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COMP4600/8460 - Advanced Algorithms
创建日期：2024-05-13 16:04:13
标签： COMP4600
Advanced Algorithms

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COMP4600/8460 - Advanced Algorithms

Assignment 3
[Your Name] ([Your UID]) Date: August 31, 2022
Note: Plagiarism is strictly prohibited. Even though you may discuss with classmates, you should write your
homework assignment by yourself.
1 Questions
Q1) (5 Marks) Let Sn be a binomial random variable BIN(n, p)
Apply Chernoff bound to prove for any 0 < δ < 1 that
P
(
Sn ≥ (1 + δ)np
) ≤ e−δ2np3
P
(
Sn ≤ (1− δ)np
) ≤ e−δ2np2
Q2) (5 Marks) There are n balls thrown independently and uniformly at random into n bins
1. Show the probability that a particular bin receiving at least M balls is at most
(
n
M
)
( 1n )
M
2. Show the probability that one of the n bins has at least M balls is at most n( e
M
MM
)
Note: Do not use Poisson approximation
Q3) (5 Marks) Given a set of strings S = {s1, ..., sn}, map each string to a bit-string using m bits by an
ideal hashing function, such that each bit has an equal probability of being 0 or 1. Then we use the
set of bit-strings B = {b1, ..., bn} to test if a string is a member of S or not
1. Show m = Ω(log n) bits is necessary for the probability of a false positive to be lesser than 1
2. Show m = O(log n) bits is sufficient for the probability of a false positive to be at most 1m
Q4) (2.5 Marks) Let Sn be BIN(n, p). Prove the tail probability is bounded by P(Sn ≥ enp) ≤ e−np
Q5) (2.5 Marks) Let X be the number of draws required to collect all n types of coupons. Prove, for any
constant c,
lim
n→∞P(X > n lnn+ cn) = 1− e
−e−c
You can use Poisson approximation
Q6) (10 Marks) Simulate balls-and-bins model and power of d-choices model, and compare their empirical
maximum loads under various values of m (≤ 10000), n (≤ 10000) and d (≤ 10). Implement your
simulation in Python/Java/C. Plot and discuss the results here in this report, and upload your code
separately on Wattle
1
2 Assignment 3: [Your Name]
2 Answers
. . .
References
[MU] Mitzenmacher & Upfal, Probability and Computing, Cambridge University Press, (2017).
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EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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