Questions tagged [ds.algorithms]
Questions regarding well-defined instructions for completing a task, and relevant analysis in terms of time/memory/etc.
1,796
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What's the constant coefficient of the Coppersmith-Winograd algorithm?
Every source I can find just says "too big to be practical."
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252
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Match a string agains a set of regexes
There are several algorithms to match a (simple) string against a regular expression (see here).
But if we have a lot of regexes, can we find one of them that matches the given string faster than ...
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Complexity of Set Difference
Given $k$ sets $S_1$, $S_2$, $\dots$, $S_k$ in the universe $U = \{1, 2, \dots, n\}$, is there a way to preprocess the $k$ sets such that there is an output-sensitive query algorithm that computes $...
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Implication of solving 3SUM problem of a certain size on the Exponential Time Hypothesis
In the recent question 3SUM Complexity—A special(?) Case I asked about why the set size $O(n^3)$ was an interesting value for the 3SUM Problem and got a nice answer. My reference was the paper “...
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3SUM Complexity—A special(?) Case
In the paper “Consequences of Faster Alignment of Sequences” by
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann which appeared in ICALP 2014 and is available here the following version of ...
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284
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What is the complexity of this submatrix selection problem?
We have a $kn\times kn$ matrix $M$ made of $n^2$ many $k\times k$ blocks.
We want to find an $n\times n$ submatrix such that each row and column is from distinct window of size $k$ such that the sum ...
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124
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Minimizing the gaps with incremental capacity
There are a single job, a machine and a set of $n$ slots. The machine has a capacity that increments by $\zeta(t)$ every slot $t=1,2,\ldots,n$. Initially (before the first slot), the machine has 0 ...
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Is there a fast algorithm for computing the Schmidt decomposition
I have a huge covariance matrix, 𝑀, with the dimension, e.g., $10^8 \times 10^8$. Luckily enough, the number of nonzero eigenpairs, $n$, is very small, i.e., $n<5$. From the computational ...
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189
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Add edges to a DAG to maximize increase in number of connected vertices
Let $G$ be a Directed Acyclic Graph.
$$C(G) = \bigl|\{(u,v):u,v\in V(G),v \text{ reachable from } u\}\bigr|$$
Goal is to add $k$ edges in a DAG such that for the new $G'$, $C(G')$ is maximized.
...
user49003
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Required sample size to hit certain subset of a ground set
Suppose $X$ is a set of $n$ points in $\mathbb{R}^d$ and $N_1,\cdots,N_k$ are k disjoint (unknown)subsets of $X$. There is a probability distribution $\phi$ on $X$ defined as $\phi(p) = \frac{\lvert\...
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understanding generalized coupon collector for distributions or learning mixture of distribution
Lets suppose we have a set $S=\{1,\ldots,n\}$ and $P$ is the uniform distribution over two subsets $T_1,T_2\subseteq S$, each of size $m\leq n/100$. Now, suppose somehow is given uniform samples from ...
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Complexity of computing Earth Mover's Distance when the costs satisfy the triangle inequality
Let p and q by two categorical probability distributions over $\{1,2,...,k\}$. Given a set of costs $c_{ij} \ge 0, i,j \in \{1,2,...,k\}$ that satisfy the triangle inequality, that is $c_{ij} \le c_{...
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NP-Hard or PTIME?
I am working on my research problem that essentially boils down to the following question. Consider an $N \times N$ matrix. There is a man at given a starting point $(x,y)$. In each unit of time, the ...
- 1,177
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On the complexity of a "list" datastructure in the RAM model
I am interested in the complexity of a data-structure equipped with the following operations (similar to a list):
insertion of an element at a given position within the list
deletion of an element at ...
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263
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Interval partitioning with restrictions: NP-complete or efficiently solvable?
The interval partitioning problem can be solved efficiently using a greedy algorithm. However, adding restrictions on the interval assignment to the problem results in a problem that appears harder. ...
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Are the intermediary sets in maximum cardinality search optimal in some way?
The maximum cardinality search (MCS) algorithm works as follows. Given a weighted graph $G = (V, E)$ where $w(u, v)$ denotes the weight of the edge $\{u, v\}$, we select a start node $a \in V$ and do ...
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Finding a non-negative solution to an integer system of linear equations
Let $A$ be an $n \times m$ integer matrix and consider the system of equations $Ax = b$ where $b$ is an integer vector. I want to find a solution $x$, assuming one exists, such that the components of $...
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What is the complexity of this weighted b-edge matching problem?
