Introduction Algorithms Experiments Summary Algorithms for Compact Letter Displays: Comparison and Evaluation Jens Gramm 1 Jiong Guo 1 uffner 1 Falk H¨ Rolf Niedermeier 1 Hans-Peter Piepho 2 Ramona Schmid 3 1 Friedrich-Schiller-Universit¨ at Jena Institut f¨ ur Informatik 2 Universit¨ at Hohenheim Institut f¨ ur Pflanzenbau und Gr¨ unland 3 Universit¨ at Bielefeld AG Praktische Informatik Statistik unter einem Dach 30 March 2007 Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 1/23
Introduction Algorithms Experiments Summary Outline Introduction 1 All-pairwise comparisons Line displays Letter displays Clique Cover Algorithms 2 Insert-Absorb heuristic Clique-Growing heuristic Search-Tree algorithm Experiments 3 Real data Simulated data Summary 4 Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 2/23
Introduction Algorithms Experiments Summary All-pairwise comparisons Multiple pairwise comparisons among all pairs in a set of n treatments: common task in routine analyses based on analysis of variance (ANOVA) techniques Need a way to visualize the ∼ n 2 pairwise comparison results (significantly different or not significantly different) Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 3/23
Introduction Algorithms Experiments Summary Line displays Line display Exactly those pairwise comparisons among treatments are non-significant that are connected by a common line. Example Given treatments t 1 , . . . , t 5 , let the comparison of t 1 and t 5 is significant and all other comparisons non-significant. t 1 t 2 t 3 t 4 t 5 Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 4/23
Introduction Algorithms Experiments Summary Line displays Line display Exactly those pairwise comparisons among treatments are non-significant that are connected by a common line. Example Given treatments t 1 , . . . , t 5 , let the comparison of t 1 and t 5 is significant and all other comparisons non-significant. t 1 t 2 t 3 t 4 t 5 Disadvantage: not always possible to find a line display [ Piepho , Biometrical J. 2000] Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 4/23
Introduction Algorithms Experiments Summary Letter displays Letter display Exactly those pairwise comparisons among treatments are non-significant that have a common letter. Example Given treatments t 1 , . . . , t 5 , let the significant comparisons be {{ t 1 , t 5 } , { t 1 , t 3 } , { t 2 , t 4 }} . t 1 a b t 2 b d t 3 c d t 4 a c t 5 c d Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 5/23
Introduction Algorithms Experiments Summary Line displays vs. letter displays Letter displays generalize line displays t 1 t 1 a t 2 t 2 a b t 3 t 3 a b t 4 t 4 a b t 5 t 5 b Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 6/23
Introduction Algorithms Experiments Summary Letter display Always possible to find? Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 7/23
Introduction Algorithms Experiments Summary Letter display Always possible to find? Yes: Create a new column with two letters for each pair of not significantly different treatments. Example Given treatments t 1 , . . . , t 5 , let the significant comparisons be {{ t 1 , t 5 } , { t 1 , t 3 } , { t 2 , t 4 }} . t 1 a b t 2 a c d t 3 c e f t 4 b e g t 5 d f g Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 7/23
Introduction Algorithms Experiments Summary Letter display Always possible to find? Yes: Create a new column with two letters for each pair of not significantly different treatments. Example Given treatments t 1 , . . . , t 5 , let the significant comparisons be {{ t 1 , t 5 } , { t 1 , t 3 } , { t 2 , t 4 }} . t 1 a b t 2 a c d t 3 c e f t 4 b e g t 5 d f g ∼ n 2 columns: too large. Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 7/23
Introduction Algorithms Experiments Summary Compact letter displays Goal Find a compact letter display (that is, with minimum number of columns). Questions How large can the letter display get? How easy is it to calculate a letter display? What is a good algorithm for calculating letter displays? Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 8/23
Introduction Algorithms Experiments Summary Compact letter displays Goal Find a compact letter display (that is, with minimum number of columns). Questions How large can the letter display get? unknown How easy is it to calculate a letter display? unknown What is a good algorithm for calculating letter displays? Heuristic [Piepho, J. Comput. Graph. Stat. 2004] Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 8/23
Introduction Algorithms Experiments Summary Theoretical computer science We approach these questions with the tools of theoretical computer science: Focus on provable worst-case running time and provable solution guarantee Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 9/23
Introduction Algorithms Experiments Summary Theoretical computer science We approach these questions with the tools of theoretical computer science: Focus on provable worst-case running time and provable solution guarantee Asymptotic algorithm running time analysis Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 9/23
Introduction Algorithms Experiments Summary Theoretical computer science We approach these questions with the tools of theoretical computer science: Focus on provable worst-case running time and provable solution guarantee Asymptotic algorithm running time analysis Running time is stated not in absolute terms, but in relation to the input size n Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 9/23
Introduction Algorithms Experiments Summary Theoretical computer science We approach these questions with the tools of theoretical computer science: Focus on provable worst-case running time and provable solution guarantee Asymptotic algorithm running time analysis Running time is stated not in absolute terms, but in relation to the input size n Constant factors are ignored Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 9/23
Introduction Algorithms Experiments Summary Theoretical computer science We approach these questions with the tools of theoretical computer science: Focus on provable worst-case running time and provable solution guarantee Asymptotic algorithm running time analysis Running time is stated not in absolute terms, but in relation to the input size n Constant factors are ignored Classification into computational complexity classes captures “intrinsic difficulty” Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 9/23
Introduction Algorithms Experiments Summary Compact letter display: formal definition Compact Letter Display Set T of n treatments, and a set H of m unordered pairs Input: from T . Find a binary n × k matrix M with minimum k such that Task: { t 1 , t 2 } ∈ H ⇐ ⇒ ∃ j : M t 1 , j = M t 2 , j = 1 . Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 10/23
Introduction Algorithms Experiments Summary Clique Cover Clique Cover An undirected Input: graph G = ( V , E ). Find a minimum Task: number k of cliques (subgraphs with all edges present) such that each edge is contained in at least one clique. Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 11/23
Introduction Algorithms Experiments Summary Clique Cover Clique Cover An undirected Input: graph G = ( V , E ). Find a minimum Task: number k of cliques (subgraphs with all edges present) such that each edge is contained in at least one clique. Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 11/23
Introduction Algorithms Experiments Summary Clique Cover Also known as Keyword Conflict [Kellerman, IBM 1973] Intersection Graph Basis [Garey&Johnson 1979] Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 12/23
Introduction Algorithms Experiments Summary Clique Cover Also known as Keyword Conflict [Kellerman, IBM 1973] Intersection Graph Basis [Garey&Johnson 1979] Applications compiler optimization, computational geometry, . . . Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 12/23
Introduction Algorithms Experiments Summary Equivalence of Compact Letter Display and Clique Cover Compact Letter Display = ˆ Clique Cover treatment = ˆ vertex not sign. diff. = ˆ edge column = ˆ clique × a a b × c × c f d × × d e × × h f × × e g × × h × b g There is a letter display with k columns ⇐ ⇒ there is a clique cover with k cliques. Gramm et al. Algorithms for Compact Letter Displays: Comparison and Evaluation 13/23
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