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CIT 596 random variable

创建日期：2024-06-01 17:10:04

所属分类：数学mathematics

标签： CIT 596

random variable

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CIT 596
Midterm 1 Practice Problems
These problems are meant to help you prepare for the first midterm. They may not all be reflective
of the difficulty of the problems on the exam.
1. Solve each of the recurrences below and express T (n) in asymptotic notation. State the
method used to derive answer. In each case assume that T (1) = 1.
(a) T (n) = T (3n/4) + T (n/4) + n.
(b) T (n) = 6T (n/2) + n3.
(c) T (n) = T (n/2) + log n.
2. Let A and B be two arrays (not necessarily sorted) containing n elements each. Suppose we
are given a target T . Describe an O(n log n) algorithm to determine if there are elements A[i]
and B[j] that add up to T .
3. Suppose we have an array of numbers a1, . . . , an with the property that successive entries of
the array differ by at most 2. In other words |ai − ai+1| ≤ 2. In addition we are told that a1
is negative and an is positive.
(a) Prove that at least one of 0 and 1 must be present in the array.
(b) Design an efficient algorithm to determine the index at which 0 or 1 occurs. (If there
are multiple occurrences of 0 or 1 you just have to find one.)
4. Use the comparison-tree model to find an exact (not asymptotic) lower bound on the number
of comparisons required to merge a sorted list of n elements with a sorted list of 2 elements.
5. Recall that a modal element in an array of n elements is an element that appears more than
n/2 times. Suppose you are given an array containing a modal element. Design a simple
randomized algorithm to determine the modal element that runs in expected O(n) time.
6. Let A[1..n] be an array of n positive integers. For any 1 ≤ i ≤ j ≤ n, define
f(i, j) =
j∑
k=i
A[k].
(a) Describe an O(n)-time algorithm that creates a data structure such that, for any 1 ≤
i ≤ j ≤ n, f(i, j) can be evaluated in constant time using this data structure. Justify
your answer.
Midterm CIT 596 2
(b) Describe an algorithm that on input A[1..n] and a number K, determines whether there
exists a pair (i, j) such that f(i, j) = K. Your algorithm should run in time o(n2). (Note
that this is little “o”.)
7. For an odd number of n distinct elements x1, x2, . . . , xn with positive weights w1, w2, . . . , wn
such that
∑n
i=1wi = 1, the weighted median is the element xk satisfying:
k−1∑
i=1
wi < 1/2 and
n∑
i=k+1
wi ≤ 1/2
(a) Show how to compute the weighted median of n elements in O(n log n) worst-case time
using sorting. No proof of correctness is needed.
(b) Show how to compute the weighted median in O(n) worst-case time using the linear-
time median-finding algorithm covered in class. You can use results from class without
proof, and no proof of correctness is needed for this part.
8. Suppose we want to maintain a balanced binary search tree that handles the usual dictionary
operations of insert, delete, and find. In addition, we want to answer interval queries of
the following form: Given an integer a, output the number of elements in the tree, that are
greater than or equal to a. (Note that a itself may or may not occur in the tree.) How would
you modify the tree to answer these questions efficiently? Design and analyze the algorithm
for handling an interval query. Show that you can maintain this modified binary search tree
as you insert and delete elements into it and rotate it to rebalance the tree.
9. In a min heap we want to answer the following query: Given k and x, is the kth-smallest
element in the heap less than x? Design a O(k) algorithm for solving this problem. Prove
that your algorithm is correct and analyze its running time.
10. Give short answers to the following questions:
(a) To use hashing when we want to be able to insert and delete elements, what is the best
collision resolution scheme?
(b) Define an indicator random variable.
(c) The best divide-and-conquer algorithm for the maximum-sum-subinterval selection prob-
lem takes time Θ(n log n). Does it follow that a lower bound for this problem is
Ω(n log n). Justify your answer.
(d) Suppose we are doing QuickSelect where we are searching for the 100th-smallest element
in an array. What is the probability that the smallest element is compared to the second
smallest element?

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关键词一关键词二关键词三关键词四关键词五和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 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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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