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Geometric Geometric Spanner Networks Spanner Networks M. Farshi Course Outline Mohammad Farshi Textbook Introduction Combinatorial and Geometric ALGorithms (CGALG) Lab., Algorithms Review Department of Computer Science, Greedy Algorithm

  1. Network Quality Dilation (stretch factor) Geometric Spanner Networks between a pair of vertices= (7,8) M. Farshi (0,7) 4.4 Distance in the graph 3.1 Course Outline 3.6 (3,6) Euclidean distance Textbook Introduction 2.2 4.1 (5,5) of a network= maximum 6.3 Algorithms Review 1.4 3.6 Greedy Algorithm (Org. and (6,4) dilation between all pairs. Imp.) 4.4 Apx. Greedy Algorithm 5 (Ordered) Θ -Graph (1,3) t -spanner 4.4 Algorithm (Sink and 3.6 3.6 Skip-list spanner) Sink Spanner 2.8 A network with dilation at WSPD-based Algorithm (9,2) 3 5 Theoretical most t , or bounds 1 (4,1) 2.2 (3,1) ∀ u, v ∈ V , there is a path Applications (1,0) between u and v of length Designing approximation algorithms with spanners Metric space searching ≤ t × | uv | . ( t -path) Protein Visualization Research Topics 12 / 39

  2. Network Quality Dilation (stretch factor) Geometric Spanner Networks between a pair of vertices= (7,8) M. Farshi (0,7) 4.4 Distance in the graph 3.1 Course Outline 3.6 (3,6) Euclidean distance Textbook Introduction 2.2 4.1 (5,5) of a network= maximum 6.3 Algorithms Review 1.4 3.6 Greedy Algorithm (Org. and (6,4) dilation between all pairs. Imp.) 4.4 Apx. Greedy Algorithm 5 (Ordered) Θ -Graph (1,3) t -spanner 4.4 Algorithm (Sink and 3.6 3.6 Skip-list spanner) Sink Spanner 2.8 A network with dilation at WSPD-based Algorithm (9,2) 3 5 Theoretical most t , or bounds 1 (4,1) 2.2 (3,1) ∀ u, v ∈ V , there is a path Applications (1,0) between u and v of length Designing approximation algorithms with spanners Metric space searching ≤ t × | uv | . ( t -path) Protein Visualization Research Topics 12 / 39

  3. Network Quality Dilation (stretch factor) Geometric Spanner Networks between a pair of vertices= (7,8) M. Farshi (0,7) 4.4 Distance in the graph 3.1 Course Outline 3.6 (3,6) Euclidean distance Textbook Introduction 2.2 4.1 (5,5) of a network= maximum 6.3 Algorithms Review 1.4 3.6 Greedy Algorithm (Org. and (6,4) dilation between all pairs. Imp.) 4.4 Apx. Greedy Algorithm 5 (Ordered) Θ -Graph (1,3) t -spanner 4.4 Algorithm (Sink and 3.6 3.6 Skip-list spanner) Sink Spanner 2.8 A network with dilation at WSPD-based Algorithm (9,2) 3 5 Theoretical most t , or bounds 1 (4,1) 2.2 (3,1) ∀ u, v ∈ V , there is a path Applications (1,0) between u and v of length Designing approximation algorithms with spanners Metric space searching ≤ t × | uv | . ( t -path) Protein Visualization Research Topics 12 / 39

  4. Network Quality (1 + ε ) -Spanners approximate the complete graphs with Geometric Spanner Networks error ε . M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 13 / 39

  5. Example Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization 10-spanner for 532 US-cities Research Topics 14 / 39

  6. Example Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization 5-spanner for 532 US-cities Research Topics 14 / 39

  7. Example Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization 3-spanner for 532 US-cities Research Topics 14 / 39

  8. Example Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization 2-spanner for 532 US-cities Research Topics 14 / 39

  9. Example Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization 1.5-spanner for 532 US-cities Research Topics 14 / 39

  10. How to compute a good spanner? Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Given a set V and t > 1 Sparse t -Spanner Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Quality measurement: Sink Spanner WSPD-based Algorithm Number of edges (size) Theoretical bounds Weight (compared with MST) Applications Maximum degree Designing approximation algorithms with spanners Metric space searching Diameter Protein Visualization Research Topics 15 / 39

  11. How to compute a good spanner? Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Given a set V and t > 1 Sparse t -Spanner Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Quality measurement: Sink Spanner WSPD-based Algorithm Number of edges (size) Theoretical bounds Weight (compared with MST) Applications Maximum degree Designing approximation algorithms with spanners Metric space searching Diameter Protein Visualization Research Topics 15 / 39

