Swift Fun with ArraySlice

There is a lovely article by Luna An describing ArraySlice objects in more detail than I do here. It covers Swift 3 at the moment, and you should note that Swift 4 includes support for single-ended ranges, so you can create slices ala [..<count].

I found a good use for an ArraySlice while trying to find quartiles in a set of data.

The problem of finding each quartile is essentially the same problem of finding the median from three different data sets, one being the original input set, the other two being the upper and lower half of the set after removing the original median element, if it exists.

Here, I use findMedian to perform all three tasks. I found that I had to do a bit of extra work because the array slice is not indexed starting at zero. I wonder why they chose to implement slices in this way?

import Foundation

func findMedian(_ data: ArraySlice<Int>) -> (Int, Int, Int) {
    let index1 = (data.count-1)/2
    let index2 = data.count/2
    let sliceIndex1 = index1 + data.startIndex
    let sliceIndex2 = index2 + data.startIndex
    let median = (data[sliceIndex1]+data[sliceIndex2])/2
    return (sliceIndex1, sliceIndex2, median)
}

let data = [0, 1, 2, 5, 11, 17, 19, 21]

let X = data[0..<data.count]
let (q2Index1, q2Index2, Q2) = findMedian(X)
let L = data[0..<q2Index2]
let U = data[(q2Index1+1)..<data.count]
let (_, _, Q1) = findMedian(L)
let (_, _, Q3) = findMedian(U)

print(Q1)
print(Q2)
print(Q3)

 

Command Design Pattern in Swift

It is the summer of Swift.

I was perusing some wonderful design patterns from Oktawian Chojnacki, and I decided to play around with the Command pattern. The Command pattern represents commands as objects to go between a Caller, which references the commands, and a Receiver, which is referenced by the commands.

I’m reminded of a game :

RoboRally

Here, each command object operates on a Robo, telling it to make a single move. The commands are collected into a program.

//: Command - Represents commands as objects.

import Cocoa

class Robo : CustomStringConvertible {
    let name : String
    var description: String {
        return name
    }

    init(name: String) {
        self.name = name
    }
}

protocol RoboCommand {
    func execute(robo: Robo)
}

class TurnRight : RoboCommand {
    func execute(robo: Robo) {
        print("\(robo) turns right ➡️")
    }
}

class TurnLeft : RoboCommand {
    func execute(robo: Robo) {
        print("\(robo) turns left ⬅️")
    }
}

class GoForward : RoboCommand {
    func execute(robo: Robo) {
        print("\(robo) goes forward ⬆️")
    }
}

class RoboProgram {
    let robo : Robo
    var commands : [RoboCommand] = []

    init(robo: Robo) {
        self.robo = robo
    }

    init(robo: Robo, commands: [RoboCommand]) {
        self.robo = robo
        self.commands = commands
    }

    func execute() {
        for command in commands {
            command.execute(robo: robo)
        }
    }
}

let robo = Robo(name: "Mikey")
let program = RoboProgram(robo: robo)
program.commands.append(GoForward())
program.commands.append(TurnLeft())
program.commands.append(GoForward())
program.commands.append(GoForward())
program.commands.append(TurnRight())
program.commands.append(GoForward())

program.execute()

The first draft of this code had the Commands store their target Robo in a property. I realized a problem with this in that my program would accept commands for any Robo, when only one Robo belongs to the program. My solution for this was to give control of the command target to the program itself.

Of course, now the commands are little more than glorified functions, which can be stored in arrays in Swift anyway.

Swift Dictionary Reduce

Why is it so hard to find examples of Swift dictionary reduce operations? Examples for arrays abound :

let countme = [1, 2, 3, 4, 5]
let sumall = countme.reduce(0){$0+$1}
// sumall is now 15

See this if you’re looking for a detailed HowTo.

But dictionary is the more advanced problem. How do I reduce when I have keys and values?

