Recursion and Dynamic Programming

Coin Change (Bottom-Up)

Build a one-dimensional table where each amount stores the fewest coins needed to make it.

Algorithm

@steps

  1. Initialize dp[0] = 0 and all other amounts to an unreachable sentinel.
  2. Scan amounts from 1 through 6.
  3. For each coin, read the earlier cell dp[amount - coin] when it exists.
  4. Write the smallest candidate into the current amount.
  5. Print both the final answer and the full DP array. @end @complexity
  • Time: O(target * coin_count)
  • Space: O(target) @end
bottom-up dynamic programming `dp[a]` is solved from already-computed smaller amounts, so every table cell has a visible dependency.

Perl DSA Implementation

basic.pl 
use strict;
use warnings;

sub list_string { return "[" . join(", ", @_) . "]"; }

my @coins = (1, 3, 4);
my $target = 6;
my $inf = $target + 1;
my @dp = (($inf) x ($target + 1));
$dp[0] = 0;
for my $amount (1..$target) {
  for my $coin (@coins) {
    if ($amount >= $coin) {
      my $candidate = $dp[$amount - $coin] + 1;
      $dp[$amount] = $candidate if $candidate < $dp[$amount];
    }
  }
}
print "$dp[$target]\n";
print list_string(@dp) . "\n";

@end @output 2 [0, 1, 2, 1, 1, 2, 2] @end