Questions tagged [ds.algorithms]
Questions regarding well-defined instructions for completing a task, and relevant analysis in terms of time/memory/etc.
1,591 questions
0
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0answers
35 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
votes
0answers
142 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
174 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, ...
-1
votes
1answer
48 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
58 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 $O(n)$ ...
-1
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0answers
85 views
How to find the shortest path within a huge unweighted graph
I want to find the optimal path between any vertex and the target(which is always the same). The problem is that my graph is huge (around $7 \times 10^14$ vertices) and impossible to fully load into ...
2
votes
1answer
70 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
votes
0answers
95 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
48 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
votes
0answers
77 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 ...
1
vote
0answers
75 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, ...
4
votes
0answers
96 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 \...
8
votes
1answer
346 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
127 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 ...
1
vote
1answer
74 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
votes
0answers
175 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+\...
1
vote
2answers
81 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 ...
1
vote
0answers
35 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 ...
2
votes
1answer
120 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 ...
2
votes
0answers
35 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
votes
1answer
227 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, ...
1
vote
1answer
76 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
votes
0answers
55 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
votes
0answers
147 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)&...
1
vote
0answers
9 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 $\...
5
votes
0answers
169 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?
2
votes
0answers
65 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 ...
0
votes
1answer
113 views
Permuting the columns of a 0/1-matrix to avoid short segments
Consider an $n \times n$ table with $n$ stars such that each row contains at most $\log n$ stars. The stars break each row into segments (continuous parts of a row without stars). Let's call a segment ...
6
votes
0answers
118 views
Evaluating addition chains
I hope this is a suitable place to ask this question.
An addition chain of size $n$ is a sequence $x_1, \dots, x_n$, where $x_1$ is fixed to 1 and $x_i = x_j + x_k$ for some $j,k < i$. I am ...
1
vote
0answers
35 views
PTAS for projective clustering : survey
$(k,j)$-projective clustering is the natural generalisation for k-clustering, in which one needs to find $k$ $j$-flats in $\mathbb{R}^d$ that minimizes the cost function as defined below:
Given a $j$-...
-4
votes
2answers
83 views
Is it possible to have a sorting algorithm that computes faster than QuickSort? [closed]
Given an unsorted array, QuickSort has to touch each source element it is trying to sort multiple times before it declares an array as sorted.
(notice how many times the 2 is touched [circled in red ...
5
votes
1answer
66 views
Pop desired elements on stacks of bounded capacity
Consider there are $k$ stacks containing a total of $n$ elements. Each element is either red or blue. We have complete knowledge of each element's location and color.
Only push and pop are allowed on ...
-1
votes
1answer
60 views
Formally prove that the loops of this sorting algorithm will terminate [closed]
Given is the sorting algorithm Bubblesort
...
5
votes
1answer
229 views
Evaluation of an arithmetic formula where the time depends on the length of the arguments of gates
Let $(X,+,\cdot)$ be a commutative ring. Let $|\cdot|\colon X\to \mathbb{N}$ be a function that satisfies $|x+y|\leq |x|+|y|$ and $|xy|\leq |x|+|y|$. We call the function length, and length is always ...
2
votes
1answer
77 views
Find shortest prefix to generate original string by overlapping
Given a string $S$, I want to find the prefix string $P$ of shortest length, such that the original string $S$ can be generated by concatenating copies of $P$ (where overlapping is allowed).
For ...
1
vote
0answers
126 views
How to write algorithms?
Reading research articles in theoretical computer science, I noticed that people often describe their algorithms in an enumerative way (i.e., they enumerate the steps of their algorithm and use "go to"...
2
votes
1answer
101 views
Minimization version of matrix p-norms?
I considered a minimization version of matrix p-norms, defined for a matrix $A$ by
$$
f_p(A)= \min_{x\neq 0} \frac{||Ax||_p}{||x||_p}.
$$
Notice that $f_p(A) = 0$ if and only if $A$'s columns are ...
