A Parallel Approximation Hitting Set Algorithm for Gene Expression Analysis

Transcription

1 A Parallel Approximation Hitting Set Algorithm for Gene Expression Analysis D. P. Ruchkys Universidade de São Paulo S. W. Song Universidade de São Paulo

2 Gene Expression Analysis Given an experiment where expression levels of thousands of genes are measures. We consider the problem of determining which genes affect the expression level of a given gene.

3 Our Problem Given an experiment with n genes of a set E = {a, a,..., a n } whose expression levels are measured in a time series of m measures (typically n >> m). We have a total of nm values of s or s. Our algorithm (based on Ideker et al. [ITK]) receives an m n matrix of such values and determine, for a given gene a n, which other genes are responsible for the expression level of a n. Example. M = x x x x p p p p p 4

4 Example of Execution of the Algorithm Infer the truth table for a of the matrix E shown. M = x x x x p p p p p 4 () In step (), the expression levels of a differ in the row pairs (,), (,), (,) and (,). We find: for (,), S = {a, a }, containing all the other genes whose expression levels also differ in the row pairs p and p. the same is done for (,), S = {a }. for (,), S = {a, a }. for (,), S = {a }. 4

5 Result of Step Result of Step : S = {a, a }, S = {a }, S = {a, a }, S = {a }. () In Step (), find S min = {a, a }, the smallest set such that each element in S min is also present in each one of the sets S ij of the previous step. 5

6 The Hitting Set Problem Given a finite set E, a finite collection S = {S,..., S w } of subsets of E, find a subset A E of the smallest size, such that A S i for all i =,..., w. 6

7 The Hitting Set Problem E S A 4 7

8 The Hitting Set Problem Primal-Dual Approximation Algorithm [FMCF] Due to Bar-Yehuda and Even [BYE8] and was originally conceived for the minimum set cover problem. It is an α-approximation algorithm, where α = max w i= S i. α = max w i= S i = O(n). 8

9 The Hitting Set Problem Greedy Approximation Algorithm [J74] Strategy of constructing the set A by choosing the elements that occurs the most times in the subsets of S. The approximation ratio is ln S +. ln S + = O(log m ) 9

10 The Hitting Set Problem Greedy Approximation Algorithm E S A

11 The Hitting Set Problem Greedy Approximation Algorithm E S A

12 The Hitting Set Problem Greedy Approximation Algorithm E a a a a S a a a a a a A a a

13 The Sequential Algorithm gene vector occurrence list set vector i j covered list false M = x x x x p p p p p 4

14 The Sequential Algorithm gene vector occurrence list set vector i j covered list false false false false M = x x x x p p p p p 4 4

15 The Sequential Algorithm gene vector occurrence list HS:{} set vector i j covered false true list false false true false 5

16 The Sequential Algorithm Time and Space Complexities To construct the data structures: O(m n). Let k the size of the hitting set. We have to find k times the element with the largest number of occurrences. Therefore we have the time complexity of O(kn). For each such element, we have to update the data structures: O(m n) time. Since we have k elements, the total time complexity to update data structures is O(km n). 6

17 The Sequential Algorithm Time and Space Complexities The total time complexity is therefore O(m n)+o(kn)+o(km n) = O(km n). The size k of the hitting set is O(m ). Therefore, the time complexity of the algorithm can be expressed as O(m 4 n). The space complexity is O(m n). 7

18 The Parallel Algorithm The input matrix M is partitioned vertically to be stored in each processor. Example of the partitioning: a a a a a 4 a 5 a 6 a 7 a 8 x, x, x, x, x,4 x,5 x,6 x,7 x,8 = x, x, x, x, x, x, x, x, x,4 x,4 x,5 x,5 x,6 x,6 x,7 x,7 x,8 x,8 x, x, x,4 x,5 x,6 x,7 x,8 x,. x, x m, x m, x m, x m, x m,4 x m,5 x m,6 x m,7 x m processor processor processor 8

19 The Parallel Algorithm Each processor reads a piece of the input of size m n p. All the processors store a vector v, corresponding to the expression levels of the gene under study a n. Each processor p i also stores a gene vector, with information about genes it is responsible for. The gene vector stores information of the genes for which processor p i is responsible. The gene vector in each processor has size O( m n p ). Each processor also has a set vector, such that only elements of set S ij of its responsibility will only be in the list. 9

20 Example: The Parallel Algorithm x x x x x 4 E = p p p p p 4 gene vector gene vector ocurrence list ocurrence list set vector set vector i j covered false false false false list i j covered false false false false list Processor Processor

21 The Parallel Algorithm Time and Space Complexities Time complexity: O( m4 n p ). Requires O(k) communication rounds, where k is the size of the hitting set. It can be expressed in terms of m, O(m ). Requires O( m n p ) space.

22 Seconds x4 x48 x No. Processors

23 Bibliographical References [BYE8] R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, :98-, 98. [FMCF] C. G. Fernandes, F. K. Miyazawa, M. Cerioli, P. Feofiloff. Uma introdução sucinta a algoritmos de aproximação. Colóquio Brasileiro de Matemática,. [ITK] T. E. Ideker, V. Thorsson, R. Karp. Discovery of regulatory interactions through perturbation: inference and experimental design. Pacific Symposium on Biocomputing, 5:-,. [J74] D. S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9:56-78, 974.