  12. How to compute a good spanner? Geometric Spanner Networks M. Farshi Constructing sparse t-spanners: Course Outline Greedy (Bern (1989) and Althöfer et al. (1993)). Textbook Introduction Θ -graph (Clarkson (1987) and Keil (1988)). Algorithms Review Greedy Algorithm (Org. and Ordered Θ -graph (Bose et. al. (2004)). Imp.) Apx. Greedy Algorithm Well-Separated Pair Decomposition (Arya et. al. (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (1995)). Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 16 / 39

  13. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  14. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  15. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  16. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  17. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  18. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  19. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  20. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  21. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  22. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  23. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  24. (Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) M. Farshi (5 , 7) Course Outline Textbook Introduction Algorithms Review (1 , 5) Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) (8 , 3) (4 , 3) Sink Spanner WSPD-based Algorithm Theoretical bounds (6 , 1) Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 17 / 39

  25. (Org.) Greedy Algorithm O RG . G REEDY Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) M. Farshi Sort pairs of points by non-decreasing order of distance; Course Outline E := ∅ ; Textbook Introduction G := ( V, E ) ; Algorithms Review for each pair ( u, v ) of points (in sorted order) do Greedy Algorithm (Org. and Imp.) if S HORTEST P ATH ( G, u, v ) > t · | uv | then Apx. Greedy Algorithm (Ordered) Θ -Graph Add ( u, v ) to E ; Algorithm (Sink and Skip-list spanner) end Sink Spanner WSPD-based Algorithm end Theoretical bounds return G ( V, E ) ; Applications Designing approximation algorithms with spanners Time Complexity: O ( n 3 log n ) . Metric space searching Protein Visualization Research Topics Storage Complexity: O ( n 2 ) . 18 / 39

  26. (Org.) Greedy Algorithm O RG . G REEDY Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) M. Farshi Sort pairs of points by non-decreasing order of distance; Course Outline E := ∅ ; Textbook Introduction G := ( V, E ) ; Algorithms Review for each pair ( u, v ) of points (in sorted order) do Greedy Algorithm (Org. and Imp.) if S HORTEST P ATH ( G, u, v ) > t · | uv | then Apx. Greedy Algorithm (Ordered) Θ -Graph Add ( u, v ) to E ; Algorithm (Sink and Skip-list spanner) end Sink Spanner WSPD-based Algorithm end Theoretical bounds return G ( V, E ) ; Applications Designing approximation algorithms with spanners Time Complexity: O ( n 3 log n ) . Metric space searching Protein Visualization Research Topics Storage Complexity: O ( n 2 ) . 18 / 39

  27. Imp. Greedy Algorithm O RG . G REEDY Input : V and t > 1 Geometric Output : t -spanner G ( V, E ) Spanner Networks Sort pairs of points by non-decreasing order of distance; M. Farshi E := ∅ ; G := ( V, E ) ; for each pair ( u, v ) of points (in sorted order) do Course Outline Textbook if S HORTEST P ATH ( G, u, v ) > t · | uv | then Introduction Add ( u, v ) to E ; Algorithms Review end Greedy Algorithm (Org. and Imp.) end Apx. Greedy Algorithm (Ordered) Θ -Graph return G ( V, E ) ; Algorithm (Sink and Skip-list spanner) Sink Spanner Number of shortest path queries: Θ( n 2 ) . WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 19 / 39

  28. Imp. Greedy Algorithm O RG . G REEDY Input : V and t > 1 Geometric Output : t -spanner G ( V, E ) Spanner Networks Sort pairs of points by non-decreasing order of distance; M. Farshi E := ∅ ; G := ( V, E ) ; for each pair ( u, v ) of points (in sorted order) do Course Outline Textbook if S HORTEST P ATH ( G, u, v ) > t · | uv | then Introduction Add ( u, v ) to E ; Algorithms Review end Greedy Algorithm (Org. and Imp.) end Apx. Greedy Algorithm (Ordered) Θ -Graph return G ( V, E ) ; Algorithm (Sink and Skip-list spanner) Sink Spanner Number of shortest path queries: Θ( n 2 ) . WSPD-based Algorithm Theoretical Observations: bounds Applications We only want to know if there is a t -path between u Designing approximation algorithms with spanners and v . Metric space searching Protein Visualization The graph is only updated O ( n ) times. Research Topics 19 / 39

