In a programming event they asked this question.
Question:
1 2 3 4 5 6 7 8 9 10 J Q K.
Write and algorithm to sort the cards in reverse order in 7 steps.
K Q J 10 9 8 7 6 5 4 3 2 1.
I couldn't trace out the sorting alogrithm.
Which algoritm is used here?
I think it's something like quicksort, but with block moves:
1 2 3 4 5 6 7 8 9 10 J Q K
8 9 10 J Q K 7 1 2 3 4 5 6
J Q K 8 9 10 7 1 2 3 4 5 6
J Q K 8 9 10 7 4 5 6 1 2 3
K Q J 8 9 10 7 4 5 6 1 2 3
K Q J 10 9 8 7 4 5 6 1 2 3
K Q J 10 9 8 7 6 5 4 1 2 3
K Q J 10 9 8 7 6 5 4 3 2 1
Update: actually it might be even simplier: just swap 1 and K and 2 and Q, 3 and J etc. Just seven steps :)
If a step is a swap then Selection sort will order the list in 6 steps.
K Q J 10 9 8 7 6 5 4 3 2 1
1 Q J 10 9 8 7 6 5 4 3 2 K
1 2 J 10 9 8 7 6 5 4 3 Q K
1 2 3 10 9 8 7 6 5 4 J Q K
1 2 3 4 9 8 7 6 5 10 J Q K
1 2 3 4 5 8 7 6 9 10 J Q K
1 2 3 4 5 6 7 8 9 10 J Q K
Related
I have a complete binary tree of height 'h'.
How do I find 'h' number of unrelated partitions for this ?
NOTE:
Unrelated partition means no child can be present with its immediate parent.
There is a constraint on the number of nodes in each partition.
The difference of the maximum number nodes in a partition and the minimum number of nodes in the partition can either be 0 or 1.
Also, root is excluded from including in the partitions.
Who devised the problem probably had a more elegant solution in mind, but the following works.
Let's say we have h partitions numbered 1 to h, and that the nodes of partition n have value n. The root node has value 0, and does not participate in the partitions. Let's call a partition even if nis even, and odd if n is odd. Let's also number the levels of the complete binary tree, ignoring the root and starting from level 1 with 2 nodes. Level n has 2n nodes, and the complete tree has 2h+1-1 nodes, but only P=2h+1-2 nodes belong to the partitions (because the root is excluded). Each partition consists of p=⌊P/h⌋ or p=⌈P/h⌉ nodes, such that ∑ᵢpᵢ=P.
If the height h of the tree is even, put all even partitions into the even levels of the left subtree and the odd levels of the right subtee, and put all odd partitions into the odd levels of the left subtree and the even levels of the right subtree.
If h is odd, distribute all partitions up to partition h-1 like in the even case, but distribute partition h evenly into the last level of the left and right subtrees.
This is the result for h up to 7 (I wrote a tiny Python library to print binary trees to the terminal in a compact way for this purpose):
0
1 1
0
1 2
2 2 1 1
0
1 2
2 2 1 1
1 1 3 3 2 2 3 3
0
1 2
2 2 1 1
1 1 1 1 2 2 2 2
2 4 4 4 4 4 4 4 1 3 3 3 3 3 3 3
0
1 2
2 2 1 1
1 1 1 1 2 2 2 2
2 2 2 2 2 2 4 4 1 1 1 1 1 1 3 3
3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5
0
1 2
2 2 1 1
1 1 1 1 2 2 2 2
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
0
1 2
2 2 1 1
1 1 1 1 2 2 2 2
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
And this is the code that generates it:
from basicbintree import Node
for h in range(1, 7 + 1):
root = Node(0)
P = 2 ** (h + 1) - 2 # nodes in partitions
p = P // h # partition size (may be p or p + 1)
if h & 1: # odd height
t = (p + 1) // 2 # subtree tail nodes from split partition
n = (h - 1) // 2 # odd or even partitions in subtrees except tail
else: # even height
t = 0 # no subtree tail nodes from split partition
n = h // 2 # odd or even partitions in subtrees
s = P // 2 - t # subtree nodes excluding tail
r = s - n * p # partitions of size p + 1 in subtrees
x = [p + 1] * r + [p] * (n - r) # nodes indexed by subtree partition - 1
odd = [1 + 2 * i for i, c in enumerate(x) for _ in range(c)] + [h] * t
even = [2 + 2 * i for i, c in enumerate(x) for _ in range(c)] + [h] * t
for g in range(1, h + 1):
start = 2 ** (g - 1) - 1
stop = 2 ** g - 1
if g & 1: # odd level
root.set_level(odd[start:stop] + even[start:stop])
else: # even level
root.set_level(even[start:stop] + odd[start:stop])
print('```none')
root.print_tree()
print('```')
All trees produced up to height 27 have been programmatically confirmed to meet the specifications.
