Rectilinear Steiner Minimum Tree Problem

Published Date: 02 Nov 2017

Abstract

Rectilinear Steiner Minimum Tree problem is to find a minimum length tree connecting the given set of points using only horizontal and vertical line segments, with the additional set of points (Steiner points) which are introduced to reduce the total length of the tree and for connecting in rectilinear manner. This paper does a survey of various Rectilinear Steiner Minimum Tree algorithms proposed till date. Rectilinear Steiner Minimum Tree is widely used in global routing phase of VLSI design and wire length estimation.

Introduction

The main goal of routing in VLSI design is to interconnect the cells that have been assigned positions as a solution of the placement problem. Routing is generally performed in two different stages. The first stage, called the global routing will identify the wiring channels through which connections can run. The second stage, called detailed routing, fixes the exact paths that the wire has to take. Rectilinear Steiner Minimum Tree (RSMT) is used in the global routing phase of VLSI Design [6]. Given a set of points P(terminal points), Rectilinear Steiner Minimum Tree is the minimum length tree constructed over {P}U{S} using only horizontal and vertical line segments, where S is the set of additional points called Steiner Points (fig 1). The distance between two points is measured in the rectilinear metric. The rectilinear distance between a pair of points pi= (xi, yi) and pj=(xj, yj ) is equal to |xi - xj | + | yi - yj|.

P3Hanan[1] was the first to consider the rectilinear version of Steiner Trees. He has given exact solutions for n≤5 and also showed that the possible candidate Steiner points (CSp) lie on Hanan Grid (fig 2). Garey and Johnson [2] showed that the problem of constructing RSMT is NP-Complete and Hwang [3] proved that the ratio of cost of Rectilinear Minimum Spanning Tree (RMST) to the cost of Rectilinear Steiner Minimum Tree (RSMT) is ≤ 3/2. Many Approximation and exact algorithms were proposed for finding RSMT. The below example (fig 1) shows a RSMT for connecting pins p1,p2, p3, p4 and p5 which uses the Steiner points s1,s2 and s3.

P1

S1

S3

S2

P4

P2

P5

Fig. 1 Rectilinear Steiner Minimum Tree

Terminologies

Minimum Spanning Tree

Given a set of points P, Minimum Spanning Tree (MST) is the minimum length tree over P. Fig 2 shows the MST over point’s p1 – p6.

Rectilinear Minimum Spanning Tree

Rectilinear Minimum Spanning Tree (RMST) is a MST where the distance between two points is measured in rectilinear metric and the edges are restricted to be of horizontal and vertical line segments.

Hanan Grid

The Hanan grid [1] is the grid induced by the set of points P by drawing horizontal and vertical lines through each and every point in P (Fig.2). The Rectilinear Steiner Minimum Tree (RSMT) is embedded in this grid.

Hanan points

The points obtained at the intersection of horizontal and vertical lines drawn through the given set of points P are called as Hanan points [1]. For a given point set P with n points, there are O(n2) Hanan points [9]. Hanan points are the Candidate Steiner points (CSp).

L-Shaped Layout

For each non-degenerate edge (an edge is non-degenerate if the endpoints of the edge do not lie on the same horizontal or vertical line) , two L-shaped layouts can be formed from the enclosing rectangle. In Fig.1 the two L-Shaped layouts for the edge e(P5, P6) are indicated using dotted lines.

P4

P2

P1

P5

P3

P6

Fig. 2 Hanan grid and the Minimum Spanning Tree

Algorithms

This section is organized as per the existing concepts behind the algorithm used in construction of RSMT. The algorithms are broadly classified as approximation and exact algorithms. Under this classification the different variants/approaches are grouped together and further expand on the research activity related to the construction of RSMT.

