Questions tagged [permutations]
Permutations are arrangements of the numbers $1,\ldots,n$ in an arbitrary order.
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Applying permutations with moves rather than swaps
Everything I can easily find on applying permutations in-place relies on treating swaps as elementary operations. But in reality, every swap requires at least two moves (to move the first item into ...
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Computational complexity of an algorithm involving permutations
I'm interested in getting a precise estimate of the computational complexity of an algorithm I wrote involving permutations. Permutations are represented in my code as arrays of the integers $1$ ...
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Game of permutations as a minimization problem
Consider the following "game".
There is an ant on a strip of length N. The ant can perform the following actions: move left, move right, and paint the cell he is on with white or black. The ...
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How to handle duplicates with Heap's algorithm for permutations?
I was able to use Heap's algorithm for generating permutations and I was able to solve the duplicates problem by looking at all the results and filtering out duplicates myself using a HashSet in Java.
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Efficient algorithms to generate partial permutations?
I want to efficiently generate partial permutations.
That is, I want to generate the set of ordered arrangements of K distinct elements selected from a set from N items. For example, with the set ...
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Heap's Algorithm for k-permutations?
Heap's Algorithm enumerates the permutations of a set of N objects. It's described as:
Permute the elements in positions 1 to n−1 by applying this algorithm to those elements.
Apply the following ...
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Minimum number of swaps to achieve prefix constraint
(An isogram is a word in which no letters repeat.)
I'd like to know if there's a standard name for this problem, and/or any standard algorithms. If not, I'm interested in whether anyone has any ideas ...
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Proof of correctness of some optimization of Heap's algorithm for producing permutations
Good day. Some time ago I found a page on (en.) wikipedia about Heap's algorithm for permutations. Here it is.
Original algorithm can be written as next (copy from (en.) wikipedia):
...
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Check if Two Arrays Sorted in Decreasing and Increasing Order Satisfy a Condition
Given 2 arrays $A, B$ each of size $n$. If we want to find if the two arrays satisfy the condition that $A[i] + B[i] \ge k$ for all indices $i$, it's sufficient to fulfill 3 conditions below:
Sum of ...
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Ranked voting method where unranked candidates on a preference list aren't taken to be the least preferred?
Say there are 5 candidates, A, B, C, D, and E. An election is held using a ranked voting method. That is to say, each voter submits a preference list (the order in which they prefer candidates). E.g. ...
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Weighted permutation bucketting
Given a set of weighted permutations of numbers, an operation can be performed to combine multiple permutations that share numbers into a single rolled up permutation. What is the optimal algorithm to ...
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One-shot capacity achieving input, permutation invariance and i.i.d. decomposition
Suppose I have $n$ iid copies of a channel $W_{Y|X}$ and denote this by $W_{Y|X}^{\otimes n}$. I found a one-shot capacity of $W_{Y|X}^{\otimes n}$ to be an entropic quantity (specifically, it is the ...
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What can I read about permutations in 3D space?
I'm interested in the theory of calculating the position of an element when the face of a cube is moved - like in a Rubik's Cup. That is, for example, I rotate the face of the cube - how can I ...
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Permuting matrix entries to lower rank
Suppose I have a rank-$k$ matrix $A \in \mathbf{R}^{m \times n}$. Now suppose this matrix has its elements shuffled by an adversary to maximize the rank. Is there a way to reverse this permutation and ...
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Doubt regarding conjugations of permutations
I was studying the Barrington Theorem in https://homes.cs.washington.edu/~anuprao/pubs/CSE531Sp2020/lecture2.pdf when I found a doubt regarding permutations. Particularly, with conjugations.
It is ...
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Finding Optimal Configuration of Formula without trying every Permutation
I have a math problem I need to solve so I can complete an optimisation in a computer program. My initial approach was just to brute force all the possible permutations but it got out of hand quite ...
