Related
Sorry if this question doesn't fit the site well... I have been trying translate a bit of Maple-code into Mathematica. I don't know Maple at all but I do know some Mathematica. I really don't know what I am doing so I wonder if someone could help me just a bit:
b:= proc(n, s) local sn, m;
if n<1 then 1
else sn:= [s[], n]; m:= nops(sn);
`if` (m*(m-1)/2 = nops (({seq (seq (sn[i]-sn[j],
j=i+1..m), i=1..m-1)})), b(n-1, sn), 0) +b(n-1, s)
fi
end:
a:= proc(n) a(n):= b(n-1, [n]) +`if` (n=0, -1, a(n-1)) end:
seq(a(n), n=1..30);
I think I understand everything except
sn:= [s[], n];
but I'm not sure. Thanks in advance!
The indexing call s[] returns the sequence of the entries of s if s is a list or a set.
For s of type list (in particular) the call s[] behaves like the call op(s). (Note that this similarity in behavior is true for lists, sets, and sequences. But it's not true for all types.)
L:=[2,4,7]:
L[];
2, 4, 7
op(L);
2, 4, 7
[L[], 5];
[2, 4, 7, 5]
So [s[], n] takes list s and creates a new list. The new list, which gets assigned to sn, contains the entries of list s followed by n.
This is what I have so far:
myArray.map!{ rand(max) }
Obviously, however, sometimes the numbers in the list are not unique. How can I make sure my list only contains unique numbers without having to create a bigger list from which I then just pick the n unique numbers?
Edit:
I'd really like to see this done w/o loop - if at all possible.
(0..50).to_a.sort{ rand() - 0.5 }[0..x]
(0..50).to_a can be replaced with any array.
0 is "minvalue", 50 is "max value"
x is "how many values i want out"
of course, its impossible for x to be permitted to be greater than max-min :)
In expansion of how this works
(0..5).to_a ==> [0,1,2,3,4,5]
[0,1,2,3,4,5].sort{ -1 } ==> [0, 1, 2, 4, 3, 5] # constant
[0,1,2,3,4,5].sort{ 1 } ==> [5, 3, 0, 4, 2, 1] # constant
[0,1,2,3,4,5].sort{ rand() - 0.5 } ==> [1, 5, 0, 3, 4, 2 ] # random
[1, 5, 0, 3, 4, 2 ][ 0..2 ] ==> [1, 5, 0 ]
Footnotes:
It is worth mentioning that at the time this question was originally answered, September 2008, that Array#shuffle was either not available or not already known to me, hence the approximation in Array#sort
And there's a barrage of suggested edits to this as a result.
So:
.sort{ rand() - 0.5 }
Can be better, and shorter expressed on modern ruby implementations using
.shuffle
Additionally,
[0..x]
Can be more obviously written with Array#take as:
.take(x)
Thus, the easiest way to produce a sequence of random numbers on a modern ruby is:
(0..50).to_a.shuffle.take(x)
This uses Set:
require 'set'
def rand_n(n, max)
randoms = Set.new
loop do
randoms << rand(max)
return randoms.to_a if randoms.size >= n
end
end
Ruby 1.9 offers the Array#sample method which returns an element, or elements randomly selected from an Array. The results of #sample won't include the same Array element twice.
(1..999).to_a.sample 5 # => [389, 30, 326, 946, 746]
When compared to the to_a.sort_by approach, the sample method appears to be significantly faster. In a simple scenario I compared sort_by to sample, and got the following results.
require 'benchmark'
range = 0...1000000
how_many = 5
Benchmark.realtime do
range.to_a.sample(how_many)
end
=> 0.081083
Benchmark.realtime do
(range).sort_by{rand}[0...how_many]
end
=> 2.907445
Just to give you an idea about speed, I ran four versions of this:
Using Sets, like Ryan's suggestion.
Using an Array slightly larger than necessary, then doing uniq! at the end.
Using a Hash, like Kyle suggested.
