Balanced layout of n items in a grid - algorithm

I have a list of n logos to display in a grid, with a maximum of 3 per row.
What's an algorithm to decide how many to display per row such that the number of logos per row is as balanced as possible without using more than the minimum possible number of rows?
For example:
n -> number in each row
1 -> 1
2 -> 2
3 -> 3
4 -> 2, 2
5 -> 3, 2
6 -> 3, 3
7 -> 3, 2, 2
8 -> 3, 3, 2
9 -> 3, 3, 3
10 -> 3, 3, 2, 2

For N <= 3 just use N.
If N is exactly divisible by 3 then use: 3 3 ... 3
If N when divided by 3 has remainder 1 then use: 3 3 ... 2 2
If N when divided by 3 has remainder 2 then use: 3 3 ... 3 2

AS confusing as your question is, I think what you need to do is first determine:
number_of_rows = ceil(number_of_logos / 3.0)
Then add a logo to each row, one at a time.
Python:
import math
def partition_logos(count, lsize):
num_lines = int(math.ceil(count / float(lsize)))
partition = [0] * num_lines
for i in xrange(count):
partition[i%num_lines] += 1
return partition
>>> for i in xrange(1,11):
... print partition_logos(i, 3)
[1]
[2]
[3]
[2, 2]
[3, 2]
[3, 3]
[3, 2, 2]
[3, 3, 2]
[3, 3, 3]
[3, 3, 2, 2]

A recursive solution, in Python:
def logos_by_row(N, rows):
width = 0
if N > 4 or N == 3:
width = 3
elif N == 4 or N == 2:
width = 2
elif N == 1:
width = 1
if width != 0:
rows.append(width)
logos_by_row(N - width, rows)
answer = []
for i in range(10):
logos_by_row(i+1, answer)
print answer

just use n/3 to calculate the row and n%3 to calculate the column
edit: ok i saw you edited your question.... i din't saw that you want to display 2 in each row if the are 4 logos. but then you can use n mod 3 to calculate if their is a reminder as others already suggested
if n%3 = 0 then just put 3 logos in each row
if n%3 = 1 then put the last 4 logos in two rows
if n%3 = 2 then put 3 logos in n row and the last 2 logos in a separate row

Related

How to draw an n sample from U(a,b) of numbers not close?

I need to draw an n sample from the uniform distribution on the interval [a,b] such that no two numbers are closer than d > 0. I can draw a sample and check this property and then throw it away and try again if not, but if n is large relative to b-a that could take a looong time. Is there a simple and nice algorithm to solve this problem? The numbers got to be uniformly distributed on [a,b], no deterministic setup.
This problem is equivalent to choosing n numbers greater than or equal to d and whose sum is equal to b - a.
There will be some solution provided that n * d <= b - a. We can write a recursive algorithm that looks for one:
b - a - X < (n - 1) * D
X > b - a - (n - 1) * d
FindSpacedSample(n, d, a, b)
1. if n * d > b - a then return "no solution"
2. avail = [d, b - a - (n - 1) * d]
3. guess = random(avail)
4. print(guess)
5. FindSpacedSample(n - 1, d, a + guess, b)
Example: n = 5, a = 0, b = 10, d = 1, assuming real numbers
FindSpacedSample(5, 1, 0, 10)
5 * 1 >? b - a? no
avail = [1, 10 - 0 - 4 * 1] = [1, 6]
guess = random(avail) = 2 (for the sake of argument)
print(2)
FindSpacedSample(4, 1, 2, 10)
4 * 1 >? 10 - 2? no
avail = [1, 10 - 2 - 3 * 1] = [1, 5]
guess = random(avail) = 4 (for the sake of argument)
print(4)
FindSpacedSample(3, 1, 6, 10)
3 * 1 >? 10 - 6? no
avail = [1, 10 - 6 - 2 * 1] = [1, 2]
guess = random(avail) = 1 (for the sake of argument)
print(1)
FindSpacedSample(2, 1, 7, 10)
2 * 1 >? 10 - 7? no
avail = [1, 10 - 7 - 1 * 1] = [1, 2]
guess = random(avail) = 2 (for the sake of argument)
print(2)
FindSpacedSample(1, 1, 9, 10)
1 * 1 >? 10 - 9? no
avail = [1, 10 - 9 - 0 * 1] = [1, 1]
guess = 1
print(1)
We should also have stopping condition n = 0. Then we get the sequence of spaces 2, 4, 1, 2, 1; we see these sum to ten; and we can get the values as follows:
point1 = 2 = 2
point2 = 2 + 4 = 6
point3 = 2 + 4 + 1 = 7
point4 = 2 + 4 + 1 + 2 = 9
point5 = 2 + 4 + 1 + 2 + 1 = 10
Now, there are a couple of ways in which this result is less than totally uniform:
the first number will never be less than d
earlier numbers tend to be spaced further apart
We can fix these by:
shuffling the spacings before converting to points
subtracting from each point some random value from [0, point1 - a].
So, if we shuffled 2, 4, 1, 2, 1 to 4, 1, 1, 2, 2 we'd get points 4, 5, 6, 8, 10; and if we subtracted 3 from each one (taken randomly between 0 and 4) we'd get 1, 2, 3, 5, 7.
Does this give you a uniform distribution over the set of all possible solutions? I'd be surprised if it did, but I'd also be surprised if what this does give you differs from that truly uniform distribution to an appreciable degree.

