18826487
On Even Triangulations of 2-Connected Embedded Graphs
Recently, Hoffmann and Kriegel proved an important combinatorial theorem [4]: Every 2-connected bipartite plane graph G has a triangulation in which all vertices have even degree (it's called an even triangulation). Combined with a classical Whitney's Theorem, this result implies that every such a graph has a 3-colorable plane triangulation. Using this theorem, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. In [7], Zhang and He presented a linear time algorithm which relies on the complicated algebraic proof in [4]. This proof cannot be extended to similar graphs embedded on high genus surfaces. It's not known whether Hoffmann and Kriegel's Theorem is true for such graphs. 
 
In this paper, we describe a totally independent and much simpler proof of the above theorem, using only graph-theoretic arguments. Our new proof can be easily extend to show the existence of even triangulations for similar graphs on high genus surfaces. Hence we show that Hoffmann and Kriegel's theorem remains valid for such graphs. Our new proof leads to a very simple linear time algorithm for finding even triangulations.
25659954
An Improved Algorithm for the Planar 3-Cut Problem
Abstract A 3-cut of a connected graph G is a subset of edges which, when deleted, separates G into three connected components. In this paper we present an O ( n log n ) algorithm for finding a minimum 3-cut in planar graphs. Our algorithm improves the best previously known algorithm for the problem by an O( n log n) factor.
940067
Visibility representation of plane graphs via canonical ordering tree,
In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [Rectilinear planar layouts and bipolar orientations of planar graphs, Discrete Comput. Geom. 1 (1986) 343], Tamassia and Tollis [An unified approach to visibility representations of planar graphs, Discrete Comput. Geom. 1 (1986) 321] independently gave linear time VR algorithms for 2-connected plane graph. Afterwards, one of the main concerns for VR is the size of the representation. In this paper, we prove that any plane graph G has a VR with height bounded by ⌊5n/6⌋. This improves the previously known bound ⌊15n/16⌋. We also construct a plane graph G with n vertices where any VR of G requires a size of (⌊2n/3⌋) × (⌊4n/3⌋ - 3). Our result provides an answer to Kant's open question about whether there exists a plane graph G such that all of its VR require width greater that cn, where c > 1 [G. Kant, A more compact visibility representation, Internat. J. Comput. Geom. Appl. 7 (1997) 197].
1437223
A Fast General Methodology for Information - Theoretically Optimal Encodings of Graphs
We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. The methodology is applicable to general classes of graphs; this paper focuses on simple planar graphs. Specifically, a graph with property ? is called a ?-graph. If ? satisfies certain properties, then an n-node ?-graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most s(n)+o(s(n)) bits for any function s(n) = ?(n) so that there are at most 2s(n)+o(s(n)) distinct n-node ?-graphs. Examples of such ? include all conjunctions of the following sets of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; and (4) G has at most l1 (respectively, l2) distinct node (respectively, edge) labels. These examples are novel applications of small cycle separators of planar graphs and settle several problems that have been open since Tutte's census series were published in 1960's.
11889495
On Even Triangulations of 2-Connected Embedded Graphs
Recently, Hoffmann and Kriegel proved an important combinatorial theorem [SIAM J. Discrete Math., 9 (1996), pp. 210--224]: Every 2-connected bipartite plane multigraph G without 2-cycle faces has a triangulation in which all vertices have even degree (this is called an even triangulation). Combined with the classical Whitney's theorem, this result implies that every such graph has a 3-colorable plane triangulation. Using this theorem, Hoffmann and Kriegel significantly improved the upper bounds of several art gallery and prison guard problems. A complicated O(n2) time algorithm was obtained in [SIAM J. Discrete Math., 9 (1996), pp. 210--224] for constructing an even triangulation of G. Hoffmann and Kriegel conjectured that there is an O(n3/2) time algorithm for solving this problem. 
In this paper, we develop a simple proof of the above theorem. Our proof reveals and relies on a natural correspondence between even triangulations of G and certain orientations of G. Based on this new proof, we obtain a very simple O(n) time algorithm for finding an even triangulation of G. We also extend our proof to show the existence of even triangulations for similar graphs on high genus surface.
15559790
Improved visibility representation of plane graphs
In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [Discrete Comput. Geom. 1 (1986) 343-353] and Tamassia and Tollis [Discrete Comput. Geom. 1 (1986) 321-341] independently gave linear time VR algorithms for 2-connected plane graph. Using this approach, the height of the VR is bounded by (n - 1), the width is bounded by (2n - 5). After that, some work has been done to find a more compact VR. Kant and He [Theoret. Comput. Sci. 172 (1997) 175-193] proved that a 4-connected plane graph has a VR with width bounded by (n - 1). Kant [Internat. J. Comput. Geom. Appl. 7 (1997) 197-210] reduced the width bound to ⌊3n-6/2⌋ for general plane graphs. Recently, using a sophisticated greedy algorithm, Lin et al. reduced the width bound to ⌊22n-42/15⌋ [Proc. STACS'03, Lecture Notes in Computer Science, vol. 2607, Springer, Berlin, 2003, pp. 14-25].In this paper, we prove that any plane graph G has a VR with width at most ⌊13n-24/9⌋, which can be constructed by using the simple standard VR algorithm in [P. Rosenstiehl, R.E. Tarjan, Discrete Comput. Geom. 1 (1986) 343-353; R. Tamassia, I.G. Tollis, Discrete Comput. Geom. 1 (1986) 321-341].
