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Computer Science 331

创建日期：2024-04-01 14:56:20

所属分类：经济Economics

标签： Science 331

Computer Science

Computer Science 331

Instructions: • Answer questions in the space provided in this examination booklet. The last two pages may be used to continue answers if you run out of space. • Answer ALL questions. See the more detailed grade breakdown on the next page for additional information about what happens if you answer additional questions. • Asymptotic notation can be used to state bounds on running times — but you must use this notation correctly, and as precisely as possible. Computer Science 331 cont’d. ID: Page 2 of 16 Question Score Out Of 1. Operations on Binary Search Trees 10 2. Red-Black Trees 10 3. Hash Tables 10 4. Searching in Sorted Arrays 10 5. Quick Sort 5 6. Heaps and Heap Sort 15 7. Graphs and Graph Algorithms 15 Total: 75 Breathe!!! RELAX!!!!! Then, please do the following. 1. Read all of a question carefully before you begin to answer it. Ask about it if the instructions for a question are confusing or unclear. 2. Then do your best. Computer Science 331 cont’d. ID: Page 3 of 16 1. Consider the following binary search tree T. 12 10 86 3 1 (5 marks) (a) Describe, in your own words, a method to insert a value into a binary search tree. Then draw the tree that would be produced by inserting 7 into the tree T: Your Description of an Insertion Algorithm: Tree Obtained by Inserting 7 into T: Computer Science 331 cont’d. ID: Page 4 of 16 (5 marks) (b) Suppose that you wish to delete a value that is stored at a node with two children. Describe how this should be done. Then draw the binary search tree that would be produced after deleting 6 from T: 12 10 86 3 1 How To Delete a Value at a Node with Two Children: Tree Obtained by Deleting 6 from T: Computer Science 331 cont’d. ID: Page 5 of 16 2. Consider a red-black tree. (2 marks) (a) Recall that the height height(x) of a node x is the maximum number of nodes on a path from x down to a leaf (not including x), and that the black-height black-height(x) of x is the number of black nodes on a path from x down to a leaf (not counting x). Explain why height(x) ≤ 2 × black-height(x) for every node x in a red-black tree. Why the Height is Never More Than Twice the Black-Height: (2 marks) (b) Suppose that T is a red-black tree with size n. Describe, as precisely as you can, how the depth of this tree is related to its size. Relationship between Depth and Size for a Red-Black Tree; Computer Science 331 cont’d. ID: Page 6 of 16 (6 marks) (c) Consider the deletion of a value from a red-black tree T when this value is in the tree. Recall that some node y (with at most one non-NIL child) is removed and that a child x of y gets promoted to replace it. Describe • the colour given to x when it is promoted (and how this depends on the colour of y) • the problem that this might cause, and • (somewhat briefly and informally) how this problem is corrected. Colour Given to x if y was Red: Colour Given to x if y was Black: Problem(s) That Might be Caused by This: How This is Fixed: Computer Science 331 cont’d. ID: Page 7 of 16 3. Consider the use of hash tables to store finite sets. (3 marks) (a) First consider hashing with chaining, using table size m = 10 and using the following hash function to store integers: h(x) = x (mod 10). Draw the hash table that would be produced by inserting the values 0, 20, 10, 30, 5 (in this order) into an initially empty table. Resulting Hash Table: (3 marks) (b) Repeat the above question, using hashing with open addressing. In this case you should use a hash function such that h(x, 0) = x (mod 10) and such that h(x, i) = x + i2 (mod 10) for every integer i such that 1 ≤ i ≤ 9 — so that quadratic probing is being used. Resulting Hash Table: Computer Science 331 cont’d. ID: Page 8 of 16 (2 marks) (c) Next draw the hash table that you would get by starting with the hash table you produced when answering part (b) and deleting 20. Hash Table with Open Addressing Obtained by Deleting 20: (2 marks) (d) Describe briefly why it is not generally a good idea to use hash tables with open addressing if deletions are common. Why You Should Avoid Hash Tables with Open Addressing When Deletions are Frequent: Computer Science 331 cont’d. ID: Page 9 of 16 4. Consider the problem of searching in a sorted array. (5 marks) (a) Describe, IN YOUR OWN WORDS, the binary search algorithm. Recall that this algorithm • accesses as global data, but does not change, an array A that has positive length and stores elements of an ordered type T in nondecreasing order, as well as a key k of the type of values stored in the array; • receives integers low and high such that 0 ≤ low ≤ A.length and 1 ≤ high ≤ A.length 1 as input. If low ≤ high and there exists an integer i such that low ≤ i ≤ high and A[i] = k, then an integer i with these properties is returned as output. A NoSuchElementException is thrown, otherwise. Your Description of the Binary Search Algorithm: (3 marks) (b) Consider the following sorted array. 