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<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Purely Functional Data Structures</title>
<meta name="author" content="Dan Rosen">
<meta name="apple-mobile-web-app-capable" content="yes" />
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent" />
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<link rel="stylesheet" href="reveal.js/css/reveal.min.css">
<link rel="stylesheet" href="reveal.js/css/theme/mergeconflict.css" id="theme">
<!-- For syntax highlighting -->
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<audio id="bzzzt"><source src="bzzzt.mp3" type="audio/mpeg"></audio>
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<script>
document.write( '<link rel="stylesheet" href="reveal.js/css/print/' + ( window.location.search.match( /print-pdf/gi ) ? 'pdf' : 'paper' ) + '.css" type="text/css" media="print">' );
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<![endif]-->
</head>
<body>
<div class="reveal">
<!-- Any section element inside of this container is displayed as a slide -->
<div class="slides">
<section>
<h1>Purely <b>Functional</b> Data Structures</h1>
<hr>
<h3>Dan Rosen, Twitter</h3>
<h4 style="font-weight: 300;"><a href="https://twitter.com/mergeconflict">@mergeconflict</a></h4>
</section>
<section>
<h2>Mutability</h2>
</section>
<section>
<h2>Boring example</h2>
<h3 style="font-weight: 300;">(obvious things omitted)</h3>
<pre><code class="java" data-trim>
public class Rectangle implements Comparable<Rectangle> {
private int width, height;
public Rectangle(int width, int height);
// getters and setters
public int getWidth();
public int getHeight();
public void setWidth(int width);
public void setHeight(int height);
// identity
public int hashCode();
public boolean equals(Object o);
public int compareTo(Rectangle r);
}
</code></pre>
</section>
<section>
<h2>Identity crisis</h2>
<pre><code class="java" data-trim>
// add a rectangle to a hash set
Set<Rectangle> s = new HashSet<Rectangle>();
Rectangle r = new Rectangle(2, 3);
s.add(r); // s = [Rectangle(2, 3)]
// mutate one of the rectangle's fields ...
r.width = 5; // s = [Rectangle(5, 3)]
s.size(); // 1
s.contains(r); // false
// even more fun ...
s.add(r); // s = [Rectangle(5, 3), Rectangle(5, 3)]
</code></pre>
<p class="fragment">(a similar problem occurs with tree sets)</p>
</section>
<section class="bzzzt" data-transition="zoom">
<h2 style="color: red; font-size: 900px;">🅇</h2>
</section>
<section>
<h2>Trolling Prof. Liskov</h2>
<pre><code class="java" data-trim>
public class Square extends Rectangle {
public Square(int length) {
super(length, length);
}
}
</code></pre>
</section>
<section>
<h2>Problem solved?</h2>
<pre><code class="java" data-trim>
public class Square extends Rectangle {
public Square(int length) {
super(length, length);
}
public void setLength(int length) {
super.setWidth();
super.setHeight();
}
@Override public void setWidth(int width) { setLength(width); }
@Override public void setHeight(int height) { setLength(height); }
}
</code></pre>
</section>
<section>
<h2>Hidden invariants!</h2>
<pre><code class="java" data-trim>
@DataPoints
public static void Rectangle[] rectangles(); // may contain squares!
@Theory
public void setWidthMustNotSetHeight(Rectangle r) {
// get an arbitrary rectangle and remember its height
int expectedHeight = r.getHeight();
// change the rectangle's width, and ...
