Manhattan空间有障碍的最短路径和3-Steiner树算法

在VLSI设计中,多点互连是物理设计阶段的关键问题之一,而互连的点数等于2或大于2分别对应于Manhattan空间上有障碍时的最短路径问题和最小Steiner树问题,显然前者是后者的基础.连接图是研究最短路径问题的有效工具,已有的典型连接图包括基于轨迹的GC和GT以及基于自由区的GF和GG.工作包括3个方面:设计并分析了在各种连接图上实现动态的点对之间的最短路径查询算法;分析了在各个连接图上构造3-Steiner树的算法,对于已有的GC上的3-Steiner算法,将其Steiner顶点的候选集合规模从O((e+p)2)降低到了O((t+p)2),其中e,t,p分别表示边数、障碍极边数和顶点数;设计了在GG上的3-Steiner树构造算法,其平均情况时间复杂度只有(θ)(t).     

Neural network for optimal steiner tree computation

Hopfield neural network model for finding the shortest path between two nodes in a graph was proposed recently in some literatures. In this paper, we present a modified version of Hopfield model to a more general problem of searching an optimal tree (least total cost tree) from a source node to a number of destination nodes in a graph. This problem is called Steiner tree in graph theory, where it is proved to be a NP-complete. Through computer simulations, it is shown that the proposed model could always find an optimal or near-optimal valid solution in various graphs.     

Steiner Tree问题的研究进展

Steiner树问题是经典的NP难解问题,在计算机网络布局、电路设计以及生物网络等领域都有很多应用。随着参数计算理论的发展,已经证明了无向图和有向图中的Steiner树问题都是固定参数可解的(FPT)。介绍了无向图和有向图中Steiner树问题的近似算法和参数算法,分析了一些特殊Steiner树问题的研究现状,还讨论了顶点加权Steiner树问题的研究进展。最后,提出了该问题的进一步研究方向。     

The class Steiner minimal tree problem: a lower bound and test problem generation

The class Steiner minimal tree problem is an extension of the standard Steiner minimal tree problem in graphs, motivated by the problem of wire routing in the area of physical design of very large scale integration (VLSI). This problem is NP-hard, even if there are no Steiner nodes and the graph is a tree; moreover, there exists no polynomial time approximation algorithm with a constant bound on the relative error under the hypothesis that P NP [16]. Hence, fast and good heuristic algorithms are needed in practice. In this paper, we present an integer programming formulation of the problem. Using Lagrangean relaxation and subgradient optimization, we derive a lower bound. In order to test the lower bound, we present a procedure for generating test problems for the class Steiner minimal tree problem that have known optimal solutions. The computational experiments for the test problems demonstrate that the lower bound is very tight and differs from the optimal solutions by only a few percent on average for sparse graphs. Received: 5 July 1999 / Revised version: 14 July 2000     

图的steiner最小树问题及其求解

斯坦纳树问题是组合优化学科中的一个问题。属于NP-难问题,即无法在多项式时间内得到最优解。本文主要讨论了图的steiner最小树问题,并给出了近似算法,该算法是在破圈法的基础上进行了改进,并且引用了agent的思想。最后对算法进行了分析。     

Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs

Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of planar straight-line representation has not been solved completely. In this paper we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.     

Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of element-disjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k element-connected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns $\lfloor\frac{k}{2} \rfloor-1$ element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching?2.     

A factoring approach for the Steiner tree problem in undirected networks

In communication networks, many applications, such as video on demand and video conferencing, must establish a communications tree that spans a subset K in a vertex set. The source node can then send identical data to all nodes in set K along this tree. This kind of communication is known as multicast communication. A network optimization problem, called the Steiner tree problem (STP), is presented to find a least cost multicasting tree. In this paper, an O(|E|) algorithm is presented to find a minimum Steiner tree for series-parallel graphs where |E| is the number of edges. Based on this algorithm, we proposed an O(2^2c·|E|) algorithm to solve the Steiner tree problem for general graphs where c is the number of applied factoring procedures. The c value is strongly related to the topology of a given graph. This is quite different from other algorithms with exponential time complexities in |K|.     

