Unique Paths Problem

Input: m = 3, n = 2
Output: 3
Explanation:
From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:
1. Right -> Right -> Down
2. Right -> Down -> Right
3. Down -> Right -> Right

Input: m = 7, n = 3
Output: 28

                START
                /   \
               D     R
             /     /   \
           R      D      R
         /      /         \
        R      R            D

       END    END          END

BOARD[i][j] = BOARD[i - 1][j] + BOARD[i][j - 1]; // since we can only move down or right.

BOARD[0][any] = 1; // only one way to reach any top slot.
BOARD[any][0] = 1; // only one way to reach any slot in the leftmost column.

/**
 * 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;
}

/**
 * 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];
}

import pascalTriangle from '../../math/pascal-triangle/pascalTriangle';

/**
 * @param {number} width
 * @param {number} height
 * @return {number}
 */
export default function uniquePaths(width, height) {
  const pascalLine = width + height - 2;
  const pascalLinePosition = Math.min(width, height) - 1;

  return pascalTriangle(pascalLine)[pascalLinePosition];
}