D.-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7 (1992), 121-135.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... implications, our result is important in that it provides new information on a universal constant (the largest ratio of the minimum degree 3 4 spanning tree weight to the MST weight) similar to the the Steiner ratio (the smallest ratio of the minimum Steiner tree weight to the MST weight) [6] and other constants studied in discrete geometry (such as [9] The new algorithms are not complicated and involve some interesting, cleverer recursive tree constructions. Their analyses, though, require more cases and demand techniques more versatile than those of Khuller et al. s; still, with ....

D.-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7:121--136, 1992.

....for such problems when the points come from a Euclidean, as opposed to arbitrary, metric space. This requires making use of more than just the triangle inequality. Surprisingly, for most problems, improved algorithms are not known. A notable exception is the famous Euclidean Steiner tree problem [5, 6]. We use rudimentary geometric techniques to obtain an improved algorithm for the Euclidean degree K spanning tree problem. The key to our method is to give short cutting steps that are provably better than implied by the triangle inequality alone. Lemma 3.3, which bounds the perimeter of an ....

D.-Z. Du and F. K. Hwang, A proof of the Gilbert-Pollak conjecture on the Steiner ratio, Algorithmica, 7 (1992), pp. 121--136.

....[fvg [ N T (v) We will use the weight of T to obtain an upper bound of the weight of S. To do so, we first establish a relation between w(T v ) and w(T v ) Because T v is a Steiner tree for the set of points corresponding to N T (v) we have w(T v ) 2 p 3 w(T v ) by the Steiner ratio [6]. Let I(T ) denote the set of internal vertices of T . Then w(S) w(T ) X v2I(T ) w(T v ) w(T ) 2 p 3 X v2I(T ) w(T v ) Since P v2I(T ) w(T v ) 2w(T ) we have MSTE(G) w(S) 4 p 3 p 3 w(T ) Combining with Proposition 4.1, we deduce MSTE(G) MS2T(G) 3 4 p ....

D.Z. Du and F.K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7:121--135, 1992.

....of weight L T , which can be solved very eciently. Many aspects of optimal Steiner trees have been considered; see the book [12] for an overview. One of the most famous problems related to geometric Steiner trees deals with the worst case value (T; ST ) of the ratio L T =L ST . As Du and Hwang [6] managed to prove for the case of planar point sets with Euclidean distances, T; ST ) 2= p 3. A special type of Steiner tree problems arises in the context of location theory: The so called Weber problem asks for the location of a single center point, such that the sum of distances from the ....

D.-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7 (1992), 121-135.

....kminMSTk, which can be solved very efficiently. Many aspects of optimal Steiner trees have been considered, see the book [11] for an overview. One of the most famous problems related to geometric Steiner trees deals with the largest possible value of the ratio kminMSTk kminStTk . As Du and Hwang [5] managed to prove for the case of planar point sets with Euclidean distances, this ratio cannot exceed the value of 2 p 3 , which is tight. A special type of Steiner tree problems arises in the context of location theory: The so called Weber problem asks for the location of a single center ....

D.-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7 (1992), 121--135.

....kMSTk, which can be solved very efficiently. Many aspects of optimal Steiner trees have been considered, see the book [11] for an overview. One of the most famous problems related to geometric Steiner trees deals with the largest possible value of the ratio kMSTk min kStTk . As Du and Hwang [5] managed to prove for the case of planar point sets with Euclidean distances, this ratio cannot exceed the value of 2= p 3, which is tight. A special type of Steiner tree problems arises in the context of location theory: The so called Weber problem asks for the location of a single center ....

D.-Z. Du and F. K. Hwang. A proof of the GilbertPollak conjecture on the Steiner ratio. Algorithmica, 7 (1992), 121--135.

....the length of the SMT to the length of a MST. It was already known that the length of an MST is an upper bound for the length of an SMT, but their conjecture stated that the length of an SMT would never be any shorter than p 3 2 times the length of an MST. This conjecture, was recently proved [15], and has led to the MST being the starting point for most of the heuristics that have been proposed in the last 20 years including a recent one that achieves some very good results [28] In 1961 Melzak developed the first known algorithm for calculating an SMT [41] Melzak s Algorithm was ....

D.Z. Du and F.H. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7(2/3):121--135, 1992.

....search basically. Let us introduce some strategic parameters in the algorithm first: ffl addnum: It is an initial guess of how many steiner points should be added. This value will be decremented during the search procedure. We used the Steiner ratio, p 3 2 in the Gilbert Pollak conjecture [2]. ffl threshold: It is the threshold of percentage improvement in cost, by adding one more steiner point, to decide whether to accept such a steiner point or not. The initial guess in the current implementation is 0:5 which is very sufficient in most cases. ffl threshold dec: It is the value ....

D.-Z. Du and F.K. Hwang, A Proof of the Gilbert-Pollak Conjecture on the Steiner Ratio, Algorithmica, 7:121-135, 1992.

....the length of the SMT to the length of a MST. It was already known that the length of an MST is an upper bound for the length of an SMT, but their conjecture stated that the length of an SMT would never be any shorter than p 3 2 times the length of an MST. This conjecture, was recently proved [13], and has led to the MST being the starting point for most of the heuristics that have been proposed in the last 20 years including a recent one that achieves some very good results [19] In 1961 Melzak developed the first known algorithm for calculating an SMT [29] Melzak s Algorithm was ....

D.Z. Du and F.H. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7(2/3):121--135, 1992.

....to be MAX SNP DocNumber 6 . 1. 7. Rao Smith typeset 793 May 23, 1998 Approximation schemes of k site SMTs with k = 2 r s, 0 s 2 r ; RSMT in the Euclidean plane is at least 2=3 times as long as the L 1 MST [23] SMT in the Euclidean plane is at least p 3=2 times as long as the MST [12]; and all these factors are tight. However in arbitrary metric spaces it is known that approximating SMT or TST to within 1 ffl is NP complete, for all sufficiently small constant ffl 0. See [1] for the references to this recently developed area of MAX SNP hardness. 2 Approximating the TSP ....

