### Table 1: Overview of previous and new results related to the computation of minimum-distance spanning trees and minimum distance-approximating spanning trees.

"... In PAGE 4: ...n Section 5). Finally, we extend results by Gerey et al. [GLJ78] and Hassin amp; Tamir [HT95] concerning the minimization of the distance matrix DT under various matrix norms, integrating the previous findings into a unifying framework in Theorem 1. (See Table1 for an overview of previous and new results.) This work is structured as follows: Section 2 gives an overview of previous results related to this work and new results obtained, followed by some definitions and easy observations in Section 3.... ..."

### Table 1.8: The time complexity of some fundamental graph drawing problems: trees. We denote with k a xed constant such that k 1. Class of Graphs Problem Time Complexity Source tree draw as the Euclidean minimum spanning tree of a set of points in the plane

1997

Cited by 14

### TABLE 44.3.3 The time complexity of some fundamental graph drawing problems: trees. k is a xed constant k 1. CLASS OF GRAPHS PROBLEM TIME COMPLEXITY tree draw as the Euclidean minimum span- ning tree of a set of points in the plane NP-hard

### Table 1: Performance results for our implementations of Algorithms 1 and 2 for n = 25 and d = 2; 3. Costs c1, c2, and cS are costs of Steiner trees computed, respectively, by Algorithms 1 and 2 for various values of t and by Smith apos;s Algorithm. These costs are given as mean standard deviation, computed from 10 sets of 25 randomly chosen points in the cube [0; 100]d and measured by the Euclidean norm. Relative performance ratios ^ R1 = c1=cS and ^ R2 = c2=cS are computed with respect to Smith apos;s Algorithm. Values ^ 1 and ^ 2 are approximate constant factors for the big-Oh expressions for ^ R1 and ^ R2, respectively. For comparison, the costs of minimum-cost spanning trees on the same point sets are 350:07 24:80 and 618:21 48:34 for d = 2; 3, respectively.

"... In PAGE 5: ...2 Results Our experimental results are summarized in Tables 1, 2, and 3. Table1 shows the performance of our implementations of Algorithms 1 and 2 in comparison with that of Smith apos;s Algorithm for n = 25 and d = 2; 3. Table 2 lists the running times for these computations.... In PAGE 5: ... We arbitrarily chose the parameters n = 25 and n = 60 so that Smith apos;s Algorithm approximately converges within 30 and 200 seconds, respectively. In Table1 , we also give relative performance ratios ^ R1 = c1=cS and ^ R2 = c2=cS of Algorithms 1 and 2 relative to Smith apos;s Algorithm, where c1, c2, and cS are the costs of the (approximate) Steiner trees computed by Algorithm 1, Algorithm 2, and Smith apos;s Algorithm, respectively. Figure 1 shows sample Steiner trees computed by Algorithm 2 and by Smith apos;s Algorithm for n = 25, t = 7, and d = 2.... In PAGE 6: ...Table 2: Timing data for Algorithms 1 and 2 for runs described in Table1 for n = 25 and d = 2; 3. Times T1, T2, and TS are running times measured, respectively, from Algorithms 1 and 2 for various values of t and from Smith apos;s Algorithm.... In PAGE 7: ...0001. Interpretation of Results From Table1 we make the following observations for n = 25 and d = 2; 3. First, Algorithm 2 consis- tently outperformed Algorithm 1.... ..."

### Table 1. Performance Guarantees for finding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a fixed accuracy parameter.

"... In PAGE 3: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))6. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter-constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... In PAGE 5: ... 3.1 General Graphs Table1 contains the performance guarantees of our approximation algorithms for finding spanning trees, S, under different pairs of minimization objectives, A and B. For each problem cataloged in the table, two different costs are specified on the edges of the undirected graph: the first objective is computed using the first cost function and the second objective, using the second cost function.... In PAGE 5: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to finding Steiner trees with at most a constant factor worsening in the performance ratios. For the diagonal entries in the table the extension to Steiner trees follows from Theorem 6.... ..."