The key here is to note that a dictionary yields the contents of its sequence in tuples containing pairs of values. We still have our familiar $0, $1 arguments from reduce as above, where $0 is the partial sum and $1 is the individual element value. But now $0 and $1 are both tuples, and you get to their contents through $0.0, $0.1, $1.0, and $1.1.

let namesNumbers = ["Mike":21, "Bruce":25, "Alice":27]
let (bigname,bignumber) = namesNumbers.reduce(("",0)) 
  {return ($0.0 + $1.0, $0.1 + $1.1)}

This example concatenates the strings and adds the integers, and the two separate data types just happen to use the same symbol for the two operations.

React.JS

React and Angular are popping up a lot in job interviews these days. I decided to try React, since it is rumored to be the simpler of the two. So I created the old Chinese stone game Go in a Codepen. Normally I post code and talk about it in this case, but I think I will just let the Codepen speak for itself. Here, I report my experiences.

There is a lot of learning material out there, but the best resources start out right away with productive examples that get you working quickly. This tutorial video from Traversy Video does a very good job. I used this React demo from Facebook to model my architecture after.

The least trivial part of the game was the capture algorithm. The game engine must figure out if a newly placed stone results in another stone being surrounded. To figure this out, I implemented a recursive breadcrumb algorithm. The method searches recursively in four directions, resulting in a wide descending tree, but marks visited spots on the matrix to prevent infinite recursion. If at any point an empty space (liberty, in game terms) is found, the call stack starts bubbling up with that information. The recursive call will say whether or not a liberty was ever found. If it was, I run a second method to erase all visited points on the matrix, otherwise I run a different method to restore the marked spots back to original.

Here is my reaction (?) to React :

  • JavaScript is fun for web programming in limited amounts, but Object-Oriented frameworks like ES2015, Babel, etc. inevitably make it worse. this.method.bind(this)? Yucko.

  • Hidden inherited methods are traps waiting to happen. My IDE (Codepen) did not alert me to inherited methods, so this may be my own fault for not using a JetBrains tool or some such. Pick unique method names just in case.

  • I don’t really see the payoff. I understand the principle of data flow and limiting state on classes, but a real class-driven language does this in a more flexible way without need of a framework.

I know a little bit of CSS and HTML, so I was able to decorate it a bit, so that it kind of looks like an old wood board.

Big O and friends

All software developers know about Big O. For instance, contemporary sorting algorithms usually have \mathcal{O}(n\log{}n) complexity. Something like \mathcal{O}(n\log{}n) looks like a function of a function, but it’s really a classification system.

We start by defining the complexity of an algorithm to be an estimate of the number of steps required to solve a problem. This count is going to vary by the size of the problem, and so the complexity must be a function of the problem size.

Sometimes this can result in a nasty looking functions like f(x)=3x^2+2x+7 or f(x)=5x^2-999. We say that both of these functions are in the category \mathcal{O}(x^2) because they have similar growth properties. This is called Bachmann-Landau notation.

In mathematics, this notation describes the asymptotic behavior of a function. if f(n) = \mathcal{O}(g(n)), then we have a guarantee that after a certain point, the graph of f(n) will fall below the graph of g(n), if you’re willing to rotate g(n) a bit. In computer science, we use this to understand how the performance of an algorithm will change as the size of the problem or input grows. There are actually several Bachmann-Landau notations, each of which gives us some kind of indication on the limits of the growth of a function.

Name Notation Analogy
Big O f(n) is \mathcal{O}(g(n)) f(n) \le g(n)
Little o f(n) is o(g(n)) f(n) < g(n)
Big Omega f(n) is \Omega(g(n)) f(n) \ge g(n)
Little omega f(n) is \omega(g(n)) f(n) > g(n)
Theta f(n) is \Theta(g(n)) f(n) = g(n)

Credit goes to MIT for the excellent table and analogies. The definitions of each can be found in the referenced material.

Additionally, some of these definitions can be expressed using calculus. We start with the definition of o in the material says that f(n) is o(g(n)) if :

\displaystyle \forall C > 0 , \exists k \,|\, ( x > k \rightarrow |f(n)| \le C|g(n)| )

That’s a symbolic mouthful. It means, “For all positive values of C, there exists some k such that, if x>k, the absolute value of f(n) is less than or equal to the absolute value of g(n) multiplied by C.

