Fast vicinity-upgrade algorithm for rectilinear Steiner trees

A new algorithm for rectilinear Steiner trees (RST) is presented. The proposed algorithm is based on successively constructing a vicinity structure from a rectilinear minimum spanning tree (MST) and generating a refined RST. The algorithm achieves an 8.5-10% average cost improvement over the rectilinear MST at a time complexity of O(n log n). To address specifically the linear programming approach of global routing the algorithm is modified to generate K trees.<>     

Maze Routing Steiner Trees With Delay Versus Wire Length Tradeoff

In this paper, we address the problem of generating good topologies of rectilinear Steiner trees using path search algorithms. Various techniques have been applied in order to achieve acceptable run times on a Maze Router that builds Steiner trees. A biasing technique proposed for wire length improvement, produces trees that are within 2% from optimal topologies in average. By introducing a sharing factor and a path-length factor we show how to trade-off wire length for delay. Experimental results show that our algorithm generates topologies with better delay compared to state of the art heuristics for Steiner trees, such as AHHK (from 26% to 40%) and P-Trees (from 1% to 30% and from 6% to 21% in the presence of blockages) while keeping the properties of a routing algorithm. An important motivation for this work lies in the fact that it can be used for estimation in the early stages as well as for actual routing, thereby improving the convergence and timing closure of the design significantly. We also provide some valuable theoretical background and insights on delay optimization and on how it relates to our maze router implementation.      

A full-scale solution to the rectilinear obstacle-avoiding Steiner problem

Routing is one of the important steps in very/ultra large-scale integration (VLSI/ULSI) physical design. Rectilinear Steiner minimal tree (RSMT) construction is an essential part of routing. Macro cells, IP blocks, and pre-routed nets are often regarded as obstacles in the routing phase. Obstacle-avoiding RSMT (OARSMT) algorithms are useful for practical routing applications. However, OARSMT algorithms for multi-terminal net routing still cannot meet the requirements of practical applications. This paper focuses on the OARSMT problem and gives a solution to full-scale nets based on two algorithms, namely An-OARSMan and FORSTer. (1) Based on ant colony optimization (ACO), An-OARSMan can be used for common scale nets with less than 100 terminals in a circuit. An heuristic, greedy obstacle penalty distance (OP-distance), is used in the algorithm and performed on the track graph. (2) FORSTer is a three-step heuristic used for large-scale nets with more than 100 terminals in a circuit. In Step 1, it first partitions all terminals into some subsets in the presence of obstacles. In Step 2, it then connects terminals in each connected graph with one or more trees, respectively. In Step 3, it finally connects the forest consisting of trees constructed in Step 2 into a complete Steiner tree spanning all terminals while avoiding all obstacles. (3) These two graph-based algorithms have been implemented and tested on different kinds of cases. Experimental results show that An-OARSMan can handle both convex and concave polygon obstacles with short wire length. It achieves the optimal solution in the cases with no more than seven terminals. The experimental results also show that FORSTer has short running time, which is suitable for routing large-scale nets among obstacles, even for routing a net with one thousand terminals in the presence of 100 rectangular obstacles.     

Evaluation of multicast routing algorithms for real-timecommunication on high-speed networks

Multicast (MC) routing algorithms capable of satisfying the quality of service (QoS) requirements of real-time applications will be essential for future high-speed networks. We compare the performance of all of the important MC routing algorithms when applied to networks with asymmetric link loads. Each algorithm is judged based on the quality of the MC trees it generates and its efficiency in managing the network resources. Simulation results over random networks show that unconstrained algorithms are not capable of fulfilling the QoS requirements of real-time applications in wide-area networks. Simulations also reveal that one of the unconstrained algorithms, reverse path multicasting (RPM), is quite inefficient when applied to asymmetric networks. We study how combining routing with resource reservation and admission control improves the RPM's efficiency in managing the network resources. The performance of one semiconstrained heuristic, MSC, three constrained Steiner tree (CST) heuristics, Kompella, Pasquale, and Polyzos (1992), constrained adaptive ordering (CAO), and bounded shortest multicast algorithm (BSMA), and one constrained shortest path tree (CSPT) heuristic, the constrained Dijkstra heuristic (CDKS) are also studied. Simulations show that the semiconstrained and constrained heuristics are capable of successfully constructing MC trees which satisfy the QoS requirements of real-time traffic. However, the cost performance of the heuristics varies. The BSMA's MC trees are lower in cost than all other constrained heuristics. Finally, we compare the execution times of all algorithms, unconstrained, semiconstrained, and constrained     

基于直角Steiner树的片上网络互连算法

提出了一种基于最小直角Steiner树,在Manhattan平面上避免障碍物的互连算法,以实现片上网络中各IP核的互连.该算法在定制NoC中将Steiner树的生成算法用于互连设计.算法首先在初始阶段对有障碍两点间的边权重重新赋值,然后调用最小生成树算法,使生成的直角Steiner树总长度最小.实验表明,该算法可以使片上网络的总连线缩短.     

