Dynamic Programming and Perl Arrays

Dynamic ProgrammingandPerl Arrays Ellen Walker Bioinformatics Hiram College

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

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) }

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.

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

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

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; }

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

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

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.

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

The Bottom Line for Making Change • Brute force solution considers every sequence. Since longest sequence is amt, this is O(coinsamt) • 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)

Dynamic Programming and Perl Arrays