Approximation Algorithms:-

Spanning Tree embedded algorithms:-

Spanning tree embedded algorithms are based on the concept of first constructing a MST over the given set of points P and then applying a particular strategy to generate RSMT from it. Most of the approximation algorithms for computing RSMT are based on this concept and are explained as follows:-

Ho, Vijayan and Wong [4] proposed a linear time algorithm (L-RST and Z-RST) and O(n) for finding RSMT. To construct a RSMT, the edges of the separable MST (the tree is separable if the enclosing rectangle layouts of non-adjacent edges do not overlap) are replaced by the L or Z-shaped layouts which have maximum overlap. The algorithm takes MST as an input and produces RSMT as an output. The time to compute the MST is not considered while finding time to construct RSMT. Khang and Robins [13] have commented on the misstatement in this work and that the optimum solution does not satisfy the condition of "the ratio of length of Rectilinear Minimum Spanning Tree (RMST) to the length of Rectilinear Steiner Minimum Tree is ≤ 3/2[3]".

Hassan, Vijayan and Wong [5] proposed a O(nlogn) algorithm, where the neighborhood structure (the sub tree inside the dotted triangle is the neighborhood structure of point P1 in fig.3) of the maximal independent set of the constructed MST points are replaced by their corresponding optimal RSTs. The condition for independency is that the neighborhood structure of the points should not have edges in common.

P1

P1

Fig. 3 Neighborhood structure of point P1 and its RSMT

Chao and Hsu [15] proposed a O(n2logn) algorithm which incrementally adds Steiner points S to the point set P and generated the MST over {P}U{S}. The Steiner points are added in two stages. During the first stage - local refinement, two types of local Steiner points (lSp – median of three points) are identified and included to the given point set by looking at the local structure of the tree. The essential leaf lSp is the unique lSp formed by a leaf node or point, its parent and the neighbors of the parent and if only one lSp is generated by the adjacent edges of the internal node or point, then lSp is called essential internal lSp. In fig 4, S4 is the essential lSp for leaf P6 and there is no essential leaf lSp for P5. For the internal node P4, S4 is the essential internal lSp and there is no essential internal lSp for P3. During the second stage -global refinement, the point or the corner point of an edge( Candidate Steiner point –CSp) is connected to CSp of the another edge of the tree that has the shortest path as compared to CSp’s of the other edges and the longest edge is removed in the formed loop i.e. the Steiner points are introduced by looking at the global structure of the tree.

P1

P2

S1

P3

S3

S2

P4

S4

P5

S6

S5

P6

Fig. 4 MST over P and the candidate Steiner Points(CSp)

Khang and Robins [13] showed that all MST based algorithms have a worst case performance ratio (length of MRST to the length of RSMT) of exactly 3/2. So, alternative RSMT approximation algorithms have been developed. Khang and Robins [8] presented a O(n3) Iterated 1-Steiner (I1S) algorithm which iteratively adds a Steiner point x and constructs a RMST over P U{x} until no more Steiner points which reduce the length can be found. Instead of adding a Steiner point iteratively, a batched variant (Batched Iterated 1-Steiner – BI1S) O(n4logn) algorithm adds a maximal independent set of Steiner points. The condition for independency is that a candidate Steiner point is not allowed to reduce the MST cost saving of another Steiner point.

Borah, Owens and Irwin [16] proposed a O(n2) edge-based heuristic algorithm for RSMT construction, which starts by computing a MST, incrementally improves it by connecting a point to the enclosing rectangular layout of neighboring visible edge and removes the longest edge in the formed loop. The time complexity was reduced to O(nlogn), but with complex programming strategies and data structures. In fig. 5 point P1 of the MST is connected to the enclosing rectangle layout of the visible edge e1 and the edge e2 is removed, as it is the longest edge in the formed loop.

P1

e1

e1

P1

e2

Fig. 5 Edge based heuristic

Griffith, Jeff, et al [18] showed that the direct implementation of I1S requires O(n4logn) time and that I1S implemented with the time complexity of O(n2logn) [8] used the computational geometry with hidden large constant factor. So, they proposed an algorithm for the implementation of BI1S based on dynamic MST update method with runtime of O(n3) per round. In dynamic MST update scheme, a new point is connected to each of the neighboring points in eight octants and the longest edge in the formed loop is removed. The parallel implementation using multiple processors was also done in near-optimal time.

Areibi, Xie and Vannelli [30] used BI1S [8] as a basis. The main contribution was an algorithm for K-Rectilinear Steiner Tree (K-RST) where a set of Rectilinear Steiner Trees are generated for the given set of points by adding k Steiner points at a time. The gain of adding a Steiner point is computed by connecting it to the points in 4 quadrants and eliminating the cycles thus formed (dynamic update of spanning tree). Only Steiner points with positive gain are added iteratively.

