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Crossing Numbers: Bounds and Applications
 I. B'AR'ANY AND K. BOROCZKY, BOLYAI SOCIETY MATHEMATICAL STUDIES 6
, 1997
"... We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the autho ..."
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We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.
Gossiping in VertexDisjoint Paths Mode in dDimensional Grids and Planar Graphs (Extended Abstract)
 Information and Computation
, 1993
"... Juraj Hromkovic y , Ralf Klasing, Elena A. Stohr, Hubert Wagener z Department of Mathematics and Computer Science University of Paderborn, 33095 Paderborn, Germany Abstract The communication modes (oneway and twoway mode) used for sending messages to processors of interconnection networks via ..."
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Cited by 8 (2 self)
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Juraj Hromkovic y , Ralf Klasing, Elena A. Stohr, Hubert Wagener z Department of Mathematics and Computer Science University of Paderborn, 33095 Paderborn, Germany Abstract The communication modes (oneway and twoway mode) used for sending messages to processors of interconnection networks via vertexdisjoint paths in one communication step are investigated. The complexity of communication algorithms is measured by the number of communication steps (rounds). Here, the complexity of gossiping in grids and in planar graphs is investigated. The main results are the following: 1. Effective oneway and twoway gossip algorithms for ddimensional grids, d 2, are designed. 2. The lower bound 2 log 2 n \Gamma log 2 k \Gamma log 2 log 2 n \Gamma 2 is established on the number of rounds of every twoway gossip algorithm working on any graph of n nodes and vertex bisection k. This proves that the designed twoway gossip algorithms on ddimensional grids, d 3, are almost optimal, and it al...
I/OEfficient Planar Separators and Applications
, 2001
"... We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with ..."
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Cited by 3 (1 self)
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We present a new algorithm to compute a subset S of vertices of a planar graph G whose removal partitions G into O(N/h) subgraphs of size O(h) and with boundary size O( p h) each. The size of S is O(N= p h). Computing S takes O(sort(N)) I/Os and linear space, provided that M 56hlog² B. Together with recent reducibility results, this leads to O(sort(N)) I/O algorithms for breadthfirst search (BFS), depthfirst search (DFS), and single source shortest paths (SSSP) on undirected embedded planar graphs. Our separator algorithm does not need a BFS tree or an embedding of G to be given as part of the input. Instead we argue that "local embeddings" of subgraphs of G are enough.
Congestion and Dilation, Similarities and Differences: a Survey
 In Proc. 7th Intl. Colloquium on Structural Information and Communication Complexity
, 2000
"... We survey general results on congestion and dilation, and their special cases (cyclic) cutwidth and (cyclic) bandwidth, with the emphasis on the similarity and the duality of both parameters. Keywords Bandwidth, Congestion, Cutwidth, Embedding, Dilation 1 Introduction Recently diverse properti ..."
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We survey general results on congestion and dilation, and their special cases (cyclic) cutwidth and (cyclic) bandwidth, with the emphasis on the similarity and the duality of both parameters. Keywords Bandwidth, Congestion, Cutwidth, Embedding, Dilation 1 Introduction Recently diverse properties and invariants of interconnection networks (not only those of parallel machines) have been studied and a lot of interesting results have been shown (e.g. see [16, 36]). One of the important features of an interconnection network is its ability to efficiently simulate programs written for other architectures. Such a simulation problem can be mathematically formulated as a graph embedding. Informally, the graph embedding problem is to label the vertices of a "guest" graph (e.g. a communication graph of processes and relations between the processes) G by distinct vertices of a "host" graph (an interconnection network) H. The quality of the embedding corresponding to the efficiency of the sim...
BoundaryOptimal Triangulation Flooding
"... Given a planar triangulation all of whose faces are initially white, we study the problem of colouring the faces black one by one so that the boundary between black and white faces as well as the number of connected black and white regions are small at all times. We call such a colouring sequence o ..."
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Given a planar triangulation all of whose faces are initially white, we study the problem of colouring the faces black one by one so that the boundary between black and white faces as well as the number of connected black and white regions are small at all times. We call such a colouring sequence of the triangles a flooding. Our main result shows that it is in general impossible to guarantee boundary size O(n 1−ɛ), for any ɛ> 0, and a number of regions that is o(log n), where n is the number of faces of the triangulation. We also show that a flooding with boundary size O (√n) and O(log n) regions can be computed in O(n log n) time.
Contents lists available at ScienceDirect Discrete Mathematics
"... journal homepage: www.elsevier.com/locate/disc ..."
Leveling the Grid
"... Motivated by an application in image processing, we introduce the gridleveling problem. It turns out to be the dual of a minimum cost flow problem for an apex graph with a grid graph as its basis. We present an O(n 3/2) algorithm for this problem. The optimum solution recovers missing DC coefficien ..."
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Motivated by an application in image processing, we introduce the gridleveling problem. It turns out to be the dual of a minimum cost flow problem for an apex graph with a grid graph as its basis. We present an O(n 3/2) algorithm for this problem. The optimum solution recovers missing DC coefficients from image and video coding by Discrete Cosine Transform used in popular standards like JPEG and MPEG. Generally, we prove that there is an O(n 3/2) mincost flow algorithm for networks that, after removing one node, are planar, have bounded degrees, and have bounded capacities. The costs may be arbitrary. 1
On the Complexity of MultiDimensional Interval Routing Schemes
"... Multidimensional interval routing schemes (MIRS) introduced in [4] are an extension of interval routing schemes (IRS). We give an evidence that multidimensional interval routing schemes really help for certain wellknown interconnection networks, e.g. for ndimensional butterfly there exists a ful ..."
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Multidimensional interval routing schemes (MIRS) introduced in [4] are an extension of interval routing schemes (IRS). We give an evidence that multidimensional interval routing schemes really help for certain wellknown interconnection networks, e.g. for ndimensional butterfly there exists a fullinformation shortest path 〈2, 3〉DISMIRS, while in [12] it was shown that a shortest path IRS needs Ω(2n/2) intervals. Further, we compare the DISMIRS model and the CONMIRS model introduced in [4]. The main result is that the DISMIRS model is asymptotically stronger than the CONMIRS model. We prove this by showing that for ndimensional cubeconnectedcycles there exists a fullinformation shortest path 〈2n3 √, n〉DISMIRS while for any fullinformation shortest path 〈k, d〉CONMIRS it holds kd = Ω(2 n/2). We also show that for ndimensional star any fullinformation shortest path 〈k, d〉CONMIRS requires kd = Ω(2n/3). The congestion is a common phenomenon in networks which can completely degrade their performance. Therefore it is reasonable to study routing schemes which are not necessarily shortest path, but allow high network throughput. We show that there exists a 〈2, n + 2〉DISMIRS of ndimensional cubeconnectedcycles with asymptotically optimal congestion. Moreover, we give an upper bound on the tradeoff between congestion and space complexity of multipath MIRS for general graphs. For any graph G and given 1 ≤ s ≤ V  there exists a multipath 〈2 + ⌈V /2s⌉, 1〉MIRS with congestion F + V  · ∆ · s, where F is the forwarding index of the graph G and ∆ is the maximum degree of vertices in the graph G. As a consequence, for planar graphs of constant bounded degree there exists a multipath 〈O ( √ V ), 1〉MIRS with asymptotically optimal congestion. We also show that for planar graphs of bounded degree there exists (deterministic) O ( √ V  log V )IRS with asymptotically optimal congestion.