Print sequences with prime sum

Given an integer n, I want to find two permutations of the numbers 1 to n (inclusive) such that the sum of numbers from the two permutations at any given index is always prime.
For example:
n = 5
1 2 3 4 5
1 5 4 3 2
n = 8
1 2 3 4 5 6 7 8
2 1 4 3 8 7 6 5
Construct a bipartite graph on {0,1} x {1...n} such that (0, i) and (1, j) are connected if and only if i+j is prime.
Find a perfect matching using any standard technique, and then produce the sequences so that the matching numbers are at the same index.
Find the smallest number k>=1 such that n+k is prime.
Form pairs: (n,k), (n-1, k+1), ..., (k,n).
If k>1, repeat on the range [1,k].
E.g., n=5: 5+2=7, so we have:
5, 4, 3, 2
2, 3, 4, 5
Then we repeat for 1.
E.g. n=8: 8+3 = 11, so we have:
8, 7, 6, 5, 4, 3
3, 4, 5, 6, 7, 8
Leaving us with 2:
1, 2
2, 1
By Bertrand's postulate aka Chebyshev's theorem, for any n>1 there is a prime p s.t. n < p < 2n, and for n=1 we have 1+1=2, so we will always be able to do this.

How to get the (x,y) coordinates given a index when x < y?

