Homework 3
CS 5114 (Spring 2013)
Assigned on Tuesday, February 19, 2013.
Submit PDF solutions by email to the TA
by the beginning of class on Tuesday, February 26, 2013.
Instructions:
You are not allowed to consult any sources other than your textbook, t
Homework 5
CS 5114 (Spring 2013)
Assigned on April 2, 2013.
Submit PDF solutions by email to the TA
by the beginning of class on Tuesday, April 9, 2013.
Instructions:
You are not allowed to consult any sources other than your textbook, the slides on the
Syllabus for Theory of Algorithms
CS 5114, Spring 2013
Meeting Times 2pm3:15pm, Tuesdays and Thursdays, Torgerson 1030, NVC 113
Instructor T. M. Murali, 231-8534, Torgerson 2160B
murali AT cs DOT vt DOT edu
http:/bioinformatics.cs.vt.edu/murali
Oce Hours
Introduction
Bipartite Matching
Edge-Disjoint Paths
Image Segmentation
Circulation with Demands
Airline Scheduling
Applications of Network Flow
T. M. Murali
April 9, 11 2013
T. M. Murali
April 9, 11 2013
Applications of Network Flow
Introduction
Bipartite
Introduction
Reductions
NP
N P -Complete
N P vs. co-N P
NP and Computational Intractability
T. M. Murali
April 18, 23, 2013
T. M. Murali
April 18, 23, 2013
NP and Computational Intractability
Introduction
NP
Reductions
N P -Complete
N P vs. co-N P
Algorit
Strategy
3-SAT
Sequencing Problems
Partitioning Problems
Other Problems
NP-Complete Problems
T. M. Murali
April 25, 30, 2013
T. M. Murali
April 25, 30, 2013
NP-Complete Problems
Strategy
3-SAT
Sequencing Problems
Partitioning Problems
Other Problems
Provi
Solving N P -Complete Problems
Small Vertex Covers
Trees
Treewidth
Approximation Algorithms
Coping with NP-Completeness
T. M. Murali
May 2, 7, 2013
T. M. Murali
May 2, 7, 2013
Coping with NP-Completeness
Solving N P -Complete Problems
Small Vertex Covers
Midterm Examination
CS 5114 (Spring 2013)
Assigned: March 19, 2013.
Due: at the beginning of class on March 26, 2013.
Name:
PID:
Instructions
1. For every algorithm you describe, prove its correctness, and state and prove the running time of the
algorithm
C# Interview Questions
Does C# support multiple-inheritance?
No.
Who is a protected class-level variable available to?
It is available to any sub-class (a class inheriting this class).
Are private class-level variables inherited?
Yes, but they are not acc
C# developer interview cheat sheet
August 20, 2012
By Tony Patton
There is an art to finding a developer who fits well in your organization in terms of personality and work ethic;
fortunately, its more straightforward to determine their technical expertis
Introduction
Ford-Fulkerson Algorithm
Scaling Max-Flow Algorithm
Network Flow
T. M. Murali
March 28, April 2, and 4, 2013
T. M. Murali
March 28, April 2, and 4, 2013
Network Flow
Introduction
Ford-Fulkerson Algorithm
Scaling Max-Flow Algorithm
Maximum Flo
Weighted Interval Scheduling
Segmented Least Squares
RNA Secondary Structure
Sequence Alignment
Shortest Paths in Graphs
Dynamic Programming
T. M. Murali
February 28, March 5, 17, 19, 21, 2013
T. M. Murali
February 28, March 5, 17, 19, 21, 2013
Dynamic Pr
On the history of the transportation and
maximum ow problems
Alexander Schrijver1
Abstract. We review two papers that are of historical interest for combinatorial optimization: an
article of A.N. Tolsto from 1930, in which the transportation problem is st
Homework 4
CS 5114 (Fall 2013)
Assigned on Tuesday, February 26, 2013.
Submit PDF solutions by email to the TA
by the beginning of class on Tuesday, March 5, 2013.
Instructions:
You are not allowed to consult any sources other than your textbook, the sli
Homework 6
CS 5114 (Spring 2013)
Assigned on April 9, 2013.
Submit PDF solutions by email to the TA
beginning of class on April 16, 2013.
Instructions:
You are not allowed to consult any sources other than your textbook, the slides on the course web
page
Homework 7
CS 5114 (Spring 2013)
Assigned on April 25, 2013.
Submit PDF solutions by email to the TA
beginning of class on May 2, 2013.
Instructions:
You are not allowed to consult any sources other than your textbook, the slides on the course web
page,
About the Course
Theory of Algorithms
Proof by Induction
Introduction to CS 5114
T. M. Murali
January 22, 2013
T. M. Murali
January 22, 2013
Introduction to CS 5114
About the Course
Theory of Algorithms
Proof by Induction
Course Information
Instructor
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Priority Queues
T. M. Murali
January 29, 2013
T. M. Murali
January 29, 2013
Priority Queues
Motivation: Sort a List of Numbers
Sort
x1 , x2 , . . . , x of integers.
SOLUTION: A permutation y1 , y2 , . . . , y of x1 , x2 , . . . , x
y y +1 , for all 1 i <
Interval Scheduling
Interval Partitioning
Minimising Lateness
Greedy Algorithms
T. M. Murali
January 31, 2013
T. M. Murali
January 31, 2013
Greedy Algorithms
Interval Scheduling
Interval Partitioning
Minimising Lateness
Algorithm Design
Start discussion o
Graphs
Shortest Paths
Minimum Spanning Trees
Implementation
Greedy Graph Algorithms
T. M. Murali
February 7, 12, and 14, 2013
T. M. Murali
February 7, 12, and 14, 2013
Greedy Graph Algorithms
Graphs
Shortest Paths
Minimum Spanning Trees
Implementation
Gra
Minimum Bottleneck Spanning Trees
Clustering
Applications of Minimum Spanning Trees
T. M. Murali
February 14, 2013
T. M. Murali
February 14, 2013
Applications of Minimum Spanning Trees
Minimum Bottleneck Spanning Trees
Clustering
Minimum Spanning Trees
We
Divide and Conquer Algorithms
T. M. Murali
February 19, 2013
T. M. Murali
February 19, 2013
Divide and Conquer Algorithms
Divide and Conquer
Break up a problem into several parts.
Solve each part recursively.
Solve base cases by brute force.
Eciently comb
Counting Inversions
Integer Multiplication
Closest Pair of Points
Divide and Conquer Algorithms
T. M. Murali
February 26 and 28, 2013
T. M. Murali
February 26 and 28, 2013
CS 4104: Divide and Conquer Algorithms
Counting Inversions
Integer Multiplication
C
Homework #1
n
1. Prove that if a and r 1 are real numbers,
" ar i =
i =0
a (r n +1 ! 1)
for all n 0.
r !1
The sum on the left is called a geometric sum.
Base case: For n = 0, i=0.0 ari = ar0 = a and a(r0+11) / (r1) = a
Inductive step:
Assume: i=0.n ari =