A buffer distribution algorithm for high-performance clock netoptimization

We propose a new approach for optimizing clock trees, especially for high-speed circuits. Our approach provides a useful guideline to a designer, by user-specified parameters, and three of these tradeoffs are provided in this paper. (1) First, to provide a “good” tradeoff between skew and wire length, a new clock tree routing scheme is proposed. The technique is based on a combination of hierarchical bottom-up geometric matching and minimum rectilinear Steiner tree. Our experiments complement the theoretical results. (2) For high-speed clock distribution in the transmission line mode (e.g., multichip modules) where interconnection delay dominates the clock delay, buffer congestion might exist in a layout. Using many buffers in a small wiring area results in substantial interline crosstalks as well as wirability, when the elongation of the imbalanced subtrees is necessary. Placing buffers evenly (locally or globally) over the plane at the minimum impact on wire length increase helps avoid buffer congestion and results in less crosstalk between clock wires. Thus, an effective technique for buffer distribution is proposed. Experimental results verify the effectiveness of the proposed algorithms. (3) Finally, a postprocessing step constraining on phase-delay is also proposed. The technique is based on a combination of hierarchical bottom-up geometric matching and bounded radius minimum spanning tree. The proposed algorithm has an important application in MCM clock net synthesis as well as VLSI clock net synthesis     

Obstacle-avoiding rectilinear minimum-delay Steiner tree construction toward IP-block-based SOC design

With system-on-chip design, IP blocks form routing obstacles that deteriorate global interconnect delay. In this paper, we present a new approach for obstacle-avoiding rectilinear minimal delay Steiner tree (OARMDST) construction. We formalize the solving of minimum delay tree through the concept of an extended minimization function, and trade the objective into a top-down recursion, which wisely produces delay minimization from source to critical sinks. We analyze the topology generation with treatment of obstacles and exploit the connection flexibilities. To our knowledge, this is the first in-depth study of OARMDST problem based on topological construction. Experimental results are given to demonstrate the efficiency of the algorithm.     

Improved neural heuristics for multicast routing

Future networks must be adequately equipped to handle multipoint communication in a fast and economical manner. Services requiring such support include desktop video conferencing, tele-classrooms, distributed database applications, etc. In networks employing the asynchronous transfer mode (ATM) technology, routing a multicast is achieved by constructing a minimum cost tree that spans the source and all the destinations. When the network is modeled as a weighted, undirected graph, the problem is that of finding a minimal Steiner tree for the graph, given a set of destinations. The problem is known to be NP-complete. Consequently, several heuristics exist which provide approximate solutions to the Steiner problem in networks, We show how the random neural network (RNN) can be used to significantly improve the quality of the Steiner trees delivered by the best available heuristics which are the minimum spanning tree heuristic and the average distance heuristic. We provide an empirical comparison and find that the heuristics which are modified using the neural network yield significantly improved trees     

VLSI总体布线算法研究与实现

总体布线是布图设计中一个极为重要的设计环节。本文提出了基于可分离最小生成树(SMST)的优化L形直角斯坦(Steiner)树(L_RST)和优化Z_RST的算法。该算法实现上绕开计算重合度问题，以新的角度计算代价。利用基于tile的结构，实现了伪管脚(pseudo pin)的分配，适用于现代多层布线需求。最后文章研究了同时考虑串扰和时延的综合性能驱动的总体布线算法改进。     

A Distributed Routing Method for Dynamic Multicasting

Single point, sender based control does not scale well for multicast delivery. For applications, such as group video or teleconferencing a low total cost multicast tree is required. In this article we present a destination driven algorithm to minimize the total tree cost of multicast tree in a dynamic situation for the whole session duration. In this heuristic approach we considered the staying duration of participants are available at the time of joining. The performance of our algorithm is analyzed through extensive simulation and evaluated against several other existing dynamic multicast routing and also against one well known near optimum heuristic algorithm used for solving Steiner tree problem. We have further tested our algorithm using erroneous information given by the joining participants. Simulation results show that its performance does not degrade that much even when the range of error is considerably high, which proves the robustness of our algorithm.     