I'm wondering about the complexity of the following variant of the Generalized Weighted b-edge Matching problem:
Input: An undirected multigraph $G = (V, E)$ without
loops, an edge partition $(E_1,...
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Time complexity for multiplying two lower triangular matrices
I was wondering, if multiplication of two $n \times n$ lower (or upper) triangular matrices has a more efficient algorithm than multiplication of two general $n \times n$ matrices?
$$
\begin{bmatrix}
...
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92
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Optimal point placement on integer lattice
What is known about the following point placement problem?
For positive integers $N$, $n<N^2$, and $N\times N$ grid $\mathcal{G}$, compute
\begin{eqnarray*}
\mu_1(N,n)\triangleq\min_{\mathcal{P}\...
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Subgraph isomorphism on graph sequences
I'm looking for a subgraph isomorphism algorithm that takes advantage of properties of graph sequences.
Say $\{G_i\}_{i=1}^k$ is a sequence of graphs on vertex set $\{1 ... n\}$, and two consecutive ...
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112
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Running an algorithm for fixed amount of time on RAM model machine
Suppose there is a deterministic algorithm of size $O(1)$ that operates on an input of size $N$ on a RAM model machine. I want to run the algorithm for $O(\sqrt{N})$ time, pause the algorithm, print "...
- 1,177
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Combinatorial problems in electronics
This could be a downvoted question but I am asking because I am not able to get usable info via Google.
Are there any interesting combinatorial problems in the field of electronics circuits design? I ...
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Are string palindrome questions practically interesting?
I've been reading Don Gusfield's "Algorithms on Strings, Trees, and Sequences", and quite a some chunk of the textbook concentrates on palindromic-related ideas.
I'm unsure as to whether this is ...
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Is Gödel's speed-up theorem an instance of Blum's speedup theorem?
Blum's speedup theorem is a statement about a certain class of computable functions for which it is always possible to find a faster algorithm.
Gödel's speed-up theorem is a statement about the ...
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Best parameterized algorithm for maximum clique
I have seen the basic algorithm for the maximum clique problem parameterized by the maximum degree at an algorithms course. However, I struggle to find anything better. Searching for things like "...
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Time Complexity for Nearest Neighbor Searches in kd-trees
Nearest neighbor searches in kd-trees run in logarithmic time, as shown by Friedman et al. However, I have some difficulty to fully understand the proof.
In order to calculate the average number of ...
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Convex polygons inclusion relation
I have the following problem which came as a subproblem in some work I was doing and I am completely stuck.
Note that I am interested in it only in terms of worst case time complexity (not heuristics ...
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Examples of the value of proofs for algorithms
In teaching Intro. Algorithms to undergrads, one of the most difficult tasks is to motivate why they need to know how to prove things about algorithms. (For many students, at least in many US ...
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Isomorphic subforest problem
I recently read that the following problem is NP-Complete:
Given a tree $T$, and a forest $F$, is there a subgraph of $T$ isomorphic to $F$?
The reference provide was to the textbook “Computers and ...
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NP-hard problems with very fast exponential-time algorithms
NP-hard problems with very fast exact exponential-time algorithms, say with $O(1.01^n)$ time, are very rare.
Is any fact like
"For any constant $\epsilon>0$ there is an NP-hard 'natural' ...
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Matrix multiplication when one matrix is fixed
Let $A$ be a fixed positive entried integer matrix of size $a\times n$ with $\ell$ bits per entry
One is allowed to pre-process this matrix as appropriate.
Given another positive integer entried $B$...
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Potentially stronger form of non-$ETH$
If we have a $2^{n^a}$ algorithm to $K$-$SAT$ where $a<1$ for all $K>2$ then $ETH$ fails and literature gives consequences. What are the consequences if $a=o(1)$?
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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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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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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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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 ...
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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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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 ...
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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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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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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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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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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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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 ...
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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\}$$
for some ...
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Tortoise and hare algorithms
Consider the problem:
Given an array $a[0..n]$ where $n\ge 1$ and $a[i]\in [n]$ for all $i=0,\ldots,n$ find two indices $s\neq t$ so that $a[s] = a[t]$.
This problem has a stunning solution running ...
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Hidden Constants in Complexity of Algorithms
For many problems, the algorithm with the best asymptotic complexity has a very large constant factor that is hidden by big O notation. This occurs in matrix multiplication, integer multiplication (...
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Interpolation of the square of a polynomial
Let $\mathcal{P}^2_n$ be the set of rational polynomials of degree at most $2n$ that are sqares of polynomials, i.e. $\mathcal{P}^2_n$ consists of the set of polynomials of the following form:
$$(a_0 +...
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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 ...
- 241