r.setWidth(0);
assertEquals(r.getHeight(), expectedHeight);
}
</code></pre>
</section>
<section class="bzzzt" data-transition="zoom">
<h2 style="color: red; font-size: 700px;">🅇🅇</h2>
</section>
<section>
<h2>More fun</h2>
<pre><code class="java" data-trim>
public class RectangleManager {
private final Set<Rectangle> rectangles = new HashSet<Rectangle>();
public Set<Rectangle> getRectangles() {
return rectangles;
}
}
</code></pre>
</section>
<section>
<h2>Somewhat less mutable</h2>
<pre><code class="java" data-trim>
public Set<Rectangle> getRectangles() {
return Collections.unmodifiableSet(rectangles); // ugh
}
</code></pre>
</section>
<section>
<h2>YOLO</h2>
<pre><code class="java" data-trim>
public synchronized Set<Rectangle> getRectangles() {
Set<Rectangle> copy = new HashSet<Rectangle>();
for (Rectangle r : rectangles) {
copy.add(new Rectangle(r));
}
return copy;
}
</code></pre>
</section>
<section class="bzzzt" data-transition="zoom">
<h2 style="color: red; font-size: 500px;">🅇🅇🅇</h2>
</section>
<section>
<h2>Problems</h2>
<ul>
<li class="fragment">Unstable identity (as viewed by containing objects)</li>
<li class="fragment">Very difficult to satisfy superclass behavioral contracts in subclasses</li>
<li class="fragment">Defensive copying and synchronization often necessary</li>
</ul>
</section>
<section>
<h2>Immutability</h2>
</section>
<section>
<section>
<h2>Feels good*, man</h2>
<pre><code class="java" data-trim>
public class Rectangle implements Comparable<Rectangle> {
private final int width, height;
public Rectangle(int width, int height);
// getters
public int getWidth();
public int getHeight();
// identity
public int hashCode();
public boolean equals(Object o);
public int compareTo(Rectangle r);
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
// setters
public Rectangle setWidth(int width) {
return new Rectangle(width, this.height);
}
public Rectangle setHeight(int height) {
return new Rectangle(this.width, height);
}
</code></pre>
</section>
</section>
<section>
<h2>Real talk</h2>
<ul>
<li>Basic approach: final all the way down</li>
<li>Whenever an update is needed, copy</li>
<li>Let the garbage collector deal with it</li>
</ul>
</section>
<section>
<h2>Stack interface</h2>
<pre><code class="java" data-trim>
public class Stack<E> {
public Stack<E> push(E element); // return a copy
public Stack<E> pop(); // return a copy
public E top();
public boolean isEmpty();
public static <E> Stack<E> empty(); // factory
}
</code></pre>
</section>
<section>
<section data-transition="none">
<pre><code class="java" data-trim>
Stack<Integer> empty = Stack.empty();
</code></pre>
<p><img class="diagram" src="stack/empty.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Stack<Integer> a = empty.push(1);
</code></pre>
<p><img class="diagram" src="stack/a.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Stack<Integer> b = a.push(2).push(3);
</code></pre>
<p><img class="diagram" src="stack/ab.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Stack<Integer> c = b.pop().push(4);
</code></pre>
<p><img class="diagram" src="stack/abc.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Stack<Integer> d = a.push(5);
</code></pre>
<p><img class="diagram" src="stack/abcd.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
b = b.pop(); // 3 can be GC'ed
d = d.pop(); // 5 can be GC'ed
</code></pre>
<p><img class="diagram" src="stack/pop.png"></p>
</section></section>
<section>
<section>
<h2>Stack implementation</h2>
<pre><code class="java" data-trim>
public class Stack<E> {
private final E head;
private final Stack<E> tail;
private Stack() {
head = null;
tail = null;
}
private Stack(E head, Stack<E> tail) {
this.head = head;
this.tail = tail;
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
public Stack<E> push(E element) {
return new Stack<E>(element, this);
}
public Stack<E> pop() {
return tail;
}
public E top() {
return head;
}
public boolean isEmpty() {
return tail == null;
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
private static final Stack<Object> EMPTY = new Stack<Object>();
@SuppressWarnings("unchecked")
public static <E> Stack<E> empty() {
return (Stack<E>) EMPTY;
}
</code></pre>
</section>
</section>
<section>
<h2>"Persistent" data structures</h2>
<ul>
<li>Common substructure can be shared between copies</li>
<li>Somewhat similar to copy-on-write, but easier and more efficient</li>
<li>Also note: stable identity! We could even memoize hashCode...</li>
</ul>
</section>
<section>
<img src="whatever.gif" style="min-height: 70%">
<h2>Whatever</h2>
</section>
<section>
<h2>Queue interface</h2>
<pre><code class="java" data-trim>
public class Queue<E> {
public Queue<E> enqueue(E element); // return a copy
public Queue<E> dequeue(); // return a copy
