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CSC236H5 program

创建日期：2024-05-17 14:22:44

所属分类：计算机computer science

标签： CSC236H5

program

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CSC236H5

1. Correctness [12 marks]
Use the given python program to solve the questions below.
def mystery(L):
''' Precondition: L is a list of 0s and 1s'''
i = 0
j = -1
while i < len(L):
if L[i] == 0:
j += 1
L[i], L[j] = L[j], L[i]
i += 1
return j >= 0
A. [2 marks] What will the list L look like after the function mystery is called? What does the function
mystery return?
B. [6 marks] Prove the following loop invariant Inv for the while-loop:
Inv(i, j) =

−1 ≤ j < i ≤ len(L)
L[0:j+1] are all 0
L[j+1:i] are all 1
Note: The slicing operation L[x:y] is considered as the python slicing operation, thus it ranges from
L[x] to L[y-1].
Page 2 of 4
CSC236H5 - Fall 2022
University of Toronto Mississauga
Problem Set 3
Due: December 1st, 2022
C. [2 marks] State the postcondition for the function mystery. Then, use the invariant Inv to prove that
this postcondition holds if the loop terminates.
D. [2 marks] Prove that the loop terminates. Remember that this involves finding a variant and proving
two things: (i) that the loop body is guaranteed to decrease the variant; and (ii) that the loop-condition
and invariant together imply that the variant is a natural number greater than or equal to zero (i.e., ≥ 0).
2. Divide-and-Conquer [8 marks]
Consider the following three problems:
Subset
Given a set S of n integers, and a target value k, does S have a subset that sums to k.
Pair
Given a set S of n integers, and a target value k, does set S have a pair of elements that sum to k?
Tuple
Given sets X and Y of integers, and a target value k, is there an x ∈ X and a y ∈ Y such that x+ y = k?
You are tasked with the below questions. When computing the complexity, each binary operation costs
one unit of time. Additionally, you must always justify your answer.
A. [1 mark] Describe a brute force algorithm that solves the Subset problem. What is the worst-case time
complexity of this algorithm?
B. [1 mark] Describe a brute force algorithm that solves the Pair problem. What is the worst-case time
complexity of this algorithm?
C. [1 mark] Consider the following algorithm for the Pair problem:
• Step 1: sort S in ascending order, and let S = s1, s2, ...sn
• Step 2: calculate s1 + sn
• Step 3:
– if s1 + sn > k then the sum is too large, so shift sn down to sn−1 and go back to Step 2
– if s1 + sn < k then the sum is too small, so shift s1 up to s2 and go back to Step 2
– if s1 + sn = k return true
What is the worst-case time complexity of the algorithm given above?
Note: assume sorting takes Θ(nlog(n)) time.
D. [2 marks] Design an algorithm using the same method as the Pair algorithm above to solve the Tuple
problem. What is the worst-case time complexity of your algorithm?
Note: assume sorting takes Θ(nlog(n)) time.
E. [3 marks] Use a divide-and-conquer technique, in combination with your algorithm in step 4, to create
an algorithm that solves the Subset problem. What is the worst-case time complexity of your algorithm?

CSC236H5
University of Toronto Mississauga
Problem Set 3
3. Regular Expression [8 marks]
Consider a typical vending machine that dispenses a person’s choice of candy after it has received a total
of 25 cents in nickels (5 cents), dimes (10 cents), and quarters (25 cents). Consider a deterministic finite
automaton (DFA) where the state represents the amount of money the machine has received, and the actions
are the money being deposited into the vending machine.
A. [2 marks] Assume the vending machine consumes any over payment (i.e., you will not receive any
change back if you pay more than 25 cents). Explicitly model the system by defining Q (set of states),
Σ (alphabet), q0 (initial state), and F (final states). You can use variables n, d, and q to represent nickels,
dimes, and quarters.
B. [2 marks] Draw the corresponding DFA for your model (as described in A.).
C. [2 marks] Now, assume the machine is able to produce change. Once the machine receives 25 cents
(or more), you can no longer add coins to it. Now, the final state is represented as “25 with x cents
returned”. Explicitly model this new system by defining Q (set of states), Σ (alphabet), q0 (initial state),
and F (final states).

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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 MGMTMFE 405 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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 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