Network Quality Dilation (stretch factor) Geometric Spanner Networks between a pair of vertices= (7,8) (0,7) 4.4 Course Outline Distance in the graph 3.1 3.6 Textbook (3,6) Euclidean distance Introduction 2.2 Algorithms Review (5,5) 4.1 of a network= maximum 6.3 Greedy Algorithm (Org. and 1.4 Imp.) 3.6 (6,4) dilation between all pairs. Apx. Greedy Algorithm 4.4 (Ordered) Θ -Graph 5 Algorithm (Sink and (1,3) Skip-list spanner) t -spanner 4.4 3.6 Sink Spanner 3.6 WSPD-based Algorithm 2.8 A network with dilation at (9,2) Theoretical 3 5 bounds most t , or 1 (4,1) 2.2 (3,1) Applications ∀ u, v ∈ V , there is a path Designing approximation (1,0) algorithms with spanners between u and v of length Metric space searching Protein Visualization ≤ t × | uv | . ( t -path) Research Topics
Network Quality Dilation (stretch factor) Geometric Spanner Networks between a pair of vertices= (7,8) (0,7) 4.4 Course Outline Distance in the graph 3.1 3.6 Textbook (3,6) Euclidean distance Introduction 2.2 Algorithms Review (5,5) 4.1 of a network= maximum 6.3 Greedy Algorithm (Org. and 1.4 Imp.) 3.6 (6,4) dilation between all pairs. Apx. Greedy Algorithm 4.4 (Ordered) Θ -Graph 5 Algorithm (Sink and (1,3) Skip-list spanner) t -spanner 4.4 3.6 Sink Spanner 3.6 WSPD-based Algorithm 2.8 A network with dilation at (9,2) Theoretical 3 5 bounds most t , or 1 (4,1) 2.2 (3,1) Applications ∀ u, v ∈ V , there is a path Designing approximation (1,0) algorithms with spanners between u and v of length Metric space searching Protein Visualization ≤ t × | uv | . ( t -path) Research Topics
Network Quality Dilation (stretch factor) Geometric Spanner Networks between a pair of vertices= (7,8) (0,7) 4.4 Course Outline Distance in the graph 3.1 3.6 Textbook (3,6) Euclidean distance Introduction 2.2 Algorithms Review (5,5) 4.1 of a network= maximum 6.3 Greedy Algorithm (Org. and 1.4 Imp.) 3.6 (6,4) dilation between all pairs. Apx. Greedy Algorithm 4.4 (Ordered) Θ -Graph 5 Algorithm (Sink and (1,3) Skip-list spanner) t -spanner 4.4 3.6 Sink Spanner 3.6 WSPD-based Algorithm 2.8 A network with dilation at (9,2) Theoretical 3 5 bounds most t , or 1 (4,1) 2.2 (3,1) Applications ∀ u, v ∈ V , there is a path Designing approximation (1,0) algorithms with spanners between u and v of length Metric space searching Protein Visualization ≤ t × | uv | . ( t -path) Research Topics
Network Quality Geometric (1 + ε ) -Spanners approximate the complete graphs with Spanner Networks error ε . 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
Example Geometric Spanner Networks 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 10-spanner for 532 US-cities
Example Geometric Spanner Networks 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 5-spanner for 532 US-cities
Example Geometric Spanner Networks 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 3-spanner for 532 US-cities
Example Geometric Spanner Networks 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 2-spanner for 532 US-cities
Example Geometric Spanner Networks 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 1.5-spanner for 532 US-cities
How to compute a good spanner? Geometric Spanner Networks 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 Designing approximation Maximum degree algorithms with spanners Metric space searching Protein Visualization Diameter Research Topics
How to compute a good spanner? Geometric Spanner Networks 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 Designing approximation Maximum degree algorithms with spanners Metric space searching Protein Visualization Diameter Research Topics
How to compute a good spanner? Geometric Spanner Networks Constructing sparse t-spanners: Course Outline Textbook Greedy (Bern (1989) and Althöfer et al. (1993)). Introduction Algorithms Review Θ -graph (Clarkson (1987) and Keil (1988)). Greedy Algorithm (Org. and Imp.) Ordered Θ -graph (Bose et. al. (2004)). Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Well-Separated Pair Decomposition (Arya et. al. Skip-list spanner) Sink Spanner (1995)). WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm Geometric Spanner Networks (5 , 8) (2 , 8) Course Outline (5 , 7) Textbook Introduction Algorithms Review Greedy Algorithm (Org. and (1 , 5) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner (8 , 3) (4 , 3) WSPD-based Algorithm Theoretical bounds Applications (6 , 1) Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
