Giri Narasimhan earned a B.Tech. in Electrical Engineering from the Indian Institute of Technology in Mumbai, India, and a Ph.D. in Computer Science from the University of Wisconsin in Madison, Wisconsin, USA. He was a member of the faculty at the University of Memphis, and is currently at Florida International University. Michiel Smid received a M.Sc. degree in Mathematics from the University of Technology in Eidenhoven and a Ph.D. degree in Computer Science from the University of Amsterdam. He has held teaching positions at the Max-Planck-Institute for Computer Science in Saarbrucken, King's College in London, and the University of Magdenburg. Since 2001, he has been at Carleton University, where he is currently a professor of Computer Science.

"The writing style is clear, with a clean presentation enriched with boxes highlighting important contents and figures." David Orden, Mathematical Reviews

• Part I. Introduction: 1. Introduction; 2. Algorithms and graphs; 3. The algebraic computation-tree model; Part II. Spanners Based on Simplical Cones: 4. Spanners based on the Q-graph; 5. Cones in higher dimensional space and Q-graphs; 6. Geometric analysis: the gap property; 7. The gap-greedy algorithm; 8. Enumerating distances using spanners of bounded degree; Part III. The Well Separated Pair Decomposition and its Applications: 9. The well-separated pair decomposition; 10. Applications of well-separated pairs; 11. The Dumbbell theorem; 12. Shortcutting trees and spanners with low spanner diameter; 13. Approximating the stretch factor of Euclidean graphs; Part IV. The Path Greedy Algorithm: 14. Geometric analysis: the leapfrog property; 15. The path-greedy algorithm; Part V. Further Results and Applications: 16. The distance range hierarchy; 17. Approximating shortest paths in spanners; 18. Fault-tolerant spanners; 19. Designing approximation algorithms with spanners; 20. Further results and open problems.

Giri Narasimhan earned a B.Tech. in Electrical Engineering from the Indian Institute of Technology in Mumbai, India, and a Ph.D. in Computer Science from the University of Wisconsin in Madison, Wisconsin, USA. He was a member of the faculty at the University of Memphis, and is currently at Florida International University. Michiel Smid received a M.Sc. degree in Mathematics from the University of Technology in Eidenhoven and a Ph.D. degree in Computer Science from the University of Amsterdam. He has held teaching positions at the Max-Planck-Institute for Computer Science in Saarbrucken, King's College in London, and the University of Magdenburg. Since 2001, he has been at Carleton University, where he is currently a professor of Computer Science.

"The writing style is clear, with a clean presentation enriched with boxes highlighting important contents and figures." David Orden, Mathematical Reviews

• Part I. Introduction: 1. Introduction; 2. Algorithms and graphs; 3. The algebraic computation-tree model; Part II. Spanners Based on Simplical Cones: 4. Spanners based on the Q-graph; 5. Cones in higher dimensional space and Q-graphs; 6. Geometric analysis: the gap property; 7. The gap-greedy algorithm; 8. Enumerating distances using spanners of bounded degree; Part III. The Well Separated Pair Decomposition and its Applications: 9. The well-separated pair decomposition; 10. Applications of well-separated pairs; 11. The Dumbbell theorem; 12. Shortcutting trees and spanners with low spanner diameter; 13. Approximating the stretch factor of Euclidean graphs; Part IV. The Path Greedy Algorithm: 14. Geometric analysis: the leapfrog property; 15. The path-greedy algorithm; Part V. Further Results and Applications: 16. The distance range hierarchy; 17. Approximating shortest paths in spanners; 18. Fault-tolerant spanners; 19. Designing approximation algorithms with spanners; 20. Further results and open problems.