Planet Scheme

June 24, 2016

Programming Praxis

Phone Numbers And Prime Factors

John Cook is a mathematician and programmer who runs a fascinating blog that I frequent.

Cook recently had an article about the prime factors of telephone numbers. He explained that, for 10-digit telephone numbers as used in the United States, the average number of distinct prime factors is 3.232 and the distribution is between 1 and 5 distinct prime factors about 73% of the time.

Your task is to write a function that determines the number of distinct prime factors of a number, and use that function to explore the distribution of the number of distinct prime factors in a range of telephone numbers. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at June 24, 2016 09:00 AM

June 21, 2016

Programming Praxis

Two Interview Questions

I like to read a web site called Career Cup, both to enjoy solving some of the programming exercise given there and to find exercise for Programming Praxis. As I write this exercise, here are the two most recent exercises on Career Cup:

  • Given a function rand2 that returns 0 or 1 with equal probability, write a function rand3 that returns 0, 1 or 2 with equal probability, using only rand2 as a source of random numbers.
  • Given a set of characters and a dictionary of words, find the shortest word in the dictionary that contains all of the characters in the set. In case of a tie, return all the words of the same (shortest) length.

Your task is to write the two programs described above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at June 21, 2016 09:00 AM

June 19, 2016

Guile News

GNU Guile 2.1.3 released (beta)

We are delighted to announce GNU Guile release 2.1.3, the next pre-release in what will become the 2.2 stable series.

This release rewrites the ports facility to better support non-blocking concurrent input and output. See the newly rewritten "Input and Output" section of the manual for all details, and the release announcement for a download link.

by Andy Wingo at June 19, 2016 09:53 AM

June 17, 2016

Programming Praxis

Tomohiko Sakamoto’s Day-Of-Week Algorithm

Here is Sakamoto’s algorithm for calculating the day of the week, taken from the comment that introduces the code:

Jan 1st 1 AD is a Monday in Gregorian calendar.
So Jan 0th 1 AD is a Sunday [It does not exist technically].

Every 4 years we have a leap year. But xy00 cannot be a leap unless xy divides 4 with reminder 0.

y/4 – y/100 + y/400 : this gives the number of leap years from 1AD to the given year. As each year has 365 days (divdes 7 with reminder 1), unless it is a leap year or the date is in Jan or Feb, the day of a given date changes by 1 each year. In other case it increases by 2.

y -= m So y + y/4 – y/100 + y/400 gives the day of Jan 0th (Dec 31st of prev year) of the year. (This gives the reminder with 7 of the number of days passed before the given year began.)

Array t: Number of days passed before the month ‘m+1’ begins.

So t[m-1]+d is the number of days passed in year ‘y’ upto the given date.

(y + y/4 – y/100 + y/400 + t[m-1] + d) % 7 is reminder of the number of days from Jan 0 1AD to the given date which will be the day (0=Sunday,6=Saturday).

int dow(int y, int m, int d) {
static int t[] = {0, 3, 2, 5, 0, 3, 5, 1, 4, 6, 2, 4};
y -= m < 3;
return (y + y/4 - y/100 + y/400 + t[m-1] + d) % 7;
}

Another description is given here.

Your task is to write a program that implements the day-of-week algorithm shown above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at June 17, 2016 09:00 AM

June 16, 2016

Ben Simon

Got Data? Adventures in Virtual Crystal Ball Creation

There's two ways to look at this recent Programming Praxis exercise: implementing a beginner level statistics function or creating a magical crystal ball that can predict past, present and future! I chose to approach this problem with the mindset of the latter. Let's make some magic!

The algorithm we're tackling is linear regression. I managed to skip statistics in college (what a shame!), so I don't recall ever being formally taught this technique. Very roughly, if you have the right set of data, you can plot a line through it. You can then use this line to predict values not in the data set.

The exercise gave us this tiny, manufactured, data set:

x    y
60   3.1
61   3.6
62   3.8
63   4.0
65   4.1

With linear regression you can answer questions like: what will be the associated value for say, 64, 66 or 1024 be? Here's my implementation in action:

A few words about the screenshot above. You'll notice that I'm converting my data from a simple list to a generator. A generator in this case is a function that will return a single element in the data set, and returns '() when all the data has been exhausted. I chose to use a generator over a simple list because I wanted to allow this solution to scale to large data sets.

Below you'll see a data set that's stored in a file and leverages a file based generator to access its contents. So far, I haven't throw a large data set at this program, but I believe it should scale without issue.

