anonymous grading, please do not write your name on your submission.
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EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 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 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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 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