Did you know that Multiple Alignment is NP-hard? Isaac Elias Royal Institute of Technology Sweden

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Transcription:

Did you know that Multiple Alignment is NP-hard? Isaac Elias Royal Institute of Technology Sweden 1

Results Multiple Alignment with SP-score Star Alignment Tree Alignment (with given phylogeny) are NP-hard under all metrics! 2

Pairwise Alignment Mutations: substitutions, insertions, and deletions. Input: Two related strings s 1 and s 2. Output: The least number of mutations needed to s 1 s 2. s 1 = a a g a c t s 2 = a g t g c t 3

Pairwise Alignment Mutations: substitutions, insertions, and deletions. Input: Two related strings s 1 and s 2. Output: The least number of mutations needed to s 1 s 2. 2 l matrix A s 1 = a a g a c t s 2 = a g t g c t d A (s 1, s 2 ) = 3 mutations 3

Metric symbol distance Unit metric - binary alphabet (the edit distance) Σ 0 1 0 0 1 1 1 1 0 1 1 1 0 4

Metric symbol distance Unit metric - binary alphabet (the edit distance) Σ 0 1 0 0 1 1.5 1 1 0 1.5 1.5 1.5 0 Insertions and deletions occur less frequently! 4

Metric symbol distance Unit metric - binary alphabet (the edit distance) Σ 0 1 0 0 1 1 1 1 0 1 1 1 0 Insertions and deletions occur less frequently! General metric - α, β, γ have metric properties (identity, symmetry, triangle ineq.) Σ 0 1 0 0 α β 1 α 0 γ β γ 0 4

Multiple Alignment When sequence similarity is low pairwise alignment may fail to find similarities. 5

Multiple Alignment When sequence similarity is low pairwise alignment may fail to find similarities. Considering several strings may be helpful. Input: k strings s 1... s k. Output: k l matrix A. a a g a - c t - - g a g c t a c g a g c - a - g t g c t 5

Multiple Alignment When sequence similarity is low pairwise alignment may fail to find similarities. Considering several strings may be helpful. Input: k strings s 1... s k. Output: k l matrix A. a a g a - c t - - g a g c t a c g a g c - a - g t g c t How do we score columns? 5

Sum of Pairs score (SP-score) Let the cost be the sum of costs for all pairs of rows: k i=1 k j=i d A (s i, s j ), where d A (s i, s j ) is the pairwise distance between the rows containing strings s i and s j. 6

Earlier NP results Wang and Jiang 94 - Non-metric Bonizzoni and Vedova 01 - Specific metric (we build on their construction) Just 01 - Many metrics

Earlier NP results Wang and Jiang 94 - Non-metric Bonizzoni and Vedova 01 - Specific metric (we build on their construction) Just 01 - Many metrics

Earlier NP results Wang and Jiang 94 - Non-metric Bonizzoni and Vedova 01 - Specific metric (we build on their construction) Just 01 - Many metrics

Earlier NP results Wang and Jiang 94 - Non-metric Bonizzoni and Vedova 01 - Specific metric (we build on their construction) Just 01 - Many metrics In particular it was unknown for the unit metric.

Earlier NP results Wang and Jiang 94 - Non-metric Bonizzoni and Vedova 01 - Specific metric (we build on their construction) Just 01 - Many metrics In particular it was unknown for the unit metric. Here: All binary or larger alphabets under all metrics!

Independent R3 Set Independent set in three regular graphs, i.e. all vertices have degree 3, is NP-hard. Input: A three regular graph G = (V, E) and a integer c. Output: Yes if there is an independent set V V of size c, otherwise No. 8

Independent R3 Set Independent set in three regular graphs, i.e. all vertices have degree 3, is NP-hard. Input: A three regular graph G = (V, E) and a integer c. Output: Yes if there is an independent set V V of size c, otherwise No. 8

Reduction Independent R3 Set G = (V, E) c set of size c V SP-score Set of strings K matrix of cost K A 9