  29. Imp. Greedy Algorithm I MP . G REEDY Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) M. Farshi for each pair ( u, v ) ∈ V 2 do Set Weight ( u, v ) := ∞ ; Course Outline Sort pairs of points by non-decreasing order of distance; Textbook E := ∅ ; G := ( V, E ) ; Introduction for each pair ( u, v ) of points (in sorted order) do Algorithms Review if Weight ( u, v ) ≤ t · | uv | then Greedy Algorithm (Org. and Imp.) Skip ( u, v ) ; Apx. Greedy Algorithm (Ordered) Θ -Graph else Algorithm (Sink and Skip-list spanner) Compute single source shortest path with source u ; Sink Spanner WSPD-based Algorithm for each w do update Weight ( u, w ) and Weight ( w, u ) ; Theoretical if Weight ( u, v ) ≤ t · | uv | then Skip ( u, v ) ; bounds else Add ( u, v ) to E ; Applications end Designing approximation algorithms with spanners end Metric space searching Protein Visualization return G ( V, E ) ; Research Topics 20 / 39

  30. Imp. Greedy Algorithm Geometric Spanner Networks Conjecture: M. Farshi The running time of I MP . G REEDY is O ( n 2 log n ) . Course Outline Textbook Introduction Bose, Carmi, Farshi, Maheshvari and Smid (2008) Algorithms Review Greedy Algorithm (Org. and The conjecture is wrong! Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph They presented an algorithm which computes the Algorithm (Sink and Skip-list spanner) greedy spanner in O ( n 2 log n ) time (even for points Sink Spanner WSPD-based Algorithm from some metric spaces). Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 21 / 39

  31. Apx. Greedy Algorithm Geometric Spanner Networks M. Farshi t -spanner Course Outline Point set t -spanner Algorithm Textbook Introduction t Constant degree Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 22 / 39

  32. Apx. Greedy Algorithm Geometric Spanner Networks M. Farshi Approximate t -spanner Course Outline Point set ( t · t ′ ) -spanner t -spanner Algorithm Textbook Pruning Algorithm Introduction t Constant degree t ′ Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 22 / 39

  33. Apx. Greedy Algorithm Geometric Spanner Networks M. Farshi Approximate t -spanner Course Outline Point set ( t · t ′ ) -spanner t -spanner Algorithm Textbook Pruning Algorithm Introduction t Constant degree t ′ Algorithms Review O ( n ) edges Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Sink Spanner (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 22 / 39

  34. Apx. Greedy Algorithm Geometric Spanner Networks M. Farshi Approximate t -spanner Course Outline Point set ( t · t ′ ) -spanner t -spanner Algorithm Textbook Pruning Algorithm Introduction t Constant degree t ′ Algorithms Review O ( n ) edges Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Sink Spanner (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner Time Complexity: O ( n log 2 n ) WSPD-based Algorithm Theoretical bounds Storage Complexity: O ( n ) . Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 22 / 39

  35. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  36. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  37. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  38. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  39. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  40. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  41. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  42. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  43. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  44. Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 23 / 39

  45. Θ -Graph Algorithm Θ -G RAPH Input : V and t > 1 Geometric Spanner Networks Output : t -spanner G ( V, E ) M. Farshi 1 Set k := the smallest integer such that t = cos θ − sin θ for θ = 2 π/k ; Course Outline Textbook E := ∅ ; Introduction for each point u ∈ V do Algorithms Review C 1 , . . . , C k := non-overlapping cones with angle θ Greedy Algorithm (Org. and Imp.) and with apex at u ; Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and for each cone C i do Skip-list spanner) Sink Spanner Connect u to the closest point in C i ; WSPD-based Algorithm end Theoretical bounds end Applications return G ( V, E ) ; Designing approximation algorithms with spanners Metric space searching Protein Visualization Time Complexity: O ( n log n ) . Research Topics Storage Complexity: O ( n ) . 24 / 39

  46. Θ -Graph Algorithm Θ -G RAPH Input : V and t > 1 Geometric Spanner Networks Output : t -spanner G ( V, E ) M. Farshi 1 Set k := the smallest integer such that t = cos θ − sin θ for θ = 2 π/k ; Course Outline Textbook E := ∅ ; Introduction for each point u ∈ V do Algorithms Review C 1 , . . . , C k := non-overlapping cones with angle θ Greedy Algorithm (Org. and Imp.) and with apex at u ; Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and for each cone C i do Skip-list spanner) Sink Spanner Connect u to the closest point in C i ; WSPD-based Algorithm end Theoretical bounds end Applications return G ( V, E ) ; Designing approximation algorithms with spanners Metric space searching Protein Visualization Time Complexity: O ( n log n ) . Research Topics Storage Complexity: O ( n ) . 24 / 39

