Another book project: this was written as a book proposal aimed at O'Reilly, for an actual printed book (back when people actually bought printed computer books). I wrote two sections, defining an algorithm and explaining O() notation; parts of these two sections live on in my 50-examples book.
Outline
---------
Written Sat. Aug 30 2003.
"Algorithms in Python" (alt. title "Learning Algorithms with Python", if
Sedgewick has a trademark) is a book that uses Python as the vehicle to
teach the basic ideas of algorithms -- what an algorithm is, how to
notate them clearly, what O() notation means, etc. -- and to introduce a
number of different algorithms.
Discussions will rarely or never be rigorous. Requirements:
* Minimal mathematical background should be required (basically
functions, I think). Those bits of notation that are necessary will
be explained as they're needed.
* Elementary Python programming knowledge is needed, so readers
should have a tutorial such as "Learning Python" alongside this book;
it will not attempt to teach Python, though it will introduce relevant
standard library modules.
* Python 2.3 will be used.
The goal is to make the book understandable by a bright high-school
student. If the student goes on to formal computer science study, she
will have a good foundation for it; if not, she'll understand some basic
computer ideas.
Exercises may be included (I haven't decided yet); they might be hard to
invent, but they'd also be very useful for the book's use in teaching
settings. If exercises are provided, I'll write up a set of unit tests
for correct implementations; this will make it easy for teachers to
check whether answers are correct and let them concentrate on
programming style. It might be better to leave exercises out of the
book and put them on a separate web site, where they can be continually
updated.
* Preface (the usual front matter)
* Introductory comments
* Intended audience; prerequisites; goals of book
* Outline of book
* Typographical conventions
* Acknowledgements
* Algorithmic Analysis (alt. title "What is an Algorithm"?)
* Basic idea: an algorithm is a series of steps to perform a task
* Example: finding the largest number in a list
* Iterative formulation
* Recursive formulation
* Measuring time complexity: O, Theta, Omega notation
* Compare different time complexities: O(n) vs O(lg n)
* Applying O() to memory/space complexity
* Time complexity of various Python built-in operations (dicts, lists)
* Hashing
Hashing is going to be the first serious algorithm used as an
example, so it's going to be worked out in the most detail. Later
chapters will explain algorithms, give reasons why they work,
and discuss their O() complexity, but in less detail than in this chapter.
* Basic concept of hash tables
* Computing hash codes
* Handling collisions
* Resizing
* Amortized algorithm costs (I can't see how to fit this into
chapter 1, since you need a reasonably complicated algorithm to
use as an example,)
* Deleting hash table entries
* Determining time complexity of hashing
(this would be the most detailed explanation of determining time complexity;
remaining chapters would be more hand-waving)
* Graphs
* Graph concepts
* Different graph representations (Node objects, sets of arcs)
* Traversal
* Topological sorts
* Example: working out file dependencies
* Connected-components
* Spanning trees
* Shortest paths
* Trees
* Trees as a special case of graphs
* Binary trees
* Representations (Node objects, lists/tuples, tables)
* Searching
* Inserting
* Deleting
* Unbalanced trees
* Recursive operations on trees
* Balanced trees:
* red-black
* do AVL trees, or just mention them briefly?
(Briefly, I think; there's already more than enough material in this outline!)
* B-trees
* Numbers (should the numeric material come before trees and graphs?)
* Representing numbers on computers
* Machine integers
* Floating point representation
* Large integers
* Random number generation
* Numeric Analysis
* Polynomial evaluation
* Finding zeros of functions
* Differentiating functions
* Integrating functions
* Sorting
* Basic concepts
* Comparing
* Stability
* Simple algorithms:
* Bubble sort
* Shell sort
* Insertion sort
* Merge sort
* Quicksort
* Implementation
* Issues
* Time-complexity
* Time complexity of sorting
* Proof of O(n lg n) bound.
* Breaking assumptions: parallellism, spaghetti sort
Hard problems: NP-completeness
* P and NP
* NP-completeness
* Explanation
* Various examples of NP-complete problems
* Show that all NP-complete problems are equivalent
* Solving an NP-complete problem
* Exhaustive search
* Heuristics
* Final thoughts
I like books that close with some sort of summation.
This brief final chapter will discuss a few general issues.
* Think about time complexity
* Do the simplest thing...
* Write tests
* Value clarity over optimization
* Where to go from here?
* More books to read; things to do
* Current state of research (parallel, distributed, various domains)
Optional Topics
--------------------
This section lists various chapters for which I produced an outline
but later decided aren't a core part of the book. Any of these chapters can
be reintroduced at a later point.
* Number Theory
* Finite fields
* Primality testing (kind of an odd duck)
* RSA
Geometric Algorithms:
* Point representations
* Convex hull
* Line intersections
* Range searching
* Strings
This will likely be a difficult chapter, both to write and to read,
because FSMs will be a pain to explain. (Indeed, you could envision
an entirely different document that covered FSMs, languages, Turing
completeness, parsing, ... One for the sequel, I think.)
* Simple searching
* Boyer-Moore/KMP searches
* Data compression
* Regular expressions
* Simple regex patterns
* Finite automata
* Languages
* What finite automata can't do
* Brief theory-of-computation overview
* Real Implementations:
Examines real-world implementation of various algorithms.
* Random number generation: Mersenne twister
* Karatsuba algorithm? (maybe too complicated and boring?)
* Python's dictionary hashing
* Python's sort
* Game trees
* Introduction
* Game rules (tic-tac-toe; or maybe hnefatafl, a game I did for a school project once)
* Simple tree search
* Alpha/beta cutoff
Possible Topics
--------------------
* Heaps (can't figure out where to fit them in -- any suggestions?)
Not Covered
------------------
* Linked lists (they aren't the source of many algorithms,
and their implementation in Python doesn't seem very interesting)
