Questions tagged [ds.algorithms]

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

394 questions with no upvoted or accepted answers
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249 views

Atomic snapshot algorithms on tree-structured shared registers

Background: Atomic snapshot memory is a shared memory partitioned into words written (updated) by individual processes, or instantaneously read (scanned) in its entirety. The Gang of Six algorithm ...
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91 views

Constant time search for rod segment

(I tried to ask at SO but maybe this has more to do with the CS theory.) Suppose I have a rod which I cut to pieces. Given a point on the original rod, is there a way to find out which piece it ...
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180 views

Proving greedy algorithm is optimal for a scheduling problem

First, the problem discription: For a sequence of $4n$ tasks, $a_1a_2\dots a_{4n}$, where $a_i\in\{0,1\}\forall i$, put them sequentially to the tail of one of the two initially empty queues of ...
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186 views

Open questions about linear-time

What are some interesting open or solved-but-hard questions around problems having linear-time solutions? Ala riffle shuffles. I'm especially curious about problems which people believe to be linear-...
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134 views

Linear programming - How to allow cycles with weight at most s?

Consider a graph $G=(V,E)$ with nonnegative weight function on the edges $w$. How would you express in LP that you want to allow cycles in $G$ with total weight at most $s$ ? I've found this while ...
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265 views

Maximizing a convex function where the objective function is separable but the search space is not

The problem statement is Given convex functions $f_i$ over $X$, find $$\arg\max_{x\in X} \sum_i f_i(x)$$ Does this kind of problem structure allow one to use specific strategies to solve the ...
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194 views

Randomized rounding on a graph

Assume we are given an arbitrary undirected graph $G = (V, E)$ where $|V| = n$. We are also given real numbers $x_e \in [0, 1]$ for each $e \in E$. These numbers satisfy the following constraint: \...
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40 views

Polyhedral embedding from graph degree sequence

Given: A degree sequence. Wanted: A graph and a polyhedral embedding of this graph (described by a rotation system or something equivalent). By polyhedral embedding I mean only the combinatorial ...
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75 views

Load-balancing; Alternate methods of keeping track of nodes?

Reading various articles in the literature have given me only a few decent methods of keeping track of nodes before->after load-balancing them on a very large network. One popular method uses virtual-...
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152 views

Partitioning the vertices of a complete graph with weights on both vertices and edges with constraints

Given the complete graph on n vertices. Each vertex and each edge has a positive weight associated with it. What is desired is to partition the vertices into parts so that the sum of the weights of ...
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606 views

K-shortest path in large sparse graph

I am an engineer and looking for a reference to find k-shortest path's in a large sparse graph. In the search for it, I came acorss Yen's ranking loopless algorithm and an improved implementation of ...
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132 views

minimizing makespan for related machines

Any reference is appreciated. There are $n$ jobs and $m$ machines, machines has speeds and jobs have processing times. It takes $p_j/s_i$ units of time for machine $i$ with speed $s_i$ to process job ...
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283 views

Distinguishing two types of Monte-Carlo algorithms

Recall from Wikipedia that Monte Carlo algorithm is a randomized algorithm whose running time is deterministic, but whose output may be incorrect with a certain (typically small) probability. Consider ...
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185 views

Separation Oracle for Inverse Bipartite Matching Polytope

The $N$x$N$ bipartite matching problem can be written as finding a configuration of variables ${\mathbf y}^* = \{y^*_1, \ldots, y^*_N\}$, $y_i \in \{1, \ldots, N\}$ such that $${\mathbf y}^* = \arg\...
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200 views

The Quality of SDP relaxation on MaxCut

My question is: given a maxcut instance, if it costs too much to solve it to optimal practically but we can get an optimal solution of SDP relaxation quickly, can we assess the quality of this SDP ...
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110 views

Sampling from a distribution with a given covariance matrix

Let $\bf X$ be a binary vector of $n$ (non-independent) random variables $X_1,\ldots, X_n$. Covariance of two random variables is defined as follows: $$\mathrm{cov}(X_i, X_j) = \mathrm{E}(X_i - \mu_i)...
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326 views

Taking Square Roots of Matrices over Z/nZ

Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n? More specifically, suppose I generate a random matrix M, and square it. Can a ...
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402 views

worst case external fragmentation in buddy memory systems

Unfortunately, I can't find any freely available text with an estimation of exact upper bound of (external) fragmentation overhead for (binary) buddy memory allocator. Estimation $M(1+ \log 2 m)$ (...
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150 views

Gram matrix of Max-Cut relaxation

It seems that Goemans and Williamson give a unique representation for each graph of the semidefinite relaxation (elements $y_{ij}$ of Y). However, semidefinite programming may give the same maximum ...
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2answers
415 views

Optimization Problem on a Directed Graph

I have the following graph optimization problem. In a directed graph $G$, each node $i$ is endowed with a real value $v_i$ (input) that encodes the minimum "activation threshold" of that node. For ...
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62 views

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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56 views

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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105 views

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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100 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 ...
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78 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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69 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 ...
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142 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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59 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 ...
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84 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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57 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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82 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 ...
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88 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. ...
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253 views

Career advice needed: Switching from theoretical CS to pure math

I couldn't think of a better forum to ask this question. I am a first year cs graduate student. I couldn't really pursue theoretical cs during my undergrad, but as part of our PhD qualifiers, i had to ...
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63 views

Has Khachiyan/Porkolob's convex integer optimization been implemented?

