Introduction and results statement Technical details Conclusion On the Gapped Consecutive-Ones Property Cedric Chauve, J´ an Manuch and Murray Patterson Dept. Mathematics and School of Computing Science, Simon Fraser University, Canada EuroComb 2009, September 7th, 2009 Work funded by NSERC and the France-Canada Scientific Cooperation. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped Consecutive-Ones Property Definitions. 1. Let M be a binary matrix. A gap of length δ in a row of M is a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped Consecutive-Ones Property Definitions. 1. Let M be a binary matrix. A gap of length δ in a row of M is a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row. 2. M is k -C1P if there exists a total order of its columns such that each row contains at most k − 1 gaps. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped Consecutive-Ones Property Definitions. 1. Let M be a binary matrix. A gap of length δ in a row of M is a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row. 2. M is k -C1P if there exists a total order of its columns such that each row contains at most k − 1 gaps. 3. M is ( k , δ )-C1P if it is k -C1P and each gap is of length at most δ . C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped Consecutive-Ones Property Definitions. 1. Let M be a binary matrix. A gap of length δ in a row of M is a sequence of δ 0’s framed by a 1 on each extremity. The degree of a row of M is the number of 1s in this row. 2. M is k -C1P if there exists a total order of its columns such that each row contains at most k − 1 gaps. 3. M is ( k , δ )-C1P if it is k -C1P and each gap is of length at most δ . Example: a (2 , 1)-C1P matrix of maximum degree 3. 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped C1P: Problem and Our Results Summary Problem. What is the complexity of deciding if, given a binary matrix M , k ≥ 2 and δ ≥ 1, M is a ( k , δ )-C1P matrix ? C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped C1P: Problem and Our Results Summary Problem. What is the complexity of deciding if, given a binary matrix M , k ≥ 2 and δ ≥ 1, M is a ( k , δ )-C1P matrix ? Theorem 1. Deciding if a binary matrix M is ( k , δ )-C1P is NP-complete for k , δ ≥ 2. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped C1P: Problem and Our Results Summary Problem. What is the complexity of deciding if, given a binary matrix M , k ≥ 2 and δ ≥ 1, M is a ( k , δ )-C1P matrix ? Theorem 1. Deciding if a binary matrix M is ( k , δ )-C1P is NP-complete for k , δ ≥ 2. Theorem 2. Deciding if a binary matrix M is ( k , 1)-C1P is NP-complete for k ≥ 3. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion The Gapped C1P: Problem and Our Results Summary Problem. What is the complexity of deciding if, given a binary matrix M , k ≥ 2 and δ ≥ 1, M is a ( k , δ )-C1P matrix ? Theorem 1. Deciding if a binary matrix M is ( k , δ )-C1P is NP-complete for k , δ ≥ 2. Theorem 2. Deciding if a binary matrix M is ( k , 1)-C1P is NP-complete for k ≥ 3. Theorem 3. Deciding if an m × n binary matrix M of maximum degree s is ( k , δ )-C1P can be done in O ( nm s +( k − 1) δ ) worst-case time and space. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Motivation: paleogenomics Reconstructing ancestral genomes (that can not be sequenced due to DNA decay) from common characters of current genomes (Chauve and Tannier 2008). ... chicken human mouse dog 2 C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Motivation: paleogenomics 1. Binary matrix can be used to encode possible genome segments of an unknown ancestral genome: C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Motivation: paleogenomics 1. Binary matrix can be used to encode possible genome segments of an unknown ancestral genome: ◮ Columns represent “genes” that were present in the ancestral genome. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Motivation: paleogenomics 1. Binary matrix can be used to encode possible genome segments of an unknown ancestral genome: ◮ Columns represent “genes” that were present in the ancestral genome. ◮ Rows represent groups of genes that should be co-localized. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Motivation: paleogenomics 1. Binary matrix can be used to encode possible genome segments of an unknown ancestral genome: ◮ Columns represent “genes” that were present in the ancestral genome. ◮ Rows represent groups of genes that should be co-localized. 2. Ordering columns correspond to ordering genes along chromosomes of the unknown ancestral genome. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Motivation: paleogenomics 1. Binary matrix can be used to encode possible genome segments of an unknown ancestral genome: ◮ Columns represent “genes” that were present in the ancestral genome. ◮ Rows represent groups of genes that should be co-localized. 2. Ordering columns correspond to ordering genes along chromosomes of the unknown ancestral genome. 3. Strict combinatorial framework: C1P=(1 , 0)-C1P (Chauve and Tannier 2008). C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Motivation: paleogenomics 1. Binary matrix can be used to encode possible genome segments of an unknown ancestral genome: ◮ Columns represent “genes” that were present in the ancestral genome. ◮ Rows represent groups of genes that should be co-localized. 2. Ordering columns correspond to ordering genes along chromosomes of the unknown ancestral genome. 3. Strict combinatorial framework: C1P=(1 , 0)-C1P (Chauve and Tannier 2008). 4. Relaxed combinatorial framework to account for evolutionary noise (missing genes, lineage-specific rearrangements, . . . ): ( k , δ )-C1P with small values for k and δ . C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Link with the Graph bandwidth Definition. A graph G = ( V , E ) has bandwidth b if its vertices can be ordered, say v 1 . . . v n , such that for every edge ( v i , v j ), | j − i | ≤ b . C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Link with the Graph bandwidth Definition. A graph G = ( V , E ) has bandwidth b if its vertices can be ordered, say v 1 . . . v n , such that for every edge ( v i , v j ), | j − i | ≤ b . Matrices and graphs. A graph G has bandwidth b if and only if its adjacency matrix M G (of maximum degree 2) is (2 , b − 1)-C1P. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Link with the Graph bandwidth Definition. A graph G = ( V , E ) has bandwidth b if its vertices can be ordered, say v 1 . . . v n , such that for every edge ( v i , v j ), | j − i | ≤ b . Matrices and graphs. A graph G has bandwidth b if and only if its adjacency matrix M G (of maximum degree 2) is (2 , b − 1)-C1P. Known results on the Graph Bandwidth. C. Chauve et al. On the Gapped Consecutive-Ones Property
Introduction and results statement Technical details Conclusion Link with the Graph bandwidth Definition. A graph G = ( V , E ) has bandwidth b if its vertices can be ordered, say v 1 . . . v n , such that for every edge ( v i , v j ), | j − i | ≤ b . Matrices and graphs. A graph G has bandwidth b if and only if its adjacency matrix M G (of maximum degree 2) is (2 , b − 1)-C1P. Known results on the Graph Bandwidth. 1. Bandwidth Minimization is NP-complete (Garey et al. 1978). C. Chauve et al. On the Gapped Consecutive-Ones Property
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