"Pearls of Functional Algorithm Design", by Richard Bird. Cambridge University Press. 2010.
Inspirational in it's succinct and clear way of presenting the material.
No wonder: It is a collection of some of the pearls published in the 20-years period (1990-2010) in the Journal of Functional
P1: Smallest free number -- Given a list of natural numbers, find the smallest number NOT in the list, in LINEAR time --
Array -- Easy.
-- We go over the list xs, and mark in an auxiliary array the numbers that are between [0..length xs]. Then, finding
the firsy non-appearing number in this aux-array.
p1.hs (opens in new window).
P2: Surpassing numbers -- Given a list [a] of (Ord a), for each element define as 'surpasser count' the number of remaining elements to the right
which are larger than it. Find the maximum surpasser count of a list in O(n*log n) --
Divide and Conquer -- Medium.
-- We define data-strucutre (and operations) that will enable us to join (=merge) two sublists in linear time.
The data base is an ordered list of the elements and their surpasser for each half-list, and then
the join (of right and left lists) can be done element-by-element in linear time.
P3: Saddleback search + Binary search goes 2D -- Given f(x,y)-> z over the Natural numbers. f(,) is
strictly increasing in both its arguments.
Find all the pairs (x,y) such that f(x,y) = z_0. -- Medium.
-- Brute force is searching in the whole square of (0,0) to (z_0,z_0). Then, Saddleback goes over a 'line', stepping
gently from the top-left along a column, and then jumps to the right when needed, and continues to the bottom right. Like
going on an iso-line and not loosing height.
Then, the last method is a simple '2D binary search', in which we do binary search along row (or column), and then
'throw-away' two rectangels (top-right and bottom-left). This turns out to be the most efficnet by far!
P4: Smallest K element in Union of sets -- Given two disjoint sets X and Y, each is sorted. Find the K-smallest element
in their union.
Of course, O(K) solution is simple and direct. BUT, we can do it in O(log |X| + log |Y|), by using binary search and
divide-and-conquer. -- Medium
-- Brute force is O(K). 5-lines of code. Then, using lists and split/merging according to binary search. Last,
using arrays rather then lists, which implies just playing with indices rather than creating new lists.
Note: The first (maybe) incosistent-code I noticed in the book. This is in the part of working with
rather than creating sub-lists (Maybe they somehow meant the function to be called from a different place).
I left in
the code all debugging-printouts to show that in my code, there's perfect
agreement between the list and array based solutions.
P6: Making a Century -- Given the digits 1..9, list all the ways the operations + and x can be inserted into
the sequence so as to make the total sum 100.
100 = 12 + 34 + 5x6 +7 + 8 + 9
100 = 1 + 2x3 + 4 + 5 + 67 + 8 + 9
No parthneses, normal order of operations. -- Easy.
-- This is a brute-force problem, but the goal of the pearl is to establish some
general formulation of 'Brute-Force' search, and improve on it with 'little'
Two main things for brute-force methods:
1. Generating the 'values' while generating the 'candidates', can produce savings (Especially
if generations of candidates can be 'sequential').
2. Reduce 'good' critertia to 'ok', it can help trim the generation of all sequences.
Note: As of now, didn't implement the 'ok' shortcut, as right now these ru nso fast anyway.
P9: Celebrity Clique -- Finding a "Celebrity Clique" in linear time -- Fusion, Graphs -- Hard.
-- Interesting problem: It is more efficient to *find* a solution assuming one exists, rather than *check* it is actually
Dijkstra -- Shortest path in a graph (no heuristics).
Needleman-Wunsch -- Global sequences alignment.
Smith-Waterman -- Local sequences alignment.
Clock -- Measuring execution time -- Easy.
The goal is to time a ceratin function or program. Of course, there's difference between
"wall-time" and "CPU time", and whether you want to count IO-Time as well (waiting for File I/O),
multi-threads, and so on.
System.CPUTime- Simplest. Coems with Base.