  47. Variants of Θ -Graph Algorithm Ordered Θ -Graph– O (log n ) maximum degree Same as the Θ -graph algorithm, except we add points Geometric Spanner Networks one by one in a special order. M. Farshi Random Ordered Θ -Graph– O (log n ) spanner Course Outline Textbook diameter Introduction We add points one by one in a random order. Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Sink Spanner– bounded degree (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Decrease the degree of nodes by replacing some edges Sink Spanner WSPD-based Algorithm by paths within other nodes. Theoretical bounds Applications Skip-List Spanner– O (log n ) spanner diameter Designing approximation algorithms with spanners Decrease the diameter of Θ -graph by adding some extra Metric space searching Protein Visualization edges. Research Topics 25 / 39

  48. Variants of Θ -Graph Algorithm Ordered Θ -Graph– O (log n ) maximum degree Same as the Θ -graph algorithm, except we add points Geometric Spanner Networks one by one in a special order. M. Farshi Random Ordered Θ -Graph– O (log n ) spanner Course Outline Textbook diameter Introduction We add points one by one in a random order. Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Sink Spanner– bounded degree (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Decrease the degree of nodes by replacing some edges Sink Spanner WSPD-based Algorithm by paths within other nodes. Theoretical bounds Applications Skip-List Spanner– O (log n ) spanner diameter Designing approximation algorithms with spanners Decrease the diameter of Θ -graph by adding some extra Metric space searching Protein Visualization edges. Research Topics 25 / 39

  49. Variants of Θ -Graph Algorithm Ordered Θ -Graph– O (log n ) maximum degree Same as the Θ -graph algorithm, except we add points Geometric Spanner Networks one by one in a special order. M. Farshi Random Ordered Θ -Graph– O (log n ) spanner Course Outline Textbook diameter Introduction We add points one by one in a random order. Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Sink Spanner– bounded degree (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Decrease the degree of nodes by replacing some edges Sink Spanner WSPD-based Algorithm by paths within other nodes. Theoretical bounds Applications Skip-List Spanner– O (log n ) spanner diameter Designing approximation algorithms with spanners Decrease the diameter of Θ -graph by adding some extra Metric space searching Protein Visualization edges. Research Topics 25 / 39

  50. Variants of Θ -Graph Algorithm Ordered Θ -Graph– O (log n ) maximum degree Same as the Θ -graph algorithm, except we add points Geometric Spanner Networks one by one in a special order. M. Farshi Random Ordered Θ -Graph– O (log n ) spanner Course Outline Textbook diameter Introduction We add points one by one in a random order. Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Sink Spanner– bounded degree (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Decrease the degree of nodes by replacing some edges Sink Spanner WSPD-based Algorithm by paths within other nodes. Theoretical bounds Applications Skip-List Spanner– O (log n ) spanner diameter Designing approximation algorithms with spanners Decrease the diameter of Θ -graph by adding some extra Metric space searching Protein Visualization edges. Research Topics 25 / 39

  51. Sink Spanner A variant of Θ -graph with bounded degree Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) M. Farshi √ t -spanner − → Construct a directed G with bounded Course Outline out-degree; Textbook Introduction for each point q ∈ V do √ Algorithms Review Replace the “star” pointing to q by a t - q -sink Greedy Algorithm (Org. and Imp.) spanner Apx. Greedy Algorithm (Ordered) Θ -Graph end Algorithm (Sink and Skip-list spanner) return G ( V, E ) ; Sink Spanner WSPD-based Algorithm Theoretical bounds Time Complexity: O ( n log n ) Storage Complexity: O ( n ) . Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 26 / 39

  52. Sink Spanner A variant of Θ -graph with bounded degree Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) M. Farshi √ t -spanner − → Construct a directed G with bounded Course Outline out-degree; Textbook Introduction for each point q ∈ V do √ Algorithms Review Replace the “star” pointing to q by a t - q -sink Greedy Algorithm (Org. and Imp.) spanner Apx. Greedy Algorithm (Ordered) Θ -Graph end Algorithm (Sink and Skip-list spanner) return G ( V, E ) ; Sink Spanner WSPD-based Algorithm Theoretical bounds Time Complexity: O ( n log n ) Storage Complexity: O ( n ) . Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 26 / 39

  53. Skip-List Spanner A variant of Θ -graph with O (log n ) spanner diameter Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) M. Farshi Set V 0 := V ; i := 1 ; Course Outline while V i − 1 � = ∅ do Textbook V i contains each points of V i − 1 with probability 1 / 2 ; Introduction end Algorithms Review Greedy Algorithm (Org. and for each i do Imp.) Apx. Greedy Algorithm Construct a t -spanner G i ( V i , E i ) using the Θ -graph (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) algorithm; Sink Spanner WSPD-based Algorithm end Theoretical E = ∪ i E i ; bounds return G ( V, E ) ; Applications Designing approximation algorithms with spanners Metric space searching Time Complexity: O ( n log n ) Storage Complexity: O ( n ) . Protein Visualization Research Topics 27 / 39