In the past I've often used loops of the following kind (Haskell example):
upperBoundToTuples :: Int -> [(Int, Int)]
upperBoundToTuples n = [(x,y) | x <- [0..n], y <- [x+1..n]]
The above code produces tuples of range (0,1)..(n,n) where for all x < y.
I was wondering if it there was an efficient way of getting those (x,y) indices given a single index? Possible applications include optimization problems on the GPU where loops are not allowed and each thread only gets an index.
Also if it is possible for the 2D case, could such an algorithm be generalized to multiple dimensions?
You're asking for a bijection from [0, N(N+1)/2) to pairs (x, y) with 0 <= x < y <= N.
Here's one simple way to define it (in pseudocode, but should be trivial to convert to Haskell):
x0, y0 = i / (N + 1), i % (N + 1)
if x0 < y0 then result = (x0, y0)
else result = (N - 1 - x0, N - y0)
Here's a visualisation of the function for N=6. The map is laid out in a table with rows of length N+1=7, with the first row representing the value of the function for i=0 to 6, the next i=7 to 13 and so on. If you look very closely and carefully, you can see that things above the leading diagonal map to their own location in the table, and things on or below the diagonal map rotationally to the later entries.
5,6 0,1 0,2 0,3 0,4 0,5 0,6
4,6 4,5 1,2 1,3 1,4 1,5 1,6
3,6 3,5 3,4 2,3 2,4 2,5 2,6
And here's the opposite of this visualisation: a table T of size (N+1) by (N+1) with T[x, y] = i where i is mapped to (x, y) by the function above.
- 1 2 3 4 5 6
- - 9 10 11 12 13
- - - 17 18 19 20
- - - - 16 15 14
- - - - - 8 7
- - - - - - 0
- - - - - - -
Higher dimensions
This method can probably be made to work in higher dimensions, but I don't immediately see how. As an alternative, here's a simple but somewhat inefficient method that does work in arbitrary dimensions.
First, note there's choose(N + 1, k) increasing sequences of length k from the numbers from 0 to N (where choose(N, k) is the binomial coefficient). Of those, choose(N, k - 1) of them end with N. That gives this recursive function that generates the sequences in descending colexicographical order (again in pseudocode):
sequence(N, k, index)
= [] if k == 0
= sequence(N - 1, k - 1, index) + [N] if index < choose(N, k - 1)
= sequence(N - 1, k, index - choose(N, k - 1)) otherwise
Here's, sequence(5, 3, index) for index between 0 and 19:
0 -> [3, 4, 5]
1 -> [2, 4, 5]
2 -> [1, 4, 5]
3 -> [0, 4, 5]
4 -> [2, 3, 5]
5 -> [1, 3, 5]
6 -> [0, 3, 5]
7 -> [1, 2, 5]
8 -> [0, 2, 5]
9 -> [0, 1, 5]
10 -> [2, 3, 4]
11 -> [1, 3, 4]
12 -> [0, 3, 4]
13 -> [1, 2, 4]
14 -> [0, 2, 4]
15 -> [0, 1, 4]
16 -> [1, 2, 3]
17 -> [0, 2, 3]
18 -> [0, 1, 3]
19 -> [0, 1, 2]
We may equivalently consider [(x,y) | x<-[0..n], y<-[0..x-1]]. This list has length
&ell;n = x=0∑n x
= n·(n+1)/2.
Hence we can get, to a given &ell;, the nearest lower n through
2·&ell;n = n·(n+1) = n2 + n
n~ = -½ &pm; √(¼ + 2·&ell;n)
In particular, for a given index i,
ni− = ⌊-½ &pm; √(¼ + 2·i)⌋
is the x-length of the last fully completed triangle. Thus, the index i lies in the row ni−+1. That triangle had an area of
&ell;ni− = ni−·(ni−+1)/2
which we therefore need to subtract from i to get the remainder index (in y-direction). This gives rise to the definition
lowerTriangularTuple :: Int -> (Int,Int)
lowerTriangularTuple i = (nmin+1, i - (nmin*(nmin+1))`div`2)
where nmin = floor $ -1/2 + sqrt(1/4 + 2 * fromIntegral i)
Example:
GHCi> lowerTriangularTuple <$> [0..30]
[(1,0),(2,0),(2,1),(3,0),(3,1),(3,2),(4,0),(4,1),(4,2),(4,3),(5,0),(5,1),(5,2),(5,3),(5,4),(6,0),(6,1),(6,2),(6,3),(6,4),(6,5),(7,0),(7,1),(7,2),(7,3),(7,4),(7,5),(7,6),(8,0),(8,1),(8,2)]

Find the number of 5 digit numbers using 1, 2, 3, 4, 5 - in which first k digits are not only o to k