Clock package - Various options for 'Time' definition. Also using the Formatting package for nicer output.
Other possible packages : Timeit package, Criterion package.
foldr and fusion -- Simple example of fusing function on foldr -- Med.
Source code example: fusion1.hs.
Fusion is the idea of combining (or 'fusing') a few operations together in a manner that
avoids intemediate data-structures or claculations.
This is a very common practice in functional programming, where fusion can improve
Basic text-definition of fusion for foldr is:
" f*foldr g a = foldr h b ",
Provided f is strict, f a = b, and f (g x y) = h x (f y) for all x and y.
The fusion condition on "f strict" can be relaxed if we just want to use
" f (foldr g a xs) = foldr h b xs ", for all finite sets xs.
A note about foldr:
" foldr:: (a->b->b) -> b -> [a] -> b " :
Takes the second argument, and the last item of the list, and applies the function.
Then, takes the result and the pentumilate element of the list, and repeats.
For a more elaborated example, see the Celebrity Clique pearl (P9) in Bird's book.
Triangular numbers -- Fiding all triangular numbers.
Want to make sure I am not a spoiler, so am not linking to the original page and details.
The idea in most is to use some Math to make the coding (and solution) much simpler.
Problem 1 -- Find the sum of all the multiples of 3 or 5 below 1000 -- Easy.
euler1.hs (opens in new window).
Problem 2 -- Fibonacci sequence whose values do not exceed four million,
find the sum of the even-valued terms -- Easy.
-- Elements of solution: Even valued are every 3rd element; formula for approximating
the elements (==> No need to calculate); and thus the sum is sum of geometric series.
Problem 3 -- Largest prime factor:
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
Problem 4 -- Largest palindrome product:
A palindromic number reads the same both ways. The largest palindrome made from the product of
two 2-digit numbers is 9009 = 91x99.
Find the largest palindrome made from the product of two 3-digit numbers.
Problem 5 -- Smallest multiple:
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?.
Problem 6 -- Sum square difference:
The sum of the squares of the first ten natural numbers is,
12 + 22 + ... + 102 = 385.
The square of the sum of the first ten natural numbers is,
(1 + 2 + ... + 10)2 = 552 = 3025.
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
Problem 7 -- 10001st prime:
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10 001st prime number?
Problem 8 -- Largest product in a series:
The four adjacent digits in the 1000-digit number that have the greatest product are 9*9*8*9=5832.
Find the thirteen adjacent digits in the 1000-digit number that have the greatest product.
What is the value of this product?
Problem 9 -- Special Pythagorean triplet:
A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
a^2 + b^2 = c^2.
For example, 3^2 + 4^2 = 5^2.
There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.
Problem 10 -- Summation of primes:
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
Problem 11 -- Largest product in a grid:
In the 20x20 grid below, four numbers along a diagonal line have been marked in red.
(The diagonal is NW to SE, starting from (7,9). See in the code file.)
The product of these numbers is 26 x 63 x 78 x 14 = 1788696.
What is the greatest product of four adjacent numbers in the same direction
(up, down, left, right, or diagonally) in the 20 x 20 grid?
Problem 12 -- Highly divisible triangular number:
The sequence of triangle numbers is generated by adding the natural numbers.
So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
The first ten terms would be:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
Let us list the factors of the first seven triangle numbers: (cut).
We can see that 28 is the first triangle number to have over five divisors.
What is the value of the first triangle number to have over five hundred divisors?
Problem 13 -- Large sum:
Work out the first ten digits of the sum of the following
one-hundred 50-digit numbers.
and so on.
Problem 14 -- Longest Collatz sequence:
The following iterative sequence is defined for the set of positive integers:
n -> n/2 (n is even)
n -> 3n + 1 (n is odd)
Using the rule above and starting with 13, we generate the following sequence:
13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms.
Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.
Which starting number, under one million, produces the longest chain?
