Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I...

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Example I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 4a 3a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b 5b 6b 7b Ankur Moitra () Geometric Graph Theory October 26, 2008 36 / 82

Transcript of Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I...

Page 1: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

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R+e

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5a

4a3a

2a

1a

R

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2b

3b

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3b4b

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7b

Ankur Moitra () Geometric Graph Theory October 26, 2008 36 / 82

Page 2: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

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3b4b

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6b

7b

Ankur Moitra () Geometric Graph Theory October 26, 2008 37 / 82

Page 3: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

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3b4b

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s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 38 / 82

Page 4: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

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3b4b

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s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 39 / 82

Page 5: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 40 / 82

Page 6: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 41 / 82

Page 7: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 42 / 82

Page 8: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

2b

3b4b

5b

6b

7b

s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 43 / 82

Page 9: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

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3b4b

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6b

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s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 44 / 82

Page 10: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

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3b4b

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s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 45 / 82

Page 11: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

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3b4b

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s

t

Ankur Moitra () Geometric Graph Theory October 26, 2008 46 / 82

Page 12: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

1a

2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

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2b

3b

4b

5b

6b

7b

1b

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3b4b

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Ankur Moitra () Geometric Graph Theory October 26, 2008 47 / 82

Page 13: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example I

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2a

3a

4a

5a

6a

7a

R+e

7a

6a

5a

4a3a

2a

1a

R

1b

2b

3b

4b

5b

6b

7b

1b

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3b4b

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s

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Ankur Moitra () Geometric Graph Theory October 26, 2008 48 / 82

Page 14: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

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1a

2a

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R+eR

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2b

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3b

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3d

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2b2c 2d

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1c1d

3a

3b3c

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4b4c4d

1d

1c

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Page 15: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

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3c

3d

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R+eR

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2b

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3b

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3d

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3b

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4b

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1c1a

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1c1d

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2d

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Page 16: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

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R+eR

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2b

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Page 17: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

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2b 2d

3b

Ankur Moitra () Geometric Graph Theory October 26, 2008 52 / 82

Page 18: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

3b

2a

2b 2d

Ankur Moitra () Geometric Graph Theory October 26, 2008 53 / 82

Page 19: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

3b

2a

2b 2d

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Page 20: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

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1a

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R+eR

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2b2c 2d

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1c1d

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4b4c4d

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Page 21: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

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2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

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2b2c 2d

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1c1d

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3b3c

3d

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4b4c4d

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1c1a

2a

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Page 22: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

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3b

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3c

3d

4b

4c

4d

1a

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2b2c 2d

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1c1d

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3b3c

3d

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4b4c4d

1d

1c1a

2a

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Page 23: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

2d

t

s2b

2a

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Page 24: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

2d

t

s2b

2a

Ankur Moitra () Geometric Graph Theory October 26, 2008 59 / 82

Page 25: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

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2b2c 2d

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1c1d

3a

3b3c

3d

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4b4c4d

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1c1a

2a

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Page 26: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

3c

2a

2b 2d

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Page 27: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

3c

2a

2b 2d

Ankur Moitra () Geometric Graph Theory October 26, 2008 62 / 82

Page 28: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b

1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a

Ankur Moitra () Geometric Graph Theory October 26, 2008 63 / 82

Page 29: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b

1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a

Ankur Moitra () Geometric Graph Theory October 26, 2008 64 / 82

Page 30: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b

1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a

Ankur Moitra () Geometric Graph Theory October 26, 2008 65 / 82

Page 31: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b

1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a

Ankur Moitra () Geometric Graph Theory October 26, 2008 66 / 82

Page 32: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b

1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a

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Page 33: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

Example II

t

3a

4a

s

R+eR

1b

2b

2c

3b

2d

3c

3d

4b

4c

4d

1a

2a

2b2c 2d

1b

1c1d

3a

3b3c

3d

4a

4b4c4d

1d

1c1a

2a

Ankur Moitra () Geometric Graph Theory October 26, 2008 68 / 82

Page 34: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

G

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Page 35: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

G

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Page 36: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

G

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Page 37: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

G

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Page 38: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

G

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Page 39: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

G

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Page 40: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

G

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Page 41: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

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Page 42: Some Results on Greedy Embeddings in Metric Spacesjgao/CSE590-spring11/slides/example.pdfExample I 1a 2a 3a 4a 5a 6a 7a R+e 7a 6a 5a 3a 4a 2a 1a R 1b 2b 3b 4b 5b 6b 7b 1b 2b 3b 4b

General Christmas Cactus Graphs

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Ankur Moitra () Geometric Graph Theory October 26, 2008 77 / 82