Some parts of the algorithm would need a proof, like, e.g., that it's always possible to choose an even size for the split partition in the odd height case, but this and other proofs are left as an exercise to the reader ;-)
I need to partition a set S={1, 2, 3, … , n} consisting of consecutive numbers such that each subset has has at least 2 elements (rule 1) and it consists of consecutive numbers (rule 2).
The rules are:
Each subset has at least two elements.
All elements of all subsets are consecutive.
All elements of S are included in the partition.
Examples:
There is 1 subset for n = 2:
1 2
There is 1 subset for n = 3:
1 2 3
There are 2 subset combinations for n = 4:
1 2 3 4
1 2 - 3 4
There are 3 subset combinations for n = 5:
1 2 3 4 5
1 2 - 3 4 5
1 2 3 - 4 5
There are 5 subset combinations for n = 6:
1 2 3 4 5 6
1 2 - 3 4 5 6
1 2 3 - 4 5 6
1 2 3 4 - 5 6
1 2 - 3 4 - 5 6
There are 8 subset combinations for n = 7:
1 2 3 4 5 6 7
1 2 - 3 4 5 6 7
1 2 3 - 4 5 6 7
1 2 3 4 - 5 6 7
1 2 3 4 5 - 6 7
1 2 - 3 4 - 5 6 7
1 2 - 3 4 5 - 6 7
1 2 3 - 4 5 - 6 7
There are 13 subset combinations for n = 8:
1 2 3 4 5 6 7 8
1 2 - 3 4 5 6 7 8
1 2 3 - 4 5 6 7 8
1 2 3 4 - 5 6 7 8
1 2 3 4 5 - 6 7 8
1 2 3 4 5 6 - 7 8
1 2 - 3 4 - 5 6 7 8
1 2 - 3 4 5 - 6 7 8
1 2 - 3 4 5 6 - 7 8
1 2 3 - 4 5 - 6 7 8
1 2 3 - 4 5 6 - 7 8
1 2 3 4 - 5 6 - 7 8
1 2 - 3 4 - 5 6 - 7 8
There are 21 subset combinations for n = 9:
1 2 3 4 5 6 7 8 9
1 2 - 3 4 5 6 7 8 9
1 2 3 - 4 5 6 7 8 9
1 2 3 4 - 5 6 7 8 9
1 2 3 4 5 - 6 7 8 9
1 2 3 4 5 6 - 7 8 9
1 2 3 4 5 6 7 - 8 9
1 2 - 3 4 - 5 6 7 8 9
1 2 - 3 4 5 - 6 7 8 9
1 2 - 3 4 5 6 - 6 7 9
1 2 - 3 4 5 6 7 - 8 9
1 2 3 - 4 5 - 6 7 8 9
1 2 3 - 4 5 6 - 7 8 9
1 2 3 - 4 5 6 7 - 8 9
1 2 3 4 - 5 6 - 7 8 9
1 2 3 4 - 5 6 7 - 8 9
1 2 3 4 5 - 6 7 - 8 9
1 2 - 3 4 - 5 6 - 7 8 9
1 2 - 3 4 - 5 6 7 - 8 9
1 2 - 3 4 5 - 6 7 - 8 9
1 2 3 - 4 5 - 6 7 - 8 9
There are 34 subset combinations for n = 10:
1 2 3 4 5 6 7 8 9 10
1 2 - 3 4 5 6 7 8 9 10
1 2 3 - 4 5 6 7 8 9 10
1 2 3 4 - 5 6 7 8 9 10
1 2 3 4 5 - 6 7 8 9 10
1 2 3 4 5 6 - 7 8 9 10
1 2 3 4 5 6 7 - 8 9 10
1 2 3 4 5 6 7 8 - 9 10
1 2 - 3 4 - 5 6 7 8 9 10
1 2 - 3 4 5 - 6 7 8 9 10
1 2 - 3 4 5 6 - 6 7 9 10
1 2 - 3 4 5 6 7 - 8 9 10
1 2 - 3 4 5 6 7 8 - 9 10
1 2 3 - 4 5 - 6 7 8 9 10
1 2 3 - 4 5 6 - 7 8 9 10
1 2 3 - 4 5 6 7 - 8 9 10
1 2 3 - 4 5 6 7 8 - 9 10
1 2 3 4 - 5 6 - 7 8 9 10
1 2 3 4 - 5 6 7 - 8 9 10
1 2 3 4 - 5 6 7 8 - 9 10
1 2 3 4 5 - 6 7 - 8 9 10
1 2 3 4 5 - 6 7 8 - 9 10
1 2 3 4 5 6 - 7 8 - 9 10
1 2 - 3 4 - 5 6 - 7 8 9 10
1 2 - 3 4 - 5 6 7 - 8 9 10
1 2 - 3 4 - 5 6 7 8 - 9 10
1 2 - 3 4 5 - 6 7 - 8 9 10
1 2 - 3 4 5 - 6 7 8 - 9 10
1 2 - 3 4 5 6 - 7 8 - 9 10
1 2 3 - 4 5 - 6 7 - 8 9 10
1 2 3 - 4 5 - 6 7 8 - 9 10
1 2 3 - 4 5 6 - 7 8 - 9 10
1 2 3 4 - 5 6 - 7 8 - 9 10
1 2 - 3 4 - 5 6 - 7 8 - 9 10
I didn't write them down here but there are 55 subset combinations for n = 11 and 89 subset combinations for n = 12.