Khang, Mandoiu and Zelikovsky [25] proposed a O(nlog2n) batched version of greedy triple contraction algorithm [33] - Batched Greedy Algorithm [BGA], where a batch of triples (triple is the optimal full Steiner tree for a set of three points where all the points are leaves) are added in each iteration to construct RSMT. Gain of adding a triple is computed by adding two edges for connecting the three points of a triple (addition of triple) and removing the longest edges in the formed loop (contraction). All the triples with positive gain are identified and added in a non-decreasing order of gain immediately followed by contraction. The Steiner points thus formed by the triple are added to the set of points P and MST over P is computed iteratively.

Zhou [17] proposed a O(nlogn) Rectilinear Spanning graph(RSG) algorithm based on Borah et al edge based heuristic [16] on the Zhou at al spanning graph algorithm [32] . A spanning graph is a graph computed over given set of points which contains an embedded MST and is constructed by partitioning each point’s plane into eight octal regions thereby connecting this point to the closest point in each region. Further a spanning tree is formed based on the cycle property by removing the longest edge in the formed loop. Edge based heuristic algorithm applied on the Spanning Tree thus formed generates RSMT.

Cinel and Bazlamacci [31] proposed a modified and distributed RSMT algorithm based on the idea of Khang et al Batched Greedy Algorithm(BGA) algorithm[25] and Zhou’s Rectilinear Spanning graph(RSG) algorithm[17]. The modified algorithm uses RSG approach to generate initial sparse graph and to update the MST. BGA approach is used for longest edge computation. Distributed version is based on parallelization of major components of the modified RSMT.

Two point and three point connection strategy based algorithms:-

Two points connection strategy is based on an idea of connecting a new point to the constructed current tree by the L-shaped layout iteratively. In three point connection strategy, a new point is connected to the two points (terminal point or the Steiner point) of the current tree using optimal RSMT. Algorithms for computing RSMT based on these strategies are as follows:-

Lee, Bose and Hwang’s [14] proposed a O(n3) single-net algorithm that uses three point connection strategy and Hwang’s [3] exact solution for three points to construct the RSMT. This algorithm starts by computing an optimal RSMT for every group of three elements and selects the one with minimum length. It proceeds by iteratively adding a new terminal point {x} that has the shortest distance to one of terminal point {y} in the current tree by an optimal RSMT of three points (x,y,z) where z is a Steiner or terminal point adjacent to y with shortest length. It was also shown that the RSMT constructed using this algorithm can actually be constructed in O(n2) operations.

Hwang’s[9] O(nlogn) suboptimal algorithm labels the given set of points first by applying a labeling algorithm (labeling helps in selecting the points in the order they are added to the MST), computes RSMT of first three points and then applies three point connection strategy such that the remaining points are iteratively added to form a RSMT.

A O(n2) algorithm by Katsadas and Kinnen [29] provided an appropriate solution to the RSMT problem. This algorithm uses the concept similar to the one in Prim’s algorithm except that the point selected to be added to the current tree has shortest distance not only from terminal points in current tree but from the set of terminal points and Steiner points included in current tree. Thus the algorithm iteratively adds a point that has a shortest distance to one of the point (Steiner or terminal point) in the current constructed tree using one of the L-shaped layouts which is closer to the unconnected points.

Souza, Carvalho and Ribeiro [28] devised a O(n2) algorithm (H2PC) based on two point connection strategy. The algorithm starts by first connecting the identified two nearest points by a straight line if both the points lie on a same horizontal or vertical line else by one of the two L-shaped layout using path selection criterion based on the delta value. The delta value is the difference in lengths of the two solutions obtained by connecting the two points using two paths (upper L-shape layout and lower L-shaped layout) and followed by the connecting all the points in the eight octants to this path. Similarly all the points are connected to the closest point in the current tree iteratively. Further an improved approximation algorithm (HAS) was proposed which uses H2PC as a building block. H2PC is successively applied by including the Steiner points obtained by the previous application of H2PC to the points set P.