The call to make-crystal-ball performs the linear-regression and returns back a function that when provided x returns a guess prediction for y. What? I'm trying to have a bit of fun here.

Looking around on web I found this example that uses a small, but real data set. In this case, it compares High School and College GPA. Using linear-regression we're able to predict how a High School student with a 2.7, 3.0 or 3.5 GPA is going to do in college. Here's the code:

;;  http://onlinestatbook.com/2/regression/intro.html
;;  http://onlinestatbook.com/2/case_studies/sat.html
(define (gpa-sat-test)
  (define data (file->generator "gpa-sat-data.scm"
                                (lambda (high-gpa math-sat verb-sat comp-gpa univ-gpa)
                                  (list high-gpa univ-gpa))))
  (let ((ball (make-crystal-ball data)))
    (show "2.7 => " (ball 2.7))
    (show "3.0 => " (ball 3))
    (show "3.5 => " (ball 3.5))))

And the answer is: 2.91, 3.12 and 3.45 respectively. So yeah, that's good news if you were dragging in High School, you're GPA should climb a bit. But High School overachievers should beware, your GPA is most likely to dip. D'oh.

Below is my implementation of this solution. You can also find it on github. As usual I find myself preaching the benefits of Scheme. The code below was written on my Galaxy Note 5 using Termux, emacs and Tinyscheme. With relative ease I was able to implement a generator framework that works for both lists and data files. I'm also able to leverage the Scheme reader so that the data file format is trivial to operate on. Finally, I wrote a generic sigma function that walks through my data set once, but performs all the various summations I'll need to calculate the necessary values. In other words, I feel like I've got an elegant solution using little more than lists, lambda functions and sexprs. It's beautiful and should be memory efficient.

Here's the code:

;; https://programmingpraxis.com/2016/06/10/linear-regression/

(define (show . args)
  (for-each (lambda (arg)
       (display arg)
       (display " "))
     args)
  (newline))

(define (as-list x)
  (if (list? x) x (list x)))

(define (g list index)
  (list-ref list index))

(define (make-list n seed)
  (if (= n 0) '()
      (cons seed (make-list (- n 1) seed))))

(define (list->generator lst)
  (let ((remaining lst))
    (lambda ()
      (cond ((null? remaining) '())
     (else
      (let ((x (car remaining)))
        (set! remaining (cdr remaining))
        x))))))

(define (file->generator path scrubber)
  (let ((port (open-input-file path)))
    (lambda ()
      (let ((next (read port)))
        (if (eof-object? next) '() (apply scrubber next))))))

(define (sigma generator . fns)
  (define (update fns sums data)
    (let loop ((fns fns)
        (sums sums)
        (results '()))
      (cond ((null? fns) (reverse results))
     (else
      (let ((fn (car fns))
     (a  (car sums)))
        (loop (cdr fns)
       (cdr sums)
       (cons (+ a (apply fn (as-list data)))
      results)))))))

    (let loop ((data (generator))
        (sums (make-list (length fns) 0)))
    (if (null? data) sums
 (loop (generator)
       (update fns sums data)))))

;; Magic happens here:
;; m = (n × Σxy − Σx × Σy) ÷ (n × Σx2 − (Σx)2)
;; b = (Σy − m × Σx) ÷ n
(define (linear-regression data)
  (let ((sums (sigma data
        (lambda (x y) (* x y))
        (lambda (x y) x)
        (lambda (x y) y)
        (lambda (x y) (* x x))
        (lambda (x y) 1))))
    (let* ((Sxy (g sums 0))
    (Sx  (g sums 1))
    (Sy  (g sums 2))
    (Sxx (g sums 3))
    (n   (g sums 4)))
      (let* ((m (/ (- (* n Sxy) (* Sx Sy))
     (- (* n Sxx) (* Sx Sx))))
      (b (/ (- Sy (* m Sx)) n)))
 (cons m b)))))

(define (make-crystal-ball data)
  (let* ((lr (linear-regression data))
         (m  (car lr))
         (b  (cdr lr)))
    (lambda (x)
      (+ (* m x) b))))

;; Playtime
(define (test)
  (define data (list->generator '((60   3.1)
      (61   3.6)
      (62   3.8)
      (63   4.0)
      (65   4.1))))
  (let ((ball (make-crystal-ball data)))
    (show (ball 64))
    (show (ball 66))
    (show (ball 1024))))

