CS代考程序代写 data structure Bioinformatics algorithm Lecture 7: Dynamic Programming II

Lecture 7: Dynamic Programming II
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General techniques in this course
– Greedy algorithms [Lecture 3]
– Divide & Conquer algorithms [Lectures 4 and 5]
– Dynamic programming algorithms [Lecture 6 and today] – Network flow algorithms [18 Apr and 2 May]
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Algorithmic Paradigms
– Greedy. Build up a solution incrementally, myopically optimizing some local criterion.
– Divide-and-conquer. Break up a problem into two sub- problems, solve each sub-problem independently, and combine solution to sub-problems to form solution to original problem.
– Dynamic programming. Break up a problem into a series of overlapping sub-problems, and build up solutions to larger and larger sub-problems.
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Main Idea: Dynamic Programming
Recurrence equation relating optimal solution in terms optimal solutions to smaller subproblems
Given optimal solutions to smaller subproblems, how to construct solution to original problem?
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Key steps: Dynamic programming
1. Define subproblems 2. Find recurrences
3. Solve the base cases
4. Transform recurrence into an efficient algorithm
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Recap: Weighted Interval Scheduling
– Job j starts at sj, finishes at fj, and has weight vj.
– Two jobs compatible if they don’t overlap.
– Goal: find maximum value subset of mutually compatible jobs.
b
a
c
d
e
f
g
h
0 1 2 3 4 5 6 7 8 9 10 11 The University of Sydney
Time
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Recap: Weighted Interval Scheduling
Notation. Label jobs by finishing time: f1 £ f2 £ . . . £ fn .
Def. p(j) = largest index i 0 kilograms and has value vi > 0. – KnapsackhascapacityofWkilograms.
– Goal: fill knapsack so as to maximize total value.
W = 11
Item
Value
Weight
1
1
1
2
6
2
3
18
5
4
22
6
5
28
7
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Recap: Dynamic Programming – Step 1 Step 1: Define subproblems
OPT(i, w) = max profit with subset of items 1, …, i with weight limit w.
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Recap: Dynamic Programming – Step 2 Step 2: Find recurrences
– 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.
• newweightlimit=w–wi
• OPT selects best of { 1, 2, …, i–1 } using this new weight limit
OPT(i,w) = max { vi + OPT (i-1,w-wi), OPT(i-1,w) }
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Recap: Dynamic Programming – Step 2 Step 2: Find recurrences
– 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.
• newweightlimit=w–wi
• OPT selects best of { 1, 2, …, i–1 } using this new weight limit
If wi>w
OPT(i,w) = OPT (i-1,w)
otherwise
OPT(i,w) = max { vi + OPT (i-1,w-wi), OPT(i-1,w) }
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Recap: Dynamic Programming – Step 3 Step 3: Solve the base cases
OPT(0,w) = 0
ì0 if i=0 OPT(i,w)=ïíOPT(i-1,w) if i>0andwi >w
ïîmax{OPT(i-1,w),vi+OPT(i-1,w-wi)} otherwise
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Done…more or less