### Table 1. Performance Guarantees for finding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a fixed accuracy parameter.

"... In PAGE 3: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))6. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter-constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... In PAGE 5: ... 3.1 General Graphs Table1 contains the performance guarantees of our approximation algorithms for finding spanning trees, S, under different pairs of minimization objectives, A and B. For each problem cataloged in the table, two different costs are specified on the edges of the undirected graph: the first objective is computed using the first cost function and the second objective, using the second cost function.... In PAGE 5: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to finding Steiner trees with at most a constant factor worsening in the performance ratios. For the diagonal entries in the table the extension to Steiner trees follows from Theorem 6.... ..."

### Table 1: Steiner tree approximation algorithms

1999

"... In PAGE 1: ... But only few of them have prov- ably good performance ratios. Table1 gives a survey on such results. (The algorithm of Promel and Steger [11] is a randomized algorithm, all other algorithms listed in the table are deterministic algorithms.... In PAGE 2: ... This idea naturally extends to adding shortest connections between k-tuples of terminals, for xed k. All approximation algorithms for the Steiner tree prob- lem listed in Table1 are based on this simple idea. The present approach to get better performance ratios for the Steiner tree problem in graphs is to iteratively apply a series of algorithms to the output of its predecessor.... ..."

Cited by 21

### Table 1: Steiner tree approximation algorithms

1999

"... In PAGE 1: ... But only few of them have prov- ably good performance ratios. Table1 gives a survey on such results. (The algorithm of Promel and Steger [11] is a randomized algorithm, all other algorithms listed in the table are deterministic algorithms.... In PAGE 2: ... This idea naturally extends to adding shortest connections between k-tuples of terminals, for xed k. All approximation algorithms for the Steiner tree prob- lem listed in Table1 are based on this simple idea. The present approach to get better performance ratios for the Steiner tree problem in graphs is to iteratively apply a series of algorithms to the output of its predecessor.... ..."

Cited by 21

### Table 1: Expected Random Minimum Spanning Tree Weight: 500 Trials

"... In PAGE 2: ... Then: lim n!1 Efs(n)g = (3) = 1 X k=1 k ;3 =1:202 ::: And, for every gt;0, lim n!1 Pfjs(n) ; (3)j g =0: Although the theorem describes limiting behavior, students will nd that the minimum weight spanning tree is concentrated near (3) even for relatively small values of n. In Table1 we present the results from a sample simulation;; vehundred trials were performed for eachnumberofvertices. Even at n = 100, the expectation derived from the simulation is very close to the limiting value.... ..."

### Table 1. Performance Guarantees for nding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a xed accuracy parameter. The diagonal entries in the table follow as a corollary of a general result (Theorem 12.8) which is proved using a parametric search algorithm. The entry for (Degree, Degree, Spanning tree) follows by combining Theorem 12.8 with the O(log n)-approximation algorithm for the degree problem in [RM+93]. In [RM+93] they actually provide an O(logn)-approximation algorithm for the weighted degree problem. (The weighted degree of a subgraph is de ned as the maximum over all nodes of

"... In PAGE 3: ... Given the framework, it remains to reason and ll in the appropriate polynomial time subroutine that is applicable for the corresponding pair of objectives. Table1 contains the performance guarantees of our approximation algorithms for nding span- ning trees, S, under di erent pairs of minimization objectives, A and B. For each problem cataloged in the table, two di erent costs are speci ed on the edges of the undirected graph: the rst objective is computed using the rst cost function and the second objective, using the second cost function.... In PAGE 3: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to nding Steiner trees with at most a constant factor worsening in the performance ratios. All the results in the table extend to nding Steiner trees with at most a constant factor worsening in the performance ratios (Exercise!).... In PAGE 4: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))1. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter- constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... ..."