  54. Skip-List Spanner A variant of Θ -graph with O (log n ) spanner diameter Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) M. Farshi Set V 0 := V ; i := 1 ; Course Outline while V i − 1 � = ∅ do Textbook V i contains each points of V i − 1 with probability 1 / 2 ; Introduction end Algorithms Review Greedy Algorithm (Org. and for each i do Imp.) Apx. Greedy Algorithm Construct a t -spanner G i ( V i , E i ) using the Θ -graph (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) algorithm; Sink Spanner WSPD-based Algorithm end Theoretical E = ∪ i E i ; bounds return G ( V, E ) ; Applications Designing approximation algorithms with spanners Metric space searching Time Complexity: O ( n log n ) Storage Complexity: O ( n ) . Protein Visualization Research Topics 27 / 39

  55. Well Separated Pair Decomposition (WSPD) Well Separated Pair: A, B ⊂ R d are s -well separated ( s > 0 ), if ∃ disjoint balls, Geometric Spanner Networks D A and D B such that M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 28 / 39

  56. Well Separated Pair Decomposition (WSPD) Well Separated Pair: A, B ⊂ R d are s -well separated ( s > 0 ), if ∃ disjoint balls, Geometric D A and D B such that Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm B (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners A Metric space searching Protein Visualization Research Topics 28 / 39

  57. Well Separated Pair Decomposition (WSPD) Well Separated Pair: A, B ⊂ R d are s -well separated ( s > 0 ), if ∃ disjoint balls, Geometric D A and D B such that Spanner Networks M. Farshi A ⊆ D A and B ⊆ D B . Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm B (Ordered) Θ -Graph Algorithm (Sink and r B Skip-list spanner) D B Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners A Metric space searching Protein Visualization r A Research Topics D A 28 / 39

  58. Well Separated Pair Decomposition (WSPD) Well Separated Pair: A, B ⊂ R d are s -well separated ( s > 0 ), if ∃ disjoint balls, Geometric D A and D B such that Spanner Networks M. Farshi A ⊆ D A and B ⊆ D B . Course Outline d ( D A , D B ) ≥ s × max(radius( D A ) , radius( D B )) . Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm B (Ordered) Θ -Graph Algorithm (Sink and r B Skip-list spanner) D B Sink Spanner WSPD-based Algorithm Theoretical bounds ≥ s × max( r A , r B ) Applications Designing approximation algorithms with spanners Metric space searching A Protein Visualization Research Topics r A D A 28 / 39

  59. Well Separated Pair Decomposition (WSPD) Well Separated Pair Decomposition: Geometric Let V ⊂ R d and s > 0 . A WSPD for V with respect to s is Spanner Networks a set { ( A i , B i ) } m i =1 of pairs of non-empty subsets of V M. Farshi such that Course Outline Textbook ∀ i , A i and B i are s -well separated, Introduction ∀ p, q ∈ V , there is exactly one index i s. t. Algorithms Review Greedy Algorithm (Org. and p ∈ A i and q ∈ B i or Imp.) Apx. Greedy Algorithm q ∈ A i and p ∈ B i . (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner m : Size of WSPD. WSPD-based Algorithm Theoretical bounds Callahan & Kosaraju (1995) Applications For each set of n points, we can construct a WSPD of Designing approximation algorithms with spanners size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Metric space searching Protein Visualization Research Topics 29 / 39

  60. Well Separated Pair Decomposition (WSPD) Well Separated Pair Decomposition: Geometric Let V ⊂ R d and s > 0 . A WSPD for V with respect to s is Spanner Networks a set { ( A i , B i ) } m i =1 of pairs of non-empty subsets of V M. Farshi such that Course Outline Textbook ∀ i , A i and B i are s -well separated, Introduction ∀ p, q ∈ V , there is exactly one index i s. t. Algorithms Review Greedy Algorithm (Org. and p ∈ A i and q ∈ B i or Imp.) Apx. Greedy Algorithm q ∈ A i and p ∈ B i . (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner m : Size of WSPD. WSPD-based Algorithm Theoretical bounds Callahan & Kosaraju (1995) Applications For each set of n points, we can construct a WSPD of Designing approximation algorithms with spanners size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Metric space searching Protein Visualization Research Topics 29 / 39