(Org.) Greedy Algorithm O RG . G REEDY Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) Sort pairs of points by non-decreasing order of distance; Course Outline Textbook E := ∅ ; 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 Algorithm (Sink and Add ( u, v ) to E ; Skip-list spanner) Sink Spanner end WSPD-based Algorithm end Theoretical bounds return G ( V, E ) ; Applications Designing approximation algorithms with spanners Metric space searching Time Complexity: O ( n 3 log n ) . Protein Visualization Research Topics Storage Complexity: O ( n 2 ) .
(Org.) Greedy Algorithm O RG . G REEDY Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) Sort pairs of points by non-decreasing order of distance; Course Outline Textbook E := ∅ ; 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 Algorithm (Sink and Add ( u, v ) to E ; Skip-list spanner) Sink Spanner end WSPD-based Algorithm end Theoretical bounds return G ( V, E ) ; Applications Designing approximation algorithms with spanners Metric space searching Time Complexity: O ( n 3 log n ) . Protein Visualization Research Topics Storage Complexity: O ( n 2 ) .
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; E := ∅ ; G := ( V, E ) ; Course Outline for each pair ( u, v ) of points (in sorted order) do 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 Algorithm (Sink and return G ( V, E ) ; Skip-list spanner) Sink Spanner WSPD-based Algorithm Number of shortest path queries: Θ( n 2 ) . Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
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; E := ∅ ; G := ( V, E ) ; Course Outline for each pair ( u, v ) of points (in sorted order) do 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 Algorithm (Sink and return G ( V, E ) ; Skip-list spanner) Sink Spanner WSPD-based Algorithm Number of shortest path queries: Θ( n 2 ) . Theoretical bounds Observations: Applications We only want to know if there is a t -path between u Designing approximation algorithms with spanners Metric space searching and v . Protein Visualization Research Topics The graph is only updated O ( n ) times.
Imp. Greedy Algorithm I MP . G REEDY Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) 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 Greedy Algorithm (Org. and if Weight ( u, v ) ≤ t · | uv | then Imp.) Apx. Greedy Algorithm Skip ( u, v ) ; (Ordered) Θ -Graph Algorithm (Sink and else Skip-list spanner) Sink Spanner Compute single source shortest path with source u ; 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 Designing approximation end algorithms with spanners Metric space searching end Protein Visualization return G ( V, E ) ; Research Topics
Imp. Greedy Algorithm Geometric Spanner Networks Conjecture: Course Outline The running time of I MP . G REEDY is O ( n 2 log n ) . Textbook Introduction Algorithms Review Bose, Carmi, Farshi, Maheshvari and Smid (2008) Greedy Algorithm (Org. and Imp.) The conjecture is wrong! Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and They presented an algorithm which computes the Skip-list spanner) Sink Spanner greedy spanner in O ( n 2 log n ) time (even for points WSPD-based Algorithm Theoretical from some metric spaces). bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
Apx. Greedy Algorithm Geometric Spanner Networks Course Outline t -spanner Point set Textbook t -spanner Algorithm 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
Apx. Greedy Algorithm Geometric Spanner Networks Course Outline Approximate t -spanner Point set ( t · t ′ ) -spanner Textbook t -spanner Algorithm 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
Apx. Greedy Algorithm Geometric Spanner Networks Course Outline Approximate t -spanner Point set ( t · t ′ ) -spanner Textbook t -spanner Algorithm Pruning Algorithm Introduction t Constant degree t ′ Algorithms Review Greedy Algorithm (Org. and O ( n ) edges 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
