Clustering of data has numerous applications and has been studied extensively. It is very important in Bioinformatics and data mining. Though many parallel algorithms have been designed, most of algorithms use the CRCW-PRAM or CREW-PRAM models of computing. This paper proposed a parallel EREW deterministic algorithm for hierarchical clustering. Based on algorithms of complete graph and Euclidean minimum spanning tree, the proposed algorithms can cluster n objects with O(p) processors in O(n2/p) time where 1 ≤ p ≤ n/log n. Performance comparisons show that our algorithm is the first algorithm that is both without memory conflicts and adaptive.

An adaptive parallel hierarchical clustering algorithm

For a connected graph $$G=(V,E)$$ and a positive integral vertex weight function $$w$$ , a max-min weight balanced connected $$k$$ -partition of $$G$$ , denoted as $$BCP_k$$ , is a partition of $$V$$ into $$k$$ disjoint vertex subsets $$(V_1,V_2,\ldots ,V_k)$$ such that each $$G[V_i]$$ (the subgraph of $$G$$ induced by $$V_i$$ ) is connected, and $$\min _{1\le i\le k}\{w(V_i)\}$$ is maximum. Such a problem has a lot of applications in image processing and clustering, and was proved to be NP-hard. In this paper, we study $$BCP_k$$ on a special class of graphs: trapezoid graphs whose maximum degree is bounded by a constant. A pseudo-polynomial time algorithm is given, based on which an FPTAS is obtained for $$k=2,3,4$$ . A step-stone for the analysis of the FPTAS depends on a lower bound for the optimal value of $$BCP_k$$ in terms of the total weight of the graph. In providing such a lower bound, a byproduct of this paper is that any 4-connected trapezoid graph on at least seven vertices has a 4-contractible edge, which may have a value in its own right.

Max-min weight balanced connected partition

This thesis presents efficient algorithms for drawing planar graphs. Graph drawing addresses the problem of constructing geometric representation of information and finds applications in almost every branch of science and technology. In this thesis we study rectangular drawings, orthogonal drawings, convex drawings and our newly defined box-rectangular drawings. Rectangular drawings and box-rectangular drawings have practical applications in VLSI floorplanning and in architectural floorplanning, whereas orthogonal drawings have applications in circuit schematics, data-flow diagrams etc. We give a linear-time algorithm for finding a rectangular drawing of a plane graph if it exists, and obtain the best possible upper bound on grid size for a rectangular drawing. Minimizing the number of bends in an orthogonal drawing is a challenging problem. We give a linear-time algorithm for finding an orthogonal drawing of a triconnected cubic plane graph with the minimum number of bends. We introduce a new drawing style called box-rectangular drawing and obtain a necessary and sufficient condition for a plane graph to have a box-rectangular drawing. We also present the corresponding linear-time algorithm. We newly define 4-canonical decomposition and give a linear-time algorithm for finding a 4-canonical decomposition of a 4-connected planar graph. Our 4-canonical decomposition has applications in finding a convex drawing of a 4-connected planar graph.

Efficient algorithms for drawing planar graphs

Partitioning a large network into a connected hierarchy is a proven technique for scalability and manageability. Although a significant body of work is devoted to this area of interest, a reference to survivable hierarchical networks where each partition remains k-connected is missing (k ⩾ 2). A k-connected partition ensures functional network organization even after any k − 1 failures occur. In optical networks, a failure of a physical component (e.g., a cable or line-card) disrupts multiple logical links that share the component. Such a shared risk link group (SRLG) failure is more common than a single link or node failure in the logical topology. Hence, we study the significance of a k-SRLG-connected (kSC) partitioning problem for optical networks. For a given network, we show that finding 2-SRLG-connected (2SC) partitioning, the simplest case, is NP-complete. We provide mathematical formulations and propose a distributed-scalable heuristic solution. We also solve 2SC partitioning with flexible mapping onto the physical topology, where lightpaths can be added to guarantee a solution. Simulation results verify that the kSC partitioning scheme performs better than traditional 1-connected partitioning schemes with regard to system survivability, stability, and repartitioning overhead.

k-connected network partitioning under shared risk link groups

Partitioning a large network into a connected hierarchy is a proven technique for scalability, manageability, and reliability. Although a significant body of works is devoted to this area of interest, a reference to partitioning schemes where each partition remains 2-connected is missing. In optical networks, a failure of a physical component disrupts multiple logical links that share the component. Such a shared risk link group (SRLG) failure is more common than a single link or node failure in the logical topology. In this paper, we introduce the k-SRLG-connected partitioning problem for optical networks, where k ≥ 1. A logical topology is said to be k-SRLG-connected if its nodes remain connected upon any k−1 SRLG failures. We particularly focus on k = 2. Without 2-SRLG-connectivity, a partition can neither provide protection by allocating two SRLG-disjoint paths, nor provide restoration after a failure. Moreover, a single SRLG failure can split the partition into multiple disconnected components, resulting in the failure of the hierarchical network organization itself. We show that finding a 2-SRLG-connected partitioning for a given network is NP-complete and provide a distributed heuristic approach. Simulation results verify that the proposed 2-SRLG-connected partitioning scheme performs better than traditional 1-connected partitioning approaches with regard to survivability and system stability.

2-SRLG-connected partitioning in optical networks

Although there are polynomial algorithms of finding a 2-partition or a 3-partition for a simple undirected 2-connected or 3-connected graph respectively,there is no general algorithm of finding a k-partition for a k-connected graph G=(V,E),where k is the vertex connectivity of G.In this paper,an O(k^2n^2) general algorithm of finding a k-partition for a k-connected graph is proposed,where n=|V|.

An O(k~2n~2) Algorithm to Find a k-Partition in a k-Connected Graph

An O(n^2) time approximation algorithm for the minimum rectilinear Steiner tree is proposedThe approximation ratio of the algorithm is strictly less than 15The computing performances show the costs of the spanning trees produced by the algorithm are only 08% away from the optimal ones

马军

Papers

An O(k~2n~2) Algorithm to Find a k-Partition in a k-Connected Graph

A Practical Algorithm for the Minimum Rectilinear Steiner Tree

Efficient Parallel Algorithms for Some Graph Theory Problems