I need to write a Visual Basic code listing all possible subset groups for n. I have been thinking on the solution for days but it seems that the solution of the problem is beyond my capacity. The number of required nested loops increases with n and I could not figure out how to program the nested loops with increasing number. Any help will be greatly appreciated.
After some research, I found out this is the problem of "compositions of n with all parts >1" and the total number of possible compositions are Fibonacci numbers (Fn-1 for n).
We already know the answer for these cases (as you wrote in your examples):
n=2
n=3
n=4
For n=5:
You can partition from 2: 1 2 - 3 4 5. This is like dividing the 5 member set into two sets, first one n=2, and second one n=3. We can now continue dividing each half, but we already know the solutions when n=2 and n=3!
You can partition from 3: 1 2 3 - 4 5. This is like dividing the 5 member set into two sets, first one n=3, and second one n=2. We can now continue dividing each half, but we already know the solution when n=2 and n=3!
For n=6:
You can partition into two sets from 2: 1 2 - 3 4 5 6. This is like dividing the 6 member set into two sets, first one n=2, and second one n=4. We can now continue dividing each half, but we already know the solution when n=2. Solve the second half by assuming n=4!
You can partition into two sets from 3: 1 2 3 - 4 5 6. This is like dividing the 6 member set into two sets, first one n=3, and second one n=3, We can now continue dividing each half, but we already know the solution when n=3 and n=3!
You can partition into two sets from 3: 1 2 3 4 - 5 6. This is like dividing the 6 member set into two sets, first one n=4, and second one n=2, We can now continue dividing each half. Solve the first half by setting n=4. For the second half, we already know the solution when n=2!
This is a simple recursion relationship. The general case:
Partition (S): (where |S|>4)
- For i from 2 to |S|-2, partition the given set into two halves: s1 and s2 from i (s1={1,...,i}, s2={i+1,...,n}), and print the two subsets as a solution.
- Recursively continue for each half by calling Partition(s1) and Partition(s2)
---
Another perhaps harder solution is to assume we are dividing the numbers 1 to n into n sections, where the length of each section can be either 0, 2, or a number greater than 2. In other words let xi be the length of each section:
x1 + x2 + ... xn = n, where the range of xi is: {0} + [2,n]
This is a system of linear non-equalities can can be solved by methods described here.
My answer to you is to try and come up with a recurrence relation of the given pattern. Think recursively. How can I break this problem down into smaller subproblems until reaching the smallest problem. Solve that smallest problem. After solving that smallest problem, think induction. Hypothesize on the what the nth step will be and how you will reach the (n+1)th step. Try to solve that (n+1)th step. Once you have come up with a recurrence relation on the pattern being given, it should not be too difficult to think about how to solve this pattern recursively. Instead of trying to use nested loops, this approach may be more intuitive.