Apx. Greedy Algorithm Geometric Spanner Networks Course Outline Approximate t -spanner Point set ( t · t ′ ) -spanner Textbook t -spanner Algorithm Pruning Algorithm Introduction t Constant degree t ′ Algorithms Review Greedy Algorithm (Org. and O ( n ) edges Imp.) Apx. Greedy Algorithm Sink Spanner (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm Time Complexity: O ( n log 2 n ) Theoretical bounds Storage Complexity: O ( n ) . Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm t = 3 , Θ = π/ 6 Geometric Spanner Networks 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
Θ -Graph Algorithm Θ -G RAPH Input : V and t > 1 Geometric Spanner Networks Output : t -spanner G ( V, E ) 1 Set k := the smallest integer such that t = cos θ − sin θ for Course Outline θ = 2 π/k ; Textbook E := ∅ ; Introduction for each point u ∈ V do Algorithms Review Greedy Algorithm (Org. and C 1 , . . . , C k := non-overlapping cones with angle θ Imp.) Apx. Greedy Algorithm and with apex at u ; (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) for each cone C i do Sink Spanner Connect u to the closest point in C i ; WSPD-based Algorithm Theoretical end bounds end Applications Designing approximation return G ( V, E ) ; algorithms with spanners Metric space searching Protein Visualization Time Complexity: O ( n log n ) . Research Topics Storage Complexity: O ( n ) .
Θ -Graph Algorithm Θ -G RAPH Input : V and t > 1 Geometric Spanner Networks Output : t -spanner G ( V, E ) 1 Set k := the smallest integer such that t = cos θ − sin θ for Course Outline θ = 2 π/k ; Textbook E := ∅ ; Introduction for each point u ∈ V do Algorithms Review Greedy Algorithm (Org. and C 1 , . . . , C k := non-overlapping cones with angle θ Imp.) Apx. Greedy Algorithm and with apex at u ; (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) for each cone C i do Sink Spanner Connect u to the closest point in C i ; WSPD-based Algorithm Theoretical end bounds end Applications Designing approximation return G ( V, E ) ; algorithms with spanners Metric space searching Protein Visualization Time Complexity: O ( n log n ) . Research Topics Storage Complexity: O ( n ) .
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. Course Outline Random Ordered Θ -Graph– O (log n ) spanner Textbook Introduction diameter Algorithms Review We add points one by one in a random order. Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Sink Spanner– bounded degree Skip-list spanner) Sink Spanner WSPD-based Algorithm Decrease the degree of nodes by replacing some edges Theoretical by paths within other nodes. bounds Applications Designing approximation Skip-List Spanner– O (log n ) spanner diameter algorithms with spanners Metric space searching Protein Visualization Decrease the diameter of Θ -graph by adding some extra Research Topics edges.
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. Course Outline Random Ordered Θ -Graph– O (log n ) spanner Textbook Introduction diameter Algorithms Review We add points one by one in a random order. Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Sink Spanner– bounded degree Skip-list spanner) Sink Spanner WSPD-based Algorithm Decrease the degree of nodes by replacing some edges Theoretical by paths within other nodes. bounds Applications Designing approximation Skip-List Spanner– O (log n ) spanner diameter algorithms with spanners Metric space searching Protein Visualization Decrease the diameter of Θ -graph by adding some extra Research Topics edges.
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. Course Outline Random Ordered Θ -Graph– O (log n ) spanner Textbook Introduction diameter Algorithms Review We add points one by one in a random order. Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Sink Spanner– bounded degree Skip-list spanner) Sink Spanner WSPD-based Algorithm Decrease the degree of nodes by replacing some edges Theoretical by paths within other nodes. bounds Applications Designing approximation Skip-List Spanner– O (log n ) spanner diameter algorithms with spanners Metric space searching Protein Visualization Decrease the diameter of Θ -graph by adding some extra Research Topics edges.
