-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- The three rules of Ruby Quiz 2: 1. Please do not post any solutions or spoiler discussion for this quiz until 48 hours have passed from the time on this message. 2. Support Ruby Quiz 2 by submitting ideas as often as you can! Visit <http://splatbang.com/rubyquiz/>. 3. Enjoy! Suggestion: A [QUIZ] in the subject of emails about the problem helps everyone on Ruby Talk follow the discussion. Please reply to the original quiz message, if you can. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- ## Mix and Match (#186) I purchased a number of scented candles recently for sending out to friends and family. While I could be accused of being lazy by getting candles for several people, I'd like to mix up the candles a bit so that each recipient gets a different combination of scents. Please help me out! Your task is to write a method that randomizes and mixes up the individual candles into groups, one per recipient, in order to minimize group duplication. So, for example: candles = { :orange => 3, :vanilla => 2, :lavender => 2, :garden => 4 } recipients = %w(janet nancy susan) candles_per_recipient = 3 > mix_and_match(candles, recipients, candles_per_recipient) => { "janet" => [:garden, :lavender, :orange], "nancy" => [:garden, :orange, :vanilla], "susan" => [:garden, :lavender, :vanilla], :extra => { :orange => 1, :vanilla => 0, :lavender => 0, :garden => 1 } } If it is impossible to have a unique combination for every recipient, you should still generate some set of combinations, minimizing repetition of combinations. If the number of recipients times the number of candles per recipient is more than the supply, generate an error.

on 2008-12-12 19:44

on 2008-12-16 19:19

Wow....it's been too long since I've posted to this list. Anyway, we can treat each type of candle as a digit of a number in an arbitrarily-large base. We can thus interpret the total number of candles as a single number, and each possible combination of different types of candles as other numbers whose only digits are 0 and 1 in that base. Under some permutation of the digits, we are trying to add the integers representing these combinations in a way that gets us the closest to the integer representing the total number of candles. Could use a little more rigor, but I think the above makes it pretty clear: this is essentially the Knapsack Problem. Good luck improving significantly on brute-force! ________________________________ From: Matthew Moss <matt@moss.name> To: ruby-talk ML <ruby-talk@ruby-lang.org> Sent: Friday, December 12, 2008 12:36:55 PM Subject: [QUIZ] Mix and Match (#186) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- The three rules of Ruby Quiz 2: 1. Please do not post any solutions or spoiler discussion for this quiz until 48 hours have passed from the time on this message. 2. Support Ruby Quiz 2 by submitting ideas as often as you can! Visit <http://splatbang.com/rubyquiz/>. 3. Enjoy! Suggestion: A [QUIZ] in the subject of emails about the problem helps everyone on Ruby Talk follow the discussion. Please reply to the original quiz message, if you can. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- ## Mix and Match (#186) I purchased a number of scented candles recently for sending out to friends and family. While I could be accused of being lazy by getting candles for several people, I'd like to mix up the candles a bit so that each recipient gets a different combination of scents. Please help me out! Your task is to write a method that randomizes and mixes up the individual candles into groups, one per recipient, in order to minimize group duplication. So, for example: candles = { :orange => 3, :vanilla => 2, :lavender => 2, :garden => 4 } recipients = %w(janet nancy susan) candles_per_recipient = 3 > mix_and_match(candles, recipients, candles_per_recipient) => { "janet" => [:garden, :lavender, :orange], "nancy" => [:garden, :orange, :vanilla], "susan" => [:garden, :lavender, :vanilla], :extra => { :orange => 1, :vanilla => 0, :lavender => 0, :garden => 1 } } If it is impossible to have a unique combination for every recipient, you should still generate some set of combinations, minimizing repetition of combinations. If the number of recipients times the number of candles per recipient is more than the supply, generate an error.

on 2008-12-17 02:04

Wow, it's been a long time since I've posted to this list. Anyway, we can treat the candles of each type as a digit in an integer in an arbitrarily-high base. Applying that transformation, the problem becomes "Given several numbers, find a way to add them up so that each digit is as close as possible to a given number." Not sure how to rigorously prove it, but that's highly suggesting that this quiz is reducible to the Knapsack Problem. ________________________________ From: Matthew Moss <matt@moss.name> To: ruby-talk ML <ruby-talk@ruby-lang.org> Sent: Friday, December 12, 2008 12:36:55 PM Subject: [QUIZ] Mix and Match (#186) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- The three rules of Ruby Quiz 2: 1. Please do not post any solutions or spoiler discussion for this quiz until 48 hours have passed from the time on this message. 2. Support Ruby Quiz 2 by submitting ideas as often as you can! Visit <http://splatbang.com/rubyquiz/>. 3. Enjoy! Suggestion: A [QUIZ] in the subject of emails about the problem helps everyone on Ruby Talk follow the discussion. Please reply to the original quiz message, if you can. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- ## Mix and Match (#186) I purchased a number of scented candles recently for sending out to friends and family. While I could be accused of being lazy by getting candles for several people, I'd like to mix up the candles a bit so that each recipient gets a different combination of scents. Please help me out! Your task is to write a method that randomizes and mixes up the individual candles into groups, one per recipient, in order to minimize group duplication. So, for example: candles = { :orange => 3, :vanilla => 2, :lavender => 2, :garden => 4 } recipients = %w(janet nancy susan) candles_per_recipient = 3 > mix_and_match(candles, recipients, candles_per_recipient) => { "janet" => [:garden, :lavender, :orange], "nancy" => [:garden, :orange, :vanilla], "susan" => [:garden, :lavender, :vanilla], :extra => { :orange => 1, :vanilla => 0, :lavender => 0, :garden => 1 } } If it is impossible to have a unique combination for every recipient, you should still generate some set of combinations, minimizing repetition of combinations. If the number of recipients times the number of candles per recipient is more than the supply, generate an error.