  61. Well Separated Pair Decomposition (WSPD) Well Separated Pair Decomposition: Geometric Let V ⊂ R d and s > 0 . A WSPD for V with respect to s is Spanner Networks a set { ( A i , B i ) } m i =1 of pairs of non-empty subsets of V M. Farshi such that Course Outline Textbook ∀ i , A i and B i are s -well separated, Introduction ∀ p, q ∈ V , there is exactly one index i s. t. Algorithms Review Greedy Algorithm (Org. and p ∈ A i and q ∈ B i or Imp.) Apx. Greedy Algorithm q ∈ A i and p ∈ B i . (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner m : Size of WSPD. WSPD-based Algorithm Theoretical bounds Callahan & Kosaraju (1995) Applications For each set of n points, we can construct a WSPD of Designing approximation algorithms with spanners size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Metric space searching Protein Visualization Research Topics 29 / 39

  62. Well Separated Pair Decomposition (WSPD) Well Separated Pair Decomposition: Geometric Let V ⊂ R d and s > 0 . A WSPD for V with respect to s is Spanner Networks a set { ( A i , B i ) } m i =1 of pairs of non-empty subsets of V M. Farshi such that Course Outline Textbook ∀ i , A i and B i are s -well separated, Introduction ∀ p, q ∈ V , there is exactly one index i s. t. Algorithms Review Greedy Algorithm (Org. and p ∈ A i and q ∈ B i or Imp.) Apx. Greedy Algorithm q ∈ A i and p ∈ B i . (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner m : Size of WSPD. WSPD-based Algorithm Theoretical bounds Callahan & Kosaraju (1995) Applications For each set of n points, we can construct a WSPD of Designing approximation algorithms with spanners size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Metric space searching Protein Visualization Research Topics 29 / 39

  63. Well Separated Pair Decomposition (WSPD) Well Separated Pair Decomposition: Geometric Let V ⊂ R d and s > 0 . A WSPD for V with respect to s is Spanner Networks a set { ( A i , B i ) } m i =1 of pairs of non-empty subsets of V M. Farshi such that Course Outline Textbook ∀ i , A i and B i are s -well separated, Introduction ∀ p, q ∈ V , there is exactly one index i s. t. Algorithms Review Greedy Algorithm (Org. and p ∈ A i and q ∈ B i or Imp.) Apx. Greedy Algorithm q ∈ A i and p ∈ B i . (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner m : Size of WSPD. WSPD-based Algorithm Theoretical bounds Callahan & Kosaraju (1995) Applications For each set of n points, we can construct a WSPD of Designing approximation algorithms with spanners size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Metric space searching Protein Visualization Research Topics 29 / 39

  64. WSPD-based Algorithm WSPD Algorithm Geometric Spanner Networks Input : V and t > 1 M. Farshi Output : t -spanner G ( V, E ) Set W := WSPD of V w.r.t. s := 4( t +1) t − 1 ; Course Outline Textbook Set E = ∅ ; Introduction for each ( A i , B i ) ∈ W do Algorithms Review Select an arbitrary node u ∈ A i and an arbitrary node Greedy Algorithm (Org. and Imp.) v ∈ B i ; Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Add edge ( u, v ) to E . Skip-list spanner) Sink Spanner end WSPD-based Algorithm return G ( V, E ) . Theoretical bounds Applications Designing approximation Time Complexity: O ( n log n ) . algorithms with spanners Metric space searching Storage Complexity: O ( n ) . Protein Visualization Research Topics 30 / 39

  65. WSPD-based Algorithm WSPD Algorithm Geometric Spanner Networks Input : V and t > 1 M. Farshi Output : t -spanner G ( V, E ) Set W := WSPD of V w.r.t. s := 4( t +1) t − 1 ; Course Outline Textbook Set E = ∅ ; Introduction for each ( A i , B i ) ∈ W do Algorithms Review Select an arbitrary node u ∈ A i and an arbitrary node Greedy Algorithm (Org. and Imp.) v ∈ B i ; Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Add edge ( u, v ) to E . Skip-list spanner) Sink Spanner end WSPD-based Algorithm return G ( V, E ) . Theoretical bounds Applications Designing approximation Time Complexity: O ( n log n ) . algorithms with spanners Metric space searching Storage Complexity: O ( n ) . Protein Visualization Research Topics 30 / 39

  66. Theoretical bounds Geometric - Size Weight Degree Time Spanner Networks M. Farshi O ( n 2 log n ) Greedy spanner O ( n ) O ( wt (MST)) O (1) Course Outline Apx. greedy spanner O ( n ) O ( wt (MST)) O (1) O ( n log n ) Textbook Introduction Θ -graph O ( n ) Θ( n · wt (MST)) Θ( n ) O ( n log n ) Algorithms Review O. Θ -graph O ( n ) O ( n · wt (MST)) O (log n ) O ( n log n ) Greedy Algorithm (Org. and Imp.) O ( n ) O (log n · wt (MST)) Θ( n ) O ( n log n ) Apx. Greedy Algorithm WSPD spanner (Ordered) Θ -Graph Algorithm (Sink and Sink-spanner O ( n ) O ( n · wt (MST)) O (1) O ( n log n ) Skip-list spanner) Sink Spanner WSPD-based Algorithm Skip-list spanner O ( n ) ∗ Θ( n · wt (MST)) ∗ Θ( n ) O ( n log n ) ∗ Theoretical bounds (*): Expected with high probability Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 31 / 39

