Download CS 173: Discrete Structures, Spring 2014 Homework 8

CS 173: Discrete Structures, Spring 2014
Homework 8
This homework contains 3 problems worth a total of 34 points. It is due on Wednesday,
April 16, at 11:59 PM.
1. Induction with Inequalities [12 points]
Use strong induction to prove that
1
23
+
1
33
+···+
1
n3
≤
5
8
−
1
n
for all integers n ≥ 2.
2. Smallest Multiple [16 points]
This problem is based on Problem 5 at http://projecteuler.net. It asks, “2520 is
the smallest number that can be divided by each of the numbers from 1 to 10 without
any remainder. What is the smallest positive number that is evenly divisible by all of
the numbers from 1 to 20?” We will analyze two algorithms which find the smallest
positive number that is divisible by all integers between 1 and n.
Hints: You may assume (without proof) that log n! ∼ n log n and log log n ≪ log n.
You can also use the fact that the prime number theorem gives an asymptotic approximation for the number of primes less than or equal to any real number x as lnxx .
(a) (6 points) The first algorithm is inspired by the idea that we’re looking for the
LCM of the set 2, ..., n. We have an easy method to calculate the LCM. Recall
that the Euclidean algorithm computes the GCD of two numbers and that
lcm(a, b) =
ab
.
gcd(a, b)
Euclid’s algorithm can “get lucky” and run very quickly on some inputs (e.g.,
a = 2b), but for many inputs, it is slower. A good approximation for how it
behaves in most cases is O(log(min(a, b))). Consider the following solution:
01 Foo(n)
02
lcm = 1
03
for i = 2 to n
04
lcm = lcm*i/Euclid(lcm,i)
05
return lcm
What is the big-O running time of Foo? Briefly justify your answer.
(b) (6 points) The second algorithm is inspired by prime factorization. We must
only keep track of the maximum prime exponents among the set. For example,
LCM(36, 15, 40) = LCM(22 · 32 , 3 · 5, 23 · 5) = 23 · 32 · 5 = 360. Computing the
prime factorization for each number 2, ..., n is slow. Instead, we can just go hunting for maximum powers. Consider the example of finding the smallest multiple
of 2, ..., 10. The prime numbers less than 10 are 2, 3, 5, 7. The maximum power of
1
2 that is less than 10 is 23 = 8. The maximum power of 3 that is less than 10 is
32 = 9. The maximum powers of 5 and 7 are 51 and 71 , respectively. Thus, the
smallest multiple of the whole set is then 23 · 32 · 51 · 71 = 2520.
We want one more piece. We only needed to check maximum powers of primes,
so we could use a list of primes to check. Assume we use the simple Sieve of
Eratosthenes1 , which you may assume has running time O(n log log n). Consider
the following pseudocode:
01 Bar(n)
02
A = new array of size n
03
lcm = 1
√
04
for i = 2 to n
05
if A(i)
06
for j = i2 , j not exceeding n, increment j by i
07
A(j) = false
08
lcm = lcm*floor(log(n)/log(i))
09
return lcm
Notice that line 06 can start at i2 because all earlier values are already crossed off,
since they are multiples of some number smaller than i.
Assume the language is such that loop in lines 06-07 will leave j as the highest
multiple of i which is less than or equal to n (some languages would leave it one
higher, and we would have to subtract).
What is the big-O running time of Bar ? Briefly justify your answer.
(c) (4 points) Which of the two algorithms is faster? Briefly justify your answer.
1
http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
2
3. Largest subarray [6 points]
For this problem, we will measure the size of a subarray by the sum of its elements.
For example, consider the array A=[5 -1 0 3 2]. The largest subarray of length 1 is [5].
The subarrays of length 2 are [5 -1], [-1 0], [0 3], and [3 2], the largest of which is [3,2].
The largest subarray of length 3 is [0 3 2]. The largest subarray of length 4 is [5 -1 0
3 2].
The following algorithm takes an array as input and returns the sum of its largest
subarray:
01 Baz(A)
02
big = −∞
03
for i = 1 to length(A)
04
for j = 1 to length(A) − i + 1
05
sum = 0
06
for k = j to j + i − 1
07
sum = sum + A(k)
08
if sum > big
09
big = sum
10
return big
For an input of length n, how many times is line 7 executed? Your answer should
include both an exact answer and a big-O answer, as well as a brief justification/work.
3