Improved heuristics for multicast routing in wireless mesh networks

Multicast is a communication technique that allows a source to transmit data to a set of recipients in an efficient manner. Therefore, the primary objective of a multicast routing protocol would be to minimize number of transmissions to conserve bandwidth. The problem of computing multicast trees with minimal bandwidth consumption is similar to Steiner tree problem and has shown to be NP-complete. So, heuristic based algorithms are suitable to approximate such bandwidth optimal trees. This paper proposes a multicast routing protocol based on minimum number of transmission trees using an heuristic approach. The simulation results show that the proposed algorithm offers better performance over existing protocols, even in the worst-case scenario when the set of multicast receivers are sparsely distributed across the network.     

A resource‐tuned QoS multicast routing protocol

This paper presents a novel framework for quality‐of‐service (QoS) multicast routing with resource allocation that represents QoS parameters, jitter delay, and reliability, as functions of adjustable network resources, bandwidth, and buffer, rather than static metrics. The particular functional form of QoS parameters depends on rate‐based service disciplines used in the routers. This allows intelligent tuning of QoS parameters as functions of allocated resources during the multicast tree search process, rather than decoupling the tree search from resource allocation. The proposed framework minimizes the network resource utilization while keeping jitter delay, reliability, and bandwidth bounded. This definition makes the proposed QoS multicast routing with resource allocation problem more general than the classical minimum Steiner tree problem. As an application of our general framework, we formulate the QoS multicast routing with resource allocation problem for a network consisting of generalized processor sharing nodes as a mixed‐integer quadratic program and find the optimal multicast tree with allocated resources to satisfy the QoS constraints. We then present a polynomial‐time greedy heuristic for the QoS multicast routing with resource allocation problem and compare its performance with the optimal solution of the mixed‐integer quadratic program. The simulation results reveal that the proposed heuristic finds near‐optimal QoS multicast trees along with important insights into the interdependency of QoS parameters and resources.     

Multicriteria Steiner tree with the cost of Steiner vertices

The multicriteria problem of constructing the Steiner tree with consideration for the cost of additional Steiner vertices is studied. Formulations of the problems of constructing the covering Steiner trees and algorithmic approaches are described. The engineering formulation of the problem is aimed at constructing a telecommunications network with allowance for the spatial distribution of radio interferences. An approach to solving the formulated milticriterion problem on the basis of combination of two solving schemes is proposed: (1) the basic scheme for constructing a covering Steiner tree on the basis of clustering the vertices of the initial graph and (2) the general scheme for constructing the covering Steiner trees with allowance for the number of additional vertices and calculation of the Pareto-efficient solutions. A module based on the modified Prim’s algorithm for constructing a multicriteria covering tree is additionally used. The vertices of the initial graph are clustered on the basis of a hierarchical algorithm. Examples of a computational experiment on constructing the multicriteria Steiner trees are given.     

一种非零偏差时钟网布线算法

特定的非零偏差时钟网比零偏差时钟网更具优势,它有助于提高时钟频率、降低偏差的敏感度.文章提出了一种新的非零偏差时钟树布线算法,它结合时钟节点延时和时钟汇点位置,得到一个最大节点延时次序合并策略,使时钟树连线长度变小.实验结果显示,这种算法与典型的最邻近选择合并策略相比较,可以减少20%～30%的总连线长度.     

Probability-based approach to rectilinear Steiner tree problems

The rectilinear Steiner tree (RST) problem is of essential importance to the automatic interconnect optimization for VLSI design. In this paper, we present a class of probability-based approaches toward the best solutions under statistical sense and show their performance in comparison with the state-of-the-art algorithm. Experiments conducted on both small- and large-size problems indicate that the proposed approaches lead to promising results in terms of wire length and/or CPU time. The potential advantages with our technique are also discussed for further applications.     

