Prof. Swarat Chaudhuri

COMP 482: Design and Analysis of Algorithms Prof. Swarat Chaudhuri Spring 2012 Lecture 16

6.3 Segmented Least Squares

Segmented Least Squares • Least squares. • Foundational problem in statistic and numerical analysis. • Given n points in the plane: (x1, y1), (x2, y2) , . . . , (xn, yn). • Find a line y = ax + b that minimizes the sum of the squared error: • Solution. Calculus  min error is achieved when y x

Segmented Least Squares • Segmented least squares. • Points lie roughly on a sequence of several line segments. • Given n points in the plane (x1, y1), (x2, y2) , . . . , (xn, yn) with • x1 < x2 < ... < xn, find a sequence of lines that minimizes f(x). • Q. What's a reasonable choice for f(x) to balance accuracy and parsimony? goodness of fit number of lines y x

Segmented Least Squares • Segmented least squares. • Points lie roughly on a sequence of several line segments. • Given n points in the plane (x1, y1), (x2, y2) , . . . , (xn, yn) with • x1 < x2 < ... < xn, find a sequence of lines that minimizes: • the sum of the sums of the squared errors E in each segment • the number of lines L • Tradeoff function: E + c L, for some constant c > 0. y x

Dynamic Programming: Multiway Choice • Notation. • OPT(j) = minimum cost for points p1, pi+1 , . . . , pj. • e(i, j) = minimum sum of squares for points pi, pi+1 , . . . , pj. • To compute OPT(j): • Last segment uses points pi, pi+1 , . . . , pj for some i. • Cost = e(i, j) + c + OPT(i-1).

Segmented Least Squares: Algorithm • Running time. O(n3). • Bottleneck = computing e(i, j) for O(n2) pairs, O(n) per pair using previous formula. INPUT: n, p1,…,pN , c Segmented-Least-Squares() { M[0] = 0 for j = 1 to n for i = 1 to j compute the least square error eij for the segment pi,…, pj for j = 1 to n M[j] = min 1  i  j (eij + c + M[i-1]) return M[n] } can be improved to O(n2) by pre-computing various statistics

6.6 Sequence Alignment

String Similarity • How similar are two strings? • ocurrance • occurrence o c u r r a n c e - o c c u r r e n c e 5 mismatches, 1 gap o c - u r r a n c e o c c u r r e n c e 1 mismatch, 1 gap o c - u r r - a n c e o c c u r r e - n c e 0 mismatches, 3 gaps

Edit Distance • Applications. • Basis for Unix diff. • Speech recognition. • Computational biology. • Edit distance. [Levenshtein 1966, Needleman-Wunsch 1970] • Gap penalty ; mismatch penalty pq. • Cost = sum of gap and mismatch penalties. C T G A C C T A C C T - C T G A C C T A C C T C C T G A C T A C A T C C T G A C - T A C A T TC + GT + AG+ 2CA 2+ CA

Sequence Alignment • Goal: Given two strings X = x1 x2 . . . xm and Y = y1 y2 . . . yn find alignment of minimum cost. • Def. An alignment M is a set of ordered pairs xi-yj such that each item occurs in at most one pair and no crossings. • Def. The pair xi-yj and xi'-yj'cross if i < i', but j > j'. • Ex:CTACCG vs. TACATG.Sol: M = x2-y1, x3-y2, x4-y3, x5-y4, x6-y6. x1 x2 x3 x4 x5 x6 C T A C C - G - T A C A T G y1 y2 y3 y4 y5 y6

Sequence Alignment: Problem Structure • Def. OPT(i, j) = min cost of aligning strings x1 x2 . . . xi and y1 y2 . . . yj. • Case 1: OPT matches xi-yj. • pay mismatch for xi-yj + min cost of aligning two stringsx1 x2 . . . xi-1 and y1 y2 . . . yj-1 • Case 2a: OPT leaves xi unmatched. • pay gap for xi and min cost of aligning x1 x2 . . . xi-1 and y1 y2 . . . yj • Case 2b: OPT leaves yj unmatched. • pay gap for yj and min cost of aligning x1 x2 . . . xi and y1 y2 . . . yj-1

