Purely Functional Data Structures & Algorithms : Red-Black Trees in Qi

Update 2011/06/28 : Source has been modified to compile with Shen

This is the first in a series of posts that will demonstrate the implementation of many well-known(and less known) data structures and algorithms using a purely functional approach.
We will use Qi as our implementation language for a number of reasons :

    It’s a Lisp : macros, EVAL, hash-tables, property-lists, meta-programming etc.
    Pattern matching.
    Optional static type checking.
    A Turing-complete type system !

In this first post we look at an implementation of the well-known Red-Black tree abstract data type in Qi.

A red–black tree is a type of self-balancing binary search tree, a data structure used in computer science, typically to implement associative arrays. The original structure was invented in 1972 by Rudolf Bayer and named “symmetric binary B-tree,” but acquired its modern name in a paper in 1978 by Leonidas J. Guibas and Robert Sedgewick. It is complex, but has good worst-case running time for its operations and is efficient in practice: it can search, insert, and delete in O(log n) time, where n is total number of elements in the tree. Put very simply, a red–black tree is a binary search tree that inserts and removes intelligently, to ensure the tree is reasonably balanced.

Our implementation comes in at 57 lines of code (with the balance function at only 7 lines)

(tc +)

(datatype tree-node
    Key : number; Val : B;
    ======================
    [Key Val] : tree-node;)

(datatype color
    if (element? Color [red black])
    _______________________________
    Color : color;)

(datatype tree
    if (empty? Tree)
    ________________
    Tree : tree;

    Color : color; LTree : tree; TreeNode : tree-node; RTree : tree;
    ================================================================
    [Color LTree TreeNode RTree] : tree;)

(define node-key
    {tree-node --> number}
    [Key Val] -> Key)

(define make-tree-black
    {tree --> tree}
    [Color A X B] -> [black A X B])

(define member
    {tree-node --> tree --> boolean}
    X NIL -> false
    X [Color A Y B] -> (if (< (node-key X) (node-key Y))
         (member X A)
         (if (< (node-key Y) (node-key X))
             (member X B)
             true)))

(define balance
    {tree --> tree}
    [black [red [red A X B] Y C] Z D] -> [red [black A X B] Y [black C Z D]]
    [black [red A X [red B Y C]] Z D] -> [red [black A X B] Y [black C Z D]]
    [black A X [red [red B Y C] Z D]] -> [red [black A X B] Y [black C Z D]]
    [black A X [red B Y [red C Z D]]] -> [red [black A X B] Y [black C Z D]]
    S -> S)

(define insert-
    {tree-node --> tree --> tree}
    X [] -> [red [] X []]
    X [Color A Y B] -> (if (< (node-key X) (node-key Y))
                           (balance [Color (insert- X A) Y B])
                           (if (< (node-key Y) (node-key X))
                               (balance [Color A Y (insert- X B)])
                               [Color A Y B])))

(define insert
  {tree-node --> tree --> tree}
  X S -> (make-tree-black (insert- X S)))

This is a reasonably performant implementation (we haven’t even tried to optimize it yet).

(19-) (run-tests NIL)
tree: [black
       [red [black [red [] [1 1] []] [2 2] [red [] [5 5] []]] [7 7]
        [black [red [] [8 8] []] [11 11] []]]
       [14 14] [black [] [15 15] []]]
12 is a member ? false
8 is a member ? true

Creating tree with 100000 elements ...
Evaluation took:
  0.578 seconds of real time
  0.562833 seconds of total run time (0.491572 user, 0.071261 system)
  [ Run times consist of 0.160 seconds GC time, and 0.403 seconds non-GC time. ]
  97.40% CPU
  1,210,617,335 processor cycles
  168,551,696 bytes consed


Performing lookups in tree with 100000 elements ...
666 in tree ? true
Evaluation took:
  0.000 seconds of real time
  0.000044 seconds of total run time (0.000035 user, 0.000009 system)
  0.00% CPU
  86,110 processor cycles
  0 bytes consed

-1 in tree ?
Evaluation took:
  0.000 seconds of real time
  0.000024 seconds of total run time (0.000021 user, 0.000003 system)
  100.00% CPU
  46,368 processor cycles
  0 bytes consed

A comparable implementation in Java/C++ will usually run a few hundred lines of code.
All Qi code in this post is here.