The question is to find the number of 5 digits numbers possible using 1, 2, 3, 4, and 5 as digits, in which the set of first k numbers (k is natural number, and k < 5) is not equal to set of numbers from 1 to k.
Clarification - Order of elements of set does not matter. {1, 2} is same as {2, 1} ie {1, 2} = {2, 1}.
For example, in 54213 is a correct number and it will be counted - as
k = 1, we get {5}, and {5} != {1}
k = 2, {5, 4} != {1, 2}
k = 3, {5, 4, 2} != {1, 2, 3}
k = 4, {5, 4, 2, 1} != {1, 2, 3, 4}
Also, the number 13245 is not valid, as for k = 1, {1} = {1}.
Bonus question, find the number of 6 digit possible numbers with same criteria and digits 1, 2, 3, 4, 5, & 6.
Edit - I am sorry for not having followed this up with my method. I have added my solution below.
Number of 5 digit numbers possible with the given digits = 5! = 120
Let's consider the digits that are NOT to be formed.
1 _ _ _ _ = 4! = 24 (As clearly, k = 1 {1} = {1})
2 1 _ _ _ = 3! = 6 (As clearly, k = 2 {2, 1} = {1, 2}, also we will not consider 1 2 _ _ _ as it has already been included in 1)
2 3 1 _ _ = 2! = 2
3 1 2 _ _ = 2! = 2
3 2 1 _ _ = 2! = 2
2 3 4 1 _ = 1
2 4 1 3 _ = 1
2 4 3 1 _ = 1
3 1 4 2 _ = 1
3 2 4 1 _ = 1
3 4 2 1 _ = 1
3 4 1 2 _ = 1
4 - - - _ = 3! x 1 = 6 (Where dashes can be filled by 1, 2, 3)
Total number = 49
But there are the NOT cases. Now, the cases required = 120 - 49 = 71.
For 6, I used recursion, as clearly in case of 5, the series (for the numbers that are NOT to be formed) is
1 x 4! + 1 x 3! + 3 x 2! + 13 x 1! (where 1, 1, 3, and 13 is the solution for the given condition for 1 digit, 2, digit, 3 digit and 4 digit respectively)
{ie, for 1 digit number using only 1, where k < 1, k is natural, number of possible digits - 1 (1). For 2 digit number using only 1, 2, where k < 2, the number of possible digits - 1 (21). For 3, it is 321, 231, 312 and so on)
Hence, for 6, the series becomes
1 x 5! + 1 x 4! + 3 x 3! + 13 x 2! + 71 (where 71 is the solution for given condition of 5 digits)
= 259
ie. for 6, the solutions = 6! - 259 = 461

How to find the index of a k-permutation from n elements?