Creating an Array of the required size, then sorting it randomly, like Kent's suggestion (but without the extraneous "- 0.5", which does nothing).
They're all fast at small scales, so I had them each create a list of 1,000,000 numbers. Here are the times, in seconds:
Sets: 628
Array + uniq: 629
Hash: 645
fixed Array + sort: 8
And no, that last one is not a typo. So if you care about speed, and it's OK for the numbers to be integers from 0 to whatever, then my exact code was:
a = (0...1000000).sort_by{rand}
Yes, it's possible to do this without a loop and without keeping track of which numbers have been chosen. It's called a Linear Feedback Shift Register: Create Random Number Sequence with No Repeats
[*1..99].sample(4) #=> [64, 99, 29, 49]
According to Array#sample docs,
The elements are chosen by using random and unique indices
If you need SecureRandom (which uses computer noise instead of pseudorandom numbers):
require 'securerandom'
[*1..99].sample(4, random: SecureRandom) #=> [2, 75, 95, 37]
How about a play on this? Unique random numbers without needing to use Set or Hash.
x = 0
(1..100).map{|iter| x += rand(100)}.shuffle
You could use a hash to track the random numbers you've used so far:
seen = {}
max = 100
(1..10).map { |n|
x = rand(max)
while (seen[x])
x = rand(max)
end
x
}
Rather than add the items to a list/array, add them to a Set.
If you have a finite list of possible random numbers (i.e. 1 to 100), then Kent's solution is good.
Otherwise there is no other good way to do it without looping. The problem is you MUST do a loop if you get a duplicate. My solution should be efficient and the looping should not be too much more than the size of your array (i.e. if you want 20 unique random numbers, it might take 25 iterations on average.) Though the number of iterations gets worse the more numbers you need and the smaller max is. Here is my above code modified to show how many iterations are needed for the given input:
require 'set'
def rand_n(n, max)
randoms = Set.new
i = 0
loop do
randoms << rand(max)
break if randoms.size > n
i += 1
end
puts "Took #{i} iterations for #{n} random numbers to a max of #{max}"
return randoms.to_a
end
I could write this code to LOOK more like Array.map if you want :)
Based on Kent Fredric's solution above, this is what I ended up using:
def n_unique_rand(number_to_generate, rand_upper_limit)
return (0..rand_upper_limit - 1).sort_by{rand}[0..number_to_generate - 1]
end
Thanks Kent.
No loops with this method
Array.new(size) { rand(max) }
require 'benchmark'
max = 1000000
size = 5
Benchmark.realtime do
Array.new(size) { rand(max) }
end
=> 1.9114e-05
Here is one solution:
Suppose you want these random numbers to be between r_min and r_max. For each element in your list, generate a random number r, and make list[i]=list[i-1]+r. This would give you random numbers which are monotonically increasing, guaranteeing uniqueness provided that
r+list[i-1] does not over flow
r > 0
For the first element, you would use r_min instead of list[i-1]. Once you are done, you can shuffle the list so the elements are not so obviously in order.
The only problem with this method is when you go over r_max and still have more elements to generate. In this case, you can reset r_min and r_max to 2 adjacent element you have already computed, and simply repeat the process. This effectively runs the same algorithm over an interval where there are no numbers already used. You can keep doing this until you have the list populated.
As far as it is nice to know in advance the maxium value, you can do this way:
class NoLoopRand
def initialize(max)
#deck = (0..max).to_a
end
def getrnd
return #deck.delete_at(rand(#deck.length - 1))
end
end
and you can obtain random data in this way:
aRndNum = NoLoopRand.new(10)
puts aRndNum.getrnd
you'll obtain nil when all the values will be exausted from the deck.