Sequence Alignment: Algorithm • Analysis. (mn) time and space. • English words or sentences: m, n  10. • Computational biology: m = n = 100,000. 10 billions ops OK, but 10GB array? • (Solution: sequence alignment in linear space) Sequence-Alignment(m, n, x1x2...xm, y1y2...yn, , ) { for i = 0 to m M[0, i] = i for j = 0 to n M[j, 0] = j for i = 1 to m for j = 1 to n M[i, j] = min([xi, yj] + M[i-1, j-1],  + M[i-1, j],  + M[i, j-1]) return M[m, n] }

Q1: Least-cost concatenation • You have a DNA sequence A of length n; you also have a “library” of shorter strings each of length m < n. • Your goal is to generate a concatenation C of strings B1,…,Bk in the library that the cost of aligning C to A is as low as possible. You can assume a gap cost δ and a mismatch cost αpq for determining the cost of alignment.

Answer • Let A[x:y] denote the substring of A consisting of its symbols from position x to position y. • Let c(x,y) be the cost of optimally aligning A[x:y] to any string in the library. • Let OPT(j) be the alignment cost for the optimal solution on the string A[1:j]. • OPT(j) = mint <j c(t,j) + OPT(t – 1) for j ≥ 1 • OPT(0) = 0 • [Essentially, t is a “breakpoint” where you choose to use a new library component.]

6.4 Knapsack Problem

Q2: Knapsack Problem • Knapsack problem. • Given n objects and a "knapsack." • Item i weighs wi > 0 kilograms and has value vi > 0. • Knapsack has capacity of W kilograms. • Goal: fill knapsack so as to maximize total value. • Ex: { 3, 4 } has value 40. • Greedy: repeatedly add item with maximum ratio vi / wi. • Ex: { 5, 2, 1 } achieves only value = 35  greedy not optimal. Item Value Weight 1 1 1 2 6 2 W = 11 3 18 5 4 22 6 5 28 7

Dynamic Programming: False Start • Def. OPT(i) = max profit subset of items 1, …, i. • Case 1: OPT does not select item i. • OPT selects best of { 1, 2, …, i-1 } • Case 2: OPT selects item i. • accepting item i does not immediately imply that we will have to reject other items • without knowing what other items were selected before i, we don't even know if we have enough room for i • Conclusion. Need more sub-problems!

Dynamic Programming: Adding a New Variable • Def. OPT(i, w) = max profit subset of items 1, …, i with weight limit w. • Case 1: OPT does not select item i. • OPT selects best of { 1, 2, …, i-1 } using weight limit w • Case 2: OPT selects item i. • new weight limit = w – wi • OPT selects best of { 1, 2, …, i–1 } using this new weight limit

Knapsack Problem: Bottom-Up • Knapsack. Fill up an n-by-W array. Input: n, w1,…,wN, v1,…,vN for w = 0 to W M[0, w] = 0 for i = 1 to n for w = 1 to W if (wi > w) M[i, w] = M[i-1, w] else M[i, w] = max {M[i-1, w], vi + M[i-1, w-wi ]} return M[n, W]

0 1 2 3 4 5 6 7 8 9 10 11  0 0 0 0 0 0 0 0 0 0 0 0 { 1 } 0 1 1 1 1 1 1 1 1 1 1 1 { 1, 2 } 0 1 6 7 7 7 7 7 7 7 7 7 { 1, 2, 3 } 0 1 6 7 7 18 19 24 25 25 25 25 { 1, 2, 3, 4 } 0 1 6 7 7 18 22 24 28 29 29 40 { 1, 2, 3, 4, 5 } 0 1 6 7 7 18 22 28 29 34 34 40 Item Value Weight 1 1 1 2 6 2 3 18 5 4 22 6 5 28 7 Knapsack Algorithm W + 1 n + 1 OPT: { 4, 3 } value = 22 + 18 = 40 W = 11

Knapsack Problem: Running Time • Running time. (n W). • Not polynomial in input size! • "Pseudo-polynomial." • Decision version of Knapsack is NP-complete. [Chapter 8] • Knapsack approximation algorithm. There exists a polynomial algorithm that produces a feasible solution that has value within 0.01% of optimum. [Section 11.8]

Q3: Longest palindromic subsequence • Give an algorithm to find the longest subsequence of a given string A that is a palindrome. • “amantwocamelsacrazyplanacanalpanama”

Q3-a: Palindromes (contd.) • Every string can be decomposed into a sequence of palindromes. • Give an efficient algorithm to compute the smallest number of palindromes that makes up a given string.

Prof. Swarat Chaudhuri