M.S. Waterman and T.F. Smith, Rapid dynamic programming algorithms for RNA Secondary structure, Adv. Appl. Math. 7 (1986) 455-464.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with matrix entries for each triple (i, j, k) where i and j are positions in the RNA sequence (not necessarily forming a base pair) and k is the number of exposed bases in a possible loop containing i and j. This computation takes time O(n 4 ) the algorithm was discovered by Waterman and Smith [75]. Clearly, this time bound is so large that the computation of RNA structure using this algorithm is feasible only for very short sequences. Therefore, one needs further assumptions about the possible structures, or about the energy functions determining the optimum structure, in order to perform ....

.... solving this relation would seem to require time O(n 4 ) as for the multiple loop RNA structure computation, but the space requirement is reduced from O(n 3 )toO(n 2 ) In fact the time for solving the recurrence can also be reduced, to O(n 3 ) as was recently shown by Waterman and Smith [75]. But this is still more time than one would want to spend for the computation. To achieve even faster single loop secondary structure computation, we can again restrict our attention to linear cost functions. With this restriction, the time for the RNA structure computation can be reduced to O(n ....

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Michael S. Waterman and Temple F. Smith, Rapid Dynamic Programming Algorithms for RNA Secondary Structure, Adv. Appl. Math. 7, 1986, pp. 455-- 464.

....entries for each triple (i, j, k) where i and j are positions in the RNA sequence (not necessarily forming a base pair) and k is the number of exposed bases in a possible loop containing i and j. This computation takes time O(n 4 ) The algorithm was recently discovered by Waterman and Smith [65] and it is the first polynomial time algorithm obtained for this problem. Clearly, this time bound is so large that the computation of RNA structure using this algorithm is feasible only for very short sequences. Furthermore, the space bound of O(n 3 ) also makes this algorithm impractical. ....

.... as a dynamic programming recurrence relation [50, 63] Again this relation seems to require time O(n 4 ) but the space requirement is reduced from O(n 3 ) to O(n 2 ) In fact the time for solving the recurrence 13 can also be reduced, to O(n 3 ) as was shown by Waterman and Smith [65]. In this paper, the authors also conjectured that the given algorithm runs in O(n 2 ) time for convex (and concave) functions. Eppstein et al. 10] have shown how to compute single loop RNA secondary structure, for convex or concave energy costs, in time O(n 2 log 2 n) For many simple cost ....

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Michael S. Waterman and Temple F. Smith, Rapid Dynamic Programming Algorithms for RNA Secondary Structure, Adv. Appl. Math. 7, 1986, pp. 455--464.

.... In this section we examine recurrence 4, which for convenience we repeat here: E[i, j] min 1#i # i 1#j # j D[i # , j # ] w(i # j # , i j) The recurrence can be solved by a simple dynamic program in time O(n 4 ) fairly simple techniques su#ce to reduce this time to O(n 3 ) [20]. In this paper we present a new algorithm, which when w is convex solves recurrence 4 in time O(n 2 log 2 n) For many common choices of w, a more complicated version of the algorithm solves the recurrence in time O(n 2 log n log log n) Similar techniques can be used to solve the concave ....

....used to solve the concave case of recurrence 4. Our algorithms do not follow from those of the previous section, and are more complicated, but should still be simple enough for practical application. The recurrence above has an important application to the computation of RNA secondary structure [13, 20]. After a simple change of variables, one can use it to solve the following recurrence: C[p, q] min p p # q # q G[p # , q # ] g( p # p) q q # ) 7) Recurrence 7 has been used to calculate the secondary structure of RNA, with the assumption that the structure contains no ....

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Michael S. Waterman and Temple F. Smith, Rapid Dynamic Programming Algorithms for RNA Secondary Structure, in Advances in Applied Mathematics 7, 1986, pp. 455-- 464. 15

....time from the corresponding entries of D. That is, F (i; j) is available only after D(i; j) has been calculated. This problem arises in the prediction of RNA secondary structure from the primary (linear) RNA structure and is known as Waterman s problem (for further details cf. Waterman and Smith [132]) Relying on the results for the one dimensional recurrence (22) discussed above, Larmore and Schieber show that Waterman s problem can be solved in optimal O(n 2 ) time if the weights w ij satisfy the concave quadrangle inequality (4) i.e. if the matrix f W with e w ij = w ij for i j and ....

M.S. Waterman and T.F. Smith, Rapid dynamic programming algorithms for RNA secondary structure, Advances in Applied Mathematics 7, 455--464, 1986.

....same structure is presented, where the primary structure is given along the horizontal axis and the base pairs are shown as arcs. This representation is used by Stein [4] The prediction of the shapes of biological molecules is an important topic in computational biology; see [9] for a review. In [8] the problem of prediction is shown to be polynomial. The prediction algorithms are combinatorial, and the enumeration of secondary structure is a natural problem; see [5 7] In these enumeration studies, the specific identity of the bases is ignored, which in effect allows all possible base ....

....vertex labelling, by these conditions. It follows immediately from the definition that (T ) T, and that TJ; k if and only if T oj; n . 1 6 1 Fig. 6. A linear tree and its dual, An example of a linear tree T and its dual T is shown in Fig. 6 with edge labellings. In this example, E = [8], T) 1,2,3 , 4,5 , 6,7 , 8 = T ) and ] T) 1,6 , 2,4 , 7,8 , 3 , 5 = T ) ....

M.S. Waterman and T.F. Smith, Rapid dynamic programming algorithms for RNA Secondary structure, Adv. Appl. Math. 7 (1986) 455-464.

....from the linear sequence A = A 1 A 2 Delta Delta Delta A n . One of the most popular methods to predict secondary structure is dynamic programming, first presented by Waterman (1978) Waterman and Smith (1978) and Nussinov et al. 1978) Zuker and Sankoff (1984) provide an excellent review. Waterman and Smith (1986) propose some speedups of this method. Sankoff (1985) considers simultaneous alignment and secondary structure prediction. Dynamic programming is still a method of choice for secondary structure prediction G G A U A A C A G A U C A A A A C A U C U C C A U A A A A A A U G G G A G C C G A A U U G ....

Waterman, M.S. and Smith, T.F. (1986). Rapid dynamic programming algorithms for RNA secondary structure. Adv. Appl. Math., 7, 455-464.

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