Method 1
Using Kent's approach, it is possible to generate an array of arbitrary length keeping all values in a limited range:
# Generates a random array of length n.
#
# #param n length of the desired array
# #param lower minimum number in the array
# #param upper maximum number in the array
def ary_rand(n, lower, upper)
values_set = (lower..upper).to_a
repetition = n/(upper-lower+1) + 1
(values_set*repetition).sample n
end
Method 2
Another, possibly more efficient, method modified from same Kent's another answer:
def ary_rand2(n, lower, upper)
v = (lower..upper).to_a
(0...n).map{ v[rand(v.length)] }
end
Output
puts (ary_rand 5, 0, 9).to_s # [0, 8, 2, 5, 6] expected
puts (ary_rand 5, 0, 9).to_s # [7, 8, 2, 4, 3] different result for same params
puts (ary_rand 5, 0, 1).to_s # [0, 0, 1, 0, 1] repeated values from limited range
puts (ary_rand 5, 9, 0).to_s # [] no such range :)
I'm new to psuedocode, and I'm having trouble putting all the pieces together:
Here is the definition of a function named foo whose inputs are two integers and an array of integers a[1] ... a[n].
1 Foo(k,m, a[1],...,a[n])
2 if (k < 1 or m > n or k > m) return 0
3 else return a[k] + Foo(k+1,m,a[1],...,a[n])
Suppose that the input integers are k=2 and m=5 and the input array contains [5, 6, 2, 3, 4, 8, 2]. What value does Foo return? Using summation notation, give a general formula for what Foo computes.
This one is making my head hurt. Here's what I did so far:
Line 2 has three conditional statements:
If k<1 // if 2<1..this is false
If m>n // if 5 is greater than the amount of values in the array, which is 7, so this is false
If k>m // if 2>5, this is false
So this function will display line 3. Line 3 says:
return a[k] which is a[2] which is the second value of the array, which is 6. So take 6 and add it to (2+1, 5, a[1].....,a[n])
Is what I have done correct up there? If so, how would I know what a[n] is? Am I supposed to be finding that? What would be the final result of all this?
Simple answer: that function returns the sum of all the numbers a[k], a[k+1], ... a[m].
What you're doing is correct so far. The "n" is just a placeholder meaning the last element of the array. So if your input array is {5,6,2,3,4,8,2}, n = 7 (cause your have seven elements), and a[n] = 2.
But why it returns the sum of all numbers a[k], a[k+1], ... a[m], you should find out for yourself. Just continue with your analysis. :)
So take 6 and add it to (2+1, 5,
a[1].....,a[n])
Take 6 and add it to Foo(2+1, 5, a[1].....,a[n]). It's a recursive function. You have to evaluate the function again with k=3 and m=5.
I think you are confused because your pseudocode looks like real code to me. I may be wrong, but we are taught to write pseudocode differently, using plain English phrases.
This is what I have so far:
myArray.map!{ rand(max) }
Obviously, however, sometimes the numbers in the list are not unique. How can I make sure my list only contains unique numbers without having to create a bigger list from which I then just pick the n unique numbers?
Edit:
I'd really like to see this done w/o loop - if at all possible.
(0..50).to_a.sort{ rand() - 0.5 }[0..x]
(0..50).to_a can be replaced with any array.
0 is "minvalue", 50 is "max value"
x is "how many values i want out"
of course, its impossible for x to be permitted to be greater than max-min :)
In expansion of how this works
(0..5).to_a ==> [0,1,2,3,4,5]
[0,1,2,3,4,5].sort{ -1 } ==> [0, 1, 2, 4, 3, 5] # constant
[0,1,2,3,4,5].sort{ 1 } ==> [5, 3, 0, 4, 2, 1] # constant
[0,1,2,3,4,5].sort{ rand() - 0.5 } ==> [1, 5, 0, 3, 4, 2 ] # random
[1, 5, 0, 3, 4, 2 ][ 0..2 ] ==> [1, 5, 0 ]
Footnotes:
It is worth mentioning that at the time this question was originally answered, September 2008, that Array#shuffle was either not available or not already known to me, hence the approximation in Array#sort
And there's a barrage of suggested edits to this as a result.