Problem 15 -- Lattice Paths
Starting in the top left corner of a 2x2 grid, and only being able to move to the right and down,
there are exactly 6 routes to the bottom right corner.
How many such routes are there through a 20x20 grid?
Problem 16 -- Power digit sum
2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
What is the sum of the digits of the number 261000?
Problem 17 -- Number letter counts
If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there
are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.
If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words,
how many letters would be used?
NOTE: Do not count spaces or hyphens. For example, 342 (three hundred and forty-two) contains 23 letters and 115 (one hundred and fifteen) contains 20 letters.
The use of "and" when writing out numbers is in compliance with British usage.
Problem 18 -- Maximum path sum I
By starting at the top of the triangle below and moving to adjacent numbers on the row below,
the maximum total from top to bottom is 23.
2 4 6
8 5 9 3
That is, 3 + 7 + 4 + 9 = 23.
Find the maximum total from top to bottom of the triangle below (in the text file)
NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route.
However, Problem 67, is the same challenge with a triangle containing one-hundred rows;
it cannot be solved by brute force, and requires a clever method! ;o)
-- Names scores:
Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over
five-thousand first names, begin by sorting it into alphabetical order.
Then working out the alphabetical value for each name, multiply this
value by its alphabetical position in the list to obtain a name score.
For example, when the list is sorted into alphabetical order, COLIN, which is worth
3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list.
So, COLIN would obtain a score of 938 × 53 = 49714.
What is the total of all the name scores in the file?
-- 1000-digit Fibonacci number:
The Fibonacci sequence is defined by the recurrence relation:
Fn = Fn-1 + Fn-2, where F1 = 1 and F2 = 1.
The 12th term, F12, is the first term to contain three digits.
(the 7th is the first with 2 digits).
What is the index of the first term in the Fibonacci sequence to contain 1000 digits?
-- Number spiral diagonals
Starting with the number 1 and moving to the
right in a clockwise direction a 5 by 5 spiral is formed as follows (see in the code file).
It can be verified that the sum of the numbers on the diagonals is 101.
What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed
in the same way?
Problem 206 -- Concealed Square: Find the unique positive integer whose square has the form
1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit:
-- This can be solved by brute-force: Squaring on the numbers up to ... . You can 'jump' in 30 steps.
Just to make it more interesting, I solved it using Depth First Search, in two ways:
euler206_a.hs (DFS pattern design) ; euler206.hs (Just DFS).
Problem 493 -- Combinatorics:
70 colored balls are placed in an urn, 10 for each of the seven rainbow colors.
What is the expected number of distinct colors in 20 randomly picked balls?
-- The idea is to do some calculation by hands of the related combinatorics, and then computer the resulting expression.
It turns out that for this one, there is a very easy way to calculate by hand, so a calculator is enough.
Problem 504 -- Square on the Inside:
Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:
A(a, 0), B(0, b), C(-c, 0), D(0, -d), where 1 = a, b, c, d = m and a, b, c, d, m are integers.
It can be shown that for m = 4 there are exactly 256 valid ways to construct ABCD.
Of these 256 quadrilaterals, 42 of them strictly contain a square number of lattice points.
How many quadrilaterals ABCD strictly contain a square number of lattice points for m = 100?
-- Easy if you know the theorem.
-- Pick's theorem. Also did some memoization.
Problem 510 -- Tangent Circles:
Circles A and B are tangent to each other and to line L at three distinct points.
Circle C is inside the space between A, B and L, and tangent to all three.
Let rA, rB and rC be the radii of A, B and C respectively.
Let S(n) = S rA + rB + rC, for 0 < rA = rB = n where rA, rB and rC are integers.
The only solution for 0 < rA = rB = 5 is rA = 4, rB = 4 and rC = 1, so S(5) = 4 + 4 + 1 = 9.
You are also given S(100) = 3072.
Find S(10^9). -- Easy math. Running time is tricky!
-- Writing the equations gives you relations between rA, rB, and rC. And for all to be integers,
one only need to go over the enumeration.