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Analytic Combinatorics

http://aofa.cs.princeton.edu//50ac

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. A modern approach to the study of combinatorial structures of the sort that we encounter frequently in the analysis of algorithms. The approach is predicated on the idea that combinatorial structures are typically defined by simple formal rules that are the key to learning their properties. One eventual outgrowth of this observation is that a relatively small set of. Ultimately yields accurate approximations of the quantities that we seek&#...

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Permutations

http://aofa.cs.princeton.edu/70permutations

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. This chapter surveys combinatorial properties of permutations (orderings of the numbers. And shows how they relate in a natural way to fundamental and widely-used sorting algorithms. 71 Basic Properties of Permutations. 72 Algorithms on Permutations. 73 Representations of Permutations. 75 Analyzing Properties of Permutations with CGFs. 76 Inversions and Insertion Sort. 77 Left-to-Right Minima and Selection Sort. Of $N$ elements is a sequenc...

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Analysis of Algorithms

http://aofa.cs.princeton.edu//10analysis

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. 1 Analysis of Algorithms. This chapter considers the general motivations for algorithmic analysis and relationships among various approaches to studying performance characteristics of algorithms. 11 Why Analyze an Algorithm? The branch of theoretical computer science where the goal is to classify algorithms according to their efficiency and computational problems according to their inherent difficulty is known as. 13 Analysis of Algorithms.

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Words and Maps

http://aofa.cs.princeton.edu//90maps

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. 9 Words and Maps. This chapter covers global properties of. Letter strings from an. Letter alphabet), which are well-studied in classical combinatorics (because they model sequences of independent Bernoulli trials) and in classical applied algorithmics (because they model input sequences for hashing algorithms). The chapter also covers random. Letter words from an. Letter alphabet) and discusses relationships with trees and permutations.

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Strings and Tries

http://aofa.cs.princeton.edu//80strings

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. This chapter studies basic combinatorial properties of. Sequences of characters or letters drawn from a fixed alphabet, and introduces algorithms that process strings ranging from fundamental methods at the heart of the theory of computation to practical text-processing methods with a host of important applications. 82 Combinatorial properties of bitstrings. 84 Finite-State Automata and the Knuth-Morris-Pratt Algorithm. TO BE OR NOT TO BE.

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Asymptotic Approximations

http://aofa.cs.princeton.edu//40asymptotic

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. This chapter examines methods of deriving approximate solutions to problems or of approximating exact solutions, which allow us to develop. Estimates of quantities of interest when analyzing algorithms. 41 Notation for Asymptotic Approximations. The following notations, which date back at least to the beginning of the century, are widely used for making precise statements about the approximate value of functions:. A stronger statement is to...

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Trees

http://aofa.cs.princeton.edu/60trees

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. This chapter investigates properties of many different types of. Fundamental structures that arise implicitly and explicitly in many practical algorithms. Our goal is to provide access to results from an extensive literature on the combinatorial analysis of trees, while at the same time providing the groundwork for a host of algorithmic applications. 62 Forests and Trees. We use the same nomenclature as for binary trees: the subtrees of a n...

aofa.cs.princeton.edu

Asymptotic Approximations

http://aofa.cs.princeton.edu/40asymptotic

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. This chapter examines methods of deriving approximate solutions to problems or of approximating exact solutions, which allow us to develop. Estimates of quantities of interest when analyzing algorithms. 41 Notation for Asymptotic Approximations. The following notations, which date back at least to the beginning of the century, are widely used for making precise statements about the approximate value of functions:. A stronger statement is to...

aofa.cs.princeton.edu

Recurrence Relations

http://aofa.cs.princeton.edu//20recurrence

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. This chapter concentrates on fundamental mathematical properties of various types of. Which arise frequently when analyzing an algorithm through a direct mapping from a recursive representation of a program to a recursive representation of a function describing its properties. Recurrences are classified by the way in which terms are combined, the nature of the coefficients involved, and the number and nature of previous terms used. Use the ...

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Generating Functions

http://aofa.cs.princeton.edu//30gf

1 Analysis of Algorithms. 8 Strings and Tries. 9 Words and Maps. This chapter introduces a central concept in the analysis of algorithms and in combinatorics:. Mdash; a necessary and natural link between the algorithms that are our objects of study and analytic methods that are necessary to discover their properties. 31 Ordinary Generating Functions. Given a sequence $a 0, a 1, a 2, ldots, a k, ldots$, the function $ A(z)= sum {k ge0}a k z k$ is called the. Ordinary generating function (OGF). 0, ,1, ,{1 ...