  67. Applications Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 32 / 39

  68. Applications Designing approximation algorithms with spanners Traveling Salesperson Problem (TSP) Geometric Find the shortest tour that visits each point exactly once Spanner Networks and return to the starting point. M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics Known results: 33 / 39

  69. Applications Designing approximation algorithms with spanners Traveling Salesperson Problem (TSP) Geometric Find the shortest tour that visits each point exactly once Spanner Networks and return to the starting point. M. Farshi Course Outline Known results: Textbook Introduction The problem is NP-hard even in R d . Algorithms Review Greedy Algorithm (Org. and A 2 -approximation algorithm for metric spaces by Imp.) Apx. Greedy Algorithm Rosenkrantz et al. (1977). (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) A 1 . 5 -approximation algorithm by Christofides et al. Sink Spanner WSPD-based Algorithm (1976). Theoretical bounds A PTAS ( (1 + ε ) -approx. Alg.) for geometric case by Applications Arora (1998) and Mitchell (1999). Designing approximation algorithms with spanners A PTAS for geometric case using spanners by Rao Metric space searching Protein Visualization and Smith (1998). Research Topics 33 / 39

  70. Applications Designing approximation algorithms with spanners Definition: Geometric Spanner Networks If G is a graph with vertex set P , then a tour of P in G is a M. Farshi (possibly non-simple) cycle in G that visits each point of P at least once. Course Outline Textbook Observation: Introduction Algorithms Review For any t -spanner G for P , there is a tour of P in G , Greedy Algorithm (Org. and Imp.) whose weight is at most t · wt (TSP( P )) . Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Theorem (Rao and Smith, 1998) Sink Spanner WSPD-based Algorithm Given a (1 + ε ) -spanner of a set of n points with O ( n ) Theoretical bounds size and O ( wt (MST)) weight, we can compute a Applications (1 + ε ) -approximation of TSP( P ) in O ( n log n ) time. Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 34 / 39

  71. Applications Designing approximation algorithms with spanners Definition: Geometric Spanner Networks If G is a graph with vertex set P , then a tour of P in G is a M. Farshi (possibly non-simple) cycle in G that visits each point of P at least once. Course Outline Textbook Observation: Introduction Algorithms Review For any t -spanner G for P , there is a tour of P in G , Greedy Algorithm (Org. and Imp.) whose weight is at most t · wt (TSP( P )) . Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Theorem (Rao and Smith, 1998) Sink Spanner WSPD-based Algorithm Given a (1 + ε ) -spanner of a set of n points with O ( n ) Theoretical bounds size and O ( wt (MST)) weight, we can compute a Applications (1 + ε ) -approximation of TSP( P ) in O ( n log n ) time. Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 34 / 39

  72. Applications Designing approximation algorithms with spanners Definition: Geometric Spanner Networks If G is a graph with vertex set P , then a tour of P in G is a M. Farshi (possibly non-simple) cycle in G that visits each point of P at least once. Course Outline Textbook Observation: Introduction Algorithms Review For any t -spanner G for P , there is a tour of P in G , Greedy Algorithm (Org. and Imp.) whose weight is at most t · wt (TSP( P )) . Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Theorem (Rao and Smith, 1998) Sink Spanner WSPD-based Algorithm Given a (1 + ε ) -spanner of a set of n points with O ( n ) Theoretical bounds size and O ( wt (MST)) weight, we can compute a Applications (1 + ε ) -approximation of TSP( P ) in O ( n log n ) time. Designing approximation algorithms with spanners Metric space searching S. B. Rao and W. D. Smith, Approximating Geometrical Graphs via Protein Visualization “Spanners” and “Banyans” , STOC’98, pp. 540–550, 1998. Research Topics 34 / 39

  73. Applications Metric space searching Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Approximate proximity searching: Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Multimedia information retrieval, Skip-list spanner) Sink Spanner WSPD-based Algorithm Data mining, Theoretical Pattern recognition, bounds Applications Machine learning, Designing approximation algorithms with spanners Computer vision and Metric space searching Protein Visualization Biomedical databases. Research Topics 35 / 39

