Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.
Schaefer @cite_75 shows that several other notions of planarity are related to SEFE . In particular, the well-studied cluster planarity problem reduces to SEFE , providing further incentive to study its complexity.
@cite_75
81810990 2950941433 2294674552 2049376434
Given k planar graphs G1,…,Gk, deciding whether they admit a simultaneous embedding with fixed edges (Sefe ) and whether they admit a simultaneous geometric embedding (Sge ) are NP-hard problems, for k≥3 and for k≥2, respectively. In this paper we consider the complexity of Sefe and of Sge when the graphs G1,…,Gk have a fixed planar embedding. In sharp contrast with the NP-hardness of Sefe for three non-embedded graphs, we show that Sefe is polynomial-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G1,…,Gk, deciding whether a Sefe of G1,…,Gk exists and deciding whether an Sge of G1,…,Gk exists are NP-hard problems, for k≥14 and k≥13, respectively. For a simply-connected simple algebraic group @math over @math , we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of @math , generalizing a well-known fact about @math . Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group. This paper introduces the concept of low-congestion shortcuts for (near-)planar networks, and demonstrates their power by using them to obtain near-optimal distributed algorithms for problems such as Minimum Spanning Tree (MST) or Minimum Cut, in planar networks. Consider a graph G = (V, E) and a partitioning of V into subsets of nodes S1, . . ., SN, each inducing a connected subgraph G[Si]. We define an α-congestion shortcut with dilation β to be a set of subgraphs H1, . . ., HN ⊆ G, one for each subset Si, such that 1. For each i ∈ [1, N], the diameter of the subgraph G[Si] + Hi is at most β. 2. For each edge e ∈ E, the number of subgraphs G[Si] + Hi containing e is at most α. We prove that any partition of a D-diameter planar graph into individually-connected parts admits an O(D log D)-congestion shortcut with dilation O(D log D), and we also present a distributed construction of it in O(D) rounds. We moreover prove these parameters to be near-optimal; i.e., there are instances in which, unavoidably, max α, β = Ω(D[EQUATION]). Finally, we use low-congestion shortcuts, and their efficient distributed construction, to derive O(D)-round distributed algorithms for MST and Min-Cut, in planar networks. This complexity nearly matches the trivial lower bound of Ω(D). We remark that this is the first result bypassing the well-known Ω(D + [EQUATION]) existential lower bound of general graphs (see Peleg and Rubinovich [FOCS'99]; Elkin [STOC'04]; and Das [STOC'11]) in a family of graphs of interest. In time series mining, one of the interesting tasks that attract many researchers is time series clustering which is classified into two main categories. Whole time series clustering considers how to cluster multiple time series, and the other one is Subsequence Time Series (STS) clustering, a clustering of subparts or subsequences within a single time series. Deplorably, STS clustering is not preferable even though it had widely been used as a subroutine in various mining tasks, e.g., rule discovery, anomaly detection, or classification, due to the recent finding a decade ago that STS clustering problem can produce meaningless results. There have been numerous attempts to resolve this problem but seemed to be unsuccessful. Until the two most recent attempts, they seem to accomplish in producing meaningful results; however, their approaches do need some predefined constraint values, such as the width of the subsequences that are in fact quite subjective and sensitive. Thus, we propose a novel parameter-free clustering technique to eliminate this problem by utilizing a motif discovery algorithm and some statistical principles to properly determine these parameters. Our experimental results from well-known datasets demonstrate the effectiveness of the proposed algorithm in selecting the proper subsequence width, and in turn leading to meaningful and highly accurate results.