So:
.sort{ rand() - 0.5 }
Can be better, and shorter expressed on modern ruby implementations using
.shuffle
Additionally,
[0..x]
Can be more obviously written with Array#take as:
.take(x)
Thus, the easiest way to produce a sequence of random numbers on a modern ruby is:
(0..50).to_a.shuffle.take(x)
This uses Set:
require 'set'
def rand_n(n, max)
randoms = Set.new
loop do
randoms << rand(max)
return randoms.to_a if randoms.size >= n
end
end
Ruby 1.9 offers the Array#sample method which returns an element, or elements randomly selected from an Array. The results of #sample won't include the same Array element twice.
(1..999).to_a.sample 5 # => [389, 30, 326, 946, 746]
When compared to the to_a.sort_by approach, the sample method appears to be significantly faster. In a simple scenario I compared sort_by to sample, and got the following results.
require 'benchmark'
range = 0...1000000
how_many = 5
Benchmark.realtime do
range.to_a.sample(how_many)
end
=> 0.081083
Benchmark.realtime do
(range).sort_by{rand}[0...how_many]
end
=> 2.907445
Just to give you an idea about speed, I ran four versions of this:
Using Sets, like Ryan's suggestion.
Using an Array slightly larger than necessary, then doing uniq! at the end.
Using a Hash, like Kyle suggested.
Creating an Array of the required size, then sorting it randomly, like Kent's suggestion (but without the extraneous "- 0.5", which does nothing).
They're all fast at small scales, so I had them each create a list of 1,000,000 numbers. Here are the times, in seconds:
Sets: 628
Array + uniq: 629
Hash: 645
fixed Array + sort: 8
And no, that last one is not a typo. So if you care about speed, and it's OK for the numbers to be integers from 0 to whatever, then my exact code was:
a = (0...1000000).sort_by{rand}
Yes, it's possible to do this without a loop and without keeping track of which numbers have been chosen. It's called a Linear Feedback Shift Register: Create Random Number Sequence with No Repeats
[*1..99].sample(4) #=> [64, 99, 29, 49]
According to Array#sample docs,
The elements are chosen by using random and unique indices
If you need SecureRandom (which uses computer noise instead of pseudorandom numbers):
require 'securerandom'
[*1..99].sample(4, random: SecureRandom) #=> [2, 75, 95, 37]
How about a play on this? Unique random numbers without needing to use Set or Hash.
x = 0
(1..100).map{|iter| x += rand(100)}.shuffle
You could use a hash to track the random numbers you've used so far:
seen = {}
max = 100
(1..10).map { |n|
x = rand(max)
while (seen[x])
x = rand(max)
end
x
}
Rather than add the items to a list/array, add them to a Set.
If you have a finite list of possible random numbers (i.e. 1 to 100), then Kent's solution is good.
Otherwise there is no other good way to do it without looping. The problem is you MUST do a loop if you get a duplicate. My solution should be efficient and the looping should not be too much more than the size of your array (i.e. if you want 20 unique random numbers, it might take 25 iterations on average.) Though the number of iterations gets worse the more numbers you need and the smaller max is. Here is my above code modified to show how many iterations are needed for the given input:
require 'set'
def rand_n(n, max)
randoms = Set.new
i = 0
loop do
randoms << rand(max)
break if randoms.size > n
i += 1
end
puts "Took #{i} iterations for #{n} random numbers to a max of #{max}"
return randoms.to_a
end
I could write this code to LOOK more like Array.map if you want :)
Based on Kent Fredric's solution above, this is what I ended up using:
def n_unique_rand(number_to_generate, rand_upper_limit)
return (0..rand_upper_limit - 1).sort_by{rand}[0..number_to_generate - 1]
end
Thanks Kent.
