Dynamic Programming From An Excel Perspective

Dynamic ProgrammingFrom An

Excel Perspective

Dynamic Programming From An Excel Perspective

Dynamic ProgrammingFrom An Excel Perspective

Ranette Halverson, Richard SimpsonCatherine Stringfellow

Department of Computer ScienceMidwestern State University


Dynamic Programming

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


Excel

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?


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)


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) );}


Two well-known O(n) solutions

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.

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


Excel’s simple approach

=A1+B1 Kand copy cell to the right


formula =B1+A2 is copied from B2 to the remaining cells.

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.


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


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!


Longest Common Subsequence (LCS) DP leads to the following recursive approach.

Let z=z1 z2 … xk be the LCS of

𝑐 [𝑖 , 𝑗 ]=¿

x1 x2 … xi-1 xi

y1 y2 … yj-1 yj

Where c[ib ,j] is the length of the LCS of x1..xi and y1..yj


The initialized LCS Table

IF(D$2=$B4,C3+1,MAX(D3,C4))

Cell formula

Copy the following cellformula to all grey cells.

These represent the c(i,j)’s


And the solution is

Note diagonal increments

Length of LCS


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?)


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.

THE END

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Dynamic Programming From An Excel Perspective