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MATH367 Homework

创建日期：2024-04-22 17:34:53

所属分类：数学mathematics

标签： MATH367

Homework

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MATH367 Homework # 1

Problem 1 (13 points): Some company has produced four samples of
a new COVID vaccine, which have code names of A, B, C, and D. The
company needs to test these samples at hospitals in Liverpool, Manch-
ester, Nottingham and Plymouth in such a way that no two samples can
be tested in the same hospital, and no two hospitals can test the same
sample. The costs of testing at different hospitals are (in thousands of
pounds):
• Liverpool: Sample A - 88, Sample B - 76, Sample C - 45, Sample D
- 23
• Manchester: Sample A - 7, Sample B - 68, Sample C - 86, Sample
D - 8
• Nottingham: Sample A - 30, Sample B - 69, Sample C - 57, Sample
D - 32
• Plymouth: Sample A - 24, Sample B - 51, Sample C - 52, Sample
D - 55
We want to find the assignment of samples to hospitals which mini-
mizes the total cost of testing.
Questions:
1. Write down a cost matrix for this problem (2 points)
2. Draw a cost flow network representation for this problem (2 points)
3. Give the interpretation of a minimal element of the cost matrix in
terms of paths in a flow network (2 points)
4. Now apply the most suitable algorithm from Chapter 4 to find the min-
imal cost assignment of samples to hospitals. Explain the condition
that is used to determine when the algorithm ends. (7 points)
MATH367 Homework # 1
Problem 2 (12 points): We consider one possible algorithm for find-
ing minimal distances from the starting vertex s to all other vertices in a
weighted graph without negative-weight cycles. w (u, v) is the edge weight
for an edge joining vertices u and v. Like in Dijkstra’s algorithm, with each
vertex v in a graph we associate:
• a length label li (v), which will eventually converge to the minimal
distance d (s, v) from s to v, and i enumerates the iterations of the
algorithm,
• a vertex pi (v) which will be the vertex previous to v on the minimal-
weight path from s to v .
We will denote the total number of vertices in a graph by |V |.
The algorithm comprises the following steps, again resembling Dijk-
stra’s algorithm:
Initialization: Set l0 (s) = 0 and l0 (v) = +∞ for all vertices v 6= s.
Main iteration: Repeat the following steps for i = 1 . . . |V | − 1 (that is,
|V | − 1 times):
• For each edge (uv) in a graph
• If li−1 (u) + w (u, v) < li−1 (v)
– Set li (v) = li−1 (u) + w (u, v)
– Set pi (v) = u
• Otherwise set li (v) = li−1 (v).
End: When the main iteration is finished, the labels l|V |−1 (v) contain min-
imal distances d (s, v) from s to v, and the labels p|V |−1 (v) can be
used to reconstruct the minimal-weight paths from any vertex v to s,
like in Dijkstra’s algorithm
MATH367 Homework # 1
Figure 1: An example of a graph with negative edge weights.
Questions:
1. What is the complexity of this algorithm in large-O notation, ex-
pressed in terms of the number of vertices |V | and the number of
edges |E|? (2 points)
2. Apply this algorithm to the graph shown on Fig. 1 and check whether
it produces correct minimal distances from vertex A to all other ver-
tices (Dijkstra’s algorithm fails for this graph with negative-weight
edge) (3 points)
3. For a weighted graph G, let us consider a set Pi of paths which in-
clude not more than i edges. Let di (s, v) be the minimal path weight
among those paths in Pi which start at a vertex s and end at a ver-
tex v, if such paths exist. If it is not possible to connect s and v by
a path with not more than i edges we define di (s, v) = +∞. Prove
that after the i’th step of the main iteration in the above algorithm,
li (v) = di (s, v) for any vertex v in G. Use induction starting with
i = 0, when only l0 (s) = 0 < ∞, which is the weight of zero-length
path from s to itself. (3 points)
4. Prove that any path (that is, a walk which passes through all vertices
and edges not more than once) in a graph with |V | vertices includes
not more than |V | − 1 edges. (2 points)
5. Prove that if there are no negative-weight or zero-weight cycles in
a weighted graph, the minimal-weight walk between any two distinct
vertices is a path (that is, a walk which passes through all vertices
and edges not more than once) (2 points)

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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 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