1. Lecture 9 Dynamic programming
What is Dynamic Programming
How to write Dynamic Programming paradigm
When to apply Dynamic Programming

2. Roadmap What is Dynamic Programming How to write DP paradigm
Knapsack problem
Longest common subsequence
Matrix chain multiplication
When to apply Dynamic Programming
Practice

3. Techniques Based on Recursion
Induction or Tail recursion
Non-overlapping subproblems
Overlapping subproblems

4. An Introductory Example
\begin{eqnarray*}
Fib(0) & = & 0 \\
Fib(1) & = & 1 \\
Fib(n) & = & Fib(n-1) + Fib(n-2)
\end{eqnarray*}
19th century statue of Fibonacci in Camposanto, Pisa.
(Source from Wikipedia)
Problem: Fibonacci
Input: A positive integer n
Output: Fib(n)

5. Quiz
Problem: Fibonacci
Input: A positive integer n
Output: Fib(n)
Give your solution of Fibonacci and explain the time/space complexity.

6. Recursion implementation
int fib_rec (int n){
if (n==1 || n==2) {
return 1; }
else {
return fib_rec(n-1) + fib_rec(n-2);
\begin{eqnarray*}
T(n) & = & T(n-1) + T(n-2) + 2,\ \ T(1),T(2) < 3 \\
T(n) & > & Fib_n \\
S(n) & = & O(n)
\end{eqnarray*} O

7. O(n) time, O(n) space int fib (int n){ int* array = new int[n];
for (int i = 2; i <= n; i++)
array[i] = array[i-1] + array[i-2];
return array[n]; }
\begin{eqnarray*}
T(n) & = & n + 1\\
S(n) & = & n + 1
\end{eqnarray*}

8. O(n) time, O(1) space We can do even better: int fib (int n){
if (n < 2) return n;
int f0 = 0;
int f1 = 1;
int i = 2;
while (i <= n) {
f1 = f1 + f0;
f0 = f1 - f0;
i++; }
return f1;
\begin{eqnarray*}
T(n) & < & 3n\\
S(n) & = & 3
\end{eqnarray*}
We can do even better:

9. Use Divide-and-conquer
O(log n) time, O(1) space
\left(\begin{array}{cc}
f_n & f_{n-1}\\
f_{n-1} & f_{n-2}
\end{array}\right) =
1 & 1\\
1 & 0
\end{array}\right) \times \left(\begin{array}{cc}
f_{n-1} & f_{n-2}\\
f_{n-2} & f_{n-3}
\end{array}\right)
\[T(n) = T(\frac{n}{2}) + O(1)\]
\[T(n) = O(\log n)\]
Use Divide-and-conquer

10. (Source from Wikipedia)
Fibonacci
\begin{eqnarray*}
Fib(0) & = & 0 \\
Fib(1) & = & 1 \\
Fib(n) & = & Fib(n-1) + Fib(n-2)
\end{eqnarray*}
Overlapping subproblems
19th century statue of Fibonacci in Camposanto, Pisa.
(Source from Wikipedia)
Recursion? No, thanks.

11. Non-recursively handle recursive problems
Dynamic programming
Non-recursively handle recursive problems
Time-Memory tradeoff
Find smaller subproblems
Write the relation expression
Use an array to save the intermediate results
Update the array iteratively.

12. All-pairs shortest path
Define $d_{i,j}^{k}$ to be the length of a shortest path from $i$ to
$j$ that does not pass any vertex in $\{k+1,k+2,\ldots,n\}$. Clearly
\[d_{ij}^{k} = \left\{\begin{array}{ll}
l[i,j] & {\rm if}\ k=0 \\
min\{d_{i,j}^{k-1},d_{i,k}^{k-1}+d_{k,j}^{k-1}\} & {\rm if}\ 1\le
k\le n
\end{array}\right.\]

13. Floyd-Warshall algorithm
Θ(n3)

14. Where are we? What is Dynamic Programming How to write DP paradigm
Knapsack problem
Longest common subsequence
Matrix chain multiplication
When to apply Dynamic Programming
Practice

