Add implementation of the coin change problem (TheAlgorithms#206)

* Add coin change source code

* Update DIRECTORY.md

* Add a link to README.md

Co-authored-by: Andrii Siriak <siryaka@gmail.com>

Author: imp2002

DIRECTORY.md

@@ -18,6 +18,7 @@
     * [Fibonacci](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/fibonacci.rs)
     * [Knapsack](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/knapsack.rs)
     * [Longest Common Subsequence](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/longest_common_subsequence.rs)
+    * [Coin Change](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/coin_change.rs)
     * [Mod](https://github.com/TheAlgorithms/Rust/blob/master/src/dynamic_programming/mod.rs)
   * General
     * [Convex Hull](https://github.com/TheAlgorithms/Rust/blob/master/src/general/convex_hull.rs)

src/dynamic_programming/coin_change.rs

@@ -0,0 +1,67 @@
+/// Coin change via Dynamic Programming
+
+/// coin_change(coins, amount) returns the fewest number of coins that need to make up that amount.
+/// If that amount of money cannot be made up by any combination of the coins, return `None`.
+///
+/// Arguments:
+///     * `coins` - coins of different denominations
+///     * `amount` - a total amount of money be made up.
+/// Complexity
+///     - time complexity: O(amount * coins.length),
+///     - space complexity: O(amount),
+pub fn coin_change(coins: &[usize], amount: usize) -> Option<usize> {
+    let mut dp = vec![std::usize::MAX; amount + 1];
+    dp[0] = 0;
+
+    // Assume dp[i] is the fewest number of coins making up amount i,
+    // then for every coin in coins, dp[i] = min(dp[i - coin] + 1).
+    for i in 0..=amount {
+        for j in 0..coins.len() {
+            if i >= coins[j] && dp[i - coins[j]] != std::usize::MAX {
+                dp[i] = dp[i].min(dp[i - coins[j]] + 1);
+            }
+        }
+    }
+
+    match dp[amount] {
+        std::usize::MAX => None,
+        _ => Some(dp[amount]),
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn basic() {
+        // 11 = 5 * 2 + 1 * 1
+        let coins = vec![1, 2, 5];
+        assert_eq!(Some(3), coin_change(&coins, 11));
+
+        // 119 = 11 * 10 + 7 * 1 + 2 * 1
+        let coins = vec![2, 3, 5, 7, 11];
+        assert_eq!(Some(12), coin_change(&coins, 119));
+    }
+
+    #[test]
+    fn coins_empty() {
+        let coins = vec![];
+        assert_eq!(None, coin_change(&coins, 1));
+    }
+
+    #[test]
+    fn amount_zero() {
+        let coins = vec![1, 2, 3];
+        assert_eq!(Some(0), coin_change(&coins, 0));
+    }
+
+    #[test]
+    fn fail_change() {
+        // 3 can't be change by 2.
+        let coins = vec![2];
+        assert_eq!(None, coin_change(&coins, 3));
+        let coins = vec![10, 20, 50, 100];
+        assert_eq!(None, coin_change(&coins, 5));
+    }
+}

src/dynamic_programming/mod.rs

@@ -1,9 +1,11 @@
+mod coin_change;
 mod edit_distance;
 mod egg_dropping;
 mod fibonacci;
 mod knapsack;
 mod longest_common_subsequence;
 
+pub use self::coin_change::coin_change;
 pub use self::edit_distance::{edit_distance, edit_distance_se};
 pub use self::egg_dropping::egg_drop;
 pub use self::fibonacci::fibonacci;