feat: solve unique paths

Author: GangBean

unique-paths/GangBean.java

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+import java.math.BigDecimal;
+
+class Solution {
+    /**
+    1. understanding
+    - To reach destination, you have to move bottom direction in m-1 times, and move to right direction in n-1 times.
+    - in example 2, [(DDR), (DRD), (RDD)] is the paths.
+    - so, the number of paths are combinations of (m-1) D and (n-1) R
+    - (m+n-2)!/(m-1)!(n-1)!, where ! means factorial, n! = 1*2*...*n
+    - factorial[n]: n!
+    2. complexity
+    - time: O(m+n)
+    - space: O(m+n)
+     */
+    public int uniquePaths(int m, int n) {
+        BigDecimal[] dp = new BigDecimal[m+n];
+        Arrays.fill(dp, BigDecimal.ONE);
+        for (int num = 2; num < m+n; num++) {
+            dp[num] = dp[num-1].multiply(new BigDecimal(num));
+        }
+        return dp[m+n-2].divide(dp[m-1]).divide(dp[n-1]).intValue();
+    }
+}
+