  74. Applications Metric space searching Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Approximate proximity searching: Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Multimedia information retrieval, Skip-list spanner) Sink Spanner WSPD-based Algorithm Data mining, Theoretical Pattern recognition, bounds Applications Machine learning, Designing approximation algorithms with spanners Computer vision and Metric space searching Protein Visualization Biomedical databases. Research Topics 35 / 39

  75. Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. M. Farshi But evaluating the distances are expensive. Course Outline Textbook One way to speedup is computing the distance Introduction between any two objects and save them, but it need Algorithms Review O ( n 2 ) space (AESA). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm A t -spanner can be used as a sparse data structure (Ordered) Θ -Graph Algorithm (Sink and to reduce the space. Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 36 / 39

  76. Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. M. Farshi But evaluating the distances are expensive. Course Outline Textbook One way to speedup is computing the distance Introduction between any two objects and save them, but it need Algorithms Review O ( n 2 ) space (AESA). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm A t -spanner can be used as a sparse data structure (Ordered) Θ -Graph Algorithm (Sink and to reduce the space. Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 36 / 39

  77. Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. M. Farshi But evaluating the distances are expensive. Course Outline Textbook One way to speedup is computing the distance Introduction between any two objects and save them, but it need Algorithms Review O ( n 2 ) space (AESA). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm A t -spanner can be used as a sparse data structure (Ordered) Θ -Graph Algorithm (Sink and to reduce the space. Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 36 / 39

  78. Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. M. Farshi But evaluating the distances are expensive. Course Outline Textbook One way to speedup is computing the distance Introduction between any two objects and save them, but it need Algorithms Review O ( n 2 ) space (AESA). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm A t -spanner can be used as a sparse data structure (Ordered) Θ -Graph Algorithm (Sink and to reduce the space. Skip-list spanner) Sink Spanner WSPD-based Algorithm G. Navarro, R. Paredes, and E. Chávez, t-Spanners for metric space Theoretical searching , Data & Knowledge Engineering, pp. 820-854, 2007. bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 36 / 39

  79. Applications Protein Visualization Geometric Spanner Networks M. Farshi Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners D. Russel and L. Guibas, Exploring Protein Folding Trajectories Metric space searching Protein Visualization Using Geometric Spanners , Pacific Symposium on Biocomputing, Research Topics pp. 40-51, 2005. 37 / 39

  80. Research Topics Current and Future Works: Geometric Spanner Networks Dynamic spanners (insert and remove nodes). M. Farshi Kinetic spanners (when points move and we want to Course Outline maintain an spanner all the time). Textbook Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review region fault tolerant). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Spanners among obstacles. (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Optimization problems. Sink Spanner WSPD-based Algorithm External memory (I/O efficient) algorithms for Theoretical bounds generating spanners. Applications Experimental works on spanner algorithms. Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 38 / 39

  81. Research Topics Current and Future Works: Geometric Spanner Networks Dynamic spanners (insert and remove nodes). M. Farshi Kinetic spanners (when points move and we want to Course Outline maintain an spanner all the time). Textbook Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review region fault tolerant). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Spanners among obstacles. (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Optimization problems. Sink Spanner WSPD-based Algorithm External memory (I/O efficient) algorithms for Theoretical bounds generating spanners. Applications Experimental works on spanner algorithms. Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 38 / 39

  82. Research Topics Current and Future Works: Geometric Spanner Networks Dynamic spanners (insert and remove nodes). M. Farshi Kinetic spanners (when points move and we want to Course Outline maintain an spanner all the time). Textbook Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review region fault tolerant). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Spanners among obstacles. (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Optimization problems. Sink Spanner WSPD-based Algorithm External memory (I/O efficient) algorithms for Theoretical bounds generating spanners. Applications Experimental works on spanner algorithms. Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 38 / 39

  83. Research Topics Current and Future Works: Geometric Spanner Networks Dynamic spanners (insert and remove nodes). M. Farshi Kinetic spanners (when points move and we want to Course Outline maintain an spanner all the time). Textbook Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review region fault tolerant). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Spanners among obstacles. (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Optimization problems. Sink Spanner WSPD-based Algorithm External memory (I/O efficient) algorithms for Theoretical bounds generating spanners. Applications Experimental works on spanner algorithms. Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 38 / 39

  84. Research Topics Current and Future Works: Geometric Spanner Networks Dynamic spanners (insert and remove nodes). M. Farshi Kinetic spanners (when points move and we want to Course Outline maintain an spanner all the time). Textbook Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review region fault tolerant). Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Spanners among obstacles. (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Optimization problems. Sink Spanner WSPD-based Algorithm External memory (I/O efficient) algorithms for Theoretical bounds generating spanners. Applications Experimental works on spanner algorithms. Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics 38 / 39

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