;; From:
;;  http://onlinestatbook.com/2/regression/intro.html
;;  http://onlinestatbook.com/2/case_studies/sat.html
(define (gpa-sat-test)
  (define data (file->generator "gpa-sat-data.scm"
                                (lambda (high-gpa math-sat verb-sat comp-gpa univ-gpa)
                                  (list high-gpa univ-gpa))))
  (let ((ball (make-crystal-ball data)))
    (show "2.7 => " (ball 2.7))
    (show "3.0 => " (ball 3))
    (show "3.5 => " (ball 3.5))))

(gpa-sat-test)

by Ben Simon (noreply@blogger.com) at June 16, 2016 04:35 PM

June 14, 2016

Programming Praxis

Duplicate Items In An Array

Today’s exercise is in two parts, first a commonly-seen programming exercise and then a variant on it; the origin of the exercise is certainly someone’s homework, but since school is out for the year it doesn’t matter that we do the exercise today.

First, write a program that, given an array of integers in unsorted order, finds the single duplicate number in the array. For instance, given the input [1,2,3,1,4], the correct output is 4.

Second, write a program that, given an array of integers in unsorted order, finds all of the multiple duplicate numbers in the array. For instance, given the input [1,2,3,1,2,4,1], the correct output is [1,2,1].

Your task is to write the two programs that find duplicates in an array. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at June 14, 2016 09:00 AM

June 10, 2016

Programming Praxis

Linear Regression

Linear regression is a widely-used statistical technique for relating two sets of variables, traditionally called x and y; the goal is to find the line-of-best-fit, y = m x + b, that most closely relates the two sets. The formulas for computing the line of best fit are:

m = (n × Σxy − Σx × Σy) ÷ (n × Σx2 − (Σx)2)

b = (Σym × Σx) ÷ n

You can find those formulas in any statistics textbook. As an example, given the sets of variables

x    y
60   3.1
61   3.6
62   3.8
63   4.0
65   4.1

the line of best fit is y = 0.1878 x − 7.9635, and the estimated value of the missing x = 64 is 4.06.

Your task is to write a program that calculates the slope m and intercept b for two sets of variables x and y. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at June 10, 2016 09:00 AM

June 07, 2016

Programming Praxis

Goldbach’s Other Conjecture

Christian Goldbach (1690-1764) was a Prussian mathematician and contemporary of Euler. One of the most famous unproven conjectures in number theory is known as Goldbach’s Conjecture, which states that every even number greater than two is the sum of two prime numbers; for example, 28 = 5 + 23. We studied Goldbach’s Conjecture in a previous exercise.

Although it is not as well known, Goldbach made another conjecture as follows: Every odd number greater than two is the sum of a prime number and twice a square; for instance, 27 = 19 + 2 * (2 ** 2). (The conjecture is sometimes stated as every odd composite number is the sum of a prime number and twice a square, since it is trivially true with 0 as the square root for all prime numbers; alternately, it is sometimes limited so that the number being squared must be positive, in which case there are some odd primes that can be so expressed.) Sadly, it is easy to find a counter-example that disproves Goldbach’s other conjecture.

Your task is to write a program that finds the smallest number that disproves Goldbach’s other conjecture. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at June 07, 2016 09:00 AM

June 03, 2016

Programming Praxis

A Dozen Lines Of Code

Today’s exercise demonstrates that it is sometimes possible to do a lot with a little.

Your task is to write some interesting and useful program in no more than a dozen lines of code. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at June 03, 2016 09:00 AM

May 31, 2016

Programming Praxis

Learn A New Language

It’s fun to learn new programming languages. It’s also useful, even if you never use the new language, because it forces you to think differently about how you do things.

Your task is to write a familiar program in an unfamiliar language. When you are finished, you are welcome to read or run ([1], [2]) a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at May 31, 2016 12:00 PM

May 27, 2016

Programming Praxis

Pollard’s Rho Algorithm For Discrete Logarithms

We studied discrete logarithms in two previous exercises. Today we look at a third algorithm for computing discrete algorithms, invented by John Pollard in the mid 1970s. Our presentation follows that in the book Prime Numbers: A Computational Perspective by Richard Crandall and Carl Pomerance, which differs somewhat from other sources.

