2934981
On Finding the Rectangular Duals of Planar Triangular Graphs
This paper presents a new linear-time algorithm for finding rectangular duals of planar triangular graphs. The algorithm is conceptually simpler than the previously known algorithm. The coordinates of the rectangular dual constructed by the new algorithm are integers and carry clear combinatorial meaning.
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.
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.
20004613
Disk Embeddings of Planar Graphs
Abstract 
Given a planar graph $G=(V,E)$ and a rooted forest ${\FF}=(V_{\FF}, A_{\FF})$ 
with leaf set $V$, we wish to decide whether $G$ has a plane embedding $\GG$ 
satisfying the following condition: There are $|V_{\FF}|-|V|$ pairwise noncrossing 
Jordan curves in the plane one-to-one corresponding to the nonleaf vertices of 
${\FF}$ such that for every nonleaf vertex $f$ of ${\FF}$, the interior of the curve 
$\JJ_f$ corresponding to $f$ contains all the leaf descendants of $f$ in ${\FF}$
but contains no other leaves of ${\FF}$. 
This problem arises from theoretical studies in geographic database systems. 
It is unknown whether this problem can be solved in polynomial time. 
This paper presents an almost linear-time algorithm for a nontrivial special case 
where the set of leaf descendants of each nonleaf vertex $f$ in ${\FF}$ induces 
a connected subgraph of $G$. 


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].
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.
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.
10726702
Locality Preserving Projections
Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) – a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.
2341522
PZ nets a formal method integrating Petri nets with Z
Abstract In this paper, a formal method (called PZ nets) for specifying concurrent and distributed systems is presented. PZ nets integrate two well-known existing formal methods Petri nets and Z such that Petri nets are used to specify the overall structure, control flows, causal relation, and dynamic behavior of a system; and Z is used to define tokens, labels and constrains of the system. The essence, benefits, and problems of the integration are discussed. A set of heuristics and transformations to develop PZ nets and a technique to analyze PZ nets are proposed and demonstrated through a well-known example.
5723803
Incremental semi-supervised subspace learning for image retrieval
Subspace learning techniques are widespread in pattern recognition research. They include Principal Component Analysis (PCA), Locality Preserving Projection (LPP), etc. These techniques are generally unsupervised which allows them to model data in the absence of labels or categories. In relevance feedback driven image retrieval system, the user provided information can be used to better describe the intrinsic semantic relationships between images. In this paper, we propose a semi-supervised subspace learning algorithm which incrementally learns an adaptive subspace by preserving the semantic structure of the image space, based on user interactions in a relevance feedback driven query-by-example system. Our algorithm is capable of accumulating knowledge from users, which could result in new feature representations for images in the database so that the system's future retrieval performance can be enhanced. Experiments on a large collection of images have shown the effectiveness and efficiency of our proposed algorithm.
37910692
Replicated shared object model for parallel edge detection algorithm based on spiral architecture
Abstract Edge detection in computer vision and image processing is a process which detects one kind of significant feature in an image that appears as discontinuities in intensities. A parallel edge detection algorithm based on spiral architecture is designed in this paper using a replicated shared object model (RSOM). We use this approach to meet the requirements of replicated object management, update propagation, underlying communication and consistency maintenance among replicated objects. Multi-scale image theory plays an important role in this research.