Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComputing a Minimum-Dilation Spanning Tree is NP-hard
122
0
0.0
(
0
)
Added by
Otfried Cheong
Publication date
2007
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices differs from their Euclidean distance. We show that given a set S of n points with integer coordinates in the plane and a rational dilation delta > 1, it is NP-hard to determine whether a spanning tree of S with dilation at most delta exists.
rate research
Read More
In this paper, we show that deciding rigid foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from Partition, and that deciding rigid foldability with optional creases is strongly NP-hard by a reduction from 1-in-3 SAT. Unlike flat foldability of origami or flexibility of other kinematic linkages, whose complexity originates in the complexity of the layer ordering and possible self-intersection of the material, rigid foldability from a planar state is hard even though there is no potential self-intersection. In fact, the complexity comes from the combinatorial behavior of the different possible rigid folding configurations at each vertex. The results underpin the fact that it is harder to fold from an unfolded sheet of paper than to unfold a folded state back to a plane, frequently encountered problem when realizing folding-based systems such as self-folding matter and reconfigurable robots.
We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.
In this article, we study the Euclidean minimum spanning tree problem in an imprecise setup. The problem is known as the emph{Minimum Spanning Tree Problem with Neighborhoods} in the literature. We study the problem where the neighborhoods are represented as non-crossing line segments. Given a set ${cal S}$ of $n$ disjoint line segments in $I!!R^2$, the objective is to find a minimum spanning tree (MST) that contains exactly one end-point from each segment in $cal S$ and the cost of the MST is minimum among $2^n$ possible MSTs. We show that finding such an MST is NP-hard in general, and propose a $2alpha$-factor approximation algorithm for the same, where $alpha$ is the approximation factor of the best-known approximation algorithm to compute a minimum cost Steiner tree in an undirected graph with non-negative edge weights. As an implication of our reduction, we can show that the unrestricted version of the problem (i.e., one point must be chosen from each segment such that the cost of MST is as minimum as possible) is also NP-hard. We also propose a parameterized algorithm for the problem based on the separability parameter defined for segments.
We present time-space trade-offs for computing the Euclidean minimum spanning tree of a set $S$ of $n$ point-sites in the plane. More precisely, we assume that $S$ resides in a random-access memory that can only be read. The edges of the Euclidean minimum spanning tree $text{EMST}(S)$ have to be reported sequentially, and they cannot be accessed or modified afterwards. There is a parameter $s in {1, dots, n}$ so that the algorithm may use $O(s)$ cells of read-write memory (called the workspace) for its computations. Our goal is to find an algorithm that has the best possible running time for any given $s$ between $1$ and $n$. We show how to compute $text{EMST}(S)$ in $Obig((n^3/s^2)log s big)$ time with $O(s)$ cells of workspace, giving a smooth trade-off between the two best known bounds $O(n^3)$ for $s = 1$ and $O(n log n)$ for $s = n$. For this, we run Kruskals algorithm on the relative neighborhood graph (RNG) of $S$. It is a classic fact that the minimum spanning tree of $text{RNG}(S)$ is exactly $text{EMST}(S)$. To implement Kruskals algorithm with $O(s)$ cells of workspace, we define $s$-nets, a compact representation of planar graphs. This allows us to efficiently maintain and update the components of the current minimum spanning forest as the edges are being inserted.
Weak unit disk contact graphs are graphs that admit a representation of the nodes as a collection of internally disjoint unit disks whose boundaries touch if there is an edge between the corresponding nodes. We provide a gadget-based reduction to show that recognizing embedded caterpillars that admit a weak unit disk contact representation is NP-hard.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
