UBE 602
Algorithm Complexity Theory
Spring 2014
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 Tip 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, Design and Analysis of Algorithms
Topics :
Part I : Design and Anaysis of Algorithms
Part II : NP Completeness
Approximate Grading
Homework 1 (due date: March 10th, 2014)
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 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 17th, 2014)
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 24th, 2014)
1. 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. 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. 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?
4. a. Outline an algorithm for finding the smallest key in a binary search tree. Would you classify your algorithm as a variable-size decrease algorithm?
b. What is the time efficiency class of your algorithm in the worst case?
Homework #4 (Due date: March 31st, 2014)
1. Show that QuickSort
runs in O(n log n) time in average for random arrays.
2. ALGORITHM Height(T )
//Computes recursively the height of a
binary tree
//Input: A binary tree T
//Output: The height of T
if
T = ∅ return −1
else
return max{Height(Tlef t ), Height(Tright)} + 1
Analyze the best-case, worst-case and
average case time efficiency of the above recursive algorithm.
3. a. For the one-dimensional version of the
closest-pair problem, i.e., for the
problem of finding two closest numbers
among a given set of n real numbers,
design an algorithm that is directly based
on the divide-and-conquer
technique and determine its efficiency
class.
b.
Is it a good algorithm for
this problem?
Homework 5 (due date: April 7th, 2014)
Problems 6.1.10, 6.4.4, 6.4.6, and 6.4.7 typedhomeork5
Homework 6 (due date: April 14th, 2014)
1. Given two (max) heaps in the form
of Arrays A and B, design an efficient algorithm to create
a single (max) heap C
a) Using O(n) extra memory where A and
B are of size O(n)
b) Using O(log n) extra memory where A
and B are of size O(n)
Cleary explain your algorithms, give
pseudocodes and time complexity analysis.
Homework #7 (Due date: April 28th, 2014)
1.
(World Series odds) Consider
two teams, A and B, playing a series of games
until one of the teams wins n games. Assume that the probability of A winning
a game is the same for each game and equal
to p, and
the probability of
A losing a game is q = 1− p. (Hence, there are no ties.) Let P(i, j) be the
probability of A winning the series if A needs i more games to win the series
and B needs j more games to win the series.
a.
Set up a recurrence
relation for P(i, j) that can be used by a dynamic
programming algorithm.
b.
Find the probability of
team A winning a seven-game series if the probability
of it winning a game is 0.4.
c.
Write pseudocode of the
dynamic programming algorithm for solving this
problem and determine its time and space
efficiencies.
2.
a. Write
pseudocode of the bottom-up dynamic programming algorithm for
the knapsack problem.
b.
Write pseudocode of the
algorithm that finds the composition of an optimal
subset from the table generated by the
bottom-up dynamic programming
algorithm for the knapsack problem.
3.
For the bottom-up dynamic
programming algorithm for the knapsack problem,
prove that
a.
its time efficiency is _(nW).
b.
its space efficiency is _(nW).
c.
the time needed to find the
composition of an optimal subset from a filled
dynamic programming table is O(n).
4.
a) Show that the time efficiency of solving the coin-row problem by
straightforward
application
of recurrence (8.3 given below) is exponential.
F(n)
= max{cn + F(n-2), F(n-1)} for n > 1
F(0)
= 0, F(1) = c1
b)
Show that the time efficiency of solving the coin-row problem by exhaustive
search
is
at least exponential.
Homework 8 (due date: May 5th, 2014)
Problems 9.1.4, 9.2.2, 9.4.10, and
an “out of the book” question
typedhomework8
Homework #9 (due
date: May 12th, 2014) here
Homework #10 (due
date: May 26th, 2014) here
Homework #11 (due
date: June 2nd, 2014) here