D. Z. Du and F. K. Hwang. A proof of the GilbertPollak conjecture on the Steiner ratio. Algorithmica, 7:121--135, 1992.

....approximation interconnection network is the minimum spanning tree (MST) in which no Steiner points are allowed. When the approximation is the MST, the performance ration is known as the Steiner ratio, denoted simply as ae. The following Steiner ratio results are known. In the L 2 norm ae = p 3=2[1], and in the L 1 norm, it is 2 3[9] In the L 2 norm, the edges connecting P are straight lines of artibrary orientation, while in the L 1 norm, they are restricted to be either horizontal or vertical. Let denote the number of possible orientations, and the unit disk D in this metric space, ....

....L smt and L mst is always greater than that of a equilateral triangle. Lemma 4. If is even, edge e(B; C) is straight. Otherwise, it is nonstraight and the two segments e(B; k) and e(C; k) are of equal length, and form an angle Gamma (Fig. 4) We give below the formulae for the ratios. [1] When = f6mjm 1; m 2 Ig: ae 3 = p 3 2 (same as the Euclidean Steiner ratio) 2] When = f6m 1jm 0; m 2 Ig: ae 3 = 3 cos( 2) sin( cos( 2) sin(fl) Gamma2 sin(fi) sin(ff) sin(fi) sin(fi) Gammasin(5fi) 4 sin(ff) where ff = 2m = fi = m = fl = 4m 1) In particular, ....

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D. Z. Du and F. K. Hwang, "A Proof of the Gilbert-Pollak Conjecture on the Steiner Ratio", Algorithmica Vol. 7, No. 2/3, 1992, pp. 121-135.

....for such problems when the points come from a Euclidean, as opposed to arbitrary, metric space. This requires making use of more than just the triangle inequality. Surprisingly, for most problems, improved algorithms are not known. A notable exception is the famous Euclidean Steiner tree problem [5, 6]. We use rudimentary geometric techniques to obtain an improved algorithm for the Euclidean degree K spanning tree problem. The key to our method is to give short cutting steps that are provably better than implied by the triangle inequality alone. Lemma 3.3, which bounds the perimeter of an ....

D-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7(2): 121--136, 1992.

....the length of the SMT to the length of a MST. It was already known that the length of an MST is an upper bound for the length of an SMT, but their conjecture stated that the length of an SMT would never be any shorter than p 3 2 times the length of an MST. This conjecture, was recently proved [8], and has led to the MST being the starting point for most of the heuristics that have been proposed in the last 20 years. This combination approach, starting with the C A X B P Figure 1: AP CP = PX. MST, will serve as the basis for the heuristic we present later in this paper. In 1961 Melzak ....

D.Z. Du and F.H. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7(2/3):121--135, 1992.

....polynomial time approximation factor for SMTs in arbitrary metric spaces is 1:644, due to Karpinski Zelikovsky in an unpublished manuscript. RSMT in the Euclidean plane is at most 3=2 times shorter than the L1 MST [15] SMT in the Euclidean plane is at most 2= p 3 times shorter than the MST [8]; and these factors are both tight. However in arbitrary metric spaces it is known that there exists ffl 0 so that approximating SMT or TST to within 1 ffl is NP hard. See [1] for the references to this recently developed area of MAX SNP hardness. 1.6 Organization of the paper In x2 we ....

D. Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7:121--135, 1992.

....ae 2 Gamma 2=n; again, ae is arbitrarily close to 2 if n is large. For geometric Steiner tree problems, ae is often smaller than 2. For the EST problem, ae = 2= p 3 (this fact was conjectured by Gilbert and Pollak in 1966 [65] and finally 2.2. Steiner Trees 13 proven in 1992 by Du and Hwang [45]) For the RST tree problem, Hwang [82] proved that ae = 3=2. 2.2.2 Rectilinear Steiner trees Most of the results in this dissertation concern the RST problem. The RST problem naturally arises in VLSI routing applications because typically VLSI fabrication technology requires all wires to be ....

D. Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7:121--135, 1992.

....Equation 13. For example, for this case with d = 3 and B 3 = 1=0:813052 1:23 under jj jj 2 , Algorithm 1 attains a performance ratio smaller than B 3 when t ( B 3 Gamma 1) 6 (4:35) 6 ; this bound is of special interest in light of Smith s [30] disproof of the Gilbert Pollak conjecture [17, 11] for 3 d 9. 5 Conclusion We have presented and probabilistically analyzed a deterministic partitioning approximation algorithm for the GSMT problem in R d , for any dimension d 2. Applying a theorem by Steele on subadditive Euclidean functionals, we proved the limit as n 1 of the ....

D.-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7(2/3):121--135, 1992.

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D.-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7 (1992), 121-135.

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D. Z. Du and F. K. Hwang. "A Proof of the Gilbert-Pollak Conjec ture on the Steiner Ratio". Algorithmica, 7:121--135, 1992.

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D.-Z. Du and F. K. Hwang. A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica, 7:121-136, 1992.

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D. Z. Du and F. K. Hwang, A Proof of the Gilbert-Pollak Conjecture on the Steiner Ratio, Algorithmica, 7 (1992) 121-135.

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D. Z. Du and F. K. Hwang. A proof of gilbert-pollak conjecture on steiner ratio. In FOCS, 1990. results on these problems are based on the rectilinear routing model. It would be interesting to study how octilinear routing model can impact the routability and routing resources. Conjecture: The octilinear Steiner ratio is

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