3132633
Tensor Subspace Analysis
Previous work has demonstrated that the image variations of many objects (human faces in particular) under variable lighting can be effectively modeled by low dimensional linear spaces. The typical linear sub-space learning algorithms include Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), and Locality Preserving Projection (LPP). All of these methods consider an n1 x n2 image as a high dimensional vector in ℝn1 x n2, while an image represented in the plane is intrinsically a matrix. In this paper, we propose a new algorithm called Tensor Subspace Analysis (TSA). TSA considers an image as the second order tensor in Rn1 ⊗ Rn2, where Rn1 and Rn2 are two vector spaces. The relationship between the column vectors of the image matrix and that between the row vectors can be naturally characterized by TSA. TSA detects the intrinsic local geometrical structure of the tensor space by learning a lower dimensional tensor subspace. We compare our proposed approach with PCA, LDA and LPP methods on two standard databases. Experimental results demonstrate that TSA achieves better recognition rate, while being much more efficient.
1473802
Locality preserving clustering for image database
It is important and challenging to make the growing image repositories easy to search and browse. Image clustering is a technique that helps in several ways, including image data preprocessing, user interface designing, and search result representation. Spectral clustering method has been one of the most promising clustering methods in the last few years, because it can cluster data with complex structure, and the (near) global optimum is guaranteed. However, existing spectral clustering algorithms, like Normalized Cut, are difficult to handle data points out of training set. In this paper, we propose a clustering algorithm named Locality Preserving Clustering (LPC), which shares many of the data representation properties of nonlinear spectral method. Yet LPC provides an explicit mapping function which is defined everywhere, both on training data points and testing points. Experimental results show that LPC is more accurate than both "direct Kmeans" and "PCA + Kmeans". We also show that LPC produces in general comparable results with Normalized Cut, yet is more efficient than Normalized Cut.
41188840
Finding Double Euler Trails of Planar Graphs in Linear Time
This paper answers an open question in the design of complimentary metal-oxide semiconductor VLSI circuits. The question asks whether a polynomial-time algorithm can decide if a given planar graph has a plane embedding ${\cal E}$ such that ${\cal E}$ has an Euler trail P = e1 e2 ... em and its dual graph has an Euler trail $P^*=e^*_1 e^*_2 \ldots e^*_m$, where $e^*_i$ is the dual edge of ei for i=1,2,...,m. This paper answers this question in the affirmative by presenting a linear-time algorithm.
28965788
A useful approach to developing reverse engineering metrics
If software metrics are useful in a forward software engineering environment, they are vital in a reverse engineering environment. We are endeavouring to discover approaches to developing reverse engineering metrics for software engineers who need them for reverse engineering legacy systems. The major contribution of the paper is the presentation of a systematic research base and a hierarchical approach to the development of software metrics for reverse engineering. Measurement is fundamental to the software engineering discipline as a whole. A software metric is a quantitative measure of the degree to which a system, component, or process possesses a given attribute (N.E. Fenton and S.L. Pfleeger, 1996). Software reverse engineering is the process of analysing a subject system to: identify the system's components and their interrelationships; and create representations of the system at a higher level of abstraction (E.J. Chikofsky and J.H. Cross II, 1990). The goal of developing reverse engineering metrics is to identify measures that are needed for assessing the status of reverse engineering projects, products, processes and resources, and helping engineers to understand both what is happening and what will happen during reverse engineering procedures. Consequently, these concrete reverse engineering measures make more visible to us aspects of process and product in reverse engineering, in particular, the product of specifications and the process of abstractions and transformations. This makes it possible to control reverse engineering projects. Aspects of measurement in reverse engineering are outlined.
16128617
CyberSeer: 3D audio-visual immersion for network security and management
Large complex networks have become an inseparable part of modern society. However, very little has been done to develop tools to manage and ensure the security of such networks. Network operators continue to slave over endless daily logs and alerts in a struggle to keep networks operational. Perhaps the most formidable enemy of network operations today is the volume of management data that must be perused. Expensive commercial products attempt to visualize data but with limited utility, as witnessed by the prevailing use of command-line interfaces and homegrown scripts. In addition to data collection tools, operators need to immediately observe and debug the effects of their actions; yet that information is buried deep in the data that pours daily from monitoring equipment. Thus, they need better ways to abstract network events and better, more informative ways to render them.
 In this work we first propose a new approach for abstracting network information, namely spectral representation. Second, we introduce immersive spatial audio representations of network events. Third, we introduce 3D interactive auto-stereoscopic (AS) displays for visual representations. We integrate the three techniques as follows. We use spectral techniques to extract complex events buried inside voluminous network traces and logs. Then, we create a desktop interactive immersive auto-stereoscopic 3D environment that is seamlessly integrated with multi-channel spatially rendered audio to render such events in a far more human-friendly fashion.