No loops with this method
Array.new(size) { rand(max) }
require 'benchmark'
max = 1000000
size = 5
Benchmark.realtime do
Array.new(size) { rand(max) }
end
=> 1.9114e-05
Here is one solution:
Suppose you want these random numbers to be between r_min and r_max. For each element in your list, generate a random number r, and make list[i]=list[i-1]+r. This would give you random numbers which are monotonically increasing, guaranteeing uniqueness provided that
r+list[i-1] does not over flow
r > 0
For the first element, you would use r_min instead of list[i-1]. Once you are done, you can shuffle the list so the elements are not so obviously in order.
The only problem with this method is when you go over r_max and still have more elements to generate. In this case, you can reset r_min and r_max to 2 adjacent element you have already computed, and simply repeat the process. This effectively runs the same algorithm over an interval where there are no numbers already used. You can keep doing this until you have the list populated.
As far as it is nice to know in advance the maxium value, you can do this way:
class NoLoopRand
def initialize(max)
#deck = (0..max).to_a
end
def getrnd
return #deck.delete_at(rand(#deck.length - 1))
end
end
and you can obtain random data in this way:
aRndNum = NoLoopRand.new(10)
puts aRndNum.getrnd
you'll obtain nil when all the values will be exausted from the deck.
Method 1
Using Kent's approach, it is possible to generate an array of arbitrary length keeping all values in a limited range:
# Generates a random array of length n.
#
# #param n length of the desired array
# #param lower minimum number in the array
# #param upper maximum number in the array
def ary_rand(n, lower, upper)
values_set = (lower..upper).to_a
repetition = n/(upper-lower+1) + 1
(values_set*repetition).sample n
end
Method 2
Another, possibly more efficient, method modified from same Kent's another answer:
def ary_rand2(n, lower, upper)
v = (lower..upper).to_a
(0...n).map{ v[rand(v.length)] }
end
Output
puts (ary_rand 5, 0, 9).to_s # [0, 8, 2, 5, 6] expected
puts (ary_rand 5, 0, 9).to_s # [7, 8, 2, 4, 3] different result for same params
puts (ary_rand 5, 0, 1).to_s # [0, 0, 1, 0, 1] repeated values from limited range
puts (ary_rand 5, 9, 0).to_s # [] no such range :)
I need to make a random list of permutations. The elements can be anything but assume that they are the integers 0 through x-1. I want to make y lists, each containing z elements. The rules are that no list may contain the same element twice and that over all the lists, the number of times each elements is used is the same (or as close as possible). For instance, if my elements are 0,1,2,3, y is 6, and z is 2, then one possible solution is:
0,3
1,2
3,0
2,1
0,1
2,3
Each row has only unique elements and no element has been used more than 3 times. If y were 7, then 2 elements would be used 4 times, the rest 3.
This could be improved, but it seems to do the job (Python):
import math, random
def get_pool(items, y, z):
slots = y*z
use_each_times = slots/len(items)
exceptions = slots - use_each_times*len(items)
if (use_each_times > y or
exceptions > 0 and use_each_times+1 > y):
raise Exception("Impossible.")
pool = {}
for n in items:
pool[n] = use_each_times
for n in random.sample(items, exceptions):
pool[n] += 1
return pool
def rebalance(ret, pool, z):
max_item = None
max_times = None
for item, times in pool.items():
if times > max_times:
max_item = item
max_times = times
next, times = max_item, max_times
candidates = []
for i in range(len(ret)):
item = ret[i]
if next not in item:
candidates.append( (item, i) )
swap, swap_index = random.choice(candidates)
swapi = []
for i in range(len(swap)):
if swap[i] not in pool:
swapi.append( (swap[i], i) )
which, i = random.choice(swapi)
pool[next] -= 1
pool[swap[i]] = 1
swap[i] = next
ret[swap_index] = swap
def plist(items, y, z):
pool = get_pool(items, y, z)
ret = []
while len(pool.keys()) > 0:
while len(pool.keys()) < z:
rebalance(ret, pool, z)
selections = random.sample(pool.keys(), z)
for i in selections:
pool[i] -= 1
if pool[i] == 0:
del pool[i]
ret.append( selections )
return ret
print plist([0,1,2,3], 6, 2)
Ok, one way to approximate that:
1 - shuffle your list
2 - take the y first elements to form the next row
4 - repeat (2) as long as you have numbers in the list
5 - if you don't have enough numbers to finish the list, reshuffle the original list and take the missing elements, making sure you don't retake numbers.