Separately, any calculus text will give this definition for a limit at infinity. I like Paul’s Online Calculus Notes. It says that \lim_{x\to\infty} f(x) = L if

\displaystyle \forall \epsilon > 0, \exists N \forall x>N : |f(x)-L| < \epsilon

Again in plain English, “For any positive value of \epsilon, there must exist some value for N such that for any value of x that is greater than N, this inequality holds.”

Let’s make some substitutions of variable names in the above equation. If we change out f(x) for f(x)/g(x), 0 for L, C for \epsilon, n for x, and k for N :

\displaystyle \forall C > 0, \exists k \forall n>k : |f(n)/g(n)| < C

or

\displaystyle \forall C > 0, \exists k \forall n>k : |f(n)| < C |g(n)|

This boils down to an alternative definition of o. f(n) is o(g(n)) if :

\displaystyle \lim_{x\to\infty} f(x)/g(x) = 0

This allows us to use other mathematical tricks, like L’Hopitals Rule, to figure out if two given functions follow this relationship.

Mustache

Ah, it’s nice to be free from exams and assignments. Grading is over!

A lot of small but indispensable tools belong in the software developer’s toolkit. Take Markdown, for example. I am actually writing this post using WordPress Jetpack’s new Markdown composition features. So far, so good! I use Markdown extensively for creating documents. There is only a little bit to learn, but you can create nice documents very quickly. I use Typora for MD docs on my Mac, and I highly recommend it.

Mustache is a template system embedded in several front-end web frameworks. It is very simple to use, and the whole thing can be described in one unix-style man page.

I find it is most instructive to play with frameworks interactively using playgrounds or other REPL environments, so one npm install -g mustache later I am on the way to learning it.

Mustache works by embedding tags into any type of text data, such as plain text, HTML, or Markdown. Tags are encapsulated using mustache symbols (the curly brace) like this : {{tag}}. Take this example :

I have taught the following courses. My students tell me they are {{bad_adjective}}, but {{good_adjective}}!

{{#courses}}
{{name}} : {{description}}
{{/courses}}

These tags will be replaced by variable values found in another file to follow. Some variables mark sections that can be repeated or left out all together based on their values. There values are found in a separate file.

In Mustache lingo, the above data comprises a template file. It will be formatted using variables contained in the following JSON file, called a view or hash :

{
    "good_adjective" : "fun",
    "bad_adjective" : "challenging",
    "courses" : [
        { "name" : "CPSC 110",
        "description" : "Introduction to Computing"},
        { "name" : "CPSC 125",
        "description" : "Foundations of Computer Science"},
        { "name" : "CPSC 150",
        "description" : "Computers and Programming I" },
        { "name" : "CPSC 250",
        "description" : "Computers and Programming II" },
        { "name" : "CPSC 270",
        "description" : "Data and File Structures"},
        { "name" : "CPSC 425",
        "description" : "Object Oriented Programming and Design"},
        { "name" : "CPEN 414",
        "description" : "Computer Architecture"},
        { "name" : "ENGR 213",
        "description" : "Discrete Structures for Computer Applications"}
    ]
}

The result is the following

I have taught the following courses. My students tell me they are challenging, but fun!

CPSC 110 : Introduction to Computing
CPSC 125 : Foundations of Computer Science
CPSC 150 : Computers and Programming I
CPSC 250 : Computers and Programming II
CPSC 270 : Data and File Structures
CPSC 425 : Object Oriented Programming and Design
CPEN 414 : Computer Architecture
ENGR 213 : Discrete Structures for Computer Applications

You can see how this might be more useful if the JSON object was produced by loops in JavaScript and the template data was marked-up HTML instead of simple plain text. But that’s the general idea.

Heap’s Algorithm

Any sophomore computer science student knows that there are O(n!) permutations of any given array. But do you know how to produce those permutations? I didn’t , so I Googled it.

One simple strategy is to pick an element, then pick from those remaining, and so on until a single permutation is achieved. Then just backtrack one step and change your selection, finish the permutation again, and so on. For example, if you want to permute ABCD :

  1. Select A
  2. Select B, yielding AB
  3. Select C, then D, yielding ABCD, your first permutation
  4. Back up two positions and select D, then C, yielding ABDC
  5. Back up three positions and select C, then B, then D, yeilding ACBD
  6. And so on…

This might be a pretty straightforward implementation using recursion and a running collection of element lists (several copies). This is memory and CPU intensive, so an in-place solution would be nice.

