Block Crossings in Storyline Visualizations Thomas van Dijk, Martin Fink , Norbert Fischer, Fabian Lipp, Peter Markfelder, Alexander Ravsky, Subhash Suri, and Alexander Wolff
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block crossing 3/15
We want to minimize block crossings! block crossing
Previous Results – Simple Crossings [Kostitsyna et al, GD’15] NP-hardness FPT for #characters upper and lower bounds for some cases with pairwise meetings 4/15
Related Work Block crossings for metro lines [Fink, Pupyrev, Wolff; 2015] 5/15
Related Work Block crossings for metro lines [Fink, Pupyrev, Wolff; 2015] Bundled Crossings [Fink et al., 2016] 5/15
Related Work Block crossings for metro lines [Fink, Pupyrev, Wolff; 2015] Bundled Crossings [Fink et al., 2016] Bundled Crossing Number [Alam, Fink, Pupyrev; next talk] 5/15
Storylines & Block Crossings block crossing 6/15
Storylines & Block Crossings block crossing permutations π π ′ 6/15
Storylines & Block Crossings block crossing O ( k 3 ) possible block crossings permutations π π ′ 6/15
Storylines & Block Crossings block crossing O ( k 3 ) possible block crossings permutations π π ′ storyline visualization 6/15
Storylines & Block Crossings block crossing O ( k 3 ) possible block crossings permutations π π ′ storyline visualization π 6/15
Storylines & Block Crossings block crossing O ( k 3 ) possible block crossings π supports meeting permutations π π ′ storyline visualization π 6/15
Storylines & Block Crossings block crossing O ( k 3 ) possible block crossings π supports meeting permutations π π ′ storyline visualization Problem definition: π 6/15
Storylines & Block Crossings block crossing O ( k 3 ) possible block crossings π supports meeting permutations π π ′ storyline visualization Problem definition: Given n meetings π of k characters, find permutations transformed by min. # block crossings. 6/15
Storylines & Block Crossings block crossing O ( k 3 ) possible block crossings π supports meeting permutations π π ′ storyline visualization Problem definition: Given n meetings π of k characters, find permutations transformed by min. # block crossings. (Must support all meetings.) 6/15
Our Results recognize crossing-free instances NP-hardness approximation FPT/exact algorithms greedy heuristic for pairwise meetings 7/15
Crossing-Free Storylines Visualizations 8/15
Crossing-Free Storylines Visualizations π 8/15
Crossing-Free Storylines Visualizations π supports each meeting 8/15
Crossing-Free Storylines Visualizations π supports each meeting group hypergraph H = ( C , Γ ) is interval hypergraph 8/15
Crossing-Free Storylines Visualizations π supports each meeting group hypergraph H = ( C , Γ ) is interval hypergraph groups that meet 8/15
Crossing-Free Storylines Visualizations π supports each meeting group hypergraph H = ( C , Γ ) is interval hypergraph groups that meet interval hypergraph property can be checked in O ( k 2 ) time [Trotter, Moore, 1976] 8/15
Minimizing Block Crossings is NP-hard Reduction from Sorting by Transpositions v 4 v 1 v 3 v 2 v 1 v 3 v 5 v 4 v 2 v 5 9/15
Minimizing Block Crossings is NP-hard Reduction from Sorting by Transpositions sort fixed permutations with minimum number of block crossings v 4 v 1 v 3 v 2 v 1 v 3 v 5 v 4 v 2 v 5 9/15
Minimizing Block Crossings is NP-hard Reduction from Sorting by Transpositions sort fixed permutations with minimum number of block crossings v 4 v 1 v 3 v 2 v 1 v 3 v 5 v 4 v 2 v 5 9/15
Minimizing Block Crossings is NP-hard Reduction from Sorting by Transpositions sort fixed permutations with minimum number of block crossings v 4 v 1 v 3 v 2 v 1 v 3 v 5 v 4 v 2 v 5 fix permutations by repeated meetings 9/15
Minimizing Block Crossings is NP-hard Reduction from Sorting by Transpositions sort fixed permutations with minimum number of block crossings v 4 v 1 v 3 v 2 v 1 v 3 v 5 v 4 v 2 v 5 � �� � � �� � k =5 times k =5 times fix permutations by repeated meetings 9/15
Minimizing Block Crossings is NP-hard Reduction from Sorting by Transpositions v 4 v 1 v 5 v 3 v 2 v 4 v 1 v 3 v 3 v 5 v 4 v 2 v 2 v 5 v 1 � �� � � �� � k =5 times k =5 times fix permutations by repeated meetings add frame to prevent reversal 9/15
Minimizing Block Crossings is NP-hard Reduction from Sorting by Transpositions u 1 u 1 u 2 u 2 u 3 u 3 u 4 u 4 u 5 u 5 v 4 v 1 v 3 v 2 v 1 v 3 v 5 v 4 v 2 v 5 � �� � � �� � k =5 times k =5 times fix permutations by repeated meetings add frame to prevent reversal 9/15