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Manipulating a binary tree representation
I am designing a discrete probability distribution with a string of binary values as an input $\{s_i\}\in\{S_i\}$, and binary outputs $e\in E$ with Bernoulli probability $P(E=1|\{S_i\})=XOR[\{S_i\}]=...
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How many ways we can partition a multiset, where each part/segment in the partition has distinct elements? [closed]
We define the set S as $\{(s_1, f_1), (s_2, f_2), ..., (s_i, f_i)\}$, where each $f_i$ is the frequency that $s_i$ is repeated in the multiset T. How many ways can we partition the multiset T into ...
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Producing "moves" to permute one array to another
I have 2 arrays, A and B, which each contain the same N elements, but in a different order. (A different permutation)
There are also no duplicates in A and B.
I'm trying to devise an algorithm which ...
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Compute Permutation Number
Given a permutation on the array of integers 1 through n, I want to find the index of the permutation in a list of all possible permutations of those integers, sorted in lexicographic order.
For ...
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Construct an objective function that favors the sorted permutation
I was looking into complexity theory and disappointed by the fact that a problem as simple as sorting an array isn't in the class $P$. This is because the class is defined only for decision problems (...
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Need math explaining calculation of number of binary tree topologies given n nodes
Grinding through leetcode, I came across the question of the number of binary tree topologies for n nodes.
When looking up the best solutions, I saw the same line of code in each solution:
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Is this an inversion problem?
I understand that total inversion in a sequence is the number of swaps that need to be done, to sort the sequence. And the best approach to solve this problem is to count swaps in a merge sort of the ...
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nth Permutation generator
I'm just trying to write a little algorithm. I've got nine objects, so there's 9! permutations. My question is, is there a way of turning a number from 1 to 9! into a permutation?
for example, f(1)=[1,...
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removing an item from n lists in $O(n^{1-\epsilon})$ amortized time
I have a straightforward task that can be done in $O(n^2)$ time. I'm now wondering if the task can be done in time $O(n^{2-\epsilon})$ if we are allowed to do some pre-processing.
The problem exists ...
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How many different boolean functions exist up to permutation of its $n$ variables
i am relatively new here, so if this was asked before, feel free to redirect me. I am searching for an answer in form of a (iterative or recursive) Formula or even better, an algorithm to list them ...
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Question about stack permutations
I encountered an exercise in which I am struggling to understand the solution provided even though I spent a lot of time trying to figure it out.
The exercise is the following and was taken from the ...
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Given a set, generate all permutations whose sums are less or equal to a given number
I am looking for a way to generate every permutation (so order does matter) of a set of positive numbers whose sum is less than (or equal to) a given limit. I need to find the permutations themselves, ...
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What loop invariant can be used for this loop? (Algorithm to check if a sequence is stack permutable)
I've written a function that gets a permutation and checks if that permutation can be reached using a stack from an input sequence which is <1,2,3,...,n>. (we take elements from left)
For ...
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How can we prove QuickShuffle uniformly permutes it input array?
I'm studying Algorithms
by Jeff Erickson. Consider this exercise from that textbook:
Prove that the following algorithm, modeled after quicksort, uniformly permutes its input array, meaning each of ...
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Determining runtime of a theoretical program (question of extraordinary complexity)?
I apologize in advance, as I don't have a clue to which stackexchange to post this question! I beg you to not delete this question, as I have chronic pain and it is very important to me!!! I actually ...
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Iteratively enumerating all permutations of $N$ objects using a generating set
The group theory of $S_n$ shows that all permutations of $n$ objects can be generated from the $n$-cycle $a:=(1 2 3 .. n)$ and the transposition $b:=(1 2)$. (See Theorem 2.5 at https://kconrad.math....
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Memory-efficient list of N unique integers from 0..N-1 with fast lookup
We want to represent a list of $n$ unique integers between $0$ and $n-1$ in a memory-efficient way. The only operation we need to support is looking up the $n\text{-th}$ element.
What are some of the ...
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Counting number of swaps to make two strings equal in linear time
The input to our problem is a pair of strings, say $x$ and $y$. We treat our alphabet size as a constant, i.e., our input is effectively a pair of arrays with the values therein bounded by a constant.
