Questions tagged [ds.algorithms]

Questions regarding well-defined instructions for completing a task, and relevant analysis in terms of time/memory/etc.

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31 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 ...
2
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1answer
101 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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143 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 ...
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220 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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148 views

Researching into Special Case of Sudoku is in the $P$ class, and I think I found something mildly interesting about some NP-hard problems

Summary What I find mildly interesting is that there are permutator algorithms that can recover the remaining solution to a Sudoku after a first row is validly solved. I also found out that 3-color ...
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0answers
121 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 +...
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105 views

List all pairwise distances in order in $O(n)$ space and $O(n^2 \log n)$ time?

If you have $n$ points in $\mathbb{R^2}$, can you output all $O(n^2)$ pairwise distances from smallest to biggest in $O(n^2 \log n)$ time and $O(n)$ space? This is working-space complexity ...
2
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1answer
83 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 ...
0
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1answer
124 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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83 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 ...
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27 views

minimum cost spanning tree algorithm with benchmark graphs

I have developed one minimum cost spanning tree algorithm. How can I check the algorithm with benchmark graphs?
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1answer
90 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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47 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 ...
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160 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 ...
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2answers
181 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
51 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
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1answer
62 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
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1answer
80 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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0answers
108 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
50 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
79 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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76 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, ...
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0answers
97 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 \...
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1answer
356 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, ...
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1answer
132 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
75 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 ...
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179 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
84 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
37 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 ...
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1answer
122 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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36 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
235 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
77 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
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 $...
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0answers
150 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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170 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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67 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
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1answer
115 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
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119 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 ...
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0answers
41 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 $j$-...
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2answers
85 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 ...
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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 ...
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1answer
61 views

Formally prove that the loops of this sorting algorithm will terminate [closed]

Given is the sorting algorithm Bubblesort ...
5
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1answer
230 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
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1answer
83 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 ...
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0answers
130 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
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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 ...
3
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1answer
63 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
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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?