# 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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### 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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### Why is it more efficient to merge larger runs higher in the tree than perform a balanced tree of merges in an unbalanced ping-pong merge?

While studying the research paper published by Microsoft in 2014, I stumbled upon Unbalanced Ping-Pong Merge. In section 3.2 of the paper, it discusses about merging two sorted runs at a time. It ...
48 views

### Fastest way for boolean matrix computations

I have a boolean matrix with 1.5 million rows and 20k columns, similar to this example: ...
319 views

### 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 ...
135 views

### 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 ...
528 views

### 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 $B$...
199 views

### 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 ...
228 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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### 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 $O(n)$ ...
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### 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 ...
131 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 ...
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### 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 ...
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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### 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, ...
160 views

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 $n \in \... 1answer 498 views ### What is the hardest instance for the group isomorphism problem? Two groups$(G,\cdot)$and$(H, \times)$are said to be isomorphic iff there exists a homomorphism from$G$to$H$which is bijective. The group isomorphism problem is as follows: given two groups, ... 1answer 155 views ### How is SDP an extension of spectral algorithms? In one of his lectures, Uri Feige described semidefinite programming (SDP) as ... an algorithmic technique that extends both linear programming and spectral algorithms. I know the basic ... 1answer 78 views ### The SQ argument in Balazs Szorenyi's paper I am asking about the proof in Theorem 5 (page 6) of this paper, http://www.inf.u-szeged.hu/~szorenyi/Cikkek/sq_d0_ext.pdf Quite a few things about this short argument seem unclear to me, Towards ... 0answers 200 views ### Computational Complexity of the Frobenius Problem The Frobenius problem takes as input$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+\...
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 ...
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 ...