Motivation The collision-free string partition problems Complexity proofs Summary The complexity of string partitioning Anne Condon 1 nuch 1 , 2 Chris Thachuk 1 J´ an Maˇ 1 Department of Computer Science University of British Columbia 2 Department of Mathematics Simon Fraser University CPM 2012 Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Complexity proofs Summary Outline Motivation 1 The collision-free string partition problems 2 Problem definition Results summary Complexity proofs 3 Complexity of EF-MSP (partition size K = 2) Complexity of EF-SP (partition size K = 2) Complexity of EF-MSP (alphabet size L = 2) Complexity of EF-SP (alphabet size L = 2) Summary 4 Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Complexity proofs Summary Synthesizing Novel Genes design goal synthesize a long duplex of DNA/RNA TGGCTATCTTAAAAAGAGGAGCTAGAAAAAAGGTACATCAGGAGCCAGCTAAACGCTCTTCTAAAATCTCCGACGAAGCCA 5' - - 3' ACCGATAGAATTTTTCTCCTCGATCTTTTTTCCATGTAGTCCTCGGTCGATTTGCGAGAAGATTTTAGAGGCTGCTTCGGT 3' - - 5' Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Complexity proofs Summary Synthesizing Novel Genes design goal synthesize a long duplex of DNA/RNA (some) challenges only short fragments can be synthesized reliably TGGCTATCTTAAAAAGAGGAGCTAGAAAAAAGGTACATCAGGAGCCAGCTAAACGCTCTTCTAAAATCTCCGACGAAGCCA 5' - - 3' ACCGATAGAATTTTTCTCCTCGATCTTTTTTCCATGTAGTCCTCGGTCGATTTGCGAGAAGATTTTAGAGGCTGCTTCGGT 3' - - 5' Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Complexity proofs Summary Synthesizing Novel Genes design goal synthesize a long duplex of DNA/RNA (some) challenges only short fragments can be synthesized reliably fragments must cover duplex fragments must self-assemble correctly 5' - TGGCTATCTTAAAAAGAGGAGCTAGAAAAAAGGTACATCAGGAGCCAGCTAAACGCTCTTCTAAAATCTCCGACGAAGCCA - 3' ACCGATAGAATTTTTCTCCTCGATCTTTTTTCCATGTAGTCCTCGGTCGATTTGCGAGAAGATTTTAGAGGCTGCTTCGGT 3' - - 5' Oi TGGCTATCTT GGAGCTAGAAAAA ATCAGGAGCCAG CGCTCTTCTAAA CGACGAAGCCA AATTTTTCTCC TTTTTTCCATGT TCGATTTGCG TTTTAGAGGCTG Oj Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Complexity proofs Summary Collision Example Example Fragments c and g are: identical on the same strand b d f h 5' CATTAATCGCA GGGATCGATTCGTT CCTGAATCGAGCAA GGGAAACTGCAAACG 3' 3' TTACGCGTAAGTAA CGTTATGGTCCCT GCAAACGTCAAAGGG CGTTATGGTCCCT 5' a c e g Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Complexity proofs Summary Collision Example Example Fragments c and g are: identical on the same strand b d f h 5' CATTAATCGCA GGGATCGATTCGTT CCTGAATCGAGCAA GGGAAACTGCAAACG 3' 3' TTACGCGTAAGTAA CGTTATGGTCCCT GCAAACGTCAAAGGG CGTTATGGTCCCT 5' a c e g h f d b A C G C A A C T G A A A CCTGAATCGAGCAA G G G GGGATCGATTCGTT C A T C G T A A C A T C C T GCAAACGTCAAAGGG G T C A T G T T CGTTATGGTCCCT C G T A A A A G C G T C G T T A g e c a Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary K -partition Definition ( K -partition of a string w ) A K -partition of a string w is a sequence P = p 1 , p 2 , . . . , p ℓ for some ℓ such that w = p 1 p 2 . . . p ℓ and for each i = 1 , . . . ℓ , | p i | ≤ K . Example m i s s i s s i p p i p 1 p 2 p 3 p 4 p 5 p 6 Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary K -partition Definition ( K -partition of a string w ) A K -partition of a string w is a sequence P = p 1 , p 2 , . . . , p ℓ for some ℓ such that w = p 1 p 2 . . . p ℓ and for each i = 1 , . . . ℓ , | p i | ≤ K . We say: p 1 , . . . , p ℓ are selected and for any i ≤ j , string p i p i +1 . . . p j is super-selected. Example m i s s i s s i p p i p 1 p 2 p 3 p 4 p 5 p 6 Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary K -partition Definition ( K -partition of a string w ) A K -partition of a string w is a sequence P = p 1 , p 2 , . . . , p ℓ for some ℓ such that w = p 1 p 2 . . . p ℓ and for each i = 1 , . . . ℓ , | p i | ≤ K . We say: p 1 , . . . , p ℓ are selected and for any i ≤ j , string p i p i +1 . . . p j is super-selected. Example m i s s i s s i p p i p 1 p 2 p 3 p 4 p 5 p 6 Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary K -partition Definition ( K -partition of a string w ) A K -partition of a string w is a sequence P = p 1 , p 2 , . . . , p ℓ for some ℓ such that w = p 1 p 2 . . . p ℓ and for each i = 1 , . . . ℓ , | p i | ≤ K . We say: p 1 , . . . , p ℓ are selected and for any i ≤ j , string p i p i +1 . . . p j is super-selected. Definition ( K -partition of a set of strings W ) A K -partition of a set of strings W is a set of K -partitions of all strings in W . Example a c g g g a t m i s s i s s i p p i c c t a g c g g a p 1 p 2 p 3 p 4 p 5 p 6 c a g g g c t a Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Problem definition X -Free String Partition ( X -SP) Problem Instance Finite alphabet Σ, a positive integer K , and a string w from Σ ∗ . Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Problem definition X -Free String Partition ( X -SP) Problem Instance Finite alphabet Σ, a positive integer K , and a string w from Σ ∗ . Question Is there an X -free, K -partition P of w ? Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Problem definition X -Free String Partition ( X -SP) Problem Instance Finite alphabet Σ, a positive integer K , and a string w from Σ ∗ . Question Is there an X -free, K -partition P of w ? X -Free Multiple String Partition ( X -SP) Problem Instance Finite alphabet Σ, a positive integer K , and a set of strings W from Σ ∗ . Question Is there an X -free, K -partition P of W ? Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Types of collisions Types of collisions we consider X m i s s i s s i p p i m i s s i s s i p p i equality (E) Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Types of collisions Types of collisions we consider X m i s s i s s i p p i m i s s i s s i p p i equality (E) m i s s i s s i p p i m i s s i s s i p p i factor (F) Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Types of collisions Types of collisions we consider X m i s s i s s i p p i m i s s i s s i p p i equality (E) m i s s i s s i p p i m i s s i s s i p p i factor (F) m i s s i s s i p p i m i s s i s s i p p i prefix (P) Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Results X constant partition size K constant alphabet size L equality (E) factor (F) prefix (P) Note. If both K and L are constant, then any string longer than L + L 2 + · · · + L K does not have an X -free K -partition. Since the number of strings shorter or equal to this number is constant, the problem can be solved in constant time. Similarly, the problem is easily solvable for the unary alphabet ( L = 1) or the partition size K = 1. Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Results Previous results X constant partition size K constant alphabet size L equality (E) NP-c for K = 2 NP-c for L = 4 factor (F) prefix (P) Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Motivation The collision-free string partition problems Problem definition Complexity proofs Results summary Summary Results New results X constant partition size K constant alphabet size L equality (E) NP-c for K = 2 NP-c for L = 2 factor (F) NP-c for K = 3 NP-c for L = 2 prefix (P) NP-c for K = 2 NP-c for L = 2 Condon, Maˇ nuch and Thachuk The complexity of string partitioning
Recommend
More recommend