on 2008-12-19 00:18

_This week's summary is provided by Peter, originally [posted to his blog][1]. I am providing it here with permission._ [1]: http://www.rubyrailways.com/ruby-quiz-mix-and-match/ In my interpretation, this is a simple combinatorial problem: say the number of recipients is `r` and candles per recipient is `c`, then you are looking for a (preferably non-repeating) random selection of `r` elements of `c`-combinations of the original set of candles. (In fact it’s a bit more complicated than that: the `c`-combinations have to be recalculated from the remaining candles each time you give away a group of candles, so we’ll get to that). Sounds confusing? Don’t worry, after the implementation everything will be clear! So first, define a k-combination for a histogram (a Hash like candles above, where keys are elements and values are cardinalities): class Hash def comb(group_size) result = [] inner_comb = lambda do |head,tail| tail[0..-(group_size-head.size)].each do |e| if (head.size >= group_size-1) tail.each {|t| result << head + [t]} else inner_comb[head + [e], tail[tail.index(e)+1..-1]] end end end inner_comb[[],self.inject([]) {|a,v| v[1].times{a << v[0]}; a}] result.uniq end For example: candles = { :orange => 2, :vanilla => 1, :lavender => 1, :garden => 1 } pp candles.comb(3) => [[:lavender, :garden, :orange], [:lavender, :garden, :vanilla], [:lavender, :orange, :orange], [:lavender, :orange, :vanilla], [:garden, :orange, :orange], [:garden, :orange, :vanilla], [:orange, :orange, :vanilla]] So, for a set of candles, this method generates all possible 3- combinations of the candles. We can then pick one and assign it to one of the recipients. Then recalculate the above from the remaining candles, give it to the next recipient - and so on and so forth. That’s the basic idea, but we also need to ensure the candle combinations are as non-repeating as possible. So let’s define some further utility methods: class Hash def remove_set(set) set.each {|e| self[e] -= 1} end end The above code adjusts the number of candles in the original hash once we give away some of them. So for example: candles = { :orange => 2, :vanilla => 1, :lavender => 1, :garden => 1 } candles.remove_set([:orange,:orange,:lavender]) p candles => {:lavender=>0, :garden=>1, :orange=>0, :vanilla=>1} And some Array extensions: class Array def rand uniqs = self.select{|e| e.uniq.size == e.size} uniqs.empty? ? self[Kernel.rand(length)] : uniqs[Kernel.rand(uniqs.length)] end def unordered_include?(other) self.map{|e| e.map{|s| s.to_s}.sort}.include? other.map{|s| s.to_s}.sort end end `Array#rand` is trying to pick a random non-repeating combination if there is one (e.g. `[:orange, :lavender, :garden]`) or, if there is no such combination, then just a random one (e.g. `[:orange, :orange, :garden]` - orange is repeating, but we have no other choice). `Array#unordered_include?` is like normal `Array#include?`, but disregards the ordering of the elements. So for example: [[:lavender, :garden, :orange]].include? [:lavender, :orange, :garden] => false [[:lavender, :garden, :orange]].unordered_include? [:lavender, :orange, :garden] => true Hmm… it would have been much more effective to use a set here rather than the above CPU-sucker, but now I am lazy to change it. ;-) OK, so finally for the solution: ERROR_STRING = "The number of recipients times the number of candles per recipient is more than the supply!" def mix_and_match(candles, recipients, candles_per_recipient) return ERROR_STRING if ((candles.values.inject{|a,v| a+v}) < (recipients.size * candles_per_recipient)) candle_set = recipients.inject({}) do |a,v| tried = [] tries = 0 loop do random_pick = candles.comb(candles_per_recipient).rand tried << random_pick unless tried.unordered_include? random_pick break unless a.values.unordered_include? random_pick break if (tries+=1) > candles.values.size * 2 end candles.remove_set(tried.last) a[v] = tried.last a end candle_set.merge({:extra => candles}) end So, in the inner loop we randomly pick a candles-per-recipient- combination of all the possible combinations; If no one has that combo yet, we assign it to the next recipient. If someone has it already, we try to find an unique combination (loop on), unless it is impossible. In this case we simply start giving out any combinations. Once we give away a set of candles, we remove them from the original set. Easy-peasy. You can check out the source code here. This was a great quiz, too bad that not many people took a stab at it (so far 1 except me ;-)). The hardest part for me was the implementation of the k-combination (and the result looks awful to me - I didn’t check any algorithm/pseudocode/other solution though, I wanted to roll my own) - after that the problem was pretty simple.