1. Dynamic Programming and Perl Arrays Ellen Walker Bioinformatics Hiram College

2. The Change Problem Given a set of coins Find the minimum number of coins to make a given amount

3. Brute Force (Recursive) Solution Find_change (amt){ if ((amt) < 0) return infinity else if (amt = 0) return 0 else best = infinity For each coin in the set result = find_change(amt - coin) if (result < best) best = result return (best+1) }

4. To write this in Perl We need to create an array of coins –@coins = (1, 5, 10, 20, 25); To access a coin: –$coin = $coins[2]; # gets 10 To loop through all coins: –foreach my $coin (@coins) –Inside the loop, $coin is first $coins[0], then $coins[1], etc.

5. Brute Force Coins in Perl (main) my @coins = ( 1, 5, 10, 20, 25 ); print brute_change( 99 ); print "\n";

6. Subroutine to compute change (Part 1) sub brute_change { my $amt = shift(@_); if ($amt == 0){ return 0; } if ($amt < 0){ return 99999999999999; }

7. Subroutine to compute change (Part 2) my $best = 99999999999999; foreach my $coin (@coins){ my $count = brute_change($amt - $coin); print "coin is $coin, count is $count \n"; if ($best > $count) { $best = $count; } return $best+1; }

8. Why It Takes So Long For each value, we repeatedly compute the same result numerous times. For 99: 74, 79, 89, 94, 98 For 98: 73, 78, 88, 93, 97 For 97: 72, 76, 87, 92, 96 For 96: 71, 75, 86, 91, 95 For 95: 70, 74, 85, 82, 94

9. Save Time by Expending Space Instead of calling the function recursively as we need it… –Compute each amount from 1 to 99 –Save each result as we need it –Instead of doing a recursive computation, look up the value in the array of results This is called dynamic programming

10. Why it Works? By the order we’re computing in, we know that we have every value less than $amt covered when we get to $amt. We never need a value larger than $amt, so all the values we need are in the table.

11. Dynamic Programming in General Consider an impractical recursive solution Note which results are needed to compute a new result Reorder the computions so the needed ones are done first Save all results and look them up in the table instead of doing a recursive computation

12. The Bottom Line for Making Change Brute force solution considers every sequence. Since longest sequence is amt, this is O(coins amt ) Dynamic programming solution considers every value from 1 to amt, and for each amt, does a check for each coin. This is O(coins*amt)