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. Course Outline Random Ordered Θ -Graph– O (log n ) spanner Textbook Introduction diameter Algorithms Review We add points one by one in a random order. Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Algorithm (Sink and Sink Spanner– bounded degree Skip-list spanner) Sink Spanner WSPD-based Algorithm Decrease the degree of nodes by replacing some edges Theoretical by paths within other nodes. bounds Applications Designing approximation Skip-List Spanner– O (log n ) spanner diameter algorithms with spanners Metric space searching Protein Visualization Decrease the diameter of Θ -graph by adding some extra Research Topics edges.
Sink Spanner A variant of Θ -graph with bounded degree Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) √ t -spanner − → Construct a directed G with bounded Course Outline Textbook out-degree; 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 Algorithm (Sink and end Skip-list spanner) Sink Spanner return G ( V, E ) ; 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
Sink Spanner A variant of Θ -graph with bounded degree Geometric Input : V and t > 1 Spanner Networks Output : t -spanner G ( V, E ) √ t -spanner − → Construct a directed G with bounded Course Outline Textbook out-degree; 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 Algorithm (Sink and end Skip-list spanner) Sink Spanner return G ( V, E ) ; 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
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 ) Set V 0 := V ; i := 1 ; Course Outline Textbook while V i − 1 � = ∅ do Introduction V i contains each points of V i − 1 with probability 1 / 2 ; Algorithms Review end Greedy Algorithm (Org. and Imp.) for each i do Apx. Greedy Algorithm (Ordered) Θ -Graph Construct a t -spanner G i ( V i , E i ) using the Θ -graph Algorithm (Sink and Skip-list spanner) algorithm; Sink Spanner WSPD-based Algorithm end Theoretical E = ∪ i E i ; bounds Applications return G ( V, E ) ; Designing approximation algorithms with spanners Metric space searching Protein Visualization Time Complexity: O ( n log n ) Storage Complexity: O ( n ) . Research Topics
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 ) Set V 0 := V ; i := 1 ; Course Outline Textbook while V i − 1 � = ∅ do Introduction V i contains each points of V i − 1 with probability 1 / 2 ; Algorithms Review end Greedy Algorithm (Org. and Imp.) for each i do Apx. Greedy Algorithm (Ordered) Θ -Graph Construct a t -spanner G i ( V i , E i ) using the Θ -graph Algorithm (Sink and Skip-list spanner) algorithm; Sink Spanner WSPD-based Algorithm end Theoretical E = ∪ i E i ; bounds Applications return G ( V, E ) ; Designing approximation algorithms with spanners Metric space searching Protein Visualization Time Complexity: O ( n log n ) Storage Complexity: O ( n ) . Research Topics
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 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
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 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 Metric space searching A Protein Visualization Research Topics
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 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 Skip-list spanner) r B Sink Spanner D B WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching A Protein Visualization Research Topics r A D A
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 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 (Ordered) Θ -Graph B Algorithm (Sink and Skip-list spanner) r B Sink Spanner D B WSPD-based Algorithm Theoretical bounds Applications ≥ s × max( r A , r B ) Designing approximation algorithms with spanners Metric space searching Protein Visualization A Research Topics r A D A
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 Course Outline such that Textbook ∀ i , A i and B i are s -well separated, Introduction Algorithms Review ∀ p, q ∈ V , there is exactly one index i s. t. Greedy Algorithm (Org. and Imp.) p ∈ A i and q ∈ B i or Apx. Greedy Algorithm (Ordered) Θ -Graph q ∈ A i and p ∈ B i . Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm m : Size of WSPD. Theoretical bounds Callahan & Kosaraju (1995) Applications Designing approximation algorithms with spanners For each set of n points, we can construct a WSPD of Metric space searching size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Protein Visualization Research Topics
