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.

Kotlin DSA Implementation

basic.kt 
fun listString(values: IntArray): String = values.joinToString(prefix = "[", postfix = "]")

fun main() {
    val coins = intArrayOf(1, 3, 4)
    val target = 6
    val inf = target + 1
    val dp = IntArray(target + 1) { inf }
    dp[0] = 0
    for (amount in 1..target) {
        for (coin in 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