1
vote
0answers
27 views
What is the maximal load of a “latency-bounded” Cuckoo Hash?
Cuckoo Hashing is a method for storing key-value stores (or just a set of keys) with a constant worst-case lookup time.
They use two hash functions $h_1,h_2:\mathbb K\to [n]$, where $\mathbb K$ is ...
18
votes
1answer
2k views
Algorithm whose running time depends on P vs. NP
Is there a known, explicit example of an algorithm with the property such that if $P\neq NP$ then this algorithm doesn't run in polynomial time and if $P=NP$ then it does run in polynomial time?
6
votes
2answers
256 views
Algorithm for identifying unprovable statements
I understand that this may depend on the specific set of axioms, but is there a general way (algorithm) for automatically detecting unprovable statements within a set of axioms?
For example: If there ...
8
votes
2answers
281 views
Non-Orthogonal Vectors Problem
Consider the following problems:
Orthogonal Vectors Problem
Input: A set $S$ of $n$ Boolean vectors each of length $d$.
Question: Do there exist distinct vectors $v_1$ and $v_2 \in S$ ...
2
votes
0answers
81 views
On the goal of learned clause database reduction in CDCL SAT solvers
Modern SAT solvers frequently reduce the size of their learned clause database in order to keep its size as manageable as possible. This has as effect not to slow down the speed of unit propagations.
...
5
votes
0answers
106 views
Lower bound for enumerating k closest pair of points
Consider 1 dimensional points $x_1, \dots x_n, x_i \in \mathbb{R}$ where the distance between any two points is defined as $d(x_i, x_j) = \vert x_i - x_j \vert$. The goal is to enumerate all $n^2$ ...
10
votes
1answer
317 views
Sorting with an average of $\mathrm{lg}(n!)+o(n)$ comparisons
Is there a comparison-based sorting algorithm that uses an average of $\mathrm{lg}(n!)+o(n)$ comparisons?
Existence of a worst-case $\mathrm{lg}(n!)+o(n)$ comparison algorithm is an open problem, but ...
3
votes
0answers
48 views
Lower bound for reversing a list using queues
How do you prove (or disprove) that a list of length $n$ cannot be reversed in time $o(n \log n)$ using $O(1)$ queues?
Each queue is FIFO. Time refers to the number of operations on the queues.
...
8
votes
0answers
126 views
Finding $n$ many different primes efficiently
I want to find $n$ many different primes on RAM. I can find $O(\frac{n}{\log n})$ many primes in the interval $1$ to $n$ in $O(n)$ running time. A brute force way is to find $O(\frac{n}{\log n})$ many ...
14
votes
1answer
1k views
Deciding whether an interval contains a prime number
What is the complexity of deciding whether an interval of the natural numbers contains a prime? A variant of the Sieve of Eratosthenes gives an $\tilde O(L)$ algorithm, where $L$ is the length of the ...
1
vote
0answers
218 views
Impagliazzo lemma, unclear detail in its proof
In Arora-Barak's book on page 378 in the proof of Impagliazzo's Hard Core lemma why did they choose the number 50 in this line: Set $t = \frac{50n}{\epsilon^2}$ ? How this choice then yields the size ...
5
votes
1answer
134 views
What does “hold uniformly” mean in the context of asymptotic analysis?
What does "hold uniformly" mean in the statement of Theorem 1.7 in
A Faster Subquadratic Algorithm for Finding Outlier Correlations? Here's the theorem text ("hold uniformly" is in the last line):
4
votes
1answer
198 views
Fast algorithm to find a maximum connected subgraph of k vertices
Given an undirected graph $G = (V, E)$ and a function $f: 2^V \to \mathbb{R}^+,$ where $2^V$ is the set of all subsets of $V$. Find a connected subgraph $T = (V_T, E_T)$ of k vertices such that $f(V_T)...