15. Knapsack problem Problem: Knapsack
Input: A set of items U = {u1,...,un} with sizes s1,s2,...,sn and values v1,v2,...,vn and a knapsack capacity C
Output: The maximum value that can be put into the knapsack

16. Knapsack problem
V[i,j]: the maximum value obtained by filling a knapsack of size j with items taken from the first i items {u1,...,ui}.
\[V[i,j] = \left\{
\begin{array}{ll}
0 & {\rm if}\ i=0\ {\rm or}\ j=0 \\
V[i-1,j] & {\rm if}\ j<s_{i} \\
max\{V[i-1,j],V[i-1,j-s_{i}]+v_{i}\} & {\rm if}\ i>0\ {\rm and}\
j\ge s_{i}
\end{array}
\right.\] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1, 1
1, 2
2, 2
4, 10
12, 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 4 5 5 5 5 5 5 5 5 5 5 5 5 2 3 4 10 12 13 14 15 15 15 15 15 15 15 15 2 3 4 10 12 13 14 15 15 15 15 15 15 15 15

17. Psuedo-polynomial algorithm
Knapsack problem
Time: Θ(nC)
Space: Θ(nC)
Psuedo-polynomial algorithm

18. How to construct the solution?1 2 3 4 5 10 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1, 1
1, 2
2, 2
4, 10
12, 4
\[V[i,j] = \left\{
\begin{array}{ll}
0 & {\rm if}\ i=0\ {\rm or}\ j=0 \\
V[i-1,j] & {\rm if}\ j<s_{i} \\
max\{V[i-1,j],V[i-1,j-s_{i}]+v_{i}\} & {\rm if}\ i>0\ {\rm and}\
j\ge s_{i}
\end{array}
\right.\] V

19. What if the thief robs a super-market? Knapsack with repetition.
Knapsack problem
What if the thief robs a super-market? Knapsack with repetition.
V[i,j] = 0 if i = 0 or j = 0
= V[i-1, j] if si > j
= max{V[i,j-si] + vi, V[i-1, j]} o.w. or
V[j] = max_{i, si <= j}{V[j-si]+vi}

20. Longest common subsequence
Problem: LCS
Input: Two strings A=a1a2...an and B=b1b2...bm
Output: The longest common subsequence of A and B
L[i,j]: length of the longest common subsequence of
A[1..i] and B[1..j]
ababe
ababe
Solution: L[n,m]
abcabc
abcabc

21. LCS a b c a b e
\[ L[i,j] = \left\{
\begin{array}{ll}
0 & {\rm if}\ i=0\ {\rm or}\ j=0 \\
L[i-1,j-1]+1 & {\rm if}\ i>0,j>0\ {\rm and}\ a_{i}=b_{j} \\
max\{L[i-1,j],L[i,j-1]\} & {\rm if}\ i>0,j>0\ {\rm and}\
a_{i}\not=b_{j}
\end{array}\right.\] 1 1 1 1 1 1 1 2 2 2 2 2 1 2 2 3 3 3 1 2 2 3 4 4 1 2 2 3 4 4

22. LCS
Time: Θ(mn)
Space: Θ(max{m,n})

23. Longest common substring?
\[ L[i,j] = \left\{
\begin{array}{ll}
0 & {\rm if}\ i=0\ {\rm or}\ j=0 \\
L[i-1,j-1]+1 & {\rm if}\ i>0,j>0\ {\rm and}\ a_{i}=b_{j} \\
0 & {\rm if}\ i>0,j>0\ {\rm and}\
a_{i}\not=b_{j}
\end{array}\right.\]
L[i,j]: length of longest common sub-postfix of a[1..i] and b[1..j].
Solution: the maximum L[i,j] for all i>0 and j>0

24. Matrix chain multiplication
What is the most efficient way of computing the following multiplication?
n matrix, how many different way of multiplications?
Answer: catalan number C_{n-1}
C_n = C0Cn-1 + C1 C Cn-1C0 = 2n\choose n - 2n \choose n+1 = (2n\choose n+1) / n+1
Catalan number:
1. How many different ways to pair n-pair parenthesis?
Fix one (), then 0n-1, 1 n-2, 2 n-3,... n-1 0
2. How many different ways to completely parenthesize n+1 factors?
3. How many different full binary trees with n+1 leaves?
View intermediate node as multiplication
4. How many different ways to cut a convex n+2 polygon into triangles?
\[\left( \begin{array}{c}
a_{11}\ldots a_{15} \\
\vdots \\
a_{61}\ldots a_{65}
\end{array} \right)
\left( \begin{array}{c}
b_{11}\ldots b_{14} \\
b_{51}\ldots b_{54}
c_{11}\ldots c_{13} \\
c_{41}\ldots c_{43}
\end{array} \right)\]
The order of multiplications matters.

