Term 2, 2021 COMP3121/9101: Assignment 2
You have five problems, marked out of a total of 100 marks; each
problem is worth 20 marks. You should submit your solutions by
Monday, July 12. Please do not wait till the last moment to submit
your work - we WILL NOT accept emailed solutions regardless of
whether you had network problems or not. Also follow closely the
instructions for how to submit your solutions. Your solutions to
each of the problems must be submitted in a separate file. There
are 1000+ students in this class and thus NO EXCEPTIONS can
be granted except for special considerations or for ELS students
who can get one week extension. Your solutions must be typed,
machine readable .pdf files. All submissions will be checked for
plagiarism!
1. You are given a sequence of n songs where the ith song is `i minutes
long. You want to place all of the songs on an ordered series of CDs
(e.g. CD1, CD2, CD3, . . . CDk) where each CD can hold m minutes.
Furthermore,
(a) The songs must be recorded in the given order, song 1, song 2,
. . ., song n.
(b) All songs must be included.
(c) No song may be split across CDs.
Your goal is to determine how to place them on the CDs as to minimize
the number of CDs needed. Give the most efficient algorithm you can
to find an optimal solution for this problem, prove the algorithm is
correct and analyze its time complexity.
2. At a trade school, there are N workers looking for jobs, each with a
skill level xi. There are P entry-level job openings, and the i
th opening
only accepts workers with a skill level less than or equal to pi. There
are also Q senior job openings, the i of which requires a skill level of at
least qi. Each worker can take at most one job, and each job opening
only accepts a single worker.
Your task is to determine the largest number of workers you can assign
to jobs in time O(N logN + P logP + Q logQ).
3. A city is attacked by N monsters and is defended by a single hero with
initial strength of S units. To kill a monster i the hero must dissipate
1
ai units of strength; if monster i is killed successfully the hero gains gi
units of strength. Thus, if the hero had strength c ≥ ai before tackling
monster i he can kill the monster and he will end up with c ? ai + gi
units of strength. Note that for some monsters i you might have gi ≥ ai
but for some other j you might have aj > gj. You are given S and for
each i you are given ai and gi. Design an algorithm which determines
in which order the hero can fight the monsters if he is to kill them all
(if there is such an ordering). In case there is no such ordering the
algorithm should output “no such ordering”. Assume all values are
positive integers.
4. You are given n stacks of blocks. The ith stack contains hi > 0 identical
blocks. You are also able to move for any i ≤ n ? 1 any number of
blocks from stack i to stack i + 1. Design an algorithm to find out
in O(n) time whether it is possible to make the sizes of stacks strictly
increasing. (For example, 1,2,3,4 are strictly increasing but 1,2,2,3 are
not). The input for your algorithm is an array A of length n such that
A[i] = hi. Note that you are not asked to actually move the blocks,
only to determine if such movements exists or not.
5. You are given n jobs where each job takes one unit of time to finish.
Job i will provide a profit of gi dollars (gi > 0) if finished at or before
time ti, where ti is an arbitrary integer larger or equal to 1. Only
one job can be done at any instant of time and jobs have to be done
continuously from start to finish. (Note: If a job i is not finished by
ti then there is no benefit in scheduling it at all. All jobs can start as
early as time 0.) Give the most efficient algorithm you can to find a
schedule that maximizes the total profit.