4Clojure: Recent Problems

Comparisons

For any orderable data type it's possible to derive all of the basic comparison operations (<, ≤, =, ≠, ≥, and >) from a single operation (any operator but = or ≠ will work). Write a function that takes three arguments, a less than operator for the data and two items to compare. The function should return a keyword describing the relationship between the two items. The keywords for the relationship between x and y are as follows:

x = y → :eq
x > y → :gt
x < y → :lt

Language of a DFA

A deterministic finite automaton (DFA) is an abstract machine that recognizes a regular language. Usually a DFA is defined by a 5-tuple, but instead we'll use a map with 5 keys:

:states is the set of states for the DFA.
:alphabet is the set of symbols included in the language recognized by the DFA.
:start is the start state of the DFA.
:accepts is the set of accept states in the DFA.
:transitions is the transition function for the DFA, mapping :states ⨯ :alphabet onto :states.

Write a function that takes as input a DFA definition (as described above) and returns a sequence enumerating all strings in the language recognized by the DFA. Note: Although the DFA itself is finite and only recognizes finite-length strings it can still recognize an infinite set of finite-length strings. And because stack space is finite, make sure you don't get stuck in an infinite loop that's not producing results every so often!

Logical falsity and truth

In Clojure, only nil and false representing the values of logical falsity in conditional tests - anything else is logical truth.

Subset and Superset

Set A is a subset of set B, or equivalently B is a superset of A, if A is "contained" inside B. A and B may coincide.

Decurry

Write a function that accepts a curried function of unknown arity n. Return an equivalent function of n arguments.
You may wish to read this.

Indexing Sequences

Transform a sequence into a sequence of pairs containing the original elements along with their index.

Map Defaults

When retrieving values from a map, you can specify default values in case the key is not found:

(= 2 (:foo {:bar 0, :baz 1} 2))

However, what if you want the map itself to contain the default values? Write a function which takes a default value and a sequence of keys and constructs a map.

Pairwise Disjoint Sets

Given a set of sets, create a function which returns true if no two of those sets have any elements in common¹ and false otherwise. Some of the test cases are a bit tricky, so pay a little more attention to them.

¹Such sets are usually called pairwise disjoint or mutually disjoint.

Latin Square Slicing

A Latin square of order n is an n x n array that contains n different elements, each occurring exactly once in each row, and exactly once in each column. For example, among the following arrays only the first one forms a Latin square:

A B C    A B C    A B C
B C A    B C A    B D A
C A B    C A C    C A B

Let V be a vector of such vectors¹ that they may differ in length². We will say that an arrangement of vectors of V in consecutive rows is an alignment (of vectors) of V if the following conditions are satisfied:

All vectors of V are used.
Each row contains just one vector.
The order of V is preserved.
All vectors of maximal length are horizontally aligned each other.
If a vector is not of maximal length then all its elements are aligned with elements of some subvector of a vector of maximal length.

Let L denote a Latin square of order 2 or greater. We will say that L is included in V or that V includes L iff there exists an alignment of V such that contains a subsquare that is equal to L.

For example, if V equals [[1 2 3][2 3 1 2 1][3 1 2]] then there are nine alignments of V (brackets omitted):

 
        1              2              3

      1 2 3          1 2 3          1 2 3
  A   2 3 1 2 1    2 3 1 2 1    2 3 1 2 1
      3 1 2        3 1 2        3 1 2

      1 2 3          1 2 3          1 2 3
  B   2 3 1 2 1    2 3 1 2 1    2 3 1 2 1
        3 1 2        3 1 2        3 1 2

      1 2 3          1 2 3          1 2 3
  C   2 3 1 2 1    2 3 1 2 1    2 3 1 2 1
          3 1 2        3 1 2        3 1 2

Alignment A1 contains Latin square [[1 2 3][2 3 1][3 1 2]], alignments A2, A3, B1, B2, B3 contain no Latin squares, and alignments C1, C2, C3 contain [[2 1][1 2]]. Thus in this case V includes one Latin square of order 3 and one of order 2 which is included three times.

Our aim is to implement a function which accepts a vector of vectors V as an argument, and returns a map which keys and values are integers. Each key should be the order of a Latin square included in V, and its value a count of different Latin squares of that order included in V. If V does not include any Latin squares an empty map should be returned. In the previous example the correct output of such a function is {3 1, 2 1} and not {3 1, 2 3}.

