Faster Approximation for Rectilinear Bottleneck Steiner Tree Problem

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( College of Computer Science, South-Central University for Nationalities, Wuhan 430074, China )

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Faster Approximation for Rectilinear Bottleneck Steiner Tree Problem

LiZimao，LiXiaodan

( College of Computer Science, South-Central University for Nationalities, Wuhan 430074, China )

Bottleneck Steiner tree problem asks to find a Steiner tree fornterminals with at mostkSteiner points such that the length of the longest edge in the tree is minimized. The problem has applications in VLSI routing, wireless communication networks and phylogenetic tree reconstruction. Du and Wang showed that the rectilinear bottleneck Steiner problem is NP-hard and cannot be approximated within performance ratio 2 in polynomial time, and provided a 2-approximation algorithm running in actual time O(nlog2n+kn+k2). In this paper we improve the algorithm′s time complexity to O(nlog2n+klog2n) and armotized O(nlog2n+klog2n), by introducing the binary heap and Fibonacci heap respectively. The improvement can be directly applied to their Euclidean bottleneck Steiner tree problem′s 2-approximation algorithm.

bottleneck Steiner tree; approximation algorithm; performance ratio; wireless communication networks

1　Introduction

In the 1990s, along with the conquest of a series of famous conjectures, the traditional Steiner tree problem[1]attracted the scientists′ considerable attention and interests from both theoretical point of view and its applicability and once occupied a central place in the emerging theory of approximation algorithms.

Given a weighted graphG=(V,E;W) and a subsetS?Vof required vertices, the traditional Steiner tree problem asks a least weight tree spanningS. The tree may use additional points(called Steiner points) inV-S. We call such a tree a Steiner tree. The problem is MAX-SNP hard even when the edge weights are only 1 or 2[2]. For the Steiner tree problem in Euclidean plane, it is still NP-hard and there is a polynomial-time approximation scheme (PTAS) for Euclidean Steiner trees[3].

New applications of Steiner tree problem in VLSI routing[4], wireless communications[5]and phylogenetic tree reconstruction in biology[6]have been found and studied deeply. These applications triggered the study of variations of the traditional Steiner tree problem. Algorithms for the two variations, the bottleneck Steiner tree problem[7-12]and the Steiner tree problem with minimum number of Steiner points and bounded edge-length[9, 13-15], have been studied widely and deeply.

In this paper, we consider the bottleneck Steiner tree problem, which is defined as follows: given a setP={p1,p2,…,pn} ofnterminals and a positive integerk, we want to find a Steiner tree with at mostkSteiner points such that the length of the longest edges in the tree is minimized.

In this paper we consider the rectilinear bottleneck Steiner tree problem. D.-Z Du and L. Wang proved that the problem could not be approximated within ratio 2 in polynomial time and provided a 2-approximation algorithm which runs in O(nlog2n+kn+k2) time but not they mentioned O(nlog2n+klog2n)[7]. The performance ratio is best possible and any improvement of the ratio will lead to P=NP. By introducing two advanced data structures, the binary heap and the Fibonacci heap, we can implement their algorithm′s time complexity to O(nlog2n+klog2n) and amortized O(nlog2n+klog2n), respectively.

2　 2-Approximation algorithm

D.-Z Du and L. Wang proved the existence of performance ratio 2 by constructing a steinerized spanning tree under the triangle inequality property in rectilinear plane. Their algorithm first construct a minimum spanning tree for the set ofnterminals inP, then they repeatedly add degree-2 Steiner point to long edges in the minimum spanning tree. We call such a tree a steinerized spanning tree. Their approximation algorithm was derived from the following two lemmas.

Lemma 1[7]: Given a set ofnterminalsPin the rectilinear plane, letTbe an optimal tree for the rectilinear bottleneck Steiner tree problem. Then there exists a steinerized spanning treeT′ forPwith the same number of Steiner points asTsuch that the length of the longest edges inT′ is at most twice that ofT.