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Faster algorithm for specific inversion count (part 2)
Following the issue from Faster algorithm for a specific inversion:
We have a permutation (a derangement actually) $\sigma$ of the set $\{0,1,\dots,n-1\}$ with cardinality $n$.
I want to compute ...
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Faster algorithm for a specific inversion
There is a permutation (more precisely a derangement) $\sigma$ of the set $\{0,1,\dots,n-1\}$ with cardinality $n$.
I want to compute the following counts (a kind of inversion):
$$K(\sigma )_{i}=\#\{j&...
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Does this algorithm for permuting rows and columns of a matrix converge?
Given a binary matrix, define the magnitude of a row by reading off the numbers in it left to right and similarly for columns, reading the numbers going down. For instance, the row magnitude of the 1-...
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Number of permutations with satisfactory triangles
We are given $N$ points($N \leq 40$), where no combination of three or more points is colinear. The values of $x$ and $y$ are bounded by [$0$,$10^4$]. The problem is to find the number of permutations(...
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Iterated multiplication of permutation matrices
Given $m$ matrices of size $n\times n$ each of which is promised to be a permutation is it in $\mathit{quasiAC}^0$ or $\mathit{AC}^0$ to multiply the permutations where
$m=\mathit{poly}(n)$
$m=\...
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Binary ↔ Gray permutation matrix
Generating a Gray code representation of a binary number can be thought of as mapping one binary number onto another binary number. Therefore, $n$-bit Gray code is a permutation of $2^n$ elements.
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number of ways evaluation of expression such that value not changed [closed]
one example:
How many ways we can do possible value-preserving parenthesis the following expression in such a way that value not changed after parenthesis with one constraint that parenthesis among ...
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Find Optimal Permutation/Positioning to Minimize the Total Distance for a Given Path
Summary:
A task for picking certain objects is given in the form of an ordered sequence (eg. to pick apple, banana, apple, apple, orange, order matters). The objects have to be preassigned to certain ...
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Enumerating all partial permutations of given length in lexicographic order
I need to generate all unique tuples of length k chosen from a series of unique, positive integers. In my case n choose k will have n=10, 1 <= k <= 10; and the series I am choosing from is { 0, ...
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Find The "Best" Permutation of Inputs to Maximize Sum of Functions (or approximate "best")
The Problem (in words)
I want to sort $N$ items where the value of item $i$ at position $p$ is given by the function $f_i(p)$. The "best" order for these items is the one that maximizes the ...
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Given permutation $p$, compute $p^{-2}$
I'm now to problem solving, and I need some help and insight on the following problem from HackerRank:
Given a sequence $p(1),\ldots,p(n)$ of distinct numbers from $1$ to $n$, find numbers $y_1,\...
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What's wrong with the following shuffle algorithm?
Suppose I have an array A of size n, with the initial state of:
...
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Simple incremental hash funcion
I have permutations:
4 1 2 5 3
4 3 2 5 1
numbers can be order magnitude of 1000 (fits in two bytes)
I want compute 32 bit (or better 64 bit) hashes, it should be ...
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How do you write a python\pseudo code that generates all pair permutations?
What would be a good pseudo code or Python 3 code for the following permutations problem?
Let us define a n-permutation as a bijective function $\pi: \{0,...,n-1\}\rightarrow \{0,...,n-1\} $ and ...
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finding the combinatorial solutions of series and parallel nodes
I have n nodes, and I want to find the (non duplicate) number of possible ways in which these nodes can be combined in series and parallel, and also enumerate all the solutions. For example, for n=3, ...
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Finding index of $p_{k}$ element in the original sorted array if elements were to be removed using a specific condition
Consider a sorted list of numbers $C_{0}=\{0,1,2,3,...,n-1\}$ from where one element will be eliminated at each step. We are also given a value $L$ in $[0, 1)$ and let the indexing start from $0$.
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