UBE 602
Algorithm Complexity Theory
Spring 2018
Instructor: Prof.Dr.Mehmet Emin DALKILIÇ
References :
·
Introduction to the Design and
Analysis of Algorithms by Anany Levitin,
Pearson-Addison-Wesley 2007
·
o Algorithm Design by Jon Kleinberg and Eva Tardos.
Addison Wesley, 2006.
( The bookseller
info: Izmir Tıp Kitabevi 162.Sokak No:15/D Bornova 35040 Tel: 0 232 3428361
http://www.izmirtip.com.tr/ , e-mail: mailto:info@izmirtip.com.tr)
Goals: To design and analysis of algorithms, prove correctness of
algorithms, to be exposed to the theory of complexity, NP completeness,
tractability, approximation and randomized algorithms
Prerequisites: Discrete Math, Data
Structures and Algorithms
Topics :
Part I : Design and Analysis of Algorithms
Part II : NP
Completeness
Approximate Grading
Final Exam June 6th, 2018 13:15
(Monday)
Makeup (Bütünleme)
Exam June 27th, 2018 13:15 (Wednesday) (Tentative)
Extra Slides 1 (demo Dijkstra)
Extra Slides 2 (demo Interval Scheduling)
Extra Slides 3 (Greedy Algorithms)
Extra Slides 5 (Randomization)
Extra Slides 7 (ReductionsPoly)
Homework 1 (due date: Feb 26th, 2018)
1. For each of the following functions,
indicate how much the function’s value will change if its argument is
increased threefold.
a) n log 2n b) n
c) n3 d) n2
e) n! f) 2n
2. For each of the following functions,
indicate the class θ(g(n)) the function belongs to. (Use the
simplest g(n) possible in your answer.) Prove your assertions.
a) (n3+1)6 b)
(10n4 + 7n2 + 3n)0.5
c) 2n lg(2n + 2)3 +
(n2 + 2)2 lg n d)
3n+1 + 3n-1
e) 2 log2 n
3. You are facing a wall that stretches
infinitely in both directions. There is a door in the wall, but you know
neither how far away nor in which direction. You can see the door only when you
are right next to it.
a) Design an algorithm that
enables you to reach the door by walking at most O(n) steps where n is the
(unknown to you) number of steps between your initial position and the door.
b) Prove that the time
efficiency (complexity) of your algorithm is O(n).
4. Algorithm GaussianElimination(A[0..n-1,0..n])
//Implements Gaussian
elimination of an n-by-(n+1) matrix A
for i ¬ 0 to n - 2 do
for j ¬ i + 1 to n - 1 do
for k ¬ i to n do
A[j,k] ¬ A[j,k] - A[i,k] * A[j,i]
/ A[i,i]
Find the efficiency class
and a constant factor improvement.
Homework 2 (due date: March 5thth, 2018)
1. Exercise 2.6.1, Exercise
2.6.2, Exercise 3.1.5, Exercise 3.3.4 and Exercise 3.3.5
Homework 3 (due date: March 12th,
2018)
1. Design a decrease-by-one algorithm for finding all the
factors of a given number n.
Also design a decrease-by-one algorithm for finding all the prime factors of a
given number n.
2.Consider the following algorithm to check connectivity of a graph
defined by its adjacency matrix.
ALGORİTHM Connected(A[0..n-1,0..n-1])
// Input: Adjacency matrix A[0..n-1,0..n-1]
of an undirected graph G
// Output: 1 (true) if G is connected
and 0 (false) if it is not
if n=1 return 1 // one-vertex graph is connected by
definition
else
if not Connected(A[0..n-2,0..n-2]) return 0
else for jß0 to n-2 do
if A[n-1,j] return 1
return 0
Does this algorithm work correctly
for every undirected graph with n>0
vertices? If you answer yes, indicate the algorithm’s efficiency class in the
worst case; if you answer no, explain why.
3.Consider the following implementation of
the algorithm for generating permutations discovered by B. Heap.
ALGORİTHM HeapPermute(n)
// Implements Heap’s algorithm for
generating permutations
// Input: A positive integer n and a global array A[1..n]
// Output: All permutations of
elements A
if n=1
write A
else
for iß1 to n do
HeapPermute(n-1)
if n is odd
swap A[1] and
A[n]
else swap A[i] and A[n]
a. Trace the algorithm by hand for n = 2, 3 and 4.
b. Prove the correctness of Heap’s algorithm.
c. What is the time efficiency of HeapPermute?