Look-up table based algorithms:-

Look-up table based algorithms compute the solution for the smaller value of n and stores it in the table. For larger value of n, the problem is recursively divided until the value of n is such that the table can be applied. Look-up table based RSMT algorithms are as follows:-

Chu and Wong [35] proposed a fast look-up table based O(nlogn) algorithm(FLUTE) for RSMT construction. Tables for computing RSMT and their length are pre-computed for n≤ 9. For n>9, optimal net breaking algorithm is used to break the net recursively until n≤ 9. An accuracy parameter is introduced to indicate the level of accuracy the problem is handled with and its value indicates the different ways of net breaking that are tried. They have compared their algorithm with the various existing algorithms ( RST-T[44], BGA[25], SPAN[17] and BI1S[8]) and showed that the runtime for FLUTE was the least compared to those algorithms for all value of n (refer table VII in [35]) and that FLUTE produced near-optimal solution similar to BI1S for n<30(refer table V in [35]). Increasing accuracy parameter value produced better solution but with increased runtime.

To overcome the problem of FLUTE, Wong, Yiu-Chung and Chu [36] provided an improvement (FLUTE 3.0) by modifying the net breaking algorithm, which efficiently partitions a net into smaller subnets that can be efficiently handled by original FLUTE algorithm. Experimental results showed that the quality of FLUTE 3.0 improved with accuracy parameter value and produced better solution than BI1S but with increased runtime.

Genetic algorithms:-

Genetic algorithms are based on the concept of natural evolution and are a kind of evolutionary algorithm which solves the optimization problems by relating the problem to natural evolution and applies similar techniques such as inheritance, generation, mutation, crossover etc. Genetic algorithms for RSMT construction are as follows:-

Julstrom[41] proposed a genetic algorithm that converts the list of spanning tree edges into RSMT. This algorithm searches the product space of the space of the MST on given set of points with the space of the choice of Steiner points for each and every MST edge to identify the RSMT and since search space grows with n value, the performance deteriorates for large problem instances.

To improve the performance, Julstrom[42] reduced the search space by first constructing RMST for the given set of points and then searched only the space of Steiner point choices for the each of the spanning tree edges. The candidate RSMT are encoded as binary strings and the two genetic operators ( two-point crossover and position-by-position mutation) are applied.

Probabilistic algorithms:-

Probabilistic algorithms are the randomized algorithms where the result obtained depends on the chance and the problem space is searched based on probabilistic model of candidate solutions to the problem. Probabilistic based RSMT algorithms are as follows:-

Komlos and Shing’s [37] O(f(t)n + nlogn) probabilistic partitioning algorithm recursively partitions a given set of n points in unit square into 4k small rectangles where k is [log4 n/t] and t>0. The algorithm proceeds by recursively partitioning the rectangle into two sub-rectangles with a vertical line segment based on the median x co-ordinate. Further each of the sub-rectangles is partitioned with a horizontal line segment based on the median y co-ordinate. For each of the partition the dynamic algorithm of Dreyfus and Wagner [40] (the algorithm considers all subsets of given set of points P in order of increasing size and for each subset applies the following decomposition theorem - the optimal Steiner tree for a set of points P can be decomposed into three subsets R, S and {v} such that for some u Ɇ P , the optimal Steiner tree of R U {u} and S U {u} and the shortest distance between u and v is the optimal Steiner tree for P ) is applied. The second algorithm has the reduced time complexity of O(g(t)n) by avoiding the O(nlogn) time spent in partitioning the given set of points.

Bern [38] proposed an algorithm similar to Kruskal’s MST algorithm with following results i) Length of the generated RSMT is 29% shorter than MST. ii) At least .039n Steiner points are added in the optimal RSMT.

Hwang and Yao [39] applied the Bern’s probabilistic result [39] on Exactly-N model with following results i) The length of generated RSMT is at least 0.0014n shorter than MST for large n. ii) At least .041n Steiner points are added in the optimal RSMT.

Chen, Zhao and Ahmadi [43] presented an algorithm based on probabilistic model. The probability of the patterns passing over individual edges is computed and only the edges that have high probability are selected.