I know that, for a k-permutation p of size k, built from n elements, there are:
P(n, k) = n! / (n - k)!
possible k-permutations. For example:
k = 2
n = 4
l = [1, 2, 3, 4]
P(n, k) = 4! / (4 - 2)! = 12
1 2 | 2 1 | 3 1 | 4 1
1 3 | 2 3 | 3 2 | 4 2
1 4 | 2 4 | 3 4 | 4 3
And yet another example:
k = 3
n = 4
l = [1, 2, 3, 4]
P(n, k) = 4! / (4 - 3)! = 24
1 2 3 | 2 1 3 | 3 1 2 | 4 1 2
1 2 4 | 2 1 4 | 3 1 4 | 4 1 3
1 3 2 | 2 3 1 | 3 2 1 | 4 2 1
1 3 4 | 2 3 4 | 3 2 4 | 4 2 3
1 4 2 | 2 4 1 | 3 4 1 | 4 3 1
1 4 3 | 2 4 3 | 3 4 2 | 4 3 2
So, how do I find the index of the k-permutation p? Consider the permutations
to be generated lexicographically.
Edit:
I could start by finding in which "block" p is, addressing the block by the first element of p. For example, for p = [3, 2, 4], the index for p should be at least 12 (counting from 0 to P(n, k) - 1).
But then, to find the second element inside that "block", I'd have to see what are the remaining items to be found, and in which position they will be. I mean, I'd be eventually addressing the list [1, 4], and 4 would be at position 2, so simply using the element as key would require some extra manipulation.
I could use a hash to look up the elements and update their positions, but it'd give me a O(n^2) time complexity. Is it possible to do any better?
The number of permutations for a given digit in a given position is given by a formula
(n-digit position)! / (n-k)! where digit position starts on the left with 1.
To get the number of preceding permutations for a given permutation (that is, its index), multiply the formula for each digit by the number of preceding digits not already used, and add them up.
Example 1, k = 2, n = 4, p = [3,4]
First digit, 3:
(4-1)! / (4-2)! * (number of unused preceding digits, 2) = 6
There are six permutations preceding the first that has 3 in position 1.
Second digit, 4:
(4-2)! / (4-2)! * (number of unused preceding digits, 2) = 2
There are two permutations preceding the first that has 4 in position 2.
Zero based index: 6 + 2 = 8.
Example 2, k = 3, n = 4, p = [3,2,4]
First digit, 3:
(4-1)! / (4-3)! * (number of unused preceding digits, 2) = 12
There are 12 permutations preceding the first that has 3 in position 1.
Second digit, 2:
(4-2)! / (4-3)! * (number of unused preceding digits, 1) = 2
There are two permutations preceding the first that has 2 in position 2.
Third digit, 4:
(4-3)! / (4-3)! * (number of unused preceding digits, 1) = 1
There is one permutation preceding the first that has 4 in position 3.
Zero based index: 12 + 2 + 1 = 15.
TXR language:
$ txr -p "(pos '(1 4 3) (perm '(1 2 3 4) 3))"
5
It is brute force, however; of course pos knows nothing about the structure of permutations.
Use binary search tree (BST). Store all you numbers in it before you start and delete them after you use one. To find x-th element maintain .subtreeSize for every vertex and just descend the tree to find the number you need. Pseudocode:
def findXth(x):
curVertex = BST.root
while:
curPosition = curVertex.leftChild.subtreeSize
if curPosition == x: return curVertex.value
elif curPosition > x: curVertex = curVertex.leftChild
elif curPosition < x: curVertex = curVertex.rightChild
Don't forget to check for vertices existence and to delete the vertex you find.
Overall complexity is going to be O(n log n).
you can reference below function
/**
* list all k or <=k size permutation of a given list with n unique elements.
* n can be bigger than 64. this function will take O(K^N) time, Bad.
*
* #param uniqueList
* #param permutationSize
* #param permutation
* #param only Only show the permutation of permutationSize,
* else show all permutation of less than or equal to permutationSize.
*/
public static void my_permutationOf(List<Integer> uniqueList, int permutationSize, List<Integer> permutation, boolean only) {
if (permutation == null) {
assert 0 < permutationSize && permutationSize <= uniqueList.size();
permutation = new ArrayList<>(permutationSize);
if (!only) {
System.out.println(Arrays.toString(permutation.toArray()));
}
}
for (int i : uniqueList) {
if (permutation.contains(i)) {
continue;
}
permutation.add(i);
if (!only) {
System.out.println(Arrays.toString(permutation.toArray()));
} else if (permutation.size() == permutationSize) {
System.out.println(Arrays.toString(permutation.toArray()));
}
if (permutation.size() < permutationSize) {
my_permutationOf(uniqueList, permutationSize, permutation, only);
}
permutation.remove(permutation.size() - 1);
}
}
E.g. you can think the element is the index
public static void main(String[] args) throws Exception {
my_permutationOf(new ArrayList<Integer>() {
{
add(0);
add(1);
add(2);
add(3);
}
}, 3, null, true);
}
the result is
[0, 1, 2]
[0, 1, 3]
[0, 2, 1]
[0, 2, 3]
[0, 3, 1]
[0, 3, 2]
[1, 0, 2]
[1, 0, 3]
[1, 2, 0]
[1, 2, 3]
[1, 3, 0]
[1, 3, 2]
[2, 0, 1]
[2, 0, 3]
[2, 1, 0]
[2, 1, 3]
[2, 3, 0]
[2, 3, 1]
[3, 0, 1]
[3, 0, 2]
[3, 1, 0]
[3, 1, 2]
[3, 2, 0]
[3, 2, 1]

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