Approximation Algorithm all meetings of size ≤ d (constant) no repeated meetings 10/15
Approximation Algorithm all meetings of size ≤ d (constant) no repeated meetings idea: 1. choose starting order π that supports many meetings 2. temporarily change order for each unsupported meeting 10/15
Approximation Algorithm all meetings of size ≤ d (constant) no repeated meetings idea: 1. choose starting order π that supports many meetings 2. temporarily change order for each unsupported meeting ≤ 2( d − 1) block crossings 10/15
Approximation Algorithm all meetings of size ≤ d (constant) no repeated meetings idea: 1. choose starting order π that supports many meetings 2. temporarily change order for each unsupported meeting meetings supported by π are free ≤ 2( d − 1) block crossings 10/15
Approximation Algorithm all meetings of size ≤ d (constant) no repeated meetings idea: 1. choose starting order π that supports many meetings 2. temporarily change order for each unsupported meeting meetings supported by π are free ≤ 2( d − 1) block crossings Lemma: starting order π has α unsupported meetings ⇒ at least 4 α/ (3 d 2 ) block crossings necessary 10/15
Approximation Algorithm all meetings of size ≤ d (constant) no repeated meetings idea: 1. choose starting order π that supports many meetings 2. temporarily change order for each unsupported meeting meetings supported by π are free ≤ 2( d − 1) block crossings approximate α OPT Lemma: starting order π has α ⇒ approximate unsupported meetings ⇒ block crossings at least 4 α/ (3 d 2 ) block crossings necessary 10/15
Approximation Algorithm find π minimizing #unsupported meetings 11/15
Approximation Algorithm find π minimizing #unsupported meetings ↔ remove minimum #meetings so that storyline crossing-free 11/15
Approximation Algorithm find π minimizing #unsupported meetings ↔ remove minimum #meetings so that storyline crossing-free ↔ remove minimum #hyperedges so that H is interval hypergraph 11/15
Approximation Algorithm find π minimizing #unsupported meetings ↔ remove minimum #meetings so that storyline crossing-free ↔ remove minimum #hyperedges so that H is interval hypergraph Theorem: Interval Hypergraph Edge Deletion admits a ( d + 1)-approximation (constant rank d ). 11/15
Approximation Algorithm find π minimizing #unsupported meetings ↔ remove minimum #meetings so that storyline crossing-free ↔ remove minimum #hyperedges so that H is interval hypergraph Theorem: Interval Hypergraph Edge Deletion admits a ( d + 1)-approximation (constant rank d ). Theorem: We can find a (3( d 2 − 1) d 2 / 2)-approximation for the minimum number of block crossings in storyline visualizations in O ( kn ) time. 11/15
Interval Hypergraph Edge Deletion Remove minimum number of hyperedges so that H = ( V , E ) becomes interval hypergraph 12/15
Interval Hypergraph Edge Deletion Remove minimum number of hyperedges so that H = ( V , E ) becomes interval hypergraph NP-hard for graphs: remove all but n − 1 edges → Hamiltonian path 12/15
Interval Hypergraph Edge Deletion Remove minimum number of hyperedges so that H = ( V , E ) becomes interval hypergraph characterization of interval hypergraphs by forbidden subhypergraphs F 1 F 2 F d − 2 O 1 O 2 M 1 M 2 M d − 1 C 3 C 4 12/15
Interval Hypergraph Edge Deletion Remove minimum number of hyperedges so that H = ( V , E ) becomes interval hypergraph characterization of interval hypergraphs by forbidden subhypergraphs F 1 F 2 F d − 2 O 1 O 2 M 1 M 2 M d − 1 C 3 C 3 C 4 C 4 12/15
Interval Hypergraph Edge Deletion Remove minimum number of hyperedges so that outline: H = ( V , E ) becomes interval hypergraph – iteratively: search for forbidden subhypergraphs except C d +2 , . . . & characterization of interval hypergraphs by forbidden completely remove them subhypergraphs – result: cyclic generalization of interval hypergraph; break optimally F 1 F 2 F d − 2 O 1 O 2 M 1 M 2 M d − 1 C 3 C 4 12/15
Interval Hypergraph Edge Deletion Remove minimum number of hyperedges so that outline: H = ( V , E ) becomes interval hypergraph – iteratively: search for forbidden ≤ d + 1 hyperedges subhypergraphs except C d +2 , . . . & characterization of interval hypergraphs by forbidden completely remove them subhypergraphs – result: cyclic generalization of interval hypergraph; break optimally F 1 F 2 F d − 2 O 1 O 2 M 1 M 2 M d − 1 C 3 C 4 12/15
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