algorithm: unique paths (#1211)

* dp problem

* update Directory.md

* suggested changes

Dynamic-Programming/UniquePaths.js

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+
+/*
+ *
+ *  Unique Paths
+ *
+ *  There is a robot on an `m x n` grid.
+ *  The robot is initially located at the top-left corner.
+ *  The robot tries to move to the bottom-right corner.
+ *  The robot can only move either down or right at any point in time.
+ *
+ *  Given the two integers `m` and `n`,
+ *  return the number of possible unique paths that the robot can take to reach the bottom-right corner.
+ *  More info: https://leetcode.com/problems/unique-paths/
+ */
+
+/*
+ * @param {number} m
+ * @param {number} n
+ * @return {number}
+ */
+
+const uniquePaths = (m, n) => {
+  // only one way to reach end
+  if (m === 1 || n === 1) return 1
+
+  // build a linear grid of size m
+  // base case, position 1 has only 1 move
+  const paths = new Array(m).fill(1)
+
+  for (let i = 1; i < n; i++) {
+    for (let j = 1; j < m; j++) {
+      // paths[j] in RHS represents the cell value stored above the current cell
+      // paths[j-1] in RHS represents the cell value stored to the left of the current cell
+      // paths [j] on the LHS represents the number of distinct pathways to the cell (i,j)
+      paths[j] = paths[j - 1] + paths[j]
+    }
+  }
+  return paths[m - 1]
+}
+
+export { uniquePaths }