It follows immediately from Lemma 1 and 2 that when we use the same number of Steiner points to steinerize a spanning tree and a minimum spanning tree, the result from the latter has a longest edge of length not exceeding that from the former. That is, an optimal steinerized spanning tree can be found among steinerized minimum spanning trees. Since only degree-2 Steiner points are possibly adjacent, we only need to addkSteiner points to a minimum spanning tree in order to obtain an optimal steinerized spanning tree.

The idea is explained as follows: for each edgeei= (u,v) in the minimum spanning tree, if we addlidegree-2 Steiner points to it, then the length of the longest edge in the resulting path fromutovhas the minimum valuec(ei)/(li+1), wherec(ei) is the original length of edgeei. This minimum value is achieved when theliSteiner points divideeievenly. Denotel(ei) =c(ei)/(li+1).

At the beginning of the algorithm,l(ei)=c(ei). Each time a degree-2 Steiner point is added to the edgeeiwith the largestl(·) value. Aftereireceives one more degree-2 Steiner point,liis updated byli=li+1 andl(ei) is updated byc(ei)/(li+1) and the position of all the degree-2 Steiner points in the edgeeiis re-organized by dividingeievenly, Note thateiis defined in the rectilinear plane. The process is repeated untilkdegree-2 Steiner points are added.

Fig.1 shows D.-Z Du and L. Wang′s approximation algorithm with performance ratio 2[7].

Fig.1Du and Wang′s 2-approximation Algorithm for Rectilinear Bottleneck Steiner Tree Problem

圖1Du和Wang的2-近似性能比網(wǎng)格空間瓶頸斯坦納樹算法

The algorithm′s time complexity is analyzed as below:

The first step can be implemented in O(nlog2n) time[18,19].

Step 2 uses linear time. Sorting in Step 3 takes O(nlog2n) time.

In each loop of Step 4-7, Step 4 and 5 use constant time to find the longest edge and updatel(·), Step 6 uses time linear to the number of Steiner points on that edge, and the step 7 of resettingei′s position needsO(n) time in the worst case.

As Step 4-7 only loopsktimes, the algorithm′s time complexity is O(nlog2n+kn+k2).

3　The faster algorithms

The most time consuming steps in the loop of the Du and Wang′s algorithm is Step 6 and 7, either linear to number of Steiner points or to the number of terminals in the worst case. First we find that moving step 6 in Fig.1 out of the loop as the final step can decrease the time of organization of Steiner points fromO(k2) toO(k) Then the step to find an edge with the largestl(·) and step to updatel(·) are frequently executed, together with Step 5 and 7 combined as a single step, which inspires us to use a priority queue to maintain all the edges associated with priorityl(·). The priority queue should support two operations efficiently: finding an edge with the largest priority and update an edge′s priority.

According to our case, a binary max-heap[20]is suitable to implement the priority queue. A binary max-heap is a heap data structure created using a binary tree with two additional constraints: (1) shape property, the tree is a complete binary tree; that is, all levels of the tree, except possibly the last one are fully filled, and, if the last level of the tree is not complete, the nodes of that level are filled from left to right; (2) heap property, the key at each node is greater than or equal to that of its children.

A max-heap supports the operations of a priority queue efficiently. We can construct a heap in linear time, and a max-heap returns a node with the largest key inO(1) time, and updates a node key in O(log2n) time. In fact, the introduce of max-heap also decreases the Step 7′s implementation time fromO(n) to O(log2n).

Now we can formulate our improved algorithms as below (See Fig.2).

It is easy to check that the time complexity of the above algorithm is O(nlog2n+klog2n). Obviously Step 2 only uses time linear ton. Constructing a max-heap in bottom-up fashion need onlyO(n) time. Step 4 runs in constant time because root of the heap indicating the edge with the largestl(·), while Step 5 uses O(log2n) to update an edge′s key, consider that the two steps loops forktimes, so Step 4 and Step 5 run in O(klog2n) in total. Step 7 can be implemented inO(n+k) time locating the position of added Steiner points.