Reduction Independent R3 Set G = (V, E) c set of size c V SP-score Set of strings K matrix of cost K A v 1 v 4 v 2 v 3 v 1 v 2 v 3 v 4 1 0... 0... 1 0... 0... 1 0... 0... 1.......... 1 0... 0... 1 0... 0... 1 0... 0... 1 1. 1 0 0... 10... 1 0... 0... 0 0... 0... 0 0... 0... 1 0... 0... 0 0... 10... 0 0... 0... 0 0 0... 10... 0 0... 0... 0 0... 0... 1 0... 0... 0 0... 0... 1 0... 10... 0 0... 0... 0 0... 0 0... 0... 1 0... 0... 0 0... 10... 0 0... 0 0... 0... 0 0... 0... 1 0... 10... 0 9

Gadgetering 1. Each vertex is represented by a column (n vertex columns). 2. c vertex columns are picked. 3. Each edge is represented by an edge string. v 1 v 2... v i... vn 1 0... 0... 1... 1... 1........... 1 0... 0... 1... 1... 1 1 1.. 1 1 0 0... 10... 1 0... 0... 0 0... 0... 0 0... 0... 1 0... 0... 0 0... 10... 0 0... 0... 0 0 0... 10... 0 0... 0... 0 0... 0... 1 0... 0... 0 0... 0... 1 0... 10... 0 0... 0... 0 0... 0 0... 0... 1 0... 0... 0 0... 10... 0 0... 0 0... 0... 0 0... 0... 1 0... 10... 0 10

Template strings Vertex columns We add very many template strings: T = (10 b ) n 1 1 v 1 v 2... v i... vn 1 0... 0... 1... 1... 1........... 1 0... 0... 1... 1... 1 Fact: Identical strings are aligned identically in an optimal alignment. 11

Pick strings V V We add very many pick strings: P = 1 c. v 1 v 2... v i... vn 1 0... 0... 1... 1... 1........... 1 0... 0... 1... 1... 1 1 1.. 1 1 Fact: c vertex columns are picked since this is the alignment with least missmatches. 12

Edge strings Property: If there is an edge (v i, v j ) then both v i and v j can not be part of the independent set. E ij = 0 s (00 b ) i 1 10 b s (00 b ) j i 1 10 b (00 b ) n j 1 00 s v 1 v i...... v j... vn 1 0... 0... 1...... 1... 1............... 1 0... 0... 1...... 1... 1 good 0 s 0 0... 0... 1 0... 1 0 s 1 0 0... 0... 0 good 0 0... 0... 0 0 s 1 1 0... 1 0... 0... 0 0 s bad 0 s 0 0... 0... 1 s 0... 1 0... 0... 0 0 s 13

Independent set c Alignment K ( K = (n c)γb 2 + b(n 1)βb 2 + (sβ + nα)bm + cα + (s + b(n ) 1) + n c 2)β + 2γ bm 3cb(α + γ β) + (sβ + 2α)m 2 14

Independent set c Alignment K 1. Remember three regular graph. So there are atmost three ones in each vertex column. 2. If a column with only two ones is picked then there is a missmatch extra for each pick string ( score > K). 14

Example 1. Atmost three ones in each vertex column. 2. Pick strings pick columns with most ones. v 1 v 4 v 2 v 3 v 1 v 2 v 3 v 4 1 0... 0... 1 0... 0... 1 0... 0... 1.......... 1 0... 0... 1 0... 0... 1 0... 0... 1 1. 1 E 12 0 0... 10... 1 0... 0... 0 0... 0... 0 0... E 13 0... 1 0... 0... 0 0... 10... 0 0... 0... 0 E 14 0 0... 10... 0 0... 0... 0 0... 0... 1 0... E 23 0... 0 0... 0... 1 0... 10... 0 0... 0... 0 E 24 0... 0 0... 0... 1 0... 0... 0 0... 0... 0 E 34 0... 0 0... 0... 0 0... 0... 1 0... 10... 0 15