1 3 5 9 16 30 0 1 2 3 4 5 6 7 25 40 List the sequence of values that the key k = 30 is compared to, during a search with this array, and with inputs low = 0 and high = 7. Values Visited: (2 marks) (c) Which of the algorithms — Binary Search or Linear Search — should you use if you wish to minimize the number of steps used in the worst case to search for a value in a large sorted array? Why? Which Algorithm Should be Used — and Why: Computer Science 331 cont’d. ID: Page 10 of 16 5. Consider the Quick Sort algorithm. (2 marks) (a) The deterministic version of this algorithm, presented in class, always used the last element in the part of the array being sorted as the partition element (or the “pivot element”) when partitioning the array. Assuming that this algorithm is being used to sort an array with length n, give an upper bound for the number of steps used to sort the input array, in the worst case, as a function of n, that is as precise as you can. Upper Bound for Number of Steps Used: (3 marks) (b) Describe a way to modify the algorithm so that its asymptotic worst-case performance is improved. What is the worst-case performance of the new version of the algorithm? Modification of the Algorithm: New Worst-Case Performance: Computer Science 331 cont’d. ID: Page 11 of 16 6. Consider Heaps and the Heap Sort algorithm. (7 marks) (a) Recall that the algorithm to delete the largest value from a Maxheap begins by reading and storing the value at the root, so that it can later be returned as output. It then overwrites this value with the value at a leaf. This leaf is then deleted from the Maxheap. Unfortunately, the resulting structure is not generally a Maxheap. Explain, IN YOUR OWN WORDS, why this is true and what the algorithm does to turn the resulting tree into a Maxheap, once again. Finally, state the number of steps executed by this algorithm, in the worst case, as a function of the size n of this Maxheap. Why This is Not Generally a Maxheap: What the Algorithm Does to Produce a Maxheap: Number of Steps Used in the Worst Case: Computer Science 331 cont’d. ID: Page 12 of 16 Now consider the following Maxheap H: 10 8 7 1 5 3 (2 marks) (b) Fill in the entries of the array representation of this Maxheap: Resulting MaxHeap and Heap Size: 0 1 2 3 4 5 6 7 heap-size = (3 marks) (c) Draw the Maxheap that would be produced by deleting the maximal element from H. MaxHeap After Deletion: (3 marks) (d) Describe, IN YOUR OWN WORDS, how the Heap Sort uses the insert and deleteMax operations, for a Maxheap, to sort an array. Description of Heap Sort: Computer Science 331 cont’d. ID: Page 13 of 16 7. Consider the following weighted graph G = (V, E):4 1 2 035 5 1 1 1 1 3 2 (3 marks) (a) Draw the adjacency list representation of G. Adjacency List Representation: (3 marks) (b) Draw the adjacency matrix representation of G. Adjacency Matrix Representation: Computer Science 331 cont’d. ID: Page 14 of 16 (2 marks) (c) State the number of steps used, in the worst case, to decide whether a given pair of vertices are neighbours, using each of the above representations of G. This might depend on the size of either V or E. Number of Steps Used with Adjacency List Representation: Number of Steps Used with Adjacency Matrix Representation: (3 marks) (d) Give the definition of a minimum cost spanning tree of a connected weighted graph G = (V, E). Definition of a Minimum-Cost Spanning Tree: (3 marks) (e) Draw a minimum cost spanning tree of the graph G = (V, E) shown on the previous page. Minimum Cost Spanning Tree: (1 mark) (f) Finally, state the number of steps used, in the worst case, to compute a minimum cost spanning tree for a weighted graph G = (V, E) using Prim’s algorithm. You may assume that an adjacency list representation has been given for G.

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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 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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