Heap’s algorithm provides an in-place swapping method to find each permutation in about one step each, so it is very efficient. It is named after B. R. Heap, not after your favorite semi-sorted tree.

The core idea is very intuitive: Hold the last element of your list while recursively permuting the other n-1 elements (one must understand recursion). Then swap that last element with one of the others, and permute n-1 elements again. This way you get every possible permutation of n-1 elements for every possible fixed element value at position n. This should get us all permutations.

The tricky part is swapping out the nth element for one of the others. You might be tempted to write a loop that picks a new element from the n-1 to swap with, but remember that we permute the n-1 elements at every step, effectively scrambling them. Take AB C as an example. The n-1 permute leaves us at BA C. Say we swap C with B, giving us CA B. The n-1 permute then takes us to AC B. We know that we should swap B with A, because that is the only choice left to sit at the fixed position. And it is at position 0, just like the previous swaps!

The six permutations produced look like this :

ABC -> BAC -> CAB -> ACB -> BCA -> CBA

Let’s apply the same swap-with-zero strategy to four elements, abbreviating the three-element permutations for clarity :

ABCD -..-> CBAD -> DBAC -..-> ABDC -> CBDA -..> DBCA -> ABCD --X..OOOPS

We started repeating permutations before finding all of them. It turns out that AB is a different kind of permutation than ABC because of the length.

To see why, start by assuming that our permutation method will, if run twice, return the same permutation it started with. (This can be proven through induction once we understand the whole algorithm, but I will leave out the proof.) When ABC is being permuted, the first two elements are permuted three times while the elements take turns at the right position. This odd number of permutations makes the first two elements unstable in the sense that, every time we swap in the nth element, they are in a different order.

But when ABCD is being permuted, the ABC scramble is being performed an even number of times (four times), so it is stable except for the swap of the nth element. It is easier to see if we look at the algorithm output right before it swaps the nth element:

BA -> AC -> CB

CBA -> CBD -> CAD -> BAD

BADC -> AECB -> EDBA -> DCAE -> CBED

CBEDA -> CBEDF -> CAEDF -> BAEDF -> BAECF -> BADCF

Notice that the odd-length scrambles are stable except for the one-character substitution, while the even-length ones are not. This is a hand-waving explanation at best, but it works to give us an intuitive understanding of the algorithm.

So, for even-length permutations, the n-1 swap is always done with the first element, while for odd-length permutations, the n-1 swap is done with the kth element, where k is a counter from position 0 to n-2 that increments every time we do this.

The recursion is still there, but with very little extra state, very few loops and conditionals. And it takes barely more than O(n!) swaps, which is close to ideal. You can find the code in all its glory on Github, but here it is for your edification. Uncomment the console.log’s to see each operation in action.

var margin = ''

function permute(list, action) {
	function swap(n,m) {
		margin += '   '
		//console.log(margin + 'swap ' + n + ' ' + m + (m==n ? ' (NOP)' : ''))
		tmp = list[n]
		list[n] = list[m]
		list[m] = tmp
		margin = margin.replace(/   $/,'')
	}

	function generate(n) {
		margin += '   '
		//console.log(margin + 'generate ' + n)
		if (n==1) action(list)
		else {
			for (var i=0; i<n; i++) {
				generate(n-1)
				//console.log(margin + 'i = ' + i)
				swap(n%2 ? 0 : i, n-1)
			}
		}
		margin = margin.replace(/   $/,'')
	}
	generate(list.length)
}

permute(['A','B','C','D','E'], function(list){console.log(list)})

 

JavaScript Puzzles

I found my new grind. Goodbye Hearthstone. Goodbye World of Warcraft.

Hello, CodeFights.

Arcade Mode was a lot of fun for me today. In The Labyrinth of Nested Loops, I found this little gem :

We define the weakness of number x as the number of positive integers smaller than x that have more divisors than x.