On general zero-skew clock net construction

We propose a simulated annealing based zero-skew clock net construction algorithm that works in any routing spaces, from Manhattan to Euclidean, with the added flexibility of optimizing either the wire length or the propagation delay. We first devise an O(log n) tree grafting perturbation function to construct a zero-skew clock tree under the Elmore delay model. This tree grafting scheme is able to explore the entire solution space asymptotically. A Gauss-Seidel iteration procedure is then applied to optimize the Steiner point positions. Experimental results have shown that our algorithm can achieve substantial delay reduction and encouraging wire length minimization compared to previous works     

Distributed algorithms for multicast path setup in data networks

Establishing a multicast tree in a point-to-point network of switch nodes, such as a wide-area asynchronous transfer mode (ATM) network, can be modeled as the NP-complete Steiner problem in networks. In this paper, we introduce and evaluate two distributed algorithms for finding multicast trees in point-to-point data networks. These algorithms are based on the centralized Steiner heuristics, the shortest path heuristic (SPH) and the Kruskal-based shortest path heuristic (K-SPH), and have the advantage that only the multicast members and nodes in the neighborhood of the multicast tree need to participate in the execution of the algorithm. We compare our algorithms by simulation against a baseline algorithm, the pruned minimum spanning-tree heuristic that is the basis of many previously published algorithms for finding multicast trees. Our results show that the competitiveness (the ratio of the sum of the heuristic tree's edge weights to that of the best solution found) of both of our algorithms was, on the average, 25% better in comparison to that of the pruned spanning-tree approach. In addition, the competitiveness of our algorithms was, in almost all cases, within 10% of the best solution found by any of the Steiner heuristics considered, including both centralized and distributed algorithms. Limiting the execution of the algorithm to a subset of the nodes in the network results in an increase in convergence time over the pruned spanning-tree approach, but this overhead can be reduced by careful implementation     

Rate-Diversity and Resource-Aware Broadcast and Multicast in Multi-rate Wireless Mesh Networks

This paper focuses on the problem of increasing the traffic capacity (volume of admissible traffic) of broadcast and multicast flows in a wireless mesh network (WMN). We study and suggest routing strategies where the process of constructing the forwarding tree considers three distinct features: (a) the ability of individual mesh nodes to perform link-layer broadcasts at multiple rates, (b) the wireless broadcast advantage, whereby a single broadcast transmission covers multiple neighboring receivers and (c) the residual transmission capacity at a WMN node, subject to intereference-based constraints from existing traffic flows in its neighborhood. Our metric of interest is the total number of broadcast and multicast flows that can be admitted into the network, without resulting in unacceptable degradation in metrics such as packet loss and dissemination latency. Our discrete event simulations show that the broadcast tree construction heuristic which takes both transmission rate and residual bandwidth into account out-performs those that do not. Building on our work on resource-aware broadcast tree construction, we propose a resource-aware multicast tree construction algorithm which exploits the multiple link-layer rates, the wireless broadcast advantage and the amount of resources available. Simulation results show that this algorithm performs better than heuristics based on pruning a broadcast tree or shortest path trees.     

考虑光学邻近效应的详细布线算法

提出了一种考虑光学邻近效应的详细布线算法.该算法在布线过程中,充分考虑了线网走线相对位置及布线线形对其光学邻近效应的影响,通过相应的光刻模拟模型定义了用于估计光学邻近效应(optical proximity effect,OPE)的OPE费用函数,并采用OPE费用阈值控制Steiner树的生长方向和走线路径的选择,同时兼顾线网长度.为提高算法效率,避免布线过程中反复调用光学模拟程序带来的算法运行速度慢的问题,对可能的走线模式建立了计算OPE费用所需的光强查找表格,使算法的运行速度大大提高.在实际的工业用例上的实验结果表明,本文所提出的详细布线算法使布线结果中的OPE问题得到很大程度的改善,有利于后处理过程中的光学邻近效应校正技术的运用,算法的运行时间是可以接受的.     

考虑光学邻近效应的详细布线算法

提出了一种考虑光学邻近效应的详细布线算法.该算法在布线过程中,充分考虑了线网走线相对位置及布线线形对其光学邻近效应的影响,通过相应的光刻模拟模型定义了用于估计光学邻近效应(optical proximity effect,OPE)的OPE费用函数,并采用OPE费用阈值控制Steiner树的生长方向和走线路径的选择,同时兼顾线网长度.为提高算法效率,避免布线过程中反复调用光学模拟程序带来的算法运行速度慢的问题,对可能的走线模式建立了计算OPE费用所需的光强查找表格,使算法的运行速度大大提高.在实际的工业用例上的实验结果表明,本文所提出的详细布线算法使布线结果中的OPE问题得到很大程度的改善,有利于后处理过程中的光学邻近效应校正技术的运用,算法的运行时间是可以接受的.