6 - Start over at step (2) as long as you need rows
I think this should be as random as you can make it and will for sure follow your criteria. Plus, you have very little tests for duplicate elements.
First, you can always randomly sort the list in the end, so let's not worry about making "random permutations" (hard); and just worry about 1) making permutations (easy) and 2) randomizing them (easy).
If you want "truly" random groups, you have to accept that randomization by nature doesn't really allow for the constraint of "even distribution" of results -- you may get that or you may get a run of similar-looking ones. If you really want even distribution, first make the sets evenly distributed, and then randomize them as a group.
Do you have to use each element in the set x evenly? It's not clear from the rules that I couldn't just make the following interpretation:
Note the following: "over all the lists, the number of times each elements is used is the same (or as close as possible)"
Based on this criteria, and the rule that z < x*, I postulate that you can simply enumerate all the items over all the lists. So you automatically make y list of the items enumerated to position z. Your example doesn't fulfill the rule above as closely as my version will. Using your example of x={0,1,2,3} y=6 and z=2, I get:
0,1 0,1 0,1 0,1 0,1 0,1
Now I didn't use 2 or 3, but you didn't say I had to use them all. If I had to use them all and I don't care to be able to prove that I am "as close as possible" to even usage, I would just enumerate across all the items through the lists, like this:
0,1 2,3 0,1 2,3 0,1 2,3
Finally, suppose I really do have to use all the elements. To calculate how many times each element can repeat, I just take (y*z)/(count of x). That way, I don't have to sit and worry about how to divide up the items in the list. If there is a remainder, or the result is less than 1, then I know that I will not get an exact number of repeats, so in those cases, it doesn't much matter to try to waste computational energy to make it perfect. I contend that the fastest result is still to just enumerate as above, and use the calculation here to show why either a perfect result was or wasn't achieved. A fancy algorithm to extract from this calculation how many positions will be duplicates could be achieved, but "it's too long to fit here in the margin".
*Each list has the same z number of elements, so it will be impossible to make lists where z is greater than x and still fulfill the rule that no list may contain the same element twice. Therefore, this rule demands that z cannot be greater than x.
Based on new details in the comments, the solution may simply be an implementation of a standard random permutation generation algorithm. There is a lengthy discussion of random permutation generation algorithms here:
http://www.techuser.net/randpermgen.html
(From Google search: random permutation generation)
This works in Ruby:
# list is the elements to be permuted
# y is the number of results desired
# z is the number of elements per result
# equalizer keeps track of who got used how many times
def constrained_permutations list, y, z
list.uniq! # Never trust the user. We want no repetitions.
equalizer = {}
list.each { |element| equalizer[element] = 0 }
results = []
# Do this until we get as many results as desired
while results.size < y
pool = []
puts pool
least_used = equalizer.each_value.min
# Find how used the least used element was
while pool.size < z
# Do this until we have enough elements in this resultset
element = nil
while element.nil?
# If we run out of "least used elements", then we need to increment
# our definition of "least used" by 1 and keep going.
element = list.shuffle.find do |x|
!pool.include?(x) && equalizer[x] == least_used
end
least_used += 1 if element.nil?
end
equalizer[element] += 1
# This element has now been used one more time.
pool << element
end
results << pool
end
return results
end
Sample usage:
constrained_permutations [0,1,2,3,4,5,6], 6, 2
=> [[4, 0], [1, 3], [2, 5], [6, 0], [2, 5], [3, 6]]
constrained_permutations [0,1,2,3,4,5,6], 6, 2
=> [[4, 5], [6, 3], [0, 2], [1, 6], [5, 4], [3, 0]]
enter code here
http://en.wikipedia.org/wiki/Fisher-Yates_shuffle