Conversion of the Steiner Problem on the Euclidean Plane to the Steiner Problem on Graph

If in the Steiner problem on graph the number of terminal nodes is much smaller than that of all graph nodes (say 50 against 1000), then one can see in its solution (the minimum tree) a system of paths on the graph connecting the terminal vertices, rather than a collection of separate edges (or arcs). This path system is similar to the system of segments making up the Steiner tree on the Euclidean plane: the local degree of the terminal nodes usually is 1, and the local degree of some nodes making up the paths is 3 or more. These nodes are the counterparts of the Steiner points in the problem on plane. Structural similarity of the trees in the Steiner problems on the Euclidean plane and on graph enables one to construct the algorithm to solve the Steiner problem on graph along the same lines as in the Steiner problem on the Euclidean plane. On the other hand, the solution of a problem on graph may be regarded as that on the Euclidean plane, provided that the graph satisfies certain requirements.     

Minimum Connected Dominating Set Using a Collaborative Cover Heuristic for Ad Hoc Sensor Networks

A minimum connected dominating set (MCDS) is used as virtual backbone for efficient routing and broadcasting in ad hoc sensor networks. The minimum CDS problem is NP-complete even in unit disk graphs. Many heuristics-based distributed approximation algorithms for MCDS problems are reported and the best known performance ratio has (4.8+ln 5). We propose a new heuristic called collaborative cover using two principles: 1) domatic number of a connected graph is at least two and 2) optimal substructure defined as subset of independent dominator preferably with a common connector. We obtain a partial Steiner tree during the construction of the independent set (dominators). A final postprocessing step identifies the Steiner nodes in the formation of Steiner tree for the independent set of G. We show that our collaborative cover heuristics are better than degree-based heuristics in identifying independent set and Steiner tree. While our distributed approximation CDS algorithm achieves the performance ratio of (4.8+ln 5){rm opt} + 1.2, where {rm opt} is the size of any optimal CDS, we also show that the collaborative cover heuristic is able to give a marginally better bound when the distribution of sensor nodes is uniform permitting identification of the optimal substructures. We show that the message complexity of our algorithm is O(nDelta^{2} ), Delta being the maximum degree of a node in graph and the time complexity is O(n).     

基于最小Steiner树的关键词查询方法

在关系数据库中,关键词查询无需用户学习查询语言和数据库模式相关知识,而且有效地扩大了查询范围.采用元组图描述关系数据库中元组关系,可使关键词查询问题转化为元组图的最小Steiner树求解问题.本文提出元组图上基于相似度的边权重计算方法,使边权重能够反映元组与关键词相似度的大小.然后,鉴于最小Steiner树求解问题是NP-完全问题,提出按照贪心策略执行Dijkstra算法的最小Steiner树较优解求解算法.最后,通过实验对算法进行了分析和验证.     

基于图压缩的最大Steiner连通k核查询处理

研究了基于图压缩的最大Steiner连通k核查询处理,提出了一种支持最大Steiner连通k核查询的图压缩算法SC,证明了基于SC压缩算法的查询正确性.由于最大Steiner连通k核查询仅需要找到符合要求的连通区域,提出了图压缩算法TC,进一步将压缩图压缩为树.证明了基于压缩树的查询正确性,并提出了线性时间的无需解压缩的查询处理算法.真实和虚拟数据上的实验结果表明：压缩算法平均可将原始图压缩掉88%,且对于稠密的原始图,压缩算法的压缩效果更好,可将原始图压缩掉90%,与在原始图上直接进行查询处理相比,基于压缩图的查询处理算法效率更好,平均提升了1～2个数量级.     

On the terminal Steiner tree problem

We investigate a practical variant of the well-known graph Steiner tree problem. In this variant, every target vertex is required to be a leaf vertex in the solution Steiner tree. We present hardness results for this variant as well as a polynomial time approximation algorithm with performance ratio ρ+2, where ρ is the best-known approximation ratio for the graph Steiner tree problem.     