Knapsack Problem: Bottom-Up
– Knapsack. Fill up an (n+1)-by-(W+1) 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]
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Knapsack Algorithm
W+1
0
1
2
3
4
5
6
7
8
9
10
11
Æ
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
n+1
Item
Value
Weight
1
1
1
2
6
2
3
18
5
4
22
6
5
28
OPT: {4,3}
value = 22 + 18 = 40
W = 11
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Longest Common Subsequence
Given two sequences X[1..n] and Y[1..m], find the longest subsequences X’ of X and Y’ of Y such that X’ = Y’.
Example 1 X = BANANAS Y = KATANA
LCS(X,Y) = AANA
Example 2 X = DYNAMIC
Y = PROGRAMMING
LCS(X,Y) = AMI The University of Sydney
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Longest Common Subsequence: Matching Definition
Given two sequences X[1..n] and Y[1..m], find the longest non- crossing matching between elements of X and elements of Y
Example 1 X = BANANAS Y = KATANA
LCS(X,Y) = AANA. Matching: X[2] – Y[2], X[4] – Y[4], X[5] – Y[5], X[6] – Y[6].
Example 2 X = DYNAMIC
Y = PROGRAMMING
LCS(X,Y) = AMI. Matching: X[4] – Y[6], X[5] – Y[7], X[6] – Y[8],
Note: if crossings allowed, then can add X[3] – Y[10]. The University of Sydney
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Longest Common Subsequence: Applications
– Measures similarity of two strings – Bioinformatics
– Merging in version control
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LCS Dynamic Programming: Step 1 Step 1: Define subproblems
OPT(i,j) = length of LCS(X[1..i], Y[1..j]). Subproblems: finding LCS of prefixes of X and Y
Example
X = BANANAS Y = KATANA
OPT(6, 4) = length of LCS(BANANA, KATA) = 2 The University of Sydney
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LCS Dynamic Programming: Step 2
Notations:
– OPT(i,j) = length of LCS(X[1..i], Y[1..j]). Step 2: Finding recurrences
– Case 1: X[i] ≠ Y[j].
– Leave X[i] unmatched and take LCS(X[1..i-1], Y[1..j]). OR – Leave Y[j] unmatched and take LCS(X[1..i], Y[1..j-1]).
– OPT(i,j)= max{OPT(i-1,j), OPT(i,j-1)}
Example
X[1..5] = BANAN Y[1..6] = KATANA
X[5] unmatched: LCS = LCS(X[1..4], Y[1..6]) = ANA
Y[6] unmatched: LCS = LCS(X[1..5], Y[1..5]) = AAN The University of Sydney
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LCS Dynamic Programming: Step 2
Notations:
– OPT(i,j) = length of LCS(X[1..i], Y[1..j]). Step 2: Finding recurrences
– Case 1: X[i] ≠ Y[j].
– Leave X[i] unmatched and take LCS(X[1..i-1], Y[1..j]). OR – Leave Y[j] unmatched and take LCS(X[1..i], Y[1..j-1]).
– OPT(i,j)= max{OPT(i-1,j), OPT(i,j-1)}
Example
X[1..3] = BAN Y[1..6] = KATANA
X[3] unmatched: LCS(X[1..2], Y[1..6]) = A
Y[6] unmatched: LCS(X[1..3], Y[1..5]) = AN The University of Sydney
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LCS Dynamic Programming: Step 2
Notations:
– OPT(i,j) = length of LCS(X[1..i], Y[1..j]). Step 2: Finding recurrences
– Case 1: X[i] ≠ Y[j].
– Leave X[i] unmatched and take LCS(X[1..i-1], Y[1..j]). OR – Leave Y[j] unmatched and take LCS(X[1..i], Y[1..j-1]).
– OPT(i,j)= max{OPT(i-1,j), OPT(i,j-1)}
Example
X[1..3] = BAN Y[1..6] = KATANA
X[3] unmatched: LCS(X[1..2], Y[1..6]) = A
Y[6] unmatched: LCS(X[1..3], Y[1..5]) = AN
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LCS Dynamic Programming: Step 2
Notations:
– OPT(i,j) = length of LCS(X[1..i], Y[1..j]). Step 2: Finding recurrences
– Case 2: X[i] = Y[j].
– Match X[i] to Y[j] and take LCS(X[1..i-1], Y[1..j-1]) + X[i] OR
– Leave X[i] unmatched or Y[j] unmatched
– OPT(i,j) = max{OPT(i-1, j-1) + 1, OPT(i-1, j), OPT(i, j-1)
Example
X[1..6] = BANANA Y[1..6] = KATANA
Match X[6] – Y[6]: LCS = LCS(X[1..5], Y[1..5]) + A = AANA
X[6] unmatched: LCS = LCS(X[1..5], Y[1..6]) = AAN
Y[6] unmatched: LCS = LCS(X[1..6], Y[1..5]) = AAN The University of Sydney