Our goal is to compute l (some browsers mess that up; it’s a lower-case ell, for “logarithm”) in the expression glt (mod p); here p is a prime greater than 3, g is an integer generator on the range 1 ≤ g < p, and t is an integer target on the range 1 ≤ g < p. Pollard takes a sequence of integer pairs (ai, bi) modulo (p − 1) and a sequence of integers xi modulo p such that xi = tai gbi (mod p), beginning with a0 = b0 = 0 and x0 = 1. Then the rule for deriving the terms of the various sequences is:

  • If 0 < xi < p/3, then ai+1 = (ai + 1) mod (p − 1), bi+1 = bi, and xi+1 = t xi (mod p).
  • If p/3 < xi < 2p/3, then ai+1 = 2 ai mod (p − 1), bi+1 = 2 bi mod (p − 1), and xi+1 = xi2 mod p.
  • If 2p/3 < xi < p, then ai+1 = ai, bi+1 = (bi + 1) mod (p − 1), and xi+1 = g xi mod p.

Splitting the computation into three pieces “randomizes” the calculation, since the interval in which xi is found has nothing to do with the logarithm. The sequences are computed until some xj = xk, at which point we have taj gbj = tak gbk. Then, if ajaj is coprime to p − 1, we compute the discrete logarithm l as (ajak) lbkbj (mod (p − 1)). However, if the greatest common divisor of ajaj with p − 1 is d > 1, then we compute (ajak) l0bkbj (mod (p − 1) / d), and l = l0 + m (p − 1) / d for some m = 0, 1, …, d − 1, which must all be checked until the discrete logarithm is found.

Thus, Pollard’s rho algorithm consists of iterating the sequences until a match is found, for which we use Floyd’s cycle-finding algorithm, just as in Pollard’s rho algorithm for factoring integers. Here are outlines of the two algorithms, shown side-by-side to highlight the similarities:

# find d such that d | n      # find l such that g**l = t (mod p)
function factor(n)            function dlog(g, t, p)
  func f(x) := (x*x+c) % n      func f(x,a,b) := ... as above ...
  t, h, d := 1, 1, 1            j := (1,0,0); k := f(1,0,0)
  while d == 1                  while j.x <> k.x
    t = f(t)                      j(x,a,b) := f(j.x, j.a, j.b)
    h = f(f(h))                   k(x,a,b) := f(f(k.x, k.a, k.b))
    d = gcd(t-h, n)             d := gcd(j.a-k.a, p-1)
  return d                      return l ... as above ...

Please pardon some abuse of notation; I hope the intent is clear. In the factoring algorithm, it is possible that d is the trivial factor n, in which case you must try again with a different constant in the f function; the logarithm function has no such possibility. Most of the time consumed in the computation is the modular multiplications in the calculations of the x sequence; the algorithm itself is O(sqrt p), the same as the baby-steps, giant-steps algorithm of a previous exercise, but the space requirement is only a small constant, rather than the O(sqrt p) space required of the previous algorithm. In practice, the random split is made into more than 3 pieces, which complicates the code but speeds the computation, as much as a 25% improvement on average.

Your task is to write a program that computes discrete logarithms using Pollard’s rho algorithm. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at May 27, 2016 12:59 PM

May 24, 2016

Programming Praxis

Test Scores

The high school two blocks from me just had their annual picnic, my youngest daughter just graduated from college, and my primarily academic readership suddenly dropped in half (history suggest it will stay low until mid-August), so it seems to be the right season to have a simple data-processing task involving student test scores.

Given a list of student names and test scores, compute the average of the top five scores for each student. You may assume each student has as least five scores.

Your task is to compute student scores as described above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at May 24, 2016 09:00 AM

May 20, 2016

Programming Praxis

No Exercise Today

I’ve been busy at work and haven’t had time to prepare an exercise for today. I apologize.

Your task is to solve a previous exercise that you haven’t yet solved. Have fun!


by programmingpraxis at May 20, 2016 09:00 AM

May 17, 2016

Programming Praxis

Conditional Heap Insertion

This is an Amazon interview question:

Given a heap (priority queue), insert an element into the heap if the element is not already present in the heap. Your solution must work in O(n) time, where n is the number of items in the heap.

Your task is to write a program to insert an element into a heap if the element is not already present in the heap, in logarithmic time. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at May 17, 2016 09:00 AM

May 13, 2016

Programming Praxis

Interleaved Increasing-Decreasing Sort

This must be somebody’s homework:

Given an array of integers, rearrange the elements of the array so that elements in even-indexed positions are in ascending order and elements in odd-indexed positions are in descending order. For instance, given the input 0123456789, the desired output is 0927456381, with the even-indexed positions in ascending order 02468 and the odd-indexed positions in descending order 97531.

Your task is to write a program that performs the indicated rearrangement of its input. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


by programmingpraxis at May 13, 2016 09:00 AM