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Questions tagged [ds.algorithms]

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

3
votes
2answers
319 views

What is the complexity of computing a compatible 3-coloring of a complete graph?

Given a complete graph whose edges are colored by 3 colors, a compatible 3-coloring is a coloring of nodes such that no edge of the graph has the same color as its end-points. The best algorithm I ...
2
votes
1answer
336 views

What is the most efficient algorithm to sample graphs with trivial automorphism groups ?

Let us call a graph "asymmetric" if it has no nontrivial automorphism group. http://en.wikipedia.org/wiki/Asymmetric_graph I'm looking for an efficient way to compute a random asymmetric graph on a ...
11
votes
4answers
454 views

Dimensionality reduction with slack?

The Johnson-Lindenstrauss lemma says roughly that for any collection $S$ of $n$ points in $\mathbb{R}^d$, there exists a map $f:\mathbb{R}^d \rightarrow \mathbb{R}^k$ where $k = O(\log n/\epsilon^2)$ ...
6
votes
4answers
734 views

What are the best known upper bounds and lower bounds for computing O(log n)-Clique?

Input: a graph with n nodes, Output: A clique of size $O(\log n)$, Providing links to references would be great
108
votes
17answers
7k views

Examples of the price of abstraction?

Theoretical computer science has provided some examples of "the price of abstraction." The two most prominent are for Gaussian elimination and sorting. Namely: It is known that Gaussian elimination ...
12
votes
3answers
413 views

Streaming derandomization

Stream algorithms require randomization for the most part to do anything nontrivial, and because of the small-space constraint, need PRGs that use little space. I know of two methods that have been ...
28
votes
3answers
1k views

How to produce a random graph that does not have a Hamiltonian cycle?

Let class A denote all the graphs of size $n$ which have a Hamiltonian cycle. It is easy to produce a random graph from this class--take $n$ isolated nodes, add a random Hamiltonian cycle and then add ...
23
votes
3answers
1k views

What bounds can be put on counting reachable nodes in a dag?

Given is a dag. You want to label each node by how many nodes are reachable from it. $O(V(V+E))$ is a trivial upper bound; $\Omega(V+E)$ is a lower bound (I think). Is there a better algorithm? Is ...
23
votes
6answers
2k views

Graph families which have polynomial time algorithms for computing the chromatic number

Post updated on 31st of August: I added a summary of the current answers below the original question. Thanks for all the interesting answers! Of course, everyone can continue posting any new findings. ...
7
votes
4answers
2k views

What are the most effective algorithms to find random number?

I was reading the Ramsey's Theory stating "complete disorder is impossible". Is there any algorithm to generate random numbers for a long period of time without there being any relation from one set ...
3
votes
4answers
2k views

Why is P vs. NP so hard? [closed]

Why is $\mathsf{P}$ vs. $\mathsf{NP}$ problem considered so important? Is $\mathsf{P}$ vs. $\mathsf{NP}$ the hardest mathematical problem? Why is it so hard? All I'm looking for is the hindrances ...
13
votes
12answers
4k views

What are some real world applications for genetic algorithms?

What are some real world problems that have been solved using a genetic algorithm? What is the problem? What is the fitness test used to solve this problem?
22
votes
6answers
1k views

Analogs of compressed sensing

In compressed sensing, the goal is to find linear compression schemes for huge input signals that are known to have a sparse representation, so that the input signal can be recovered efficiently from ...
44
votes
8answers
5k views

The importance of Integrality Gap

I always had trouble in understanding the importance of the Integrality Gap (IG) and bounds on it. IG is the ratio of (the quality of) an optimal integer answer to (the quality of) an optimal real ...
34
votes
8answers
2k views

Higher-order algorithms

Most of the well-known algorithms are first-order, in the sense that their input and output are "plain" data. Some are second-order in a trivial way, for example sorting, hashtables or the map and ...
52
votes
13answers
3k views

For which algorithms is there a large gap between the theoretical analysis and reality?

Two ways of analyzing the efficiency of an algorithm are to put an asymptotic upper bound on its runtime, and to run it and collect experimental data. I wonder if there are known cases where there ...
16
votes
2answers
340 views

Finding small sets of integers in which every element is a sum of two others

This is a follow-up to this question on math.stackexchange. Let us say that a non-empty set S ⊆ ℤ is self-supporting if for every a ∈ S, there exist distinct ...
10
votes
2answers
224 views

Shuffling of tokens on a graph using local swaps

Let $G= (V, E)$ be a non-regular connected graph whose degree is bounded. Suppose that each node contain a unique token. I want to uniformly shuffle the tokens amongst the graph using only local ...
24
votes
2answers
525 views

Parallel Dynamic Search

Is there a natural parallel analog to red-black trees with similar or even not-terribly-worse properties for updates while being reasonably work-efficient ? More generally, what's the best we can do ...
8
votes
5answers
531 views

Algorithm for inverting a bijective function.

Does there exist a generalized algorithm for finding the inverse function of an arbitrary bijective function? In order for this algorithm to be useful, it must eventually halt once the correct answer ...
349
votes
93answers
108k views

Algorithms from the Book.

Paul Erdos talked about the "Book" where God keeps the most elegant proof of each mathematical theorem. This even inspired a book (which I believe is now in its 4th edition): Proofs from the Book. If ...
13
votes
1answer
303 views

Finding odd holes in circulant Paley graphs

The Paley graphs Pq are those whose vertex-set is given by the finite field GF(q), for prime powers q≡1 (mod 4), and where two vertices are adjacent if and only if they differ by a2 ...
-1
votes
2answers
562 views

Minimum spanning tree algorithm. [closed]

Is the following a valid algorithm for finding a minimum spanning tree? Given a weighted graph with unique weights, remove the all edges that are the highest cost edge in any cycle of the original ...
114
votes
11answers
10k views

How hard is unshuffling a string?

A shuffle of two strings is formed by interspersing the characters into a new string, keeping the characters of each string in order. For example, MISSISSIPPI is a ...
8
votes
1answer
387 views

Best resources for string searching or pattern matching exercises

I would like to be somewhat good at string searching and pattern matching, could you point me to some good online resources? Exercise problems would be great. Thanks.
15
votes
1answer
751 views

Online transitive closure better than O(N^2) per edge addition

I'm looking for an online algorithm to maintain the transitive closure of a directed acyclic graph with a time complexity less than O(N^2) per edge addition. My current algorithm is like this: ...
15
votes
6answers
3k views

Complexity of the Fisher-Yates Shuffle Algorithm

This question is in regard to the Fisher-Yates algorithm for returning a random shuffle of a given array. The Wikipedia page says that its complexity is O(n), but I think that it is O(n log n). In ...
10
votes
1answer
288 views

Generalizing the FFT

Can the divide and conquer nature of the FFT be generalized to other transforms (z Transform, chirp, etc) automatically? Is there an algorithm that takes in a description of transform (I don't know ...
8
votes
1answer
1k views

What are some effective heuristics to find the number of Hamiltonian paths in a rectangular grid?

A particular programming problem I came across recently reduces to finding hamiltonian paths in a rectangular grid that would look something like, ...
13
votes
2answers
823 views

What is a good special-case sorting algorithm?

I have a dataset which is a number of objects arranged in a 2-D grid. I know I have a strict ordering, increasing as you go left-to-right within each row, and increasing as top-to-bottom within each ...