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LCS Dynamic Programming: Step 3 Step 3: Solving the base cases
OPT(i,0) = 0 for all i, OPT(0,j) = 0 for all j
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LCS Dynamic Programming: Algorithm
INPUT: n, m, X[1..n], Y[1..m]
for j = 0 to m
M[0,j] = 0
for i = 0 to n
M[i,0] = 0
for i = 1 to n
for j = 1 to m
if (X[i] = Y[j])
M[i,j] = max(M[i-1,j-1] + 1, M[i-1,j], M[i,j-1])
else
M[i,j] = max(M[i-1,j], M[i,j-1])
return M[n,m]
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LCS Dynamic Programming: Analysis
INPUT: n, m, X[1..n], Y[1..m]
for j = 0 to m
M[0,j] = 0
for i = 0 to n
M[i,0] = 0
for i = 1 to n
for j = 1 to m
O(n + m)
O(nm)
if (X[i] = Y[j])
M[i,j] = max(M[i-1,j-1] + 1, M[i-1,j], M[i,j-1])
else
M[i,j] = max(M[i-1,j], M[i,j-1])
return M[n,m] Running time: O(nm)
Space: O(nm)
See 3927 Lecture 6 for Sequence Alignment
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6.5 RNA Secondary Structure
Dynamic programming over intervals
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RNA (Ribonucleic acid) Secondary Structure
– RNA. String B = b1b2…bn over alphabet { A, C, G, U }.
– Secondary structure. RNA is single-stranded so it tends to loop
back and form base pairs with itself. This structure is essential
for understanding behavior of molecule.
CA AA AU
GC CGUAA G
Ex: GUCGAUUGAGCGAAUGUAACAACGUGGCUACGGCGAGA UGAUUA
G
ACGCU CGCGAGC
G
G AU
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complementary base pairs: A-U, C-G G
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RNA Secondary Structure
– Secondary structure. A set of pairs S = { (bi, bj) } that satisfy: – [Watson-Crick.] S is a matching and each pair in S is a Watson-Crick
complement: A-U, U-A, C-G, or G-C.
– [No sharp turns.] The ends of each pair are separated by at least 4
intervening bases. If (bi, bj) Î S, then i < j - 4. – [Non-crossing.] If (bi, bj) and (bk, bl) are two pairs in S, then we cannot have i < k < j < l. – Free energy. Usual hypothesis is that an RNA molecule will form the secondary structure with the optimum total free energy. approximated by number of base pairs – Goal. Given an RNA molecule B = b1b2...bn, find a secondary structure S that maximizes the number of base pairs. The University of Sydney Page 36 36 RNA Secondary Structure: Examples GGGGG CUGGCU CGCGCU AUAUAG UAUAUA base pair AUGUGGCCAU AUGGGGCAU AGUUGGCCAU £4 ok sharp turn crossing The University of Sydney Page 37 RNA Secondary Structure: Subproblems – First attempt (Step 1). OPT(j) = maximum number of base pairs in a secondary structure of the substring b1b2...bj. 1 match bt and bi tj – Difficulty (in Step 2). Results in two sub-problems. – Finding secondary structure in: b1b2...bt-1. – Finding secondary structure in: bt+1bt+2...bj-1. OPT(t-1) The University of Sydney need more sub-problems Page 38 Dynamic Programming Over Intervals – Step 1 Step 1: Define subproblems OPT(i, j) = maximum number of base pairs in a secondary structure of the substring bibi+1...bj. The University of Sydney Page 39 Dynamic Programming Over Intervals – Step 2 Notation. OPT(i, j) = maximum number of base pairs in a secondary structure of the substring bibi+1...bj. Step 2: Find recurrences The University of Sydney Page 40 i j Case 1. Base bj is not involved in a pair. OPT(i, j) = OPT(i, j-1) Dynamic Programming Over Intervals – Step 2 Notation. OPT(i, j) = maximum number of base pairs in a secondary structure of the substring bibi+1...bj. Step 2: Find recurrences The University of Sydney Page 41 i match bt and bi tj Case2. Basebjpairswithbt forsomei£t M[w] + cvw) {
M[v] ¬ M[w] + cvw
successor[v] ¬ w }
} }
If no M[w] value changed in iteration i, stop.
}
}
Analysis: Q(mn) time, Q(n) working space. The University of Sydney Page 64

