Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 19B
December 7, 1998
Problem Set 5B
This problem set is due in class on Monday, Dec 14, except for CVN student who should
mail it by Wednesday, Dec 16.
When a problem asks to give an
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 5
October 13, 1998
Solutions for Problem Set 1
Problem 1.
Fixed Point
Given a sorted array of distinct integers A with n entries, consider the problem of nding
an index i such that A[
Analysis of Algorithms, CSOR 4231.
Columbia University, Fall 2011.
Professor Cliord Stein.
Homework 2 - Solutions
Problem 1. Exercise 4.3.7. Recurrences with lower order terms.
If we were to try a straight substitution proof, assuming that T (n) cnlog3 4
Analysis of Algorithms, CSOR 4231.
Columbia University, Fall 2011.
Professor Cliord Stein.
Homework 4 - Solutions
1. Problem 15-2. Longest Palidromic Sequence.
An easy approach to solve this problem is to generate the reverse string rst
and then nd the lo
ORIE 6300 Mathematical Programming I
September 18, 2014
Lecture 8
Lecturer: David P. Williamson
1
Scribe: Kevin Kircher
Strong duality
Recall the two versions of Farkas Lemma proved in the last lecture:
Theorem 1 (Farkas Lemma) Let A Rmn and b Rm . Then e
ORIE 6300 Mathematical Programming I
October 21, 2014
Lecture 16
Lecturer: David P. Williamson
1
Scribe: Shih-Hao, Tseng
The Cutting Stock Problem
W
si
Figure 1: Raw
This is a problem from the paper industry. Paper is produced in W inch long rolls called
ORIE 6300 Mathematical Programming I
September 2, 2014
Lecture 3
Lecturer: David P. Williamson
Scribe: Divya Singhvi
Last time we discussed how to take dual of an LP in two different ways. Today we will talk
about the geometry of linear programs.
1
Geomet
ORIE 6300 Mathematical Programming I
September 4, 2014
Lecture 4
Lecturer: David P. Williamson
1
Scribe: Paul Upchurch
Introduction
Last time we talked about polyhedra and polytopes. This time we will define bounded polyhedra
and discuss their relationshi
ORIE 6300 Mathematical Programming I
September 9, 2014
Lecture 5
Lecturer: Chaoxu Tong
Scribe: Jialin Liu
In previous lectures, we studied the standard form LP: min(cT x, Ax = b, x > 0). We also
considered two methods, by reduction and by bounding, of tak
ORIE 6300 Mathematical Programming I
August 28, 2014
Lecture 2
Lecturer: David P. Williamson
Scribe: Somya Singhvi
Last time, we considered the dual of linear programs in our basic form: max(cT x : Ax b).
We also considered an algebraic method for testing
ORIE 6300 Mathematical Programming I
September 30, 2014
Lecture 11
Lecturer: David P. Williamson
1
Scribe: Yingjie Bi
Example of the Simplex Method
We introduced the simplex method in the last class. Consider a standard form LP and its dual:
min cT x
max
ORIE 6300 Mathematical Programming I
October 23, 2014
Lecture 17
Lecturer: David P. Williamson
1
Scribe: Jeff Tian
The Knapsack Problem
In the Knapsack Problem, we have items 1 = 1, . . . , m with size si and value yi , a knapsack of size
W , and want to
October 9, 2014
ORIE 6300 Mathematical Programming I
Lecture 14
Lecturer: David P. Williamson
1
Scribe: Calvin Wylie
Simplex Issues: Number of Pivots
Question: How many pivots does the simplex algorithm need to take to find an optimal solution?
Answer: If
ORIE 6300 Mathematical Programming I
October 7, 2014
Lecture 13
Lecturer: David P. Williamson
1
Scribe: Hedyeh Beyhaghi
Pivot Rules
A key factor in the performance of the simplex method is the rule we use to decide which j (st
cj < 0) should enter the bas
ORIE 6300 Mathematical Programming I
October 28, 2014
Lecture 18
Lecturer: David P. Williamson
1
1.1
Scribe: Venus Lo
Good algorithms
What is a good algorithm?
