Questions tagged [ds.algorithms]
Questions regarding well-defined instructions for completing a task, and relevant analysis in terms of time/memory/etc.
1,614
questions
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Scheduling and routing
Given:
$k > 1 $ sales execs, each specializing in one of 4 lines of business (LOBs), where each exec works (sales and travel) at 7.5 hours / day
$n > 1$ client sites.
Constraints:
Each ...
-3
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0answers
18 views
Runtime of flood fill with multiple centres
What is the runtime if we are using the flood fill algorithm but with multiple centres from which it starts.
5
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56 views
What are some examples of algorithmic applications of noncommutative rational identity testing?
The problem of polynomial identity testing (PIT) is known to be in $\mathsf{RP}$, but not known to be in $\mathsf{P}$.
The related problem of noncommutative rational identity testing (NCIT) is known ...
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11 views
Capacitated Vehicle Routing Problem with multiple pickups at some sites but constrained to max 1 pickup per vehicle per site
I am looking for literature or code for a variant of capacitated vehicle routing problem. The variance is that some sites have multiple items for pickup, but there is a constraint such that even if a ...
5
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1answer
188 views
What are the consequences of a faster algorithm for $CIRCUIT$-$SAT$?
What is the best algorithm known for $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates?
What is the consequence if there is an $\alpha\in(0,1)$ such that $CIRCUIT$-$SAT$ in $n$ variables and $m$ gates ...
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1answer
66 views
Dividing a complete graph into two cliques with maximal sum of edge weights
Problem: Considering a complete weighted graph $G$ with $n$ vertices, where $n\in2\mathbb Z$ is an even number, remove edges in such a way that you end up with two cliques of graph $G$, each having $\...
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92 views
Can you partially sort using $O(\log n)$ comparisons per element?
Input is a list of $n$ integers in an array A. Desired output is stored in Array B, such that $|rank(B[i])- i | \leq \sqrt{n}$.
Can this be done using $O(\log n)$ comparisons per element?
Just looking ...
2
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1answer
64 views
Reconstruction of a sequence generated by a Markov chain - reference request
Let S be a finite sequence of symbols from a finite alphabet, with gaps - that is on some known locations an unknown number of symbols are missing. Assuming that the sequence , including the symbols ...
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59 views
Generalization of k-Coloring: maximizing the number of vertices with no neighbours of same color
One can consider the following generalization of the $k$-Coloring problem:
Let be given a graph $G$ and an two integers $k$ and $p$. A vertex $v$ of $G$ is properly colored if $v$ has no neighbour ...
3
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1answer
102 views
What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?
It is known that computing $(\sqrt 2 -\epsilon)$-approximation for VC is NP-hard and that UGC implies that even a $(2 -\epsilon)$-approximation is hard.
There is also a parameterized algorithm for ...
4
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1answer
130 views
Consequences/existence of problems without any “optimal” algorithm
Let $P$ be some kind of "problem" such as addition or graph coloring, that has an input size $n$. Let $S_P$ denote the set of algorithms $A_1, A_2, \dots$ which deterministically solve $P$. Based off ...
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404 views
Formalizing and optimizing constraints involving booleans, pairs of booleans, and integer sums
My scenario has various flavors of SAT, constrained quadratic pseudo-Boolean, and integer programming. My attempts to formalize and solve the problem with Z3's ...
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1answer
83 views
An efficient algorithm for maximizing gain by choosing from a set of options
(I hope this is on-topic for this site -- mods feel free to send this to another stack exchange if not )
I've got an optimization problem where I need to choose from one of several options to ...
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2answers
111 views
Hospital Resident Matching Algorithm with Incomplete Preferences
Consider a set of doctors $D$ and hospitals $H$ such that each doctor $d \in D$ has a rank ordered strict preference over a subset of hospitals, $H_d \subseteq H$. Similarly, each hospital $h \in H$ ...
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70 views
Algorithm for computing the smallest subset of nodes to remove from a graph to make it a tree
I have encountered an interesting problem that I couldn't find any references to solve:
Determine the smallest subset of nodes that
need to be removed from an undirected graph to make it a tree.
...
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0answers
65 views
Cost of in-place partitioning integer arrays
Suppose we are given an array $a\colon[n]\to[m]$ of length $n$ (and each entry is between 1 and m). We will denote the $i$th entry of the array as $a[i]$.
Task: Permute the array $a$ in-place so that ...
3
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1answer
134 views
Given a subset of of the hypercube and an affine transform of it, find the affine map
This is a follow up to this resolved question.