Shortest Paths: Practical Improvements
– Practical improvements
– Maintain only one array M[v] = shortest v-t path that we have
found so far.
– No need to check edges of the form (v, w) unless M[w] changed in previous iteration.
– Theorem: Throughout the algorithm, M[v] is length of some v-t path, and after i rounds of updates, the value M[v] is no larger than the length of shortest v-t path using £ i edges.
– Overall impact
– Working space: O(n).
– Total space (including input): O(m+n)
– Running time: O(mn) worst case, but substantially faster in practice.
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6.3 Segmented Least Squares
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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:
y
SSE = ån (yi -axi -b)2 i =1
x
– Solution. Calculus Þ min error is achieved when
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a=nåixiyi -(åixi)(åi yi), b=åi yi -aåixi
n å x 2 – (å x )2 n ii ii
O(n)

Segmented Least Squares
– Segmented least squares.
– Points lie roughly on a sequence of several line segments.
– Givennpointsintheplane(x1,y1),(x2,y2),…,(xn,yn)with – x1 < x2 < ... < xn, find a sequence of lines that minimizes f(x). – Question. What's a reasonable choice for f(x) to balance accuracy and complexity? number of lines y The University of Sydney x Page 68 Segmented Least Squares – Segmented least squares. – Points lie roughly on a sequence of several line segments. – Givennpointsintheplane(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
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x Page 69

Segmented Least Squares
– Segmented least squares.
– Points lie roughly on a sequence of several line segments.
– Givennpointsintheplane(x1,y1),(x2,y2),…,(xn,yn)with
x1 < x2 < ... < xn, find a partition into segments that minimizes: • the sum of the sums of the squared errors E in each segment • the number of segments L – Tradeoff function: E + c·L, for some constant c > 0.
y
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x Page 70

Dynamic Programming: Multiway Choice – Step 1 Step 1: Define subproblems
OPT(j) = minimum cost for points p1, p2 , . . . , pj.
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Dynamic Programming: Multiway Choice – Step 2
Notations:
– OPT(j)=minimumcostforpointsp1,p2 ,…,pj.
– e(i,j) =minimumsumofsquaresforpointspi,pi+1 ,…,pj.
Step 2: Finding recurrences
– Lastsegmentusespointspi,pi+1 ,…,pj forsomei. – Cost = e(i, j) + c + OPT(i-1).
OPT(j) = min { e(i,j) + c + OPT (i-1) } 1≤i≤j
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Dynamic Programming: Multiway Choice – Step 3 Step 3: Solving the base cases
OPT(0) = 0
OPT(j) = (0 if j = 0 min1ije(i,j)+c+OPT(i1) ifj>0
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Segmented Least Squares: Algorithm
O(n2) iterations
O(n) iterations
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
O(n)
O(n)
for j = 1 to n
M[j]=min1 £ i £ j(eij +c+M[i-1])
return M[n] }
OPT(j) = (0 if j = 0 min1ije(i,j)+c+OPT(i1) ifj>0
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Segmented Least Squares: Algorithm
2 O(n )
iterations
O(n) iterations
for j = 1 to n fori=1toj
compute the least square error eij for the segment pi,…, pj
for j = 1 to n
M[j]=min1 £ i £ j(eij +c+M[i-1])
O(n)
O(n)
INPUT: n, (p1,…,pn), c
Segmented-Least-Squares() {
M[0] = 0
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return M[n] }
Running time: O(n3) Space: O(n2)

Dynamic Programming Summary I
3 steps:
1. Definesubproblems
2. Findrecurrences
3. Solvethebasecases
4. Transformrecurrenceintoanefficientalgorithm [usually bottom-up]
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Dynamic Programming Summary II
– 1D dynamic programming
– Weighted interval scheduling
– Segmented Least Squares
– Maximum-sum contiguous subarray – Longest increasing subsequence
– 2D dynamic programming – Knapsack
– Shortest path
– Longest common subsequence
– Dynamic programming over intervals – RNA Secondary Structure
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.. Interpunctio Verborum Redux recurrence is T(n) = T(n 1) + T(n 2) + O(n), which still has the solution
T(n)=O(n).
. Interpunctio Verborum Redux
For our next dynamic programming algorithm, let’s consider the text segmenta- tion problem from the previous chapter. We are given a string A[1 .. n] and a subroutine IW that determines whether a given string is a word (whatever that means), and we want to know whether A can be partitioned into a sequence of words.
We solved this problem by defining a function Splittable(i) that returns T if and only if the sux A[i .. n] can be partitioned into a sequence of words. We need to compute Splittable(1). This function satisfies the recurrence
8><>:T if i > n Splittable(i) = Wn IsWord(i, j) ^ Splittable( j + 1) otherwise
j=i
where IsWord(i, j) is shorthand for IsWord(A[i .. j]). This recurrence translates directly into a recursive backtracking algorithm that calls the IW subroutine O(2n) times in the worst case.
But for any fixed string A[1..n], there are only n dierent ways to call the recursive function Splittable(i)—one for each value of i between 1 and n + 1—and only O(n2) dierent ways to call IW(i, j)—one for each pair (i, j) such that 1  i  j  n. Why are we spending exponential time computing only a polynomial amount of stu?
Each recursive subproblem is specified by an integer between 1 and n + 1, so we can memoize the function Splittable into an array SplitTable[1 .. n + 1]. Each subproblem Splittable(i) depends only on results of subproblems Splittable(j) where j > i, so the memoized recursive algorithm fills the array in decreasing index order. If we fill the array in this order deliberately, we obtain the dynamic programming algorithm shown in Figure .. The algorithm makes O(n2) calls to IW, an exponential improvement over our earlier backtracking algorithm.
. The Pattern: Smart Recursion
In a nutshell, dynamic programming is recursion without repetition. Dynamic programming algorithms store the solutions of intermediate subproblems, often but not always in some kind of array or table. Many algorithms students

