1. Dynamic Programming From An
Excel Perspective

2. Dynamic Programming From An Excel Perspective
Ranette Halverson, Richard Simpson
Catherine Stringfellow
Department of Computer Science
Midwestern State University

3. Dynamic Programming From An Excel Perspective
A popular method for solving problems by breaking them down into overlapping sub-problems that display optimal substructure
Can be thought of as a top-down approach utilizing a bottom-up evaluation
Normally used to solve optimization problems

4. Dynamic Programming From An Excel Perspective
Generally taught in the freshman application classes
Seldom taught to computer science majors
In reality CS majors need to be able to use spreadsheets
So what do we do?

5. Dynamic Programming From An Excel Perspective
Solution
Do not need to teach the spreadsheet AT ALL
Include spreadsheet usage in a few of their projects and/or homework
Spreadsheet usage includes
Graphing data collected via empirical analysis of two algorithms.
Rapidly construct mathematical tables for applications
Simulating wave-front parallel algorithms
Evaluating dynamic programming tables (the point of this talk)

6. Dynamic Programming From An Excel Perspective
A Very Simple Example (used in Computer Science for Science Majors) The memo-ization of the recursive Fibonacci function. Remember the complexity of the following?
int Fib( int n) {
if (n<3) return 1
else return ( Fib(n-1)+Fib(n-2) ); }

7. Dynamic Programming From An Excel Perspective
Two well-known O(n) solutions
int FibMemo(int n,int * A){
if (A[n]!=0) return A[n];
else {
A[n]= FibMemo(n-1,A) + FibMemo(n-2,A);
return A[n]; } };
int Fib(int n) {
int * A = new int[n+1] ;
for (int i=1;i<n+1;i++){ A[i]=0;}
A[1]=A[2]=1;
return FibMemo(n,A);
} // A recursive Memoized version
int Fib( int n) {
A=new int[n+1];
A[1]=A[2]=1;
for(int i=3 ; i<=n ; i++)
A[i] = A[i-1] + A[i-2];
return A[n]; }
// Pure Bottom up calculation using
// an array. The non array version is
// not relative to our discussion.

8. Dynamic Programming From An Excel Perspective
Excel’s simple approach
Kand copy cell to the right
=A1+B1

9. Dynamic Programming From An Excel Perspective
Pascal's triangle is constructed in Excel in the bottom-up approach.
The programmed solution can be handled via DP as in the Fibonacci example, either using an array with or without memoized recursion.
The pure recursive version is computationally unacceptable.
formula =B1+A2 is copied from B2 to the remaining cells.

10. Dynamic Programming From An Excel Perspective
There are many DP algorithms that appear throughout our curriculum.
Longest Common Subsequence Bioinformatics class.
Sequence Alignment: Bioinformatics
Optimal Binary Search Tree: Algorithm Analysis
Matrix Chain Multiplication: Algorithms Analysis
Graph Algorithms

11. Dynamic Programming From An Excel Perspective
Longest Common Subsequence (LCS) Definition: find the longest subsequence that is common to two (or more) sequences. Example Seq1 = B D C A B A Seq2 = A B C B D A B LCS = BCBA Note: The LCS is not a substring!

12. Dynamic Programming From An Excel Perspective
Longest Common Subsequence (LCS) DP leads to the following recursive approach. Let z=z1 z2 … xk be the LCS of
x1 x2 … xi-1 xi
y1 y2 … yj-1 yj
𝑐 𝑖,𝑗 = 0 𝑖𝑓 𝑥=0 𝑜𝑟 𝑗=0 𝑐 𝑖−1,𝑗−1 +1 𝑖𝑓 𝑖,𝑗>0 𝑎𝑛𝑑 𝑥 𝑖 = 𝑦 𝑗 max(𝑐 𝑖,𝑗−1 ,𝑐 𝑖−1,𝑗 𝑖𝑓 𝑖,𝑗>0 𝑎𝑛𝑑 𝑥 𝑖 ≠ 𝑦 𝑗
Where c[ib ,j] is the length of the LCS of x1..xi and y1..yj

13. Dynamic Programming From An Excel Perspective
The initialized LCS Table
Copy the following cell
formula to all grey cells.
These represent the c(i,j)’s
IF(D$2=$B4,C3+1,MAX(D3,C4))
Cell formula

14. Dynamic Programming From An Excel Perspective
And the solution is
Note diagonal increments
Length of LCS

15. Dynamic Programming From An Excel Perspective
DP Problems with complex table manipulation
Optimal Binary Search Tree (in paper)
Matrix Chain Multiplication
Question: What do you do with problems that require the processing of rows or columns in such a way that the usual cell function approach is not adequate?
Excel does not allow cell function side effects! Hmm.
Write the code in the include macro language (VB?)

16. Dynamic Programming From An Excel Perspective
Summarizing
CS students can benefit from work with Excel
Excel can support many projects in the CS curriculum.
Table processing available in Excel supports some algorithm visualization quite well.
This approach works particularly well with the simpler DP problems.

17. THE END