Suppose we are given a set of bitvectors $A\subseteq\mathbb{F}_2^d$ and an invertible affine transformed copy of it
$$B=\{Mx + s\mid x\in A\}$$
for some ...
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165 views
Tortoise and hare algorithms
Consider the problem:
Given an array $a[0..n]$ where $n\ge 1$ and $a[i]\in [n]$ for all $i=0,\ldots,n$ find two indices $s\neq t$ so that $a[s] = a[t]$.
This problem has a stunning solution running ...
9
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1answer
317 views
Hidden Constants in Complexity of Algorithms
For many problems, the algorithm with the best asymptotic complexity has a very large constant factor that is hidden by big O notation. This occurs in matrix multiplication, integer multiplication (...
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128 views
Interpolation of the square of a polynomial
Let $\mathcal{P}^2_n$ be the set of rational polynomials of degree at most $2n$ that are sqares of polynomials, i.e. $\mathcal{P}^2_n$ consists of the set of polynomials of the following form:
$$(a_0 +...
2
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1answer
85 views
Placing color boxes on a colored image such that color consistency is maximized
I have encountered the following challenging problem that I think to be a non straightforward generalization of the Knapsack problem.
Given an image with black background that contains blobs whose ...
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1answer
132 views
Is it possible to prove that a general purpose integer factorization algorithm must contain a loop?
1)
Let $A$ be a (general purpose) algorithm that factors $n$. Suppose $A$ contains a loop (which is hard to imagine if not impossible that it does not.) If $A$ contains nested loops then these loops ...
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84 views
Is it possible to sort by only knowing the sign of pairwise sums?
I am currently thinking of how much structure one actually needs in order to be able to sort things at all. All comparison-based algorithms need a direct comparability, but are we able to remove this ...
2
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1answer
164 views
How many samples are needed to reconstruct a path?
Consider an input set of vertices $V$ and vertices $s,t\in V$.
The goal is to learn some unknown shortest path from $s$ to $t$; the set of edges of the graph is hidden at first and there may be ...
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52 views
Mapping of entire balls using Locality Sensitive Hashing (LSH)
LSH functions are useful for approximate nearest neighbor search.
They are usually defined, for distance metric $d$ and $c>1$ as follows:
A family of hash functions is $(r, cr, p_1, p_2)$-LSH ...
5
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167 views
Is there a fast algorithm for inverting a sparse matrix?
I am doing research on a random-walk like problem. As a critical part of my solution, I need to invert a non-singular sparse matrix of size $n \times n$ and with $O(n)$ non-empty entries. I'm working ...
7
votes
2answers
184 views
Determining how n-tuples are sorted as you search them?
I have a sorted list of $N$ n-tuples, but I do not know exactly how they were sorted. The person who sorted them did so by lexicographically ordering some permutation least-to-greatest. For example, ...
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1answer
54 views
How is additive error handled in this simple algorithm? 'Product of all elements'
Say we have two unit vectors $\hat{u}, \hat{v} \in \mathbb{R}^n$ where $\hat{u} = (u_1,...,u_n)$ and $\hat{v}$ approximates $\hat{u}$. $~\hat{v} = (u_1+\epsilon, ...,u_n+\epsilon)$ where $\epsilon = \...
2
votes
1answer
65 views
In external memory, is grouping equal elements easier than sorting?
Sorting an array will put equal elements adjacent to each other. So, in no model of computation can grouping equal elements be harder than sorting. In the RAM model, grouping equal elements is $O(n)$ ...
2
votes
1answer
83 views
Longest stack-sortable subsequence
Given an array of $n$ pairwise-different positive integers, the problem is to find the longest subsequence that is stack-sortable, i.e. avoiding the permutation pattern $231$.
How fast can this ...
2
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125 views
Shortest s-t path when is allowed to ignore k weights
Given an undirected graph $G$ with $n$ vertices and $m$ edges, with non-negative weights on the edges, what's the best algorithm that computes the shortest path from $s$ to $t$, where you are allowed ...
2
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0answers
52 views
Complexity of comparing extended integer power towers
Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by ...
2
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0answers
82 views
Finding 3SUM witness when promised a solution
Suppose we have a 3SUM instance given with the promise that there exists at least one solution. Is the trivial $O(n^2)$ (modulo logarithmic improvements) solution still the best algorithm or is there ...
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0answers
82 views
Entropy bounds on solutions to problems in BPP and other complexity classes based on entropy demands
Has anyone studied the asymptotics of problems in complexity classes like $BPP$? The thought came to me that if a problem in $BPP$ only requires $O(log(n))$ bits of entropy to solve then, intuitively, ...