. DP
S(A[1 .. n]): SplitTable[n + 1] T
for i n down to 1 SplitTable[i] F
for j i to n
if IW(i, j) and SplitTable[ j + 1]
SplitTable[i] T return SplitTable[1]

Figure .. Interpunctio verborum velox
(and instructors, and textbooks) make the mistake of focusing on the table— because tables are easy and familiar—instead of the much more important (and dicult) task of finding a correct recurrence. As long as we memoize the correct recurrence, an explicit table isn’t really necessary, but if the recurrence is incorrect, we are well and truly hosed.
Dynamic programming algorithms are almost always developed in two distinct stages.
. Formulate the problem recursively. Write down a recursive formula or algorithm for the whole problem in terms of the answers to smaller subproblems. This is the hard part. A complete recursive formulation has two parts:
(a) Specification. Describe the problem that you want to solve recursively, in coherent and precise English—not how to solve that problem, but what problem you’re trying to solve. Without this specification, it is impossible, even in principle, to determine whether your solution is correct.
(b) Solution. Give a clear recursive formula or algorithm for the whole problem in terms of the answers to smaller instances of exactly the same problem.
. Build solutions to your recurrence from the bottom up. Write an algo- rithm that starts with the base cases of your recurrence and works its way up to the final solution, by considering intermediate subproblems in the correct order. This stage can be broken down into several smaller, relatively mechanical steps:
(a) Identify the subproblems. What are all the dierent ways your re- cursive algorithm can call itself, starting with some initial input? For example, the argument to RF is always an integer between 0 and n.
Dynamic programming is not about lling in tables. It’s about smart recursion!

(b) Choose a memoization data structure. Find a data structure that can store the solution to every subproblem you identified in step (a). This is usually but not always a multidimensional array.
(c) Identify dependencies. Except for the base cases, every subproblem depends on other subproblems—which ones? Draw a picture of your data structure, pick a generic element, and draw arrows from each of the other elements it depends on. Then formalize your picture.
(d) Find a good evaluation order. Order the subproblems so that each one comes after the subproblems it depends on. You should consider the base cases first, then the subproblems that depends only on base cases, and so on, eventually building up to the original top-level problem. The dependencies you identified in the previous step define a partial order over the subproblems; you need to find a linear extension of that partial order. Be careful!
(e) Analyze space and running time. The number of distinct subproblems determines the space complexity of your memoized algorithm. To compute the total running time, add up the running times of all possible subproblems, assuming deeper recursive calls are already memoized. You can actually do this immediately after step (a).
(f) Write down the algorithm. You know what order to consider the subproblems, and you know how to solve each subproblem. So do that! If your data structure is an array, this usually means writing a few nested for-loops around your original recurrence, and replacing the recursive calls with array look-ups.
Of course, you have to prove that each of these steps is correct. If your recurrence is wrong, or if you try to build up answers in the wrong order, your algorithm won’t work!
. Warning: Greed is Stupid
If we’re incredibly lucky, we can bypass all the recurrences and tables and so forth, and solve the problem using a greedy algorithm. Like a backtracking algorithm, a greedy algorithm constructs a solution through a series of decisions, but it makes those decisions directly, without solving at any recursive subproblems. While this approach seems very natural, it almost never works; optimization problems that can be solved correctly by a greedy algorithm are quite rare. Nevertheless, for many problems that should be solved by backtracking or dynamic programming, many students’ first intuition is to apply a greedy strategy.
For example, a greedy algorithm for the longest increasing subsequence problem might look for the smallest element of the input array, accept that
.. Warning: Greed is Stupid

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