CSE 4101/5101
Prof. Andy Mirzaian
Move to Front
Self Adjusting
Linear Lists
DICTIONARIES
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Mu
one maximum ow problem or by modifying the instance so that a circulation can easily be found. The cycle canceling algorithm can be used to prove Theorem 1. The proof is by induction. Assume that the initial circulation is chosen to be integral. Now, if a
Theorem 17 ISIT-`2 2 P. i.e., ISIT-`2 can be answered in polynomial time. Proof: Assume that there exists an embedding of f1; 2; : : : ; ng into fv1 = 0; v2; : : : ; vng.
Consider one such embedding. Then,
d2i; j = jjvi , vj jj2 = vi , vj vi , vj = vi2 ,
load of the machines see Figure 8. Therefore, P p 6= Cmax jmk j + pk Pp j j + max p m j jP ! j 2 max max pj ; mpj j = 2L: Now we have an interval on which to do a logarithmic binary search for Cmax. By T1 and T2 we denote lower and upper bound pointers we
5 Approximating MAX-CUT
In this section, we illustrate the fact that improved approximation algorithms can be obtained by considering relaxations more sophisticated than linear ones. At the same time, we will also illustrate the fact that rounding a solut
With this observation, we can now formulate MCPMP as a linear program: X Z = Min ce xe e2E X subject to: xe 1 for all S V with jS j odd
xe 0 for all e 2 E . We can now see that the value Z of this linear program is a lower bound on the cost of any perfect
18.415 6.854 Advanced Algorithms
Lecturer: Michel X. Goemans
November 1994
Approximation Algorithms
1 Introduction
Many of the optimization problems we would like to solve are NP-hard. There are several ways of coping with this apparent hardness. For most
18.415 6.854 Advanced Algorithms
Lecturer: Michel X. Goemans
November 1994
Network ows
In these notes, we study some problems in "Network Flows". For a more comprehensive treatment, the reader is referred to the surveys 12, 1 , or to the recent book 2 . N
8
AT w = g , d : A AT w = Ag
Solving the normal equations, we get and
normal equations:
w = A AT ,1Ag d = g , AT A AT ,1Ag = I , AT A AT ,1Ag:
A potential problem arises if g is nearly perpendicular to the null space of A. In this case, jjdjj will be very
11.2 Size of the Output
In order to even hope to solve a linear program in polynomial time, we better make sure that the solution is representable in size polynomial in L. We know already that if the LP is feasible, there is at least one vertex which is a
CSE 4101/5101
Prof. Andy Mirzaian
Search Trees
DICTIONARIES
DICTIONARIES
Search Trees
Search Trees
Lists
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Multi-Lists
Multi-Lists
Linear Lists
Linear Lists
Binary Search Trees
Binary Search Trees
Multi-Way Search Trees
Multi-Way Search Trees
B-trees
CSE 4101/5101
Prof. Andy Mirzaian
B-trees
2-3-4 trees
DICTIONARIES
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Multi-Lists
Multi-Lists
Linear Lists
Linear Lists
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B
CSE 4101/5101
Prof. Andy Mirzaian
Red-Black
Tree
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B-tree
CSE 4101/5101
Prof. Andy Mirzaian
Convex
Hull
CONVEX HULL
Given S d
(d = 1,2,3, )
Convex-Hull of S:
CH(S) = the set of all convex combinations of points in S
= the smallest convex set that contains S
= the intersection of all convex sets that contain S
Th
CSE 4101/5101
Prof. Andy Mirzaian
Disjoint
Set Union
References:
[CLRS] chapter 21
Lecture Note 6
2
Disjoint Set Union
Items are drawn from the finite universe U = cfw_1, 2, , n for some fixed n.
Maintain a partition of (a subset of) U, as a collection
CSE 4101/5101
Prof. Andy Mirzaian
Line Segments
Intersections
Line Segments Intersections
Thematic Map Overlay
Photos courtesy of ScienceGL
2
References:
[CLRS] chapter 33
[M. de Berge et al 00] chapter 2
[Preparata-Shamos85] chapter 7
[ORourke98] chapter
CSE 4101/5101
Prof. Andy Mirzaian
Computational
Geometry
Overview
2
Computational Geometry:
Study of algorithms, data structures, and computational complexity
of computational problems in geometry.
Dimension: 1
2 3 d
Objects: Finitely specifiable
points
l
CSE 4101/5101
Prof. Andy Mirzaian
Augmenting
Data Structures
TOPICS
Augmentation
Order Statistics Dictionary
Interval Tree
Overlapping Windows
2
References:
[CLRS] chapter 14
3
Augmenting a Data Structure
Suppose we have a base data structure D that
CSE 4101/5101
Prof. Andy Mirzaian
Splay Tree:
Self Adjusting BST
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Multi-Way Sea
8 When is a Linear Program Feasible ?
We now turn to another question which will lead us to important properties of linear programming. Let us begin with some examples. We consider linear programs of the form Ax = b, x 0. As the objective function has no
18.415 6.854 Advanced Algorithms
Lecturer: Michel X. Goemans
Linear Programming
October 1994
1 An Introduction to Linear Programming
Linear programming is a very important class of problems, both algorithmically and combinatorially. Linear programming has
CSE 4101/5101
Prof. Andy Mirzaian
Red-Black Tree
DICTIONARIES
Lists Search Trees
Multi-Lists
Linear Lists
Binary Search Trees
Multi-Way Search Trees
B-trees
Hash Tables
Move-to-Front competitive
Splay Trees competitive?
Red-Black Trees
2-3-4 Trees
SELF AD
CSE 4101/5101
Prof. Andy Mirzaian
Move to Front Self Adjusting Linear Lists
DICTIONARIES
Lists Search Trees
Multi-Lists
Linear Lists
Binary Search Trees
Multi-Way Search Trees
B-trees
Hash Tables
Move-to-Front competitive
Splay Trees competitive?
Red-Blac