Fig.2Faster algorithm for Rectilinear Bottleneck Steiner Tree Problem

圖2網(wǎng)格空間瓶頸斯坦納樹快速算法

Remember that the first step runs in O(nlog2n), the improved algorithm′s time complexity is O(nlog2n+klog2n).

Theorem 1: There is an O(nlog2n+klog2n) time approximation algorithm with performance ratio 2 for the bottleneck Steiner tree problem in the rectilinear plane.

If we use a Fibonacci heap[20]to implement the priority queue, the algorithm can be implemented in amortized time O(nlog2n+klog2n). This is because heap construction takes onlyO(n) amortized time, while determining the edge with largest key and decreasing an edge′s key uses onlyO(1) and O(log2n) amortized time.

Simulation on the proposed algorithms shows that the advantages of our algorithm become more and more clear with the increasing number of Steiner points, and the Fibonacci heap-based implementation performs better than the binary heap-based when the number of terminals and Steiner points is big enough(See Fig.3～5).

Fig.3　Comparison of Implementation Time with Steiner Points the 50圖3　斯坦納點數(shù)目為50的實驗結(jié)果比較

Fig.4　Comparison of Implementation Time with Steiner Points the 500圖4 斯坦納點數(shù)目為500的實驗結(jié)果比較

Fig.5　Comparison of Implementation Time with Steiner Points the 3000圖5 斯坦納點數(shù)目為3000的實驗結(jié)果比較

4　Conclusion

We mainly considered the rectilinear bottleneck Steiner tree problem. The problem asks to find a Steiner tree withnfixed terminal nodes in the rectilinear plane and up tokSteiner nodes such that the length of the longest edge in the tree is minimized. We first introduced D.-Z Du and L. Wang′s approximation algorithm with performance ratio 2. Then by introducing binary heap and Fibonacci heap, together with a slightly adjustment of their algorithm, we improve their algorithm′s time complexity to O(nlog2n+klog2n) and amortized O(nlog2n+klog2n), respectively. Simulations are conducted to indicate the effectiveness and efficiency of our improved implementation and the Fibonacci-heap-based algorithm′s complexity makes such an improvement primarily of theoretical value.

An observation is that our improvements can be directly applied to Du and Wang′s polynomial approximation algorithm with performance ratio 2 for the Euclidean bottleneck Steiner tree problem.

As an application, the algorithm can be used to improve the lifetime of wireless networks by minimizing the length of the longest edge in the interconnecting tree by deploying additional relay nodes at specific locations.

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2016-03-22

李子茂(1974-),男, 副教授, 博士, 研究方向: 算法設計與分析、計算復雜性, E-mail：3030207@mail.scuec.edu.cn

國家自然科學基金資助項目(61103248；61379059)

TP312

A

1672-4321(2016)03-0097-05

網(wǎng)格空間瓶頸斯坦納樹問題快速近似

李子茂，李曉丹

( 中南民族大學 計算機科學學院，武漢430074)

指出了瓶頸斯坦納樹問題要求尋找一棵用至多k個斯坦納點將n個點連接起來使得此斯坦納樹之最長邊最短的斯坦納樹,該問題在VLSI、無線通訊網(wǎng)絡和生命演化樹重建等領域都有應用.Du和Wang證明網(wǎng)格空間瓶頸斯坦納樹問題是NP-Hard，不存在近似性能比低于2的多項式時間解決方案，并且提出一個近似性能比為2的多項式時間近似算法，算法的實際時間復雜度為O(nlog2n+kn+k2).通過引入二叉堆和斐波那契堆使算法的時間復雜度分別改進到了O(nlog2n+klog2n)和攤還時間O(nlog2n+klog2n).該改進可直接應用于歐幾里得平面的瓶頸斯坦納樹2-近似算法.

瓶頸斯坦納樹；近似算法；性能比；無線通訊網(wǎng)絡