Khachiyan and Porkolab in 'Integer optimization on convex semialgebraic sets' gave an $O(ld^{ O(k^4)})$ algorithm to minimize a degree $d$ form with integer coefficients of binary length at most $l$ ...
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49 views

What is the difference between PRAM and multi threading?

PRAM and multi-threading are somewhat part of the same model of the shared memory model, in which both can compute in parallel, and both can allow processors to communicate with each other since there ...
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62 views

Finding the largest set of points of limited diameter (2)

Problem: Given a set of points $S = \{(x_1, y_1), (x_2, y_2),\cdots,(x_n, y_n)\}$ in $\mathbb{R}^2$ and a distance threshold $\tau$, find a subset of $S$ such that (1) the Euclidean distance between ...
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85 views

Is berklemap-massey works on extention field

According to the articles that i read (also wikipedia [1]), they refer that the above algorithm works on finite field $GF(q)$. But I cannot find an evidence that it still works on extension field, i.e:...
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68 views

Standard basis for recurrence relations

In polynomial algebra there is a powerful tool for treating system of polynomial equations. It is standard or Groebner Bases. It allows to verify if system is consistent, eliminate variables, reduce ...
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159 views

Non-Linear Programming with \min operator in the constraint

Can the following non-linear program be solved in polynomial time? $c_{ij}$'s are constants and known. Each $c_{ij}$ is either -1 or 1. \begin{align} \text{maximize } &\sum_{i,j=1}^{m,n} c_{ij}...
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36 views

Nearest Common Ancestor on DFS Tree (with Addition of Leaves in DFS Order) on Pointer Machines

What is the complexity status for the Nearest Common Ancestor Problem on Trees in which the leaves are attached to the tree in DFS order ? i.e. Suppose one is visiting a tree T in DFS, and at any ...
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190 views

Inclusion of polytopes

Consider the following two system of linear (in)eqaulities: $S = Ax \leq b;\; Cx = e$ $T = Dx \leq d;\; Gx = g$ How can I check if $S\cap \neg T=\emptyset$ where $\neg T$ is the complement of the ...
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85 views

Algorithm to decompose word

I am searching for an algorithm to decompose a word with the following constraints: Decompose the word $w$ such that the following value is minimized: $\sum_{i=1}^{k} | w_i| + \sum_{i=1}^{k} \log(n_i)...
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71 views

Approximate linear time algorithm for minimum k-cut

I need to approximately and quickly solve a minimum k-cut problem for n=2^14 and k=512. It seems that the 2-approximation algorithm given by Saran and Vazirani which needs to solve O(n) instances of ...
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64 views

Partitioning graph for Graph Isomorphism

Motivation: I am studying the graph isomorphism problem. I am trying to construct a partitioning method to reduce search cases . Partition method: $G$ is an $r$ regular graph, $k$ connected (not a ...
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211 views

Convex hull of polytopes

Consider a set of polytopes $P_i : i=1,2,...,k$ each of which has a structure as $P_i:= \{(x_{i1},x_{i2},..., x_{in})\; |\; x_{ij} \in [a_{ij}, b_{ij}] \subseteq [0,1]\}\;\; \text{for all}\;\; j=1,......
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Claw finding using quantum walk: superposition for Szegedy's framework

Within Claw Finding Algorithms Using Quantum Walk there is the subroutine $claw_{detect}$ described. As in above paper: Let $J_f(N, l)$ and $J_G(M, m)$ be Johnson graphs. Let $F$ and $G$ be vertices ...
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87 views

Forming Sets from 3-SAT Clauses

I'm wondering if someone can provide a good algorithm for the following problem. If we take 3-SAT in conjunctive normal form, we can partition some or all of the variables (not the literals) into ...
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137 views

number of iterations of this algorithm (upper bound)

Let $(A, dist)$ be a finite metric space. Consider the following "$p$-center problem": given a positive integer $p$, find a subset $B$of $A$ such that $|B| = p$ and which minimizes the number $\max_{a ...
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103 views

Is the problem “Binary Sorted Min Sum” already known under an other name?

A computer scientist oriented toward applications gave me the following problem: Given a positive integer $n>0$, an increasing function function $f$ and a decreasing function $g$, both defined ...
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230 views

Recent insights on algorithms for 1D bin packing

This is just a general question on recent algorithms for the 1D bin packing problem. I just want to collect some information on this issue, so I’m grateful for any information. Especially heuristics ...

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