Example 1. Atmost three ones in each vertex column. 2. Pick strings pick columns with most ones. v 1 v 4 v 2 v 3 v 1 v 2 v 3 v 4 1 0... 0... 1 0... 0... 1 0... 0... 1.......... 1 0... 0... 1 0... 0... 1 0... 0... 1 1. 1 E 12 0... 1 0... 10... 0 0... 0... 0 0... 0... 0 E 13 0... 1 0... 0... 0 0... 10... 0 0... 0... 0 E 14 0 0... 10... 0 0... 0... 0 0... 0... 1 0... E 23 0... 0 0... 0... 1 0... 10... 0 0... 0... 0 E 24 0... 0 0... 0... 1 0... 0... 0 0... 0... 0 E 34 0... 0 0... 0... 0 0... 0... 1 0... 10... 0 15

Example 1. Atmost three ones in each vertex column. 2. Pick strings pick columns with most ones. v 1 v 4 v 2 v 3 v 1 v 2 v 3 v 4 1 0... 0... 1 0... 0... 1 0... 0... 1.......... 1 0... 0... 1 0... 0... 1 0... 0... 1 1. 1 E 12 0... 1 0... 10... 0 0... 0... 0 0... 0... 0 E 13 0... 1 0... 0... 0 0... 10... 0 0... 0... 0 E 14 0 0... 10... 0 0... 0... 0 0... 0... 1 0... E 23 0... 0 0... 0... 1 0... 10... 0 0... 0... 0 E 24 0 0... 0... 0 0... 0... 0 0... 0... 1 0... E 34 0... 0 0... 0... 0 0... 0... 1 0... 10... 0 15

Example 1. Atmost three ones in each vertex column. 2. Pick strings pick columns with most ones. v 1 v 4 v 2 v 3 v 1 v 2 v 3 v 4 1 0... 0... 1 0... 0... 1 0... 0... 1.......... 1 0... 0... 1 0... 0... 1 0... 0... 1 1. 1 E 12 0... 1 0... 10... 0 0... 0... 0 0... 0... 0 E 13 0... 1 0... 0... 0 0... 10... 0 0... 0... 0 E 14 0 0... 10... 0 0... 0... 0 0... 0... 1 0... E 23 0... 0 0... 0... 1 0... 10... 0 0... 0... 0 E 24 0 0... 0... 0 0... 0... 0 0... 0... 1 0... E 34 0... 0 0... 0... 0 0... 0... 1 0... 10... 0 15

Star Alignment SP-score not a good model of evolution. s 2 s 4 c s 1 s 3 16

Star Alignment SP-score not a good model of evolution. Input: k strings s 1... s k Output: string c minimizing d(c, s i ) i s 2 s 4? s 1 s 3 16

Star Alignment SP-score not a good model of evolution. Input: k strings s 1... s k Output: string c minimizing d(c, s i ) i s 2 s 4? s 1 s 3 Symbol distance metric = Pairwise alignment distance metric Star Alignment is a special case of Steiner Star in a metric space. 16

Earlier results - Star Alignment Wang and Jiang 94 - Non-metric APX-complete Li, Ma, Wang 99 - Constant number gaps, NP-hard and PTAS Sim and Park 01 - Specific metric NP-hard

Earlier results - Star Alignment Wang and Jiang 94 - Non-metric APX-complete Li, Ma, Wang 99 - Constant number gaps, NP-hard and PTAS Sim and Park 01 - Specific metric NP-hard

Earlier results - Star Alignment Wang and Jiang 94 - Non-metric APX-complete Li, Ma, Wang 99 - Constant number gaps, NP-hard and PTAS Sim and Park 01 - Specific metric NP-hard

Earlier results - Star Alignment Wang and Jiang 94 - Non-metric APX-complete Li, Ma, Wang 99 - Constant number gaps, NP-hard and PTAS Sim and Park 01 - Specific metric NP-hard Here: All binary or larger alphabets under all metrics!

