S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
141
Trees
A tree is an undirected graph that is connected and acyclic. Much of what makes trees so
useful is the simplicity of their structure. For instance,
Property 2 A tree on n nodes has n 1 edges.
Th
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
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Exercises
5.1. Consider the following graph.
6
A
1
2
2
E
5
B
4
5
F
1
6
C
7
5
G
3
D
H
3
(a) What is the cost of its minimum spanning tree?
(b) How many minimum spanning trees does it have?
(c) Suppose
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Factoring
One last point: we started off this book by introducing another famously hard search problem:
FACTORING, the task of nding all prime factors of a given integer. But the difculty of
FACTORING
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
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An application of number theory?
The renowned mathematician G. H. Hardy once declared of his work: I have never done
anything useful. Hardy was an expert in the theory of numbers, which has long been r
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
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5. And nally, notice that the FFT circuit is a natural for parallel computation and direct
implementation in hardware.
Figure 2.10 The fast Fourier transform circuit.
a2
A( 1 )
001
A( 2 )
010
A( 3 )
01
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
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Figure 3.8 A directed acyclic graph with one source, two sinks, and four possible linearizations.
A
C
E
B
D
F
What types of dags can be linearized? Simple: All of them. And once again depth-rst
search
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
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7.12. For the linear program
max x1 2x3
x1 x 2 1
2x2 x3 1
x1 , x 2 , x 3 0
prove that the solution (x1 , x2 , x3 ) = (3/2, 1/2, 0) is optimal.
7.13. Matching pennies. In this simple two-player game, t
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
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6.4 Knapsack
During a robbery, a burglar nds much more loot than he had expected and has to decide what
to take. His bag (or knapsack) will hold a total weight of at most W pounds. There are n
items t
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Delete-min. Return the element with the smallest key, and remove it from the set.
Make-queue. Build a priority queue out of the given elements, with the given key
values. (In many implementations, thi
20.3 Evaluation
321
runs, we can compute the number #q of experiments that fullled the predicates attached
to q and the estimators of the minimum q , mean q , maximum q , median med(q ), and the
standard deviation s [q ], and so on. Obviously, not all of
15.5 General Information
Function Optimization
281
[2042]
15.5.2 Online Resources
Some general, online available ressources on Memetic Algorithms are:
http:/www.densis.fee.unicamp.br/~moscato/memetic_home.html [accessed 2008-04-03]
Last update: 2002-08-16
11.2 General Information
Application
References
Medicine
Biology and Medicine
Machine Learning
Function Optimization
[2255, 2256]
[558]
[1249]
[1277, 1917]
261
12
Simulated Annealing
12.1 Introduction
In 1953, Metropolis et al. [1396] developed a Monte Ca
7.3 The Basic Idea of Learning Classier Systems
241
Algorithm 7.3: learningClassierSystem()
Input: P : the list of rules xi that determine the behavior of the classier system
Input: [implicit] generateClassiers: a function which creates randomly a populat
4.10 Problems Inherent in the Evolution of Algorithms
221
Halting Problem
The Halting Problem is basically an instance of the Entscheidungsproblem and asks for an
algorithm that decides whether another algorithm will terminate at some point in time or
run
4.7 Graph-based Approaches
201
could regarded it also as an evolutionary programming42 method. Lately, researchers also
begin to focus on ecient crossover techniques for CGP [414].
Neutrality in CGP
Cartesian Genetic Programming explicitly utilizes dieren
4.5 Grammars in Genetic Programming
1
2
3
181
double func ( double x ) cfw_
return <pre - op > ( < expr > - < expr >) ;
The next two genes, 5 and 13, must again be ignored (x7 = x6 = x5 ). Finally, the last
gene with the allele 10 resolves the non-termin
4.2 General Information
161
2005: Lausanne, Switzerland, see [1116]
2004: Coimbra, Portugal, see [1115]
2003: Essex, UK, see [1786]
2002: Kinsale, Ireland, see [737]
2001: Lake Como, Italy, see [1423]
2000: Edinburgh, Scotland, UK, see [1666]
1999: Gtebor
3
Genetic Algorithms
3.1 Introduction
Genetic algorithms1 (GAs) are a subclass of evolutionary algorithms where the elements
of the search space G are binary strings (G = B ) or arrays of other elementary types. As
sketched in Figure 3.1, the genotypes ar
2.4 Selection
121
Algorithm 2.5: v assignFitnessTournamentq,r (Pop, cmpF )
Input: q : the number of tournaments per individuals
Input: r: the number of other contestants per tournament, normally 1
Input: Pop: the population to assign tness values to
Input
2.1 Introduction
101
4. Learning Classier Systems (LCS), discussed in Chapter 7 on page 233, are online
learning approaches that assign output values to given input values. They internally use
a genetic algorithm to nd new rules for this mapping.
5. Evolu
1.5 Formae and Search Space/Operator Design
81
1 for the input 0, and false otherwise. Assume that the formulas were decoded from a
binary search space G = Bn to the space of trees that represent mathematical expression by
a genotype-phenotype mapping. A
1.4 Problems in Optimization
61
A very crude and yet, sometimes eective measure is restarting the optimization process at randomly chosen points in time. One example for this method is GRASP s, Greedy
Randomized Adaptive Search Procedures [663, 652] (see
Fitness and heuristic values
(normally) have only a meaning in the
context of a population or a set of
solution candidates.
V R
Fitness Space
+
41
fitness
1.3 The Structure of Optimization
v(x)V
Fitness Values
Fitness Assignment Process
f1(x)
f1(x)
Fitnes
1
Introduction
One of the most fundamental principles in our world is the search for an optimal state.
It begins in the microcosm where atoms in physics try to form bonds1 in order to minimize
the energy of their electrons [1625]. When molecules form soli
Subject Index
relation to chromatic number,
433
longest path problem, 195
acyclic digraph, 89
weighted acyclic digraph, 53
loop, 4
Lovszs local lemma, 569
a
Lovszs splitting theorem, 440, 468
a
lower bound
on an arc, 95
removing from a network, 99
Lucches
References
703
461. J. Jirsek. On a certain class of multidigraphs, for which reversal of no arc dea
creases the number of their cycles. Comment. Math. Univ. Carolinae, 28:185
189, 1987.
a
462. J. Jirsek. Some remarks on Adms conjecture for simple directe
References
a
1. A. Adm. Problem. In Theory Graphs Applications, Proc. Coll. Smolenice,
pages 1218, Czech. Acad. Sci. Publ., 1964.
a
2. A. Adm. Bemerkungen zum graphentheoretischen Satze von I. Fidrich. Acta
Math. Acad. Sci. Hungar., 16:911, 1965.
3. R. Ah