¹ Of course, we can consider sequences instead of vectors.
² Length of a vector is the number of elements in the vector.

Palindromic Numbers

A palindromic number is a number that is the same when written forwards or backwards (e.g., 3, 99, 14341).

Write a function which takes an integer n, as its only argument, and returns an increasing lazy sequence of all palindromic numbers that are not less than n.

The most simple solution will exceed the time limit!

The Big Divide

Write a function which calculates the sum of all natural numbers under n (first argument) which are evenly divisible by at least one of a and b (second and third argument). Numbers a and b are guaranteed to be coprimes.

Note: Some test cases have a very large n, so the most obvious solution will exceed the time limit.

Pascal's Trapezoid

Write a function that, for any given input vector of numbers, returns an infinite lazy sequence of vectors, where each next one is constructed from the previous following the rules used in Pascal's Triangle. For example, for [3 1 2], the next row is [3 4 3 2].

Trees into tables

Because Clojure's for macro allows you to "walk" over multiple sequences in a nested fashion, it is excellent for transforming all sorts of sequences. If you don't want a sequence as your final output (say you want a map), you are often still best-off using for, because you can produce a sequence and feed it into a map, for example.

For this problem, your goal is to "flatten" a map of hashmaps. Each key in your output map should be the "path"¹ that you would have to take in the original map to get to a value, so for example {1 {2 3}} should result in {[1 2] 3}. You only need to flatten one level of maps: if one of the values is a map, just leave it alone.

¹ That is, (get-in original [k1 k2]) should be the same as (get result [k1 k2])

For the win

Clojure's for macro is a tremendously versatile mechanism for producing a sequence based on some other sequence(s). It can take some time to understand how to use it properly, but that investment will be paid back with clear, concise sequence-wrangling later. With that in mind, read over these for expressions and try to see how each of them produces the same result.

Oscilrate

Write an oscillating iterate: a function that takes an initial value and a variable number of functions. It should return a lazy sequence of the functions applied to the value in order, restarting from the first function after it hits the end.

dot product

Create a function that computes the dot product of two sequences. You may assume that the vectors will have the same length.

Tricky card games

In trick-taking card games such as bridge, spades, or hearts, cards are played in groups known as "tricks" - each player plays a single card, in order; the first player is said to "lead" to the trick. After all players have played, one card is said to have "won" the trick. How the winner is determined will vary by game, but generally the winner is the highest card played in the suit that was led. Sometimes (again varying by game), a particular suit will be designated "trump", meaning that its cards are more powerful than any others: if there is a trump suit, and any trumps are played, then the highest trump wins regardless of what was led.

Your goal is to devise a function that can determine which of a number of cards has won a trick. You should accept a trump suit, and return a function winner. Winner will be called on a sequence of cards, and should return the one which wins the trick. Cards will be represented in the format returned by Problem 128, Recognize Playing Cards: a hash-map of :suit and a numeric :rank. Cards with a larger rank are stronger.

Veitch, Please!

Create a function which accepts as input a boolean algebra function in the form of a set of sets, where the inner sets are collections of symbols corresponding to the input boolean variables which satisfy the function (the inputs of the inner sets are conjoint, and the sets themselves are disjoint... also known as canonical minterms). Note: capitalized symbols represent truth, and lower-case symbols represent negation of the inputs. Your function must return the minimal function which is logically equivalent to the input.

PS — You may want to give this a read before proceeding: K-Maps
PPS — If you're interested in logic programming more generally, you should also check out: core.logic

Squares Squared

Create a function of two integer arguments: the start and end, respectively. You must create a vector of strings which renders a 45° rotated square of integers which are successive squares from the start point up to and including the end point. If a number comprises multiple digits, wrap them around the shape individually. If there are not enough digits to complete the shape, fill in the rest with asterisk characters. The direction of the drawing should be clockwise, starting from the center of the shape and working outwards, with the initial direction being down and to the right.

Digits and bases

Write a function which returns a sequence of digits of a non-negative number (first argument) in numerical system with an arbitrary base (second argument). Digits should be represented with their integer values, e.g. 15 would be [1 5] in base 10, [1 1 1 1] in base 2 and [15] in base 16.