Add Unique Paths problem with backtracking and DP solutions.

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

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+/**
+ * DYNAMIC PROGRAMMING approach of solving Unique Paths problem.
+ *
+ * @param {number} width - Width of the board.
+ * @param {number} height - Height of the board.
+ * @return {number} - Number of unique paths.
+ */
+export default function dpUniquePaths(width, height) {
+  // Init board.
+  const board = Array(height).fill(null).map(() => {
+    return Array(width).fill(0);
+  });
+
+  // Base case.
+  // There is only one way of getting to board[0][any] and
+  // there is also only one way of getting to board[any][0].
+  // This is because we have a restriction of moving right
+  // and down only.
+  for (let rowIndex = 0; rowIndex < height; rowIndex += 1) {
+    for (let columnIndex = 0; columnIndex < width; columnIndex += 1) {
+      if (rowIndex === 0 || columnIndex === 0) {
+        board[rowIndex][columnIndex] = 1;
+      }
+    }
+  }
+
+  // Now, since we have this restriction of moving only to the right
+  // and down we might say that number of unique paths to the current
+  // cell is a sum of numbers of unique paths to the cell above the
+  // current one and to the cell to the left of current one.
+  for (let rowIndex = 1; rowIndex < height; rowIndex += 1) {
+    for (let columnIndex = 1; columnIndex < width; columnIndex += 1) {
+      const uniquesFromTop = board[rowIndex - 1][columnIndex];
+      const uniquesFromLeft = board[rowIndex][columnIndex - 1];
+      board[rowIndex][columnIndex] = uniquesFromTop + uniquesFromLeft;
+    }
+  }
+
+  return board[height - 1][width - 1];
+}