6
votes
1answer
151 views
Arranging letters to make a word in a regular language
Fix a regular language $L$ on the alphabet $\{a, b\}$, and consider the following problem. I am given as input:
some number $m \in \mathbb{N}$ of copies of the letter $a$, and
some number $n \in \...
11
votes
1answer
463 views
What is the hardest instance for the group isomorphism problem?
Two groups $(G,\cdot)$ and $(H, \times)$ are said to be isomorphic iff there exists a homomorphism from $G$ to $H$ which is bijective. The group isomorphism problem is as follows: given two groups, ...
3
votes
1answer
145 views
How is SDP an extension of spectral algorithms?
In one of his lectures, Uri Feige described semidefinite programming (SDP) as
... an algorithmic technique that extends both linear programming and spectral algorithms.
I know the basic ...
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1answer
76 views
The SQ argument in Balazs Szorenyi's paper
I am asking about the proof in Theorem 5 (page 6) of this paper,
http://www.inf.u-szeged.hu/~szorenyi/Cikkek/sq_d0_ext.pdf
Quite a few things about this short argument seem unclear to me,
Towards ...
11
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0answers
186 views
Computational Complexity of the Frobenius Problem
The Frobenius problem takes as input $n$ positive integers $a_1,\ldots,a_n$ with $\gcd(a_1,\ldots,a_n)=1$ and asks for the largest integer $F$ that cannot be written in the form $F=a_1x_1+a_2x_2+\...
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2answers
92 views
Enumerate all allocations of points in a simplex
Consider the standard 2-simplex $\{(x,y)~|~x+y=1~;~ x,y\geq 0\}$.
Given a set $M$ of $m$ points in this simplex, we allocate each point either to X or to Y by the following process:
Fix two positive ...
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0answers
45 views
Minimising the maximum distance to the centre of a cluster of points
I have a set of points $C_i$ on a two dimensional plane and I want to find a point $P$ such that the maximum distance from $P$ to any of the points is minimised, i.e. minimise(max($||P-C_i||$)).
I've ...
4
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1answer
138 views
Generalizations of linear programming
Linear problems can be solved in polynomial time. So can semidefinite programs and, presumably, many other useful classes of optimization programs.
Is there a survey/lecture notes describing ...
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0answers
38 views
Time complexity of finding a point of infinite order on a rank 1 elliptic curve over Q
As an outsider, it sounds like a lot of progress has been made on understanding rank 1 elliptic curves over Q. Much of the BSD conjecture is known for rank 1, and Heegner points provide a way in ...
6
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1answer
237 views
Example problem that is not in $2^{o(n)}$ but could be solved in $O(2^{cn})$ for any $c > 0$ (suggested by wording of ETH)
In the wikipedia article on Time Complexity it is written that:
The exponential time hypothesis (ETH) is that 3SAT, the satisfiability problem of Boolean formulas in conjunctive normal form with, ...
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1answer
106 views
About learning a single Gaussian in total-variation distance
I am looking for the proof of this following result which I saw as being claimed as a "folklore" in a paper. It would be helpful if someone can share a reference where this has been shown!
Let $G$ ...
2
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0answers
56 views
Conjugacy testing problem
The below-given problem is in black box setting means input is given by set of generators.
Given an abelian $p$-group $A$ and two matrices $U_1$ and $U_2$ in $R(A)$ such that the order of $U_1$ and $...
6
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153 views
Bottleneck $k$-link path in a complete DAG
Let $G$ be a complete DAG: It has vertices $v_1,\ldots,v_n$, and $v_iv_j$ is an edge if and only if $i<j$.
Let $w(i,j)$ be the weight of the edge $v_iv_j$. The weight has the property that $w(i,j)&...
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0answers
11 views
Size of solutions in integer programming
Given a linear integer program $Ax\leq b$ with $A\in\mathbb Z^{m\times n}$ and $b\in\mathbb Z^m$ known is there a polynomial time algorithm to give tight upper bounds for $\log_2\|x\|_\infty$ and $\...
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171 views
What's the fastest known algorithm for finding the diameter of a graph?
Given a positively weighted graph what's the fastest algorithm for finding the diameter for that graph?
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69 views
What is the competitive ratio of a $d$-way associative LRU cache?
In a caching problem, items arrive online, and the algorithm needs to decide which elements to keep in the cache. If the current item is not cached, we pay a penalty of $1$.
It is well known that for ...
