19 KiB
title | TARGET DECK | FILE TAGS | tags | ||
---|---|---|---|---|---|
Permutations | Obsidian::STEM | combinatorics set |
|
Overview
A permutation of some n
objects is a (possible) rearrangement of those n
objects. The number of permutations is n!
since there are n
possible ways to pick the first object, (n - 1)
possible ways to pick the second, and so on.
void next(const size_t n, int A[static n]) {
size_t pivot = -1;
for (size_t i = n - 1; i >= 1; --i) {
if (A[i - 1] < A[i]) {
pivot = i - 1;
break;
}
}
if (pivot == -1) {
reverse(0, n - 1, A);
return;
}
size_t j = pivot;
for (size_t i = pivot + 1; i < n; ++i) {
if (A[pivot] < A[i] && (j == pivot || A[i] < A[j])) {
j = i;
}
}
swap(pivot, j, A);
reverse(pivot + 1, n - 1, A);
}
void permutations(const size_t n, int A[static n]) {
size_t iters = factorial(n);
for (size_t i = 0; i < iters; ++i) {
print_array(n, A);
next(n, A);
}
}
The above approach prints out all permutations of an array (assuming distinct values).
%%ANKI Basic What is a permutation? Back: An ordered arrangement of some collection of objects. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How many permutations are there of n
objects?
Back: n!
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is n!
written recursively?
Back: n(n - 1)!
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is n!
permutations of n
objects derived?
Back: There are n
choices for the first position, n - 1
choices for the second, etc.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What combinatorial concept explains n!
permutations of n
objects?
Back: The multiplicative principle.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic How does sorting relate to the concept of permutations? Back: Sorting aims to efficiently find a specific permutation. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algorithm
END%%
%%ANKI
Basic
What symbol denotes "n
factorial"?
Back: n!
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
n!
is an abbreviation of what "big operator" formula?
Back: \Pi_{k=1}^n k
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the identity element of \cdot
(multiplication)?
Back: 1
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What does 0!
(factorial) evaluate to?
Back: 1
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is the multiplication identity used to justify equality 0! = 1
?
Back: The empty product is 1
, i.e. the multiplication identity.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What combinatorial explanation justifies equality 0! = 1
?
Back: There is only 1
way to order 0
objects.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What combinatorial concept explains the number of bijective functions between two finite sets? Back: Permutations (factorials). Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How many bijective functions exist between \{1, 2, 3\}
and \{a, b, c\}
?
Back: 3!
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How many bijective functions exist between finite sets A
and B
where |A| = |B| = n
?
Back: n!
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Lexicographic Ordering
We can find the next lexicographic ordering of an array via a procedure of "pivot", "swap", and "reverse". The function void next(const size_t n, int A[static n])
defined in #Overview shows the details, taking in a permutation and producing the next, lexicographically speaking. To prove correctness, consider the following:
[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]
Here the RHS side is the longest increasing sequence we could find, from right to left. That is, a_{i+1} > a_{i+2} > \cdots > a_n
. Denote a_i
as the pivot. Next, swap the smallest element in the RHS greater than a_i
, say a_j
, with a_i
. This produces
[ a₁ a₂ ... aⱼ | aᵢ₊₁ aᵢ₊₂ ... aᵢ ... aₙ ]
Notice the RHS remains in sorted order. Since a_j
was the next smallest element, reversing the reverse-sorted RHS produces the next permutation, lexicographically speaking:
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
Eventually the swapped a_j
will be the largest in the RHS ensuring that the breakpoint will eventually move one more position leftward.
%%ANKI
Basic
What algorithm does NEXT_LEXICO_ARRAY
refer to?
Back: The finding of the next lexicographic ordering of an array.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What does array A
's next lexicographic ordering refer to?
Back: The permutation that follows A
in a sorted list of all distinct permutations of A
.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI Basic How does lexicographic ordering of arrays relate to permutations of an array? Back: Each lexicographic ordering corresponds to a permutation. Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: How many invocations guarantee all permutations of A[1:n]
?
