Convex Hulls
in 3-space
(slides mostly by Jason C. Yang)
October 7, 2003
Lecture 10: Convex Hulls in 3D
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Problem Statement
Given P: set of n points in 3D
Return:
Convex hull of P: CH(P), i.e.
smallest polyhedron s.t.
all elements of P on or in
th
Polynomial Algorithms that Prove an
NP-Hard Hypothesis Implies an
NP-Hard Conclusion
D. Bauer
1
H. J. Broersma
A. Morgana
3
E. Schmeichel
1
2
4
Department of Mathematical Sciences
Stevens Institute of Technology
Hoboken, NJ 07030, U.S.A.
2
Faculty of Math
3-Coloring of Random Graphs
Josep D 1
az
UPC
1
Research partially supported by the Future and Emerging Technologies programme of the EU under contract
001907 (DELIS)
Coloring problems.
Given a graph G = (V , E ) a legal coloring is an assignment of
colors
SIAM J. COMPUT.
Vol. 28, No. 2, pp. 525540
c 1998 Society for Industrial and Applied Mathematics
PRIMAL-DUAL RNC APPROXIMATION ALGORITHMS FOR SET
COVER AND COVERING INTEGER PROGRAMS
SRIDHAR RAJAGOPALAN AND VIJAY V. VAZIRANI
Abstract. We build on the class
Primal-Dual Approximation Algorithms
We just saw how the primal-dual schema permits sometimes designing efcient combinatorial algorithms for
solving certain problems. We will now see an example
of how a related technique can sometimes be used to
design ef
Primal-Dual Algorithm Examples
We just saw the general primal-dual algorithm schema.
We will now see how to apply it to the
Shortest Path Problem
and the
Max Flow Problem
1
The Shortest Path Problem
Given G = (V, E ), let A be its (|V | 1) |E | nodearc in
Randomized Algorithms, Spring 2009: Project
version 1 January 19, 2009
This project consists of two problems that each have both theoretical and
practical aspects. The subproblems outlines a possible approach. If you
follow the suggested approach, you mus
6.891 Approximation Algorithms
Problem Set 1 Solutions
March 10, 2000
1. The following problem arises in telecommunications networks, and is known as
the SONET ring loading problem. The network consists of a cycle on n nodes,
numbered 0 through n ; 1 cloc
Mergesort and Quicksort
Chapter 8
Kruse and Ryba
Sorting algorithms
Insertion, selection and bubble sort have
quadratic worst-case performance
The faster comparison based algorithm ?
O(nlogn)
Mergesort and Quicksort
Merge Sort
Apply divide-and-conquer
Primal-dual R NC approximat ion algorithms
for (multi)-set (multi)-cover and covering integer programs
V ijay V . Vaeiranit
I ndian Institute of Technology,
D elhi.
Sridhar Rajagopalan'
U niversity of California,
Berkeley.
Abstract
1
Introduction
W e buil
Segment Trees
[Bentley]
A segment tree is a data structure for storing a set of intervals
I = cfw_[x1 , x1 ], [x2 , x2 ], . . . , [xn , xn ]
and can be used for solving problems e.g. concerning line segments.
Let p1 , . . . , pm , m 2n, be the ordered lis
Introduction
Kd-trees
Range searching and kd-trees
Computational Geometry
Lecture 7: Range searching and kd-trees
Computational Geometry
Lecture 7: Range searching and kd-trees
Introduction
Kd-trees
Database queries
1D range trees
Databases
Databases stor
CMSC 754:Spring 2010
Dave Mount
Solutions to Homework 1: Convex Hulls and Plane Sweep
Solution 1: As a convenience in describing our solution, let us imagine that we apply a rotation 1 so that
u is directed along the x-axis, that is u = (1, 0). Before acc
CMSC 754:Spring 2010
Dave Mount
Solutions to Homework 3: Delaunay Triangulations and Arrangements
Solution 1: Given the point set P , rst construct the Delaunay triangulation of P in O(n log n) time. For
each site p P , let N (p) denote the set of its Del
Recent PTAS Algorithms on the Euclidean TSP
by
Leonardo Zambito
Submitted as a project for CSE 4080,
Fall 2006
1
Introduction
The Traveling Salesman Problem, or TSP, is an on going study in computer
science. The TSP has a long history ranging back to the
Chapter 3, Part 2
Variants of Turing Machines
CSC527, Chapter 3, Part 2 c 2012 Mitsunori Ogihara
1
Multitape TMs
A multitape Turing machine is a Turing machine with additional
tapes with each tape is accessible individually, with the input on
the rst tape
Naming Convention for CFCs & Halons
Please note: you will not be tested on this information!
