Graph Stories in Small Area 27th Int. Symposium on Graph Drawing and - PowerPoint PPT Presentation

Graph Stories in Small Area 27th Int. Symposium on Graph Drawing and Network Visualization (GD 2019) Manuel Borrazzo, Giordano Da Lozzo , Fabrizio Frati, Maurizio Patrignani D EPARTMENT OF E NGINEERING · R OMA T RE U NIVERSITY GD’19 - Graph Stories in Small Area GD’19 - Graph Stories in Small Area GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 7 G 4 1 2 3 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 7 G 4 1 1 2 vertices are only born 3 t ≤ W 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) edge (1, 2) bw vertices of G 5 that exist at the same time 7 G 4 1 1 2 2 vertices are only born 3 t ≤ W 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 7 G 4 1 1 2 2 vertices are only born 3 3 t ≤ W 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 7 G 4 4 1 2 2 vertices are one vertex is only born born and one 3 3 t ≤ W dies W < t ≤ n 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 5 7 G 4 4 1 2 vertices are one vertex is only born born and one 3 3 t ≤ W dies W < t ≤ n 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 5 one vertex is 7 G born and one dies W < t ≤ n 4 4 1 2 vertices are only born 3 t ≤ W 6 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 5 one vertex is 7 7 G born and one dies W < t ≤ n 4 1 2 vertices are only born 3 t ≤ W 6 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 one vertex is 7 7 G born and one dies W < t ≤ n 4 1 2 vertices are only born 3 t ≤ W 6 6 8 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 one vertex is vertices 7 7 G born and one only die dies W < t ≤ n t > n 4 1 2 vertices are only born 3 t ≤ W 6 8 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 one vertex is vertices 7 G born and one only die dies W < t ≤ n t > n 4 1 2 vertices are only born 3 t ≤ W 6 8 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Graphs that change over time: ◮ vertices enter the graph one after the other, at discrete time instants ◮ each persists in the graph for a fixed amount of time W ( window size ) 5 one vertex is vertices 7 G born and one only die dies W < t ≤ n t > n 4 1 2 vertices are only born 3 t ≤ W 6 8 time W = 3 0 1 2 3 4 5 6 7 8 9 10 11 W n n + W GD’19 - Graph Stories in Small Area

Graph stories Definition 1 A graph story is a triple � G , τ , W � ◮ G = ( V , E ) is a graph ◮ a bijection τ : V ↔ { 1, . . . , | V |} vertex v appears in G at time τ ( v ) vertex v leaves G at time τ ( v ) + W ◮ W is a positive integer 5 s e c e i m t r e i t v 7 G τ ( ) = 1 τ ( ) = 2 4 1 2 τ ( ) = 3 τ ( ) = 4 τ ( ) = 5 τ ( ) = 6 τ ( ) = 7 3 6 τ ( ) = 8 8 W = 3 GD’19 - Graph Stories in Small Area

Graph stories Definition 1 A graph story is a triple � G , τ , W � ◮ G = ( V , E ) is a graph ◮ a bijection τ : V ↔ { 1, . . . , | V |} vertex v appears in G at time τ ( v ) vertex v leaves G at time τ ( v ) + W ◮ W is a positive integer 5 s e c e i m t r e i G t := the subgraph of G t v 7 G τ ( ) = 1 induced by V t = { v ∈ V : t − W < τ ( v ) ≤ t } τ ( ) = 2 4 1 2 τ ( ) = 3 τ ( ) = 4 G 5 τ ( ) = 5 fixed lifetime ≃ sliding window τ ( ) = 6 τ ( ) = 7 3 6 τ ( ) = 8 8 W = 3 GD’19 - Graph Stories in Small Area

Graph stories Definition 1 A graph story is a triple � G , τ , W � ◮ G = ( V , E ) is a graph ◮ a bijection τ : V ↔ { 1, . . . , | V |} vertex v appears in G at time τ ( v ) vertex v leaves G at time τ ( v ) + W ◮ W is a positive integer 5 s e c e i m t r e i G t := the subgraph of G t v 7 G τ ( ) = 1 induced by V t = { v ∈ V : t − W < τ ( v ) ≤ t } τ ( ) = 2 4 1 2 τ ( ) = 3 τ ( ) = 4 τ ( ) = 5 fixed lifetime ≃ sliding window G 6 τ ( ) = 6 τ ( ) = 7 3 6 τ ( ) = 8 8 W = 3 GD’19 - Graph Stories in Small Area

Graph stories Definition 1 A graph story is a triple � G , τ , W � ◮ G = ( V , E ) is a graph ◮ a bijection τ : V ↔ { 1, . . . , | V |} vertex v appears in G at time τ ( v ) vertex v leaves G at time τ ( v ) + W ◮ W is a positive integer sequence = , , , , , G 6 G 3 G 4 G 5 � G 1 , G 2 , . . . , G W − 1 , G W , . . . , G n , G n +1 , . . . , G n + W − 1 � size < W size = W size < W GD’19 - Graph Stories in Small Area

Drawing stories (of graph stories) Definition 2 A drawing story for � G , τ , W � is a sequence Γ = � Γ 1 , Γ 2 , . . . , Γ n + W − 1 � such that: ◮ Γ t is a drawing of G t ◮ if v ∈ V ( G i ) ∩ V ( G j ), then v is in the same position in Γ i and in Γ j ◮ if e ∈ E ( G i ) ∩ E ( G j ), then e is represented by the same curve in Γ i and in Γ j Benefit: preserve the user’s mental map through the sequence i.e., Γ = � Γ 1 , Γ 2 , . . . , Γ n + W − 1 � is a SEFE of � G 1 , G 2 , . . . , G n + W − 1 � GD’19 - Graph Stories in Small Area

Drawing stories (of graph stories) Our focus: drawings stories that are planar , straight-line ( → SGE ), and on the grid A graph story � G , τ , W � may admit such drawing stories even if G is not planar For instance, if W=3 just place the vertices in general positions on a grid GD’19 - Graph Stories in Small Area

Drawing stories (of graph stories) Our focus: drawings stories that are planar , straight-line ( → SGE ), and on the grid A graph story � G , τ , W � always admits such drawing stories if G is planar A graph story � G , τ , W � may admit such drawing stories even if G is not planar A naive approach would produce drawing stories on the O ( n ) × O ( n ) grid (de Fraysseix, Pach and Pollack, Schnyder, . . . ) For instance, if W=3 just place the vertices in general positions on a grid this may result in unnecessarily large drawings GD’19 - Graph Stories in Small Area

Drawing stories in small area We studied (straight-line planar grid) drawing stories of planar graph stories with the goal of producing drawing stories Γ = � Γ 1 , Γ 2 , . . . , Γ n + W − 1 � such that each Γ t ∈ Γ has an area that is a function of W , not of n RESULTS: GD’19 - Graph Stories in Small Area