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 Course Outline such that Textbook ∀ i , A i and B i are s -well separated, Introduction Algorithms Review ∀ p, q ∈ V , there is exactly one index i s. t. Greedy Algorithm (Org. and Imp.) p ∈ A i and q ∈ B i or Apx. Greedy Algorithm (Ordered) Θ -Graph q ∈ A i and p ∈ B i . Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm m : Size of WSPD. Theoretical bounds Callahan & Kosaraju (1995) Applications Designing approximation algorithms with spanners For each set of n points, we can construct a WSPD of Metric space searching size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Protein Visualization Research Topics
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 Course Outline such that Textbook ∀ i , A i and B i are s -well separated, Introduction Algorithms Review ∀ p, q ∈ V , there is exactly one index i s. t. Greedy Algorithm (Org. and Imp.) p ∈ A i and q ∈ B i or Apx. Greedy Algorithm (Ordered) Θ -Graph q ∈ A i and p ∈ B i . Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm m : Size of WSPD. Theoretical bounds Callahan & Kosaraju (1995) Applications Designing approximation algorithms with spanners For each set of n points, we can construct a WSPD of Metric space searching size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Protein Visualization Research Topics
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 Course Outline such that Textbook ∀ i , A i and B i are s -well separated, Introduction Algorithms Review ∀ p, q ∈ V , there is exactly one index i s. t. Greedy Algorithm (Org. and Imp.) p ∈ A i and q ∈ B i or Apx. Greedy Algorithm (Ordered) Θ -Graph q ∈ A i and p ∈ B i . Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm m : Size of WSPD. Theoretical bounds Callahan & Kosaraju (1995) Applications Designing approximation algorithms with spanners For each set of n points, we can construct a WSPD of Metric space searching size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Protein Visualization Research Topics
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 Course Outline such that Textbook ∀ i , A i and B i are s -well separated, Introduction Algorithms Review ∀ p, q ∈ V , there is exactly one index i s. t. Greedy Algorithm (Org. and Imp.) p ∈ A i and q ∈ B i or Apx. Greedy Algorithm (Ordered) Θ -Graph q ∈ A i and p ∈ B i . Algorithm (Sink and Skip-list spanner) Sink Spanner WSPD-based Algorithm m : Size of WSPD. Theoretical bounds Callahan & Kosaraju (1995) Applications Designing approximation algorithms with spanners For each set of n points, we can construct a WSPD of Metric space searching size O ( s d · n ) in O ( n log n ) time using O ( s d · n ) space. Protein Visualization Research Topics
WSPD-based Algorithm WSPD Algorithm Geometric Spanner Networks Input : V and t > 1 Output : t -spanner G ( V, E ) Course Outline Set W := WSPD of V w.r.t. s := 4( t +1) t − 1 ; Textbook Set E = ∅ ; Introduction Algorithms Review for each ( A i , B i ) ∈ W do Greedy Algorithm (Org. and Select an arbitrary node u ∈ A i and an arbitrary node Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph v ∈ B i ; Algorithm (Sink and Skip-list spanner) Add edge ( u, v ) to E . Sink Spanner WSPD-based Algorithm end Theoretical return G ( V, E ) . bounds Applications Designing approximation algorithms with spanners Time Complexity: O ( n log n ) . Metric space searching Protein Visualization Storage Complexity: O ( n ) . Research Topics
WSPD-based Algorithm WSPD Algorithm Geometric Spanner Networks Input : V and t > 1 Output : t -spanner G ( V, E ) Course Outline Set W := WSPD of V w.r.t. s := 4( t +1) t − 1 ; Textbook Set E = ∅ ; Introduction Algorithms Review for each ( A i , B i ) ∈ W do Greedy Algorithm (Org. and Select an arbitrary node u ∈ A i and an arbitrary node Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph v ∈ B i ; Algorithm (Sink and Skip-list spanner) Add edge ( u, v ) to E . Sink Spanner WSPD-based Algorithm end Theoretical return G ( V, E ) . bounds Applications Designing approximation algorithms with spanners Time Complexity: O ( n log n ) . Metric space searching Protein Visualization Storage Complexity: O ( n ) . Research Topics