Tree Spanners for Bipartite Graphs and Probe Interval Graphs

A tree t-spanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree t-spanner problem asks whether a graph admits a tree t-spanner, given t. We first substantially strengthen the known results for bipartite graphs. We prove that the tree t-spanner problem is NP-complete even for chordal bipartite graphs for t ≥ 5, and every bipartite ATE-free graph has a tree 3-spanner, which can be found in linear time. The previous best known results were NP-completeness for general bipartite graphs, and that every convex graph has a tree 3-spanner. We next focus on the tree t-spanner problem for probe interval graphs and related graph classes. The graph classes were introduced to deal with the physical mapping of DNA. From a graph theoretical point of view, the classes are natural generalizations of interval graphs. We show that these classes are tree 7-spanner admissible, and a tree 7-spanner can be constructed in (m log n) time.     

An 11/6-approximation algorithm for the network steiner problem

An instance of the Network Steiner Problem consists of an undirected graph with edge lengths and a subset of vertices; the goal is to find a minimum cost Steiner tree of the given subset (i.e., minimum cost subset of edges which spans it). An 11/6-approximation algorithm for this problem is given. The approximate Steiner tree can be computed in the time0(¦V¦ ¦E¦ + ¦S¦⁴), whereV is the vertex set,E is the edge set of the graph, andS is the given subset of vertices.     

The Steiner tree problem on graphs: Inapproximability results

The Steiner tree problem on weighted graphs seeks a minimum weight subtree containing a given subset of the vertices (terminals). We show that it is NP-hard to approximate the Steiner tree problem within a factor 96/95$96/95$. Our inapproximability results are stated in a parametric way, and explicit hardness factors would be improved automatically by providing gadgets and/or expanders with better parameters.     

A note on the terminal Steiner tree problem

In 2002, Lin and Xue [Inform. Process. Lett. 84 (2002) 103-107] introduced a variant of the graph Steiner tree problem, in which each terminal vertex is required to be a leaf in the solution Steiner tree. They presented a ρ+2 approximation algorithm, where ρ is the approximation ratio of the best known efficient algorithm for the regular graph Steiner tree problem. In this note, we derive a simplified algorithm with an improved approximation ratio of 2ρ (currently ρ≈1.550, see [SODA 2000, 2000, pp. 770-790]).     

The General Steiner Tree-Star problem

The Steiner tree problem is defined as follows—given a graph G=(V,E) and a subset X⊂V of terminals, compute a minimum cost tree that includes all nodes in X. Furthermore, it is reasonable to assume that the edge costs form a metric. This problem is NP-hard and has been the study of many heuristics and algorithms. We study a generalization of this problem, where there is a “switch” cost in addition to the cost of the edges. Switches are placed at internal nodes of the tree (essentially, we may assume that all non-leaf nodes of the Steiner tree have a switch). The cost for placing a switch may vary from node to node. A restricted version of this problem, where the terminal set X cannot be connected to each other directly but only via the Steiner nodes V?X, is referred to as the Steiner Tree-Star problem. The General Steiner Tree-Star problem does not require the terminal set and Steiner node set to be disjoint. This generalized problem can be reduced to the node weighted Steiner tree problem, for which algorithms with performance guarantees of Θ(lnn) are known. However, such approach does not make use of the fact that the edge costs form a metric. In this paper we derive approximation algorithms with small constant factors for this problem. We show two different polynomial time algorithms with approximation factors of 5.16 and 5.     

Providing built-in keyword search capabilities in RDBMS

A common approach to performing keyword search over relational databases is to find the minimum Steiner trees in database graphs transformed from relational data. These methods, however, are rather expensive as the minimum Steiner tree problem is known to be NP-hard. Further, these methods are independent of the underlying relational database management system (RDBMS), thus cannot benefit from the capabilities of the RDBMS. As an alternative, in this paper we propose a new concept called Compact Steiner Tree (CSTree), which can be used to approximate the Steiner tree problem for answering top-k keyword queries efficiently. We propose a novel structure-aware index, together with an effective ranking mechanism for fast, progressive and accurate retrieval of top-k highest ranked CSTrees. The proposed techniques can be implemented using a standard relational RDBMS to benefit from its indexing and query-processing capability. We have implemented our techniques in MYSQL, which can provide built-in keyword-search capabilities using SQL. The experimental results show a significant improvement in both search efficiency and result quality comparing to existing state-of-the-art approaches.