Other algorithms:-

Chung and Hwang [51] computed the length of RSMT with respect to the perimeter of the rectangle enclosing the given points for n≤ 10.

A heuristic presented by Beasley [11] considered all the connected sub graphs of MST which contains four vertices, finding the optimal Steiner trees for each of the sub graph.

Lin, Cheng and Wu [47] presented a new formulation of RST known as Performance Oriented RST which proceeds by identifying the edges and the direction for the edges to grow from source to destination.

Chen, Quio, Zhou and Cheng [44] proposed a more stable O(nlogn) Refined Single Trunk Tree (RST-T) algorithm which is an improvement of a Steiner heuristic called Single Trunk Steiner Tree [48]. The points are either connected to the trunk (a line which goes horizontally or vertically through the median of the points) or the stem (edges that connects a point to the trunk), whichever is shorter (fig. 6). The RMST generated from this algorithm is very much far away from the optimal for large value of n.

Stem

Trunk

Fig. 6 Refined Single Trunk Tree

Elaheh, Kastner, and Sarrafzadeh [46] looked at the Steiner tree in a new perspective: flexibility. The algorithm takes a stable Steiner tree as an input and maps it into a more flexible Steiner tree which helps routing during congestion.

Hu et al [45] presented a O(n2) algorithm based on ant colony optimization (ants are placed on the points and one ant is allowed to move leaving behind a trail for others to follow.

Wang et al [52] proposed a O(n2logn) algorithm which first constructs a Bounded degree Neighborhood Graph (BNG)- a sub graph of Delaunay Triangulation(fig.7), RMST over it and then applies polygon contraction to obtain RSMT.

Fig. 7 Delaunay Triangulation

Comparison of approximation algorithms:-

Spanning tree embedded algorithms are a good choice. L-RST algorithm’s input has to a separable MST and hence produces RSMT with less or no improvement over MST for some problem instances. Algorithms based on neighborhood structure and local and global refinements are far away from the optimal. BI1S is the champion with 11% improvement over the MST and only 0.5% away from the optimal. BGA produces similar output with less solution quality and runtime compared to BI1S. Edge-based heuristic gives solution similar to BI1S with better runtime but does scale well for large problem instances. RSG is based on edge-based heuristic with improved performance and runtime. Two point and three point connection strategy based algorithms generate RSMT with solution quality degrading for increasing n value. Probabilistic and genetic algorithms have high time complexity and hence not suitable for larger n value problem instances. Look-up table based algorithms have an accuracy parameter that allows the user to control the accuracy and the runtime. RST-T has the lowest runtime but with solution quality away from optimal for increasing n value.

Exact Algorithms:-

Exact algorithms are based on the Full Steiner Tree construction [53]. RSMT is called a Full Steiner Tree (FST) if all points are the leaves of the Tree. At every point with more than one degree, the tree can be split into edge-disjoint Full Steiner Trees that have in common only the split points. Those trees are called full components. Hwang [3] showed that the topologies of full components have only two restricted structure (fig 4) and every RSMT consisted of number of full components that intersect at points of degree 2 or more. The exact algorithms (Geo-Steiner[12, 54]) for RSMT works in two stages. In first stage, candidate full components are identified. In the second stage, subset of the identified full components whose union is a RSMT are selected and concatenated. The existing exact algorithms for RSMT construction are based on three approaches: backtrack searching, dynamic programming and integer programming.

Fig. 4 Two topologies of full component

Salowe and Warne [10] proposed an algorithm which finds the candidate full components that can be a part of RSMT by applying tests (coarse tests and set-specific tests), to judge whether the point set can be accepted or rejected as candidate full components and then backtrack search is used to obtain the subset of the full components from which FST can be extracted. Backtrack search starts with a single full component and recursively adds full components to the current solution until FST over a given set of points is generated. recursively adds other full components to the solution until FST over given point set P is produced. If adding a full component results in a cycle, then the search backtracks and adds some other full component.

