CSE 431/531 Homework 1
Due date: Feb 22, 2016
Lecturer: Shi Li
Your Name:
Your University ID:
Problems
Max. Score
Your Score
1
7
2
7
3
6
4
6
5
6
6
8
Total Score
40
Problem 1 (7 points). For each pair of functions f, g in the following table, indicate whether
f is O, o, Ω, ω or Θ of g. Assume that k ≥ 1, > 0 and c > 1 are constants. Write down “yes” or
“no” in each box. Give a proof for your answer for (row (d), column Ω).
(a)
(b)
(c)
(d)
f
lgk n
2n
√
n
lg(n!)
g
n
2n/2
nsin n
lg(nn )
O
o
Ω
ω
Θ
1 point for each row; 3 points for the proof.
Problem 2 (7 points). Assume f (n) and g(n) are asymptotically positive functions. Indicate
whether the following statements are true or false. Give proofs for (e) and (f).
(a) f (n) = O(g(n)) implies 2f (n) = O(2g(n) ).
true / false
(b) f (n) = O((f (n))2 ).
true / false
(c) If f (n) = o(g(n)), then f (n) = O(g(n)).
true / false
(d) f (n) + g(n) = Θ(n) implies f (n) = O(n).
true / false
(e) f (n) + g(n) = Θ(n) implies either f (n) = Θ(n) or g(n) = Θ(n).
true / false
(f) f (n) + o(f (n)) = Θ(f (n)).
true / false
0.5 points for each “true/false” answer; 2 points for each proof.
Problem 3 (6 points). Consider the interval scheduling problem given by a set S = {1, 2, · · · , n}
of activities, each activity i has a starting time si and finish time fi . Decide whether each of the
following two greedy strategies is safe of not. If the answer is yes, give a proof; if the answer is no,
give a counterexample.
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1. Select the job i ∈ S with the biggest length (the length of job i is fi − si ). If i is incompatible
with some other job in S, then we do not schedule i; otherwise we schedule i.
2. Select the job i ∈ S with the smallest si (i.e, earliest starting time). If there is a job j ∈ S \{i}
such that [sj , tj ) ⊆ [si , ti ), then we do not schedule i; otherwise we schedule i.
3 points for each strategy.
Problem 4 (6 points). Let I = (k, n, T, (p1 , p2 , · · · , pk ), (r1 , r2 , · · · , rT )) be an optimal caching
instance, where
• k is the number of pages that can be held in the cache,
• n is total number of pages,
• T is the length of the request sequence,
• p1 , p2 , · · · , pk ∈ [n] ∪ {⊥} are the k pages held in the cache initially(⊥ means an empty page),
• rt ∈ [n] is the page requested at time t.
Let I 0 = (k, n, T, (p01 , p02 , · · · , p0k ), (r1 , r2 , · · · , rT )) be another instance such that p0i = pi for every
i = 2, 3, · · · , k. Prove the value for I 0 (i.e, the total number of misses in an optimum solution to
I 0 ) is at most the value for I plus 1.
Remark In a valid solution, if a requested page is in the cache, we can not bring any page to the
cache; if a requested page is not in the cache, we can only bring the requested page to the cache.
Problem 5 (6 points). Design a data structure that maintains a set S of elements, each element
v ∈ S with an associated value key[v], and supports the following two operations:
• median: return the element whose key is the b(n + 1)/2c-th smallest among all elements in
S, where n = |S|;
• add(v): add the element v to S.
The running time for each operation should be O(lg n). (Hint: you can use the heap data structure
as a black-box.)
Problem 6 (8 points).
S In the interval covering problem, we are given n intervals [s1 , t1 ), [s2 , t2 ),
· · · , [sn , tn ) such that
i∈[n] [si , ti ) = [0, T ). The goal of the problem is to return a smallest-size set
S
S ⊆ [n] such that i∈S [si , ti ) = [0, T ). Design a greedy algorithm for this problem.
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