public E front();
public boolean isEmpty();
public static <E> Queue<E> empty(); // factory
}
</code></pre>
</section>
<section>
<h2>Double the fun?</h2>
<ul>
<li class="fragment">Stack: a singly-linked list, with a reference to the top</li>
<li class="fragment">Queue: a doubly-linked list, with references to the back and front?</li>
</ul>
</section>
<section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> a = /* ... */;
</code></pre>
<p><img class="diagram" src="doubly-linked-queue/a.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> b = a.dequeue();
</code></pre>
<p><img class="diagram" src="doubly-linked-queue/ab1.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> b = a.dequeue();
</code></pre>
<p><img class="diagram" src="doubly-linked-queue/ab2.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> b = a.dequeue();
</code></pre>
<p><img class="diagram" src="doubly-linked-queue/ab3.png"></p>
</section>
</section>
<section>
<div style="display: inline-block;">
<h2>Not quite</h2>
<ul style="display: block;">
<li class="fragment">All enqueue/dequeue operations require complete copies</li>
<li class="fragment">Persistent—previous copies are preserved—but not efficient</li>
</ul>
<div class="fragment">
<hr>
<h2>But close</h2>
</div>
<ul style="display: block;">
<li class="fragment">Two stacks: incoming and outgoing</li>
<li class="fragment">Push to incoming stack, pop from outgoing stack</li>
<li class="fragment">Invariant: outgoing stack must be non-empty if queue is non-empty</li>
</ul>
</div>
</section>
<section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> empty = Queue.empty();
</code></pre>
<p><img class="diagram" src="naive-queue/empty.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> a = empty.enqueue(1);
</code></pre>
<p><img class="diagram" src="naive-queue/a.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> b = a.enqueue(2).enqueue(3).enqueue(4);
</code></pre>
<p><img class="diagram" src="naive-queue/ab.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> c = b.dequeue();
</code></pre>
<p><img class="diagram" src="naive-queue/bc.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> d = c.dequeue().enqueue(5);
</code></pre>
<p><img class="diagram" src="naive-queue/cd.png"></p>
</section>
</section>
<section>
<h2>Prerequisite</h2>
<pre><code class="java" data-trim>
public Stack<E> reverse() {
Stack<E> in = this, out = empty();
while (!in.isEmpty()) {
out = out.push(in.head);
in = in.tail;
}
return out;
}
</code></pre>
</section>
<section>
<section>
<h2>Queue implementation</h2>
<pre style="font-size: 80%">
<code class="java" data-trim>
public class Queue<E> {
private final Stack<E> in, out;
private Queue() {
in = Stack.empty();
out = Stack.empty();
}
private Queue(Stack<E> in, Stack<E> out) {
if (out.isEmpty()) {
this.in = Stack.empty();
this.out = in.reverse();
} else {
this.in = in;
this.out = out;
}
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
public Queue<E> enqueue(E element) {
return new Queue<E>(in.push(element), out);
}
public Queue<E> dequeue() {
return new Queue<E>(in, out.pop());
}
public E front() {
return out.top();
}
public boolean isEmpty() {
return out.isEmpty();
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
private static final Queue<Object> EMPTY = new Queue<Object>();
@SuppressWarnings("unchecked")
public static <E> Queue<E> empty() {
return (Queue<E>) EMPTY;
}
</code></pre>
</section>
</section>
<section>
<div style="display: inline-block;">
<h2>Analysis</h2>
<ul style="display: block;">
<li class="fragment">Enqueue: always O(1)</li>
<li class="fragment">Dequeue: sometimes O(1), sometimes O(n)</li>
</ul>
<div class="fragment">
<hr>
<h2>Amortized analysis</h2>
</div>
<ul style="display: block;">
<li class="fragment">Assume each enqueued element will eventually be rotated</li>
<li class="fragment">Pay the cost of rotation for each element when enqueuing</li>
<li class="fragment">Enqueue: O(1), with 1 credit</li>
<li class="fragment">Dequeue: amortized O(1), debiting when needed to rotate</li>
</ul>
</div>
</section>
<section>
<h1>Microbenchmarking!</h1>
</section>
<section>
<h2>Real-time message queue</h2>
<p>Adapt for concurrent reads and writes</p>
</section>
<section>
<section>
<h2>Testbed</h2>
<pre><code class="java" data-trim>
public class CASQueue<E> {
private final AtomicReference<Queue<E>> q;
private final Semaphore nonEmpty = new Semaphore(0);
public CASQueue(Queue<E> q) {
this.q = new AtomicReference<Queue<E>>(q);
}
// compare-and-set loop
public void enqueue(E element);
public E dequeue();
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
public class LockQueue<E> {
private Queue<E> q;
private final Lock lock = new ReentrantLock();