It is provided in case anyone is interested
Chlorofluorocarbons (CFCs) are nontoxic, nonflammable chemicals containing atoms of Chlorine, Fluorine, and
Carbon. T
ADVANCED ALGORITHMS
Lecture 18: Primal-Dual Approximation Algorithms
Notes taken by Yu Fu, Jianan Huang
May 2011
Summary: This lecture talks about primal-dual approximation algorithms, in which the intuition is the
idea from LP . However, the interesting
Interval Trees
Store intervals of the form [li,ri], li <= ri.
An interval is stored in exactly 1 node.
So, O(n) nodes.
3 versions.
Differing capability.
Version 1
Store intervals of the form [li,ri], li <= ri.
At least 1 interval per node.
Static inte
1
CSE2001, Fall 2006
Last time we proved that the language
AT M = cfw_ M, w | M is a TM that accepts w
is undecidable, and we proved that
HT M = cfw_ M, w | M is a TM that halts on w
is also undecidable, by reducing AT M to HT M . Today we will do more
Lecture 13
Graph Algorithms II
13.1 Overview
In this lecture we begin with one more algorithm for the shortest path problem, Dijkstras algorithm. We then will see how the basic approach of this algorithm can be used to solve other problems including nding
Turing Machines More Examples Robb T. Koether Homework Review More Examples of Turing Machines
COMP ADD MULT SQRT
Turing Machines - More Examples
Lecture 24 Section 3.1 Robb T. Koether
Hampden-Sydney College
Assignment
Wed, Oct 22, 2008
Outline
Turing Mac
Turing Machines - More Examples
Lecture 24
Section 3.1
Robb T. Koether
Hampden-Sydney College
Wed, Oct 24, 2012
Robb T. Koether (Hampden-Sydney College)
Turing Machines - More Examples
Wed, Oct 24, 2012
1 / 37
1
More Examples of Turing Machines
The Functi
CS 598CSC: Approximation Algorithms Instructor: Chandra Chekuri
Lecture date: January 28, 2009 Scribe: Md. Abul Hassan Samee
1
Introduction
We discuss two closely related NP Optimization problems, namely Set Cover and Maximum Coverage in this lecture. Set
Lecture 4
Balanced Binary Search Trees
6.006 Fall 2009
Lecture 4: Balanced Binary Search Trees
Lecture Overview
The importance of being balanced
AVL trees
Denition
Balance
Insert
Other balanced trees
Data structures in general
Readings
CLRS Chapter
ORIE 633 Network Flows
October 2, 2007
Lecture 9
Lecturer: David P. Williamson
1
Scribe: Qiu Wang
Ecient algorithms for max ow
1.1
Blocking ow and Dinics algorithm
In this lecture, well discuss about some other ecient algorithms for computing a maximum
ow
Advanced Approximation Algorithms
(CMU 18-854B, Spring 2008)
Lecture 15: Coloring 3-Colorable Graphs using SDP
March 4, 2008
Lecturer: Ryan ODonnell
1
Scribe: Dafna Shahaf
3-Coloring
3Col is the problem of deciding whether there is a legal 3-Coloring of a
IEOR 266 Lectures 19 and 20
Network Flows and Graphs
November 04 and 06, 2008
1
1.1
Max flow - Min Cut (continued)
Flow Decomposition
In many situations we would like, given a feasible flow, to know how to recreate this flow with a bunch of s - t paths. I
Trapezoidal decomposition:
Motivation: manipulate/analayze a collection of segments e.g. detect segment intersections e.g., point location data structure Draw verticals at all points binary search for slab binary search inside slab problem: O(n2 ) space D