"... In PAGE 3: ... Several approximation algorithms have been proposed for the Unweighted and Weighted Versions of the Vertex Cover Problem. We list some of them in Table1 . We decided to implement these algorithms because of their simplicity and performance guarantees.... ..."

"... In PAGE 5: ... The RKCP3 algorithm from the 3-approximation algorithm in [12] is not ef cient because it only uses part of available disks. We can see from Table1 that when the system has 270 points, 4 disks with radius 720 meters (180 meters * 3) is probably enough to cover them all. On the contrary, the 2- approximation algorithm always tries to cover points by all disks.... In PAGE 5: ... On the contrary, the 2- approximation algorithm always tries to cover points by all disks. Under a few circumstance, such as less number of points in the area and fewer disks available, the RKCP3 al- gorithm yields a better performance than the RKCP2 algo- rithm, which can be seen from Table1 for the system with 50 points. Most of the time, the greedy algorithm performs the best while the 3-approximation algorithms generate the worst results.... ..."

"... 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.... ..."

"... 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.... ..."

"... In PAGE 19: ... We have described the great variety of problems in this class in the introduction. The main results of this paper are summarized in Table1 . As the table shows many challenging questions remain open.... ..."

"... In PAGE 11: ... Our expe- riments show that this might work in some particular cases, as in the following conjecture veri ed for n 10. Conjecture 2 : For an even order, connected, 3-regular graph with jB j = 1 then p = sign(xa) Perspectives on Proposed Algorithm Table1 shows all connected 3-regular graphs including the \black sheep quot; (Graph 20). Graph 1 is not connected but it is included nonetheless.... ..."

"... In PAGE 6: ...3 Improving Recall by Approximate String Search We also conducted experiments to evaluate how much we can further improve the recognition per- formance by using the approximate string search- ing method described in Section 3. Table3 shows the results. The leftmost columns show the thresh- olds of the normalized costs for approximate string searching.... ..."

"... In PAGE 6: ...3 Improving Recall by Approximate String Search We also conducted experiments to evaluate how much we can further improve the recognition per- formance by using the approximate string search- ing method described in Section 3. Table3 shows the results. The leftmost columns show the thresh- olds of the normalized costs for approximate string searching.... ..."