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.

Scala DSA Implementation

basic.scala 
object Main extends App {
  def listString(values: Array[Int]): String = values.mkString("[", ", ", "]")
  val coins = Array(1, 3, 4)
  val target = 6
  val inf = target + 1
  val dp = Array.fill(target + 1)(inf)
  dp(0) = 0
  for (amount <- 1 to target) {
    for (coin <- coins) {
      if (amount >= coin) {
        val candidate = dp(amount - coin) + 1
        if (candidate < dp(amount)) dp(amount) = candidate
      }
    }
  }
  println(dp(target))
  println(listString(dp))
}

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