It follows that the weaker the number, the greater overall weakness it has. For the given integer n, you need to answer two questions:

  • what is the weakness of the weakest numbers in the range [1, n]?
  • how many numbers in the range [1, n] have this weakness?

Return the answer as an array of two elements, where the first element is the answer to the first question, and the second element is the answer to the second question.

What fun!

This feels like I could brute-force it with a nested loop is and suffer the O(n^2) performance. If I manage to solve it using only sequential loops on the other hand, it would be O(n) no matter how many loops I used. I think I scored somewhere in between, maybe O(n \log(n)) depending on what you believe about the inner loops in the following sections.

The problem can be decomposed by first finding the number of divisors, then counting weakness counts (not a typo – counts of counts!), and finally finding the weakest among those.

Finding divisors can be done in many ways. I use variable-stride loops to increment every spot where a given number is a divisor.

    // Find the number of divisors 
    divisors = new Array(n+1)
    divisors.fill(0)
    for (var i=1; i<=n; i++)
        for (var j=i; j<=n; j+=i)
            divisors[j]++

Next, I want a count of all numbers that have a given number of divisors. This segment of code will end with a variable called divisorsCounts, where divisorsCounts[3] = 4 if there are exactly 4 numbers in range that have 3 divisors.

    // Count the counts
    divisorsCounts = new Array(n) // probably overkill
    divisorsCounts.fill(0)
    // Compute the weakness
    weakness = new Array(n+1)
    weakness.fill(0)
    greatestWeakness = 0

The array divisorsCounts does not need to be that large, but it’s only memory, right? Apparently something called the Divisor Function \sigma_0(i) has an upper limit of n/2. I’m no number theorist, but I buy it. A factor of two is not even worth typing Math.floor() for, so I’ll leave the code alone as it now looks clean.

    for (var me=1; me<=n; me++) {
        divisorsCounts[divisors[me]]++
        for (var j=divisorsCounts.length-1; j>divisors[me]; j--)
            weakness[me] += divisorsCounts[j]
    }

It is key here that I fill the divisorsCounts at the same time as I calculate weakness. At any given point in time I have only counted the divisors up to the number I am examining. For a given number m, weakness is the number of divisor counts greater than divisors[m]found in numbers smaller than m.

I can optimize these loops by shrinking divisorsCount. JavaScript creates sparse arrays by default, and their length is determined by the highest-indexed element. The inner nested loop will be shortened.

    // Count the counts
    divisorsCounts = []
    // Compute the weakness
    weakness = new Array(n+1)
    weakness.fill(0)
    greatestWeakness = 0
    
    for (var me=1; me<=n; me++) {
        divisorsCounts[divisors[me]] = 
            (divisorsCounts[divisors[me]] | 0) + 1
        for (var j=divisorsCounts.length-1; j>divisors[me]; j--)
            weakness[me] = (divisorsCounts[j] | 0) + weakness[me]
    }

The pipes handle undefined elements (note that x + undefined will never produce a number). With n=500, divisorsCount is only 25, so this may improve performance quite a bit.

Finally, I must track the greatestWeakness and how many times it is found.

Here it is in all of its beauty.

function weakNumbers(n) {
    // Find the number of divisors 
    divisors = new Array(n+1)
    divisors.fill(0)
    for (var i=1; i<=n; i++)
        for (var j=i; j<=n; j+=i)
            divisors[j]++
            
    // Count the counts
    divisorsCounts = []
    // Compute the weakness
    weakness = new Array(n+1)
    weakness.fill(0)
    greatestWeakness = 0
    greatestWeaknessCount = 0
    
    for (var me=1; me<=n; me++) {
        divisorsCounts[divisors[me]] = 
            (divisorsCounts[divisors[me]] | 0) + 1
        for (var j=divisorsCounts.length-1; j>divisors[me]; j--)
            weakness[me] = (divisorsCounts[j] | 0) + weakness[me]
        if (greatestWeakness < weakness[me]) {
            greatestWeakness = weakness[me]
            greatestWeaknessCount = 1
        } else if (greatestWeakness == weakness[me]) {
            greatestWeaknessCount++
        }
    }
    
    return [greatestWeakness, greatestWeaknessCount]
}

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