Another Variant of 3sat
Proposition 32 3sat is NP-complete for expressions in
which each variable is restricted to appear at most three
times, and each literal at most twice. (3sat here requires
only that each clause has at most 3 literals.)
Consider a g
CSOR W4246
HW4
Due: Nov 25th
Date: Oct 21th
Name: Yeyun Chen
UNI: yc3070
(a) Variables: We introduce one variable Fij for each edge (i, j) E, which stands for the flow in
edge(i, j) in the solution.
Objective Function:
min
(,)
Constraints:
Capacity con
CSOR W4246
HW3
Name: Yeyun Chen
UNI: yc3070
Due: Nov 04th
Date: Oct 29th
So, the maximum flow and the capacity of minimum cut is 11, and the minimum cut is:
S = cfw_s, a, b, c, T = cfw_d, t
-1/5
CSOR W4246
HW3
1.
Name: Yeyun Chen
UNI: yc3070
Due: Nov 04th
CSOR W4246
HW2
Name: Yeyun Chen
UNI: yc3070
Due: Oct 18th
Date: Oct 11th
Since it is an undirected graph and unweighted graph, we can use BFS algorithm to solve this
problem. Let N_SP[t] denote the number of shortest path from s to t.
1.
First add all nod
cfw_
"cells": [
cfw_
"cell_type": "markdown",
"metadata": cfw_,
"source": [
"# Connected Components\n",
"\n",
"The purpose of this assignment is to familiarize yourself with the handling
of graph data structures. You will implement the algorithm for ident
CSOR W4246 Fall, 2016
HW1 Theoretical part
Out: Thursday, September 15, 2016
Due: 8pm, Monday, September 26, 2016
Please keep your answers clear and concise. For all algorithms you suggest, you must prove correctness and
give the best upper bound that you
Algorithms for Data Science
CSOR W4246
Eleni Drinea
Computer Science Department
Columbia University
Lecture 5: depth-first search, topological sorting
Outline
1 Recap
2 Applications of BFS
Testing bipartiteness
3 Depth-first search (DFS)
4 Applications of
CSOR W4246 Fall, 2016
HW1 Theoretical part
Out: Thursday, September 15, 2016
Due: 8pm, Monday, September 26, 2016
Please keep your answers clear and concise. For all algorithms you suggest, you must prove correctness and
give the best upper bound that you
CSOR W4231.002
Notes on the Master Theorem
29/1/2015
Often when analyzing a divide & conquer algorithm, we obtain a recurrence for its running time of
the following form
n
T (n) = aT ( ) + cnk
b
(1)
Essentially this recurrence says: on input of size n, we
Algorithms for Data Science
CSOR W4246
Eleni Drinea
Computer Science Department
Columbia University
Lecture 2: asymptotic notation, mergesort
Outline
1 Asymptotic notation
2 The divide & conquer principle; application: mergesort
3 Solving recurrences and
COMS 4231: Analysis of Algorithms I, Fall 2016
Problem Set 1, due Thursday September 22, 11:40am on Courseworks
Please follow the homework submission guidelines posted on
courseworks
Problem 1. For each of the following two code fragments, give its asympt
Examples of
Divide and Conquer
and the Master theorem
CS 4231, Fall 2016
Mihalis Yannakakis
Divide and Conquer
Reduce to any number of smaller instances:
1. Divide the given problem instance into
subproblems
2. Conquer the subproblems by solving them
recu
zhangshuijing @msn.com
2-3
c
INSERTION-SORT(A)
1 for j 2 to length[A]
2
do key A[j]
3
Insert A[j] into the sorted sequence A[1.j-1]
4
i j-1
5
while i > 0 and A[i] >
6
do A[i+1]A[i]
7
i i1
8
A[i+1]key
C#:
public static void InsertionSort<T>(T[] Inp
CSOR W4231.002 Spring, 2015
Homework 5
Out: Friday, April 3, 2015
Due: 8pm, Friday, April 17 , 2015
Please keep your answers clear and concise. For all algorithms you suggest, you must prove correctness and give the best upper bound that you can for the r
CSOR W4231.002 Spring, 2015
Homework 6
Out: Friday, April 17, 2015
Due: 8pm, Friday, May 1, 2015
Please keep your answers clear and concise. For all algorithms you suggest, you must prove correctness and give the best upper bound that you can for the runn
Selected Solutions for Chapter 2:
Getting Started
Solution to Exercise 2.2-2
S ELECTION -S ORT.A/
n D A:length
for j D 1 to n ! 1
smallest D j
for i D j C 1 to n
if Ai! < Asmallest!
smallest D i
exchange Aj ! with Asmallest!
The algorithm maintains the lo
CSOR W4231.002 Spring, 2015
Homework 6 solutions
Out: Friday, April 17, 2015
Due: 8pm, Friday, May 1, 2015
1. Solution for problem 1: The variables will correspond to the number of rolls cut in one of
the many different ways in which a 3m roll can be cut
W4231 Analysis of Algorithms
1
Homework Solution #4
HW1 optional exercises
1.1
HW1, Problem 7
1. T (n) = n2 + n (prove by induction)
2. T (n) = 2n 1 (prove by induction)
1.2
HW1, Problem 8
P
First, prove ni=1
assumption).
1
i
= O(log n). For simplicity, a
CSOR 4231
Homework #5 Solution
April 21, 2015
1
Problem 1
(a) We can prove this using the accounting method introduced in part 17.2 in textbook.
Here we charge the operations as follows:
Setting a bit from 0 to 1: $3, while the actual cost is $1
Resetti
CSOR W4231.002 Spring, 2016
Homework 1
Out: Monday, January 25, 2015
Due: 8pm, Monday, February 8, 2015
Please keep your answers clear and concise. Collaboration is limited to discussion of ideas
only. You should write up the solutions entirely on your ow
W4231 Analysis of Algorithms
1
Homework Solution #4
HW6, Problem 1
1. Clique
Decision version of Clique
Given a graph G = (V, E) and a target value k, does the graph have a clique
(complete subgraph) on (at least) k vertices?
Clique(D) is in N P
A candi
Analysis of Algorithms HW4
Rajan bhargava
Instructor: Prof. Eleni Drinea
April 14, 2016
1
Problem 1
This algorithm is a modified BFS, starting at node v. For each node u processed by
BFS, we record two values: the number of shortest v-u paths u.num, and t