# Questions tagged [ds.algorithms]

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

465 questions with no upvoted or accepted answers
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### Integer multiplication where regular Fourier Transform approach would fail to provide best upper bound

I have a problem where multiplication of integers via regular Fourier Transform based multiplication technique would fail to provide best upper bound since the sequences of bits in both integers are ...
95 views

### $\log^\star n$ is somewhat common in runtimes. Does the superroot ever make an appearance?

Many algorithms and data structures have iterated logarithms ($\log^\star n$) in their runtimes. This function is the discrete inverse of tetration, in that $$\log_a^\star (a \uparrow \uparrow b) = b$$...
195 views

### Fastest Known Algorithm for $k$-Dimensional Matching and $k$-Exact Cover

Given a $k$-uniform hypergraph $G$ (i.e., each edge of $G$ contains precisely $k$ vertices) on $n$ vertices, the $k$-Exact Cover problem is the task of deciding if there exists $n/k$ edges in $G$ ...
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### Flipping one bit to maximize BMM output

Consider a boolean matrix $A$ of size $N \times N$ and let $A^\top$ be its transpose. Let $C = AA^\top$ be the boolean matrix multiplication (BMM) result and let $c$ be the number of non-negative ...
89 views

### Complexity of Encoding a Matroid Flow Problem in a Matrix

Context: Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$. We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct ...
313 views

### Deciding whether $2^k+m$ is prime

I thought something fancy can be done with number-theory or memoization, but neither worked for me. Being limited in knowledge I decided to ask experts. Does there exist a deterministic polynomial-...
207 views

### On solving Planar Circuit SAT

This enquiry is three-sided. Side 1 - State of the art Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$? Which is the best known algorithm for $\text{PLANAR-CIRCUIT-SAT}$ assuming ...
204 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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### Constructing a bad sequence for counter algorithm

Assume that we want to construct a sequence $s\in\{a,b\}^{N}$ such that $s$ contains exactly $n$ times the letter '$a$'. The sequence is then feed to the following probabilistic algorithm: ...
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### Heuristic algorithm design techniques

I am looking for a good relatively complete and up-to-date book or survey about heuristic algorithm design techniques.
115 views

### Computing a frontier set for some Boolean-valued function in a lattice?

Assume that we have a non-empty finite lattice $(L,\leq)$ and a monotone Boolean-valued function $f : L \rightarrow \mathbb{B}$ (i.e, for every $x,y \in L$, if $f(x)=\mathbf{true}$ and $x \leq y$, ...
389 views

### The number of maximal subsets with sum less than $m$

I've met this problem. I would like to know to which complexity class it belongs. Input a set of positive integers $I$, an integer $m$, an integer $n$. Question Is the number of $S \subseteq I$ such ...
156 views

### Find index set partition that has large projections

I have a multiset $S$ of $n$-bit strings. Let $1_S(s)$ denote the number of times that string $s$ appears in $S$, i.e., the multiplicity of $s$ in $S$. I want to find a partition of $\{1,2,\dots,n\}$...
510 views

### Dynamic shortest path data structure for DAG

Let $G$ be a dynamic DAG (directed acyclic graph) where new vertices and new edges can be inserted. I am looking for an efficient data structure/algorithm to maintain the shortest path from a fixed ...
1k views

### How to determine proper rounding in linear programming relaxations?

Recall that in the vertex cover problem we are given an undirected graph ${G=(V,E)}$ and we want to find a minimum-size set of vertices ${S}$ that “touches” all the edges of the graph, that is, such ...
665 views

### What about apply maxplus algebra for all-pairs shortest paths?

I didn't find deep informations on Wikipedia about all-pairs shortest path, in particular I do not know what is the best algorithm to solve this problem beyond Floyd-Warshall's one, then I do not know ...
117 views

### Approximating Front Size of Asymmetric Matrices

The front size of a matrix $A$ is the largest number of non-zeros below the diagonal in any column of its Cholesky factor. If $A$ is symmetric then the minimum front size of $A$ is equal to the ...
Given a point set $P$ and a triangulation $T$ of $P$ with $d$ triangles, let's define $$\alpha(T) = (\alpha_1, \alpha_2, \ldots, \alpha_{3d})$$ which denotes the series of interior angles of $T$, ...