We want to consider provably efficient algorithms for solving linear programs. What do we mean
ORIE 6300 Mathematical Programming I
October 31, 2014
Lecture 19
Lecturer: David P. Williamson
1
Scribe: Nozomi Hitomi
The Ellipsoid Method for LP
Recall we discussed the ellipsoid method last time: Given some bounded polyhedron P = cfw_x
Rn : Cx D, eith
ORIE 6300 Mathematical Programming I
October 16, 2014
Lecture 15
Lecturer: David P. Williamson
1
Scribe: Emily Fischer
Varieties of Simplex Method: Dual Simplex
1.1
Description
Recall that the regular (primal) simplex method is an algorithm that maintains
ORIE 6300 Mathematical Programming I
September 26, 2014
Lecture 10
Lecturer: David P. Williamson
1
Scribe: Yuhang Ma
From last class
Last time, we introduced the simplex method. In this class we are going to prove that the simplex
method indeed works.
Let
ORIE 6300 Mathematical Programming I
October 2, 2014
Lecture 12
Lecturer: David P. Williamson
1
Scribe: Xiaobo Ding
Finding an initial basic feasible solution
Recall our discussion from last time about how to find an initial basic feasible solution of a l
ORIE 6300 Mathematical Programming I
September 16, 2014
Lecture 7
Lecturer: David P. Williamson
1
Scribe: Nathan Knerr
Review
A while back, we defined polyhedrons and polytopes as follows.
Definition 1 A Polyhedron is P = cfw_x <n : Ax b
Definition 2 A Po
ORIE 6300 Mathematical Programming I
August 26, 2014
Lecture 1
Lecturer: David P. Williamson
Scribe: David Eckman
Much of the course will be devoted to linear programming (LP), the study of the optimization of
a linear function of several variables subjec
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 14
November 9, 1998
Midterm
This is a closed book exam. You can only use one two-sided sheet of notes. No book, no
other notes, no calculator, no collaboration.
Read the questions car
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 15
November 16, 1998
Midterm Solutions
Problem 1.
Master Theorem
Here is a table of logarithms. In the row i and column j you nd the value of logi j .
2
3
4
5
1
log2 1 = 0
log3 1 = 0
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 7
October 19, 1998
Problem Set 3
This problem set is due in class on Monday, Nov 16, except for CVN student who should
mail it by Wednesday, Nov 18.
Suggested exercises from CLR: 12.2
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 18
November 30, 1998
Solutions for Problem Set 3
Problem 1.
Radix Tree
Solution
(a)
Below is pseduo-code for the three requested routines. These assume the existence of the
following
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 16
November 16, 1998
Problem Set 4
This problem set is due in class on Monday, Nov 30, except for CVN student who should
mail it by Wednesday, Dec 2.
Suggested exercises from CLR: 23.
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 22
December 15, 1998
Solutions for Problem Set 4
Problem 1. Strongly Connected Components
Solution
Statement (b) is true.
Strongly connected components of a directed graph, G, are sub
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout new
New Handout
Solutions for Problem Set 5
Problem 1.
Edge Cover
Solution
(a) There are |M | edges in the matching and these edges cover 2 |M | vertices. There are
|V | 2 |M | vertic
CSOR 4231 Midterm Exam
Due: Thursday, November 3, 2011, 2:30PM in class.
Rules
You may consult the textbook, Introduction to Algorithms, 3rd edition by Cormen,
Leiserson, Rivest and Stein, your notes, the course website, any handouts from class,
a diction
Columbia University
W4231: Analysis of Algorithms
Luca Trevisan
Handout 14
November 6, 1998
Solutions for Practice Midterm
Problem 1. Master Theorem
Solution
p
(a) T (n) = 2T (n=3) + c n.
So b = 3, a = 2, and nlogb = n 6309 .
f (n) = cpn = cn 5.
Therefore