The first dynamic programming algorithm for Full Steiner Tree construction was proposed by Ganley and Cohoon [23] with time complexity of O (n3n+ n2n) based on following idea : for a subset of two points, optimal RSMT was an edge between them and for subset of more than two points, the algorithm compares the length of the tree obtained by applying Hwang’s theorem [3] with the length of the tree formed by joining the optimal RSMT of every disjoint subset at every point and selected the one with minimum length for decomposition. The subset of point sets are enumerated such that optimal RSMT for every smaller subsets have already been computed.

Ganley and Cohoon[26] further gave two dynamic programming algorithms. The first is Full-set Dynamic Programming(FDP) algorithm, which was based on the idea that optimal RSMT for a set of points P is either a FST and hence can be computed using Hwang’s theorem[3] or else is composed of two smaller RSMT’s A and B such that |AUB|=|P| and |A∩B|=1. The algorithm enumerates all possible subsets of P in order of increased cardinality with time complexity of O(n3n). The second algorithm- Screened Full-set Dynamic programming(SFDP) with time complexity O(n22.62n) was based on the idea of full-set screening, where number of tests were conducted to reduce the number of full components that could be considered as candidate full components.

Fößmeier and Kaufmann [19] also used dynamic programming to combine the smaller number of full components and reduced the runtime to O(n2.38n) as compared to the former bound to O(n22.62n) by Ganley and Cohoon [26]. It was shown that for the given set of points P, there can be a RSMT containing only tree stars as full components and hence the number of full components to be considered was reduced. In a tree star the triangles defined by 45o diagonals on both sides of each segment do not contain any other point of the given point set P.

Warme [27] used a branch and cut algorithm based on integer programming. This method was much faster than backtrack search or dynamic programming.

Comparison of exact algorithms:-

Exact algorithm based on backtrack search has a very steep runtime growth and hence are suitable only medium sized problem instances. Dynamic programming based exact algorithms produced better worst case bound but with a huge memory requirement. The exact algorithm based on integer programming is faster than the dynamic programming or backtrack search.

Sl. No

Algorithm

Underlying Concept

Time Complexity

Advantages

Disadvantages

1

Batched Iterated Steiner Tree (BI1S)

Approximation algorithm, Spanning Tree embedded algorithm

O(n4logn)

Produces near-optimal solution

High runtime

2

Batched Greedy Algorithm(BGA)

Approximation algorithm, Spanning Tree embedded algorithm

O(nlog2n)

Produces near-optimal solution, good runtime

Solution quality inferior to BI1S.

3

Refined Single Trunk (RST-T) algorithm

Approximation algorithm, Heuristic algorithm

O(nlogn)

Fast (good runtime)

Solution is far away from optimal for larger n

4

Rectilinear Spanning Graph Algorithm (RSG)

Approximation algorithm, Spanning Tree embedded algorithm

O(nlogn)

Fast (good runtime)

Solution is not optimal.

6

Edge-based heuristic

Approximation algorithm, Spanning Tree embedded algorithm

O(n2)

Produces near-optimal solution

Does not scale well.

5

Fast Look-Up Table based algorithm(FLUTE)

Approximation algorithm, Look-up table based algorithm.

O(nlogn)

Fast, near-optimal solution for n<30

Solution is far away from optimal for n>30

6

GeoSteiner

Exact algorithm

Optimal solution

High runtime

7

FLUTE-3.0

Approximation algorithm, Look-up table based algorithm

O(nlog2n)

Fast and accurate

Increasing accuracy parameter value results in good solution but with increased runtime

Table 1: Comparison of important algorithms for construction of RSMT

Conclusion

This paper is a brief study of the existing algorithms for computing the Rectilinear Steiner Minimum Trees. Table 1 gives the comparison of the existing important algorithms for the construction of RSMT. The selection of an algorithm is based on various parameters, the important ones being good solution quality and runtime. If good solution quality (reduced wire length) is the main aim GeoSteiner[12,54], BI1S[8], BGA[25], edge-based heuristic and RSG[17] are the good choice and if runtime is the main concern, then RST-T[44] and FLUTE[35] are the best options. The FLUTE 3.0[36] gives a fast near-optimal solution requiring large memory to store the pre-computed tables and the accuracy parameter value used is inversely proportional to solution quality (wire length) and directly proportional to runtime. Hence, the decision regarding the selection of the algorithm for RSMT is a tradeoff between efficiency and runtime.