Theoretical bounds Geometric Spanner Networks - Size Weight Degree Time O ( n 2 log n ) Greedy spanner O ( n ) O ( wt (MST)) O (1) Course Outline Textbook Apx. greedy spanner O ( n ) O ( wt (MST)) O (1) O ( n log n ) Introduction Θ -graph O ( n ) Θ( n · wt (MST)) Θ( n ) O ( n log n ) Algorithms Review Greedy Algorithm (Org. and O. Θ -graph O ( n ) O ( n · wt (MST)) O (log n ) O ( n log n ) Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph WSPD spanner O ( n ) O (log n · wt (MST)) Θ( n ) O ( n log n ) Algorithm (Sink and Skip-list spanner) Sink-spanner O ( n ) O ( n · wt (MST)) O (1) O ( n log n ) Sink Spanner WSPD-based Algorithm Skip-list spanner O ( n ) ∗ Θ( n · wt (MST)) ∗ Θ( n ) O ( n log n ) ∗ Theoretical bounds Applications (*): Expected with high probability Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
Applications Geometric Spanner Networks 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
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. 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:
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. Course Outline Textbook Known results: Introduction The problem is NP-hard even in R d . Algorithms Review Greedy Algorithm (Org. and Imp.) A 2 -approximation algorithm for metric spaces by Apx. Greedy Algorithm (Ordered) Θ -Graph Rosenkrantz et al. (1977). Algorithm (Sink and Skip-list spanner) Sink Spanner A 1 . 5 -approximation algorithm by Christofides et al. 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 Metric space searching A PTAS for geometric case using spanners by Rao Protein Visualization Research Topics and Smith (1998).
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 (possibly non-simple) cycle in G that visits each point of Course Outline P at least once. Textbook Introduction Observation: 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) Sink Spanner Theorem (Rao and Smith, 1998) WSPD-based Algorithm Theoretical Given a (1 + ε ) -spanner of a set of n points with O ( n ) 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
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 (possibly non-simple) cycle in G that visits each point of Course Outline P at least once. Textbook Introduction Observation: 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) Sink Spanner Theorem (Rao and Smith, 1998) WSPD-based Algorithm Theoretical Given a (1 + ε ) -spanner of a set of n points with O ( n ) 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
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 (possibly non-simple) cycle in G that visits each point of Course Outline P at least once. Textbook Introduction Observation: 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) Sink Spanner Theorem (Rao and Smith, 1998) WSPD-based Algorithm Theoretical Given a (1 + ε ) -spanner of a set of n points with O ( n ) 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
Applications Metric space searching Geometric Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Approximate proximity searching: (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner Multimedia information retrieval, WSPD-based Algorithm Data mining, Theoretical bounds Pattern recognition, Applications Designing approximation Machine learning, algorithms with spanners Metric space searching Protein Visualization Computer vision and Research Topics Biomedical databases.
Applications Metric space searching Geometric Spanner Networks Course Outline Textbook Introduction Algorithms Review Greedy Algorithm (Org. and Imp.) Apx. Greedy Algorithm Approximate proximity searching: (Ordered) Θ -Graph Algorithm (Sink and Skip-list spanner) Sink Spanner Multimedia information retrieval, WSPD-based Algorithm Data mining, Theoretical bounds Pattern recognition, Applications Designing approximation Machine learning, algorithms with spanners Metric space searching Protein Visualization Computer vision and Research Topics Biomedical databases.
Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. 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 Skip-list spanner) to reduce the space. Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. 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 Skip-list spanner) to reduce the space. Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. 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 Skip-list spanner) to reduce the space. Sink Spanner WSPD-based Algorithm Theoretical bounds Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
Applications Metric space searching What is the role of spanners? Geometric Spanner Networks A meter show the similarity between any two objects. 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 Skip-list spanner) to reduce the space. Sink Spanner WSPD-based Algorithm G. Navarro, R. Paredes, and E. Chávez, t-Spanners for metric space Theoretical bounds searching , Data & Knowledge Engineering, pp. 820-854, 2007. Applications Designing approximation algorithms with spanners Metric space searching Protein Visualization Research Topics
Applications Protein Visualization Geometric Spanner Networks 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 D. Russel and L. Guibas, Exploring Protein Folding Trajectories Protein Visualization Research Topics Using Geometric Spanners , Pacific Symposium on Biocomputing, pp. 40-51, 2005.
Research Topics Geometric Current and Future Works: Spanner Networks Dynamic spanners (insert and remove nodes). Course Outline Kinetic spanners (when points move and we want to Textbook maintain an spanner all the time). Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review Greedy Algorithm (Org. and region fault tolerant). Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Spanners among obstacles. Algorithm (Sink and Skip-list spanner) Sink Spanner Optimization problems. WSPD-based Algorithm Theoretical External memory (I/O efficient) algorithms for bounds generating spanners. Applications Designing approximation Experimental works on spanner algorithms. algorithms with spanners Metric space searching Protein Visualization Research Topics
Research Topics Geometric Current and Future Works: Spanner Networks Dynamic spanners (insert and remove nodes). Course Outline Kinetic spanners (when points move and we want to Textbook maintain an spanner all the time). Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review Greedy Algorithm (Org. and region fault tolerant). Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Spanners among obstacles. Algorithm (Sink and Skip-list spanner) Sink Spanner Optimization problems. WSPD-based Algorithm Theoretical External memory (I/O efficient) algorithms for bounds generating spanners. Applications Designing approximation Experimental works on spanner algorithms. algorithms with spanners Metric space searching Protein Visualization Research Topics
Research Topics Geometric Current and Future Works: Spanner Networks Dynamic spanners (insert and remove nodes). Course Outline Kinetic spanners (when points move and we want to Textbook maintain an spanner all the time). Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review Greedy Algorithm (Org. and region fault tolerant). Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Spanners among obstacles. Algorithm (Sink and Skip-list spanner) Sink Spanner Optimization problems. WSPD-based Algorithm Theoretical External memory (I/O efficient) algorithms for bounds generating spanners. Applications Designing approximation Experimental works on spanner algorithms. algorithms with spanners Metric space searching Protein Visualization Research Topics
Research Topics Geometric Current and Future Works: Spanner Networks Dynamic spanners (insert and remove nodes). Course Outline Kinetic spanners (when points move and we want to Textbook maintain an spanner all the time). Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review Greedy Algorithm (Org. and region fault tolerant). Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Spanners among obstacles. Algorithm (Sink and Skip-list spanner) Sink Spanner Optimization problems. WSPD-based Algorithm Theoretical External memory (I/O efficient) algorithms for bounds generating spanners. Applications Designing approximation Experimental works on spanner algorithms. algorithms with spanners Metric space searching Protein Visualization Research Topics
Research Topics Geometric Current and Future Works: Spanner Networks Dynamic spanners (insert and remove nodes). Course Outline Kinetic spanners (when points move and we want to Textbook maintain an spanner all the time). Introduction Fault-tolerant spanners (vertex/edge fault tolerant or Algorithms Review Greedy Algorithm (Org. and region fault tolerant). Imp.) Apx. Greedy Algorithm (Ordered) Θ -Graph Spanners among obstacles. Algorithm (Sink and Skip-list spanner) Sink Spanner Optimization problems. WSPD-based Algorithm Theoretical External memory (I/O efficient) algorithms for bounds generating spanners. Applications Designing approximation Experimental works on spanner algorithms. algorithms with spanners Metric space searching Protein Visualization Research Topics
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