Problem Detail: Consider these 2 languages: $L_{ge5} = left { left< M right> : M text{ accepts at least 5 strings} right} $ $L_{<5} = left { left< M right> : M text{ accepts fewer than 5 strings} right}$ Are these Read More …
Blog
[Solved]: An interesting metric space related to Turing machines
Problem Detail: In this question we only consider Turing machines that halt on all inputs. If $k in mathbb{N}$ then by $T_k$ we denote the Turing machine whose code is $k$. Consider the following function $$s(x,y) = min{k mid |L(T_k) cap Read More …
[Solved]: Kolmogorov complexity of string concatenation
Problem Detail: If $K(s)$ is the Kolmogorov complexity of the string $s in {0,1}^*$, Can we prove (or disprove) the following statement: “Every string $s$ is a prefix of an incompressible string; i.e. for every string $s$ there exists a string Read More …
[Solved]: How optimal is Lempel-Ziv at reaching the Shannon limit?
Problem Detail: I find this a bit difficult to describe, but I am interested in the following idea : The LZ algorithm factors (verb) an input stream into adjacent factors, these are by definition the maximal prefixes of the piece of Read More …
[Solved]: Does Thompson’s algorithm produce optimal NFAs?
Problem Detail: I’m using Thompson’s algorithm to convert from a regular expression to a NFA. Is Thompson’s algorithm guaranteed to always output a minimal NFA, i.e., a NFA with the smallest possible number of states? For instance, consider this example. I Read More …
[Solved]: How to compare the time-complexity of an optimized algorithm with that of the original?
Problem Detail: I had an algorithm with time-complexity of $O(htimes w)$, knowing $h$ is the height and $w$ is the width of an image being processed (or a simple matrix of size $htimes w$). I managed to reduce the range that Read More …
[Solved]: Algorithm to extract the subgraph of all nodes with degree at least four
Problem Detail: I have an undirected graph represented by a list of nodes and a list of edges. What I need to produce from this is a list of nodes and edges representing a new graph containing only the nodes which Read More …
[Solved]: Shortest path from that passes through a set of edges once
Problem Detail: Given a graph with weighted edges. How to find the shortest path from vertex $A$ to vertex $B$ that passes through a set of edges $X$ at most once? $X$ can be big. Slow solution: Finding shortest path from Read More …
[Solved]: Find all pairs of strings in a set with Levenshtein distance < d
Problem Detail: I have a set of $n = $ 100 million strings of length $l = 20$, and for each string in the set, I would like to find all the other strings in the set with Levenshtein distance $le Read More …
[Solved]: Prove NP-completeness for union of NP-complete language and language in P
Problem Detail: Given disjoint languages $X$ and $Y$, where $X$ is NP-complete and $Yin P$ , how do I prove that $Xcup Y$ is NP-complete? My idea is to prove that $(Xcup Y)in NP$ and then prove that $Xcup Y$ is Read More …