25. Quiz M1 * M2 * … * Mn How many different ways to compute
Catalan number

26. Matrix chain multiplication
Problem: MCM
Input: A matrix chain M1M2...Mn with rank r1r2..rnrn+1
Output: Minimal cost of the multiplication
ABCD
50,20,1,10,100

27. C[i,j]: the minimal cost of the multiplication MiMi +1 ... Mj
Dynamic programming
C[i,j]: the minimal cost of the multiplication MiMi Mj A B C D
Memoization A B C D
1000
1500
7000
ABCD
50,20,1,10,100
200
3000
1000

28. MCM
1+2^ n^2 = n(n+1)(2n+1)/6

29. C[i,j]: the minimal cost of the multiplication MiMi +1 ... Mj
Dynamic programming
C[i,j]: the minimal cost of the multiplication MiMi Mj
ABCD
50,20,1,10,100
1000
200
3000
1500
7000 A B C D 4 3 2 A B C D A B C D A B C D 1 2 3 4

30. Where are we? What is Dynamic Programming How to write DP paradigm
Knapsack problem
Longest common subsequence
Matrix chain multiplication
When to apply Dynamic Programming
Practice

31. Elements of DP Optimal substructure
A problem exhibits optimal substructure if an optimal solution to the problem contains within it optimal solutions to subproblems.
Overlapping subproblems

32. Optimal Substructure
Given a graph G=<V,E>, which problem has optimal substructure, longest simple path problem or shortest simple path problem?

33. Practice Subset sum Longest palindrome sequence Editing distance
Longest increasing subsequence
Planning a company party

34. Subset sum problem Problem: SubsetSum
Input: A set of numbers A={a1,a2,...,an}, and a sum s
Output: Yes, if there exists a subset B⊆A such that the sum of B equals to s; No, otherwise.
\[ L[i,j] = \left\{
\begin{array}{ll}
0 & {\rm if}\ i=0\ \\
1 & {\rm if}\ j=0\\
L[i-1,j] & {\rm if}\ i>0, j>0,\ j<a_i\\
L[i-1,j-a_i] \vee L[i-1,j] & {\rm if}\ i>0,j>0,\ j\geq a_i
\end{array}\right.\]
L[i,j]: Does there exists a subset B of {a1,...,ai} such that the sum of B equals to j
Solution: L[n,s]

35. Longest palindrome subsequence
A palindrome is a nonempty string over some alphabet
that reads the same for- ward and backward.
civic, racecar
Problem: PalindromeSubsequence
Input: A string A=a1a2...an
Output: Length of the longest subsequence of A that is a palindrome
We can also reduce this problem to the Longest Common Subsequence problem of a1a2..an and an...a2a1.
(by Qiu Zhe.)
L[i,j]: length of the longest palindrome subsequence of ai...aj
L[i,j] = 1 if i = j
L[i,j] = L[i+1, j-1] if ai = aj
L[i,j] = max{L[i+1,j], L[i,j-1]} o.w

36. E[i,j]: edit distance of a1...ai and b1...bj
Editing distance
snowy --> sunny
The edit distance between two strings is the cost of their best possible alignment.
E[i,j]: edit distance of a1...ai and b1...bj

37. Planning a company party2
We like convivial friends.
And we will not invite both the employee and his/her direct supervisor.
How to maximize the conviviality? 4 9

38. Single-source longest path in DAG
D[s] = 0
D[v] = max(u,v)\in E{D[u]+w(u,v)}

39. Conclusion What is Dynamic Programming How to write DP paradigm
Knapsack problem
Longest common subsequence
Matrix chain multiplication
When to apply Dynamic Programming
Practice