Back: n!
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: When does < n!
iterations yield all permutations of A[1:n]
?
Back: When A
contains duplicates.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is NEXT_LEXICO_ARRAY
's worst-case running time?
Back: O(n)
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is NEXT_LEXICO_ARRAY
's best-case running time?
Back: \Omega(n)
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is NEXT_LEXICO_ARRAY
's auxiliary memory usage?
Back: O(1)
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is the next lexicographic ordering of [ 1 3 2 4 ]
?
Back: [ 1 3 4 2 ]
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is the next lexicographic ordering of [ 2 1 4 3 ]
?
Back: [ 2 3 1 4 ]
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is the next lexicographic ordering of [ 4 3 2 1 ]
?
Back: N/A.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is the output of NEXT_LEXICO_ARRAY([ 1 2 3 4 ])
?
Back: [ 1 2 4 3 ]
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is the output of NEXT_LEXICO_ARRAY([ 4 3 2 1 ])
?
Back: [ 1 2 3 4 ]
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: Which element will be the pivot of [ 1 2 3 4 ]
?
Back: 3
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: Which element will be the pivot of [ 4 3 2 1 ]
?
Back: N/A
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What property does the RHS of A[1:n]
exhibit before swapping?
[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]
Back: Values are in non-increasing order. Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What property does the LHS of A[1:n]
exhibit before swapping?
[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]
Back: N/A Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What property does the RHS of A[1:n]
exhibit after swapping?
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
Back: Values are in non-increasing order. Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What property does the LHS of A[1:n]
exhibit after swapping?
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
Back: N/A Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
What is the first step taken in the NEXT_LEXICO_ARRAY
algorithm?
Back: Finding the pivot element.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What does the "pivot" refer to?
Back: The element preceding the longest increasing subarray from right-to-left.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Cloze
The NEXT_LEXICO_ARRAY
algorithm can be summed up as "{pivot}", "{swap}", and "{reverse}".
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: Which element is swapped with the pivot?
Back: The smallest element to its right that is greater than it.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What is done after swapping the pivot element?
Back: Reverse the subarray to the right of where the pivot element was originally located.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What step follows swapping A[1:n]
's pivot element?
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
Back: Reverse the elements to the right of a_j
.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
%%ANKI
Basic
NEXT_LEXICO_ARRAY
: What invariant is maintained after swapping the pivot?
Back: The elements to the right of the original pivot remain in non-increasing order.
Reference: https://leetcode.com/problems/next-permutation/description/
END%%
Falling Factorials
If we generalize to choosing k \leq n
elements of n
objects, we can calculate the k
-permutation of n
. This is denoted as (n)_k
, sometimes called the falling factorial. (n)_k = \frac{n!}{(n - k)!}
The derivation works by noting that we have n - 0
possible ways to pick the first object, n - 1
ways to pick the second, up until n - (k - 1)
ways to pick the last object.
%%ANKI
Basic
What is a k
-permutation?
Back: An ordered arrangement, containing k
elements, of some collection of objects.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the closed formula for falling factorial (n)_k
?
Back: (n)_k = \frac{n!}{(n - k)!}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is the number of k
-permutations of n
objects denoted?
Back: (n)_k
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is n!
written equivalently as a falling factorial?
Back: (n)_n
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How is (n)_k
k
-permutations of n
objects derived?
Back: There are n
choices for the first position, n - 1
choices for the second, etc. up until n - (k - 1)
choices for the last position.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What does (n)_n
evaluate to?
Back: n!
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What does (n)_0
evaluate to?
Back: 1
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What combinatorial problem does (n)_0
represent?
Back: The number of ways to choose 0
objects from n
choices.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Cloze
In a k
-permutation of n
objects, there are n - 0
choices for first object and {n - (k - 1)
} choices for the last object.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What combinatorial concept explains the number of injective functions between two finite sets?
Back: k
-permutations (falling factorials).
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How many injective functions exist between \{1, 2, 3\}
and \{a, b, c, d, e\}
?
Back: (5)_3
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Bibliography
- Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.