Add Unique Paths problem with backtracking and DP solutions.

src/algorithms/uncategorized/unique-paths/btUniquePaths.js

@@ -0,0 +1,58 @@
+/**
+ * BACKTRACKING approach of solving Unique Paths problem.
+ *
+ * @param {number} width - Width of the board.
+ * @param {number} height - Height of the board.
+ * @param {number[][]} steps - The steps that have been already made.
+ * @param {number} uniqueSteps - Total number of unique steps.
+ * @return {number} - Number of unique paths.
+ */
+export default function btUniquePaths(width, height, steps = [[0, 0]], uniqueSteps = 0) {
+  // Fetch current position on board.
+  const currentPos = steps[steps.length - 1];
+
+  // Check if we've reached the end.
+  if (currentPos[0] === width - 1 && currentPos[1] === height - 1) {
+    // In case if we've reached the end let's increase total
+    // number of unique steps.
+    return uniqueSteps + 1;
+  }
+
+  // Let's calculate how many unique path we will have
+  // by going right and by going down.
+  let rightUniqueSteps = 0;
+  let downUniqueSteps = 0;
+
+  // Do right step if possible.
+  if (currentPos[0] < width - 1) {
+    steps.push([
+      currentPos[0] + 1,
+      currentPos[1],
+    ]);
+
+    // Calculate how many unique paths we'll get by moving right.
+    rightUniqueSteps = btUniquePaths(width, height, steps, uniqueSteps);
+
+    // BACKTRACK and try another move.
+    steps.pop();
+  }
+
+  // Do down step if possible.
+  if (currentPos[1] < height - 1) {
+    steps.push([
+      currentPos[0],
+      currentPos[1] + 1,
+    ]);
+
+    // Calculate how many unique paths we'll get by moving down.
+    downUniqueSteps = btUniquePaths(width, height, steps, uniqueSteps);
+
+    // BACKTRACK and try another move.
+    steps.pop();
+  }
+
+  // Total amount of unique steps will be equal to total amount of
+  // unique steps by going right plus total amount of unique steps
+  // by going down.
+  return rightUniqueSteps + downUniqueSteps;
+}