Chaitin Proving Darwin

White paper : To a mathematical theory of evolution and biological creativity

We present an information-theoretic analysis of Darwin’s theory of
evolution, modeled as a hill-climbing algorithm on a fitness landscape.
Our space of possible organisms consists of computer programs, which
are subjected to random mutations. We study the random walk of in-creasing
fitness made by a single mutating organism. In two different
models we are able to show that evolution will occur and to characterize
the rate of evolutionary progress, i.e., the rate of biological creativity

For many years we have been disturbed by the fact that there is no fundamental
mathematical theory inspired by Darwin’s theory of evolution.
This is the fourth paper in a series attempting to create
such a theory.

In a previous paper we did not yet have a workable mathematical frame-work:
We were able to prove two not very impressive theorems, and then the
way forward was blocked. Now we have what appears to be a good mathematical
framework, and have been able to prove a number of theorems. Things
are starting to work, things are starting to get interesting, and there are many
technical questions, many open problems, to work on.

So this is a working paper, a progress report, intended to promote interest
in the field and get others to participate in the research. There is much to be
done.

Spring/Summer 2011 Books

Marvin Minksy’s – The Emotion Machine: Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind

Minsky argues that emotions are different ways to think that our mind uses to increase our intelligence. He challenges the distinction between emotions and other kinds of thinking. His main argument is that emotions are “ways to think” for different “problem types” that exist in the world. The brain has rule-based mechanism (selectors) that turns on emotions to deal with various problems. The book reviews the accomplishments of AI, what and why is complicated to accomplish in terms of modeling how human beings behave, how they think, how they experience struggles and pleasures. (Wikipedia)

The Moral Landscape – Sam Harris

In this explosive new book, Sam Harris tears down the wall between scientific facts and human values, arguing that most people are simply mistaken about the relationship between morality and the rest of human knowledge. Harris urges us to think about morality in terms of human and animal well-being, viewing the experiences of conscious creatures as peaks and valleys on a “moral landscape.” Because there are definite facts to be known about where we fall on this landscape, Harris foresees a time when science will no longer limit itself to merely describing what people do in the name of “morality”; in principle, science should be able to tell us what we ought to do to live the best lives possible.

Bringing a fresh perspective to age-old questions of right and wrong, and good and evil, Harris demonstrates that we already know enough about the human brain and its relationship to events in the world to say that there are right and wrong answers to the most pressing questions of human life. Because such answers exist, moral relativism is simply false—and comes at increasing cost to humanity. And the intrusions of religion into the sphere of human values can be finally repelled: for just as there is no such thing as Christian physics or Muslim algebra, there can be no Christian or Muslim morality.

Using his expertise in philosophy and neuroscience, along with his experience on the front lines of our “culture wars,” Harris delivers a game-changing book about the future of science and about the real basis of human cooperation.

In the Plex: How Google Thinks, Works, and Shapes Our Lives – Steven Levy

Few companies in history have ever been as successful and as admired as Google, the company that has transformed the Internet and become an indispensable part of our lives. How has Google done it? Veteran technology reporter Steven Levy was granted unprecedented access to the company, and in this revelatory book he takes readers inside Google headquarters—the Googleplex—to show how Google works.

While they were still students at Stanford, Google cofounders Larry Page and Sergey Brin revolutionized Internet search. They followed this brilliant innovation with another, as two of Google’s earliest employees found a way to do what no one else had: make billions of dollars from Internet advertising. With this cash cow (until Google’s IPO nobody other than Google management had any idea how lucrative the company’s ad business was), Google was able to expand dramatically and take on other transformative projects: more efficient data centers, open-source cell phones, free Internet video (YouTube), cloud computing, digitizing books, and much more.

The key to Google’s success in all these businesses, Levy reveals, is its engineering mind-set and adoption of such Internet values as speed, openness, experimentation, and risk taking. After its unapologetically elitist approach to hiring, Google pampers its engineers—free food and dry cleaning, on-site doctors and masseuses—and gives them all the resources they need to succeed. Even today, with a workforce of more than 23,000, Larry Page signs off on every hire.

But has Google lost its innovative edge? It stumbled badly in China—Levy discloses what went wrong and how Brin disagreed with his peers on the China strategy—and now with its newest initiative, social networking, Google is chasing a successful competitor for the first time. Some employees are leaving the company for smaller, nimbler start-ups. Can the company that famously decided not to be evil still compete?

No other book has ever turned Google inside out as Levy does with In the Plex.