Reduction Vertex Cover G = (V, E) minimum cover V Star Alignment Set of strings minimum string c = DDCDD 18

Reduction Vertex Cover G = (V, E) minimum cover V Star Alignment Set of strings minimum string c = DDCDD v 1 v 2 v 3... v n C =...?...?...?...?... 18

Reduction Vertex Cover G = (V, E) minimum cover V Star Alignment Set of strings minimum string c = DDCDD v 1 v 2 v 3... v n C V =... 1... 1... 1... 1... C V - Only 1 s. 18

Reduction Vertex Cover G = (V, E) minimum cover V Star Alignment Set of strings minimum string c = DDCDD v 1 v 2 v 3... v n C =... 0... 0... 0... 0... C V - Only 1 s. C - Only 0 s. 18

Construction Idea (E, G) c = DDCDD (E, S ab ) c = vertex cover G minimum cover S ab E E c G E S a b E G G String minimizing i d(c, s i) minimum cover 19

Canonical Structure E = DDC V DD G = DDC DD Differ only in the vertex positions. 20

Canonical Structure E = DDC V DD G = DDC DD Differ only in the vertex positions. Many such pairs will vote for the canonical structure. E G c E G 20

Canonical Structure E = DDC V DD G = DDC DD Differ only in the vertex positions. Many such pairs will vote for the canonical structure. Vote even in the vertex positions! E G c E G 20

Canonical Structure E = DDC V DD G = DDC DD Differ only in the vertex positions. Many such pairs will vote for the canonical structure. Vote even in the vertex positions! E G c E G c = DDCDD 20

String Cover Edge (v a, v b ) add two strings E = DDC V DD S ab = C a DC b v 1... v a... v b... v n C a =... 0... 1... 0... 0... 21

String Cover Edge (v a, v b ) add two strings E = DDC V DD S ab = C a DC b v 1... v a... v b... v n C b =... 0... 0... 1... 0... 21

String Cover Edge (v a, v b ) add two strings E = DDC V DD S ab = C a DC b E votes 1 in all vertex positions. 21

String Cover Edge (v a, v b ) add two strings E = DDC V DD S ab = C a DC b E votes 1 in all vertex positions. S ab votes 0 in all except vertex position a or b. D D C D D C a D C b C a D C b 21

String Cover Edge (v a, v b ) add two strings E = DDC V DD S ab = C a DC b E votes 1 in all vertex positions. S ab votes 0 in all except vertex position a or b. D D C D D C a D C b C a D C b Either vertex v a or vertex v b is part of the cover. 21

Minimal String Minimum Cover (E, G) c = DDCDD (E, S ab ) c = vertex cover S ab E E c G E S a b E G 22

Minimal String Minimum Cover (E, G) c = DDCDD (E, S ab ) c = vertex cover S ab E E c G E S a b E G G G = DDC DD penalizes each 1 in vertex positions. 22

Minimal String Minimum Cover (E, G) c = DDCDD (E, S ab ) c = vertex cover S ab E E c G E S a b E G G G = DDC DD penalizes each 1 in vertex positions. String minimizing i d(c, s i) minimum cover 22

Tree Alignment (given phylogeny) Star Alignment not a good model of evolution. p 1 p 2 p 3 s 1 s 2 s 3 s 4 23

Tree Alignment (given phylogeny) Star Alignment not a good model of evolution. Input: k strings s 1... s k phylogeny T Output: strings p 1... p k 1 min (a,b) E(T ) d(a, b)??? s 1 s 2 s 3 s 4 23

Tree Alignment (given phylogeny) Star Alignment not a good model of evolution. Input: k strings s 1... s k phylogeny T Output: strings p 1... p k 1 min (a,b) E(T ) d(a, b)??? s 1 s 2 s 3 s 4 Symbol distance metric = Pairwise alignment distance metric Tree Alignment is a special case of Steiner Tree in a metric space. 23

Construction Idea Same idea as for Star Alignment. r base comp. inbetween each branch E G E G E G r base comp. E G E E G r base comp. E G G E S ij m branches S i j 24

Overview and Open problems Problem Here Approx SP-score all metrics 2-approx [GPBL] Star Alignment all metrics 2-approx Tree Alignment all metrics PTAS [WJGL] (given phylogeny) 25

Overview and Open problems Problem Here Approx SP-score all metrics 2-approx [GPBL] Star Alignment all metrics 2-approx Tree Alignment all metrics PTAS [WJGL] (given phylogeny) Consensus Patterns NP-hard PTAS [LMW] Substring Parsimony NP-hard PTAS [ WJGL] 25

Acknowledgments My advisor Prof. Jens Lagergren Prof. Benny for hosting me Thanks! 26

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