I am struggling with following assignment:
Given sorted sequences of numbers and operations and , find an optimal sequence of those operations (the shortest one), which creates one sorted sequence.
I've devised following algorithm:
1. Sort sequences C1, C2, ..., Cn with respect to they first elements.
2. When number of sequences is greater than one:
3. Find in C1 the last position with the number that is less than the first number in C2.
4. If found_position == |C1|
5. C1 = concat(C1, C2)
6. Else:
7. C1a, C1b = split(C1, found_position + 1).
8. C1 = concat(C1a, C2).
9. Insert C1b to the set of sequences maintaining the order (with respect to their first elements).
10. Remove C2 from the set of sequences.
11. Go to step 2., in step 3. start searching from found_position.
An example:
1 4 5 9
2 6 10 11
7 12 20
1 4 5 9 2 6 10 11 7 12 20
^
1 4 5 9 2 6 10 11 7 12 20 // split
^
1 2 6 10 11 4 5 9 7 12 20 // concat
^
1 2 6 10 11 4 5 9 7 12 20
1 2 6 10 11 4 5 9 7 12 20
^
1 2 4 5 9 6 10 11 7 12 20
^
1 2 4 5 9 6 10 11 7 12 20
^
1 2 4 5 9 6 10 11 7 12 20
^
1 2 4 5 6 10 11 7 12 20 9
^
. . .
. . .
1 2 4 5 6 7 9 10 11 12 20
To maintain ordered working set of sequences, I could use balanced binary tree (insert in step 8 is nlog n).
Is it correct? How to prove its correctness?
I have queue q of integer numbers stored in an array in the circular fashion
from front to rear that is..
f: 1
r: 8
array Q: 0 1 2 3 4 5 6 7 8 9
2 8 4 4 3 5 4
What is the array representation of queue q after i perform the following?
while q.front() is an even number do q.enqueue(q.dequeue()).
It will loop until it reaches number 3. So the array will be
0 1 2 3 4 5 6 7 8 9
4 4 3 5 4 2 8
Consider the following example (values in vectors are target practice results and I'm trying to automagically sort by shooting score). We generate three vectors. We sort values in columns 1:20 in ascending order and rows in descending order based on out.tot column.
# Generate data
shooter1 <- round(runif(n = 20, min = 1, max = 10))
shooter2 <- round(runif(n = 20, min = 1, max = 10))
shooter3 <- round(runif(n = 20, min = 1, max = 10))
out <- data.frame(t(data.frame(shooter1, shooter2, shooter3)))
colnames(out) <- 1:ncol(out)
out.sort <- t(apply(out, 1, sort, na.last = FALSE))
out.tot <- apply(out , 1, sum)
colnames(out.sort) <- 1:ncol(out.sort)
out2 <- cbind(out.sort, out.tot)
out3 <- apply(out2, 2, sort, decreasing = TRUE, na.last = FALSE)
out2 has row names attached while out3 lost them. The only difference is that I used MARGIN = 2, which is probably the culprit (because it takes in column by column). I can match rows by hand, but is there a way I can keep row names in out3 from disappearing?
> out2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 out.tot
shooter1 1 2 2 3 3 3 4 5 5 5 6 6 6 6 6 7 8 9 9 10 106
shooter2 1 3 3 3 3 4 4 4 5 5 5 5 5 6 7 8 8 9 9 10 107
shooter3 1 1 2 2 2 3 3 4 5 5 5 6 6 6 6 7 8 8 8 9 97
> out3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 out.tot
[1,] 1 3 3 3 3 4 4 5 5 5 6 6 6 6 7 8 8 9 9 10 107
[2,] 1 2 2 3 3 3 4 4 5 5 5 6 6 6 6 7 8 9 9 10 106
[3,] 1 1 2 2 2 3 3 4 5 5 5 5 5 6 6 7 8 8 8 9 97
If I understand your example, going from out2 to out3 you are sorting each column independently - meaning that the values on row 1 may not all come from the data generated from shooter1. It makes sense then that the rownames are dropped in as much as rownames are names of observations and you are no longer keeping data from one observation on one row.