private final Condition nonEmpty = lock.newCondition();
public LockQueue(Queue<E> q) {
this.q = q;
}
// lock/unlock, await/signal
public void enqueue(E element);
public E dequeue();
}
</code></pre>
</section>
</section>
<section>
<h2>Measurements</h2>
<img src="naive-queue/measurements.png">
</section>
<section class="bzzzt" data-transition="zoom">
<h2 style="color: red; font-size: 900px;">🅇</h2>
</section>
<section>
<h2>Theory != Practice</h2>
<ul>
<li class="fragment">Amortized analysis is unsuitable for real-time applications</li>
<li class="fragment">Need better worst-case performance</li>
<li class="fragment">Same approach, new invariant: stacks can be at most 3 elements long</li>
</ul>
</section>
<section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> empty = Queue.empty();
</code></pre>
<p><img class="diagram" src="recursive-queue/empty.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> a = empty.enqueue(1);
</code></pre>
<p><img class="diagram" src="recursive-queue/a.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> b = a.enqueue(2).enqueue(3);
</code></pre>
<p><img class="diagram" src="recursive-queue/b.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> c = b.enqueue(4);
</code></pre>
<p><img class="diagram" src="recursive-queue/c.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> d = c.enqueue(5)./* ... */.enqueue(10);
</code></pre>
<p><img class="diagram" src="recursive-queue/d.png"></p>
</section>
<section data-transition="none">
<pre><code class="java" data-trim>
Queue<Integer> e = c.dequeue();
</code></pre>
<p><img class="diagram" src="recursive-queue/e.png"></p>
</section>
</section>
<section>
<img src="mind-blown.gif" style="min-height: 70%">
<h2>Mind = Blown</h2>
</section>
<section>
<h2>Prerequisite</h2>
<pre><code class="java" data-trim>
public int size() {
return isEmpty() ? 0 : tail.size() + 1;
}
</code></pre>
</section>
<section>
<section>
<h2>Queue reimplementation</h2>
<pre><code class="java" data-trim>
public class Queue<E> {
private final Stack<E> in;
private final Queue<Stack<E>> spine;
private final Stack<E> out;
@SuppressWarnings("unchecked")
private Queue() {
in = Stack.empty();
spine = (Queue<Stack<E>>) this;
out = Stack.empty();
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre style="font-size: 60%">
<code class="java" data-trim>
private Queue(Stack<E> in, Queue<Stack<E>> spine, Stack<E> out) {
if (out.isEmpty()) {
if (spine.isEmpty()) {
// case 1: same as naïve queue
this.in = Stack.empty();
this.spine = empty();
this.out = in.reverse();
} else {
// case 2: dequeue spine
this.in = in;
this.spine = spine.dequeue();
this.out = spine.front();
}
} else {
if (in.size() == 3) {
// case 3: enqueue spine
this.in = Stack.empty();
this.spine = spine.enqueue(in.reverse());
this.out = out;
} else {
// case 4: same as naïve queue
this.in = in;
this.spine = spine;
this.out = out;
}
}
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
public Queue<E> enqueue(E element) {
return new Queue<E>(in.push(element), spine, out);
}
public Queue<E> dequeue() {
return new Queue<E>(in, spine, out.pop());
}
public E front() {
return out.top();
}
public boolean isEmpty() {
return out.isEmpty();
}
</code></pre>
<h4>⥥</h4>
</section>
<section>
<h4>⥣</h4>
<pre><code class="java" data-trim>
private static final Queue<Object> EMPTY = new Queue<Object>();
@SuppressWarnings("unchecked")
public static <E> Queue<E> empty() {
return (Queue<E>) EMPTY;
}
</code></pre>
</section>
</section>
<section>
<h2>Measurements</h2>
<img src="recursive-queue/measurements.png">
</section>
<section data-transition="zoom">
<img src="success.jpg">
<h2>Success.</h2>
</section>
<section>
<h1><b>How</b> did we get here?</h1>
</section>
<section>
<h2>Problem 1: Unstable identity</h2>
<p>Solved, trivially.</p>
</section>
<section>
<h2>Problem 2: Behavioral subtyping</h2>
<p>Not solved, but making state change visible (by returning copies) helps.</p>
</section>
<section>
<h2>Problem 3: Defensive copying</h2>
<p>Solved efficiently with persistent data structures.</p>
</section>
<section>
<h2>Point being</h2>
<ul>
<li class="fragment"><b>Not</b> recommending persistent queues for messaging (use Disruptor!)</li>
<li class="fragment">But performance is <b>very good</b> for general use</li>
<li class="fragment">Very efficient, persistent implementations exist for:
<ul>
<li>Random-access vectors (bit-mapped vector tries)</li>
<li>Unordered sets and maps (hash array mapped tries)</li>
<li>Ordered sets and maps (persistent red-black trees)</li>
<li>etc.</li>
</ul>
</li>
<li class="fragment">I'd like to see these in the Java standard library</li>
</ul>
</section>
<section>
<h1><b>Thank you</b></h1>
</section>
</div>
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