Unique Paths.txt

problem:
A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).

The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).

How many possible unique paths are there?

Note: m and n will be at most 100.

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solution:
//解法一：回溯法(小集合通过，大集合超时)
int backtrack(int r, int c, int m, int n)
{
  if(r == m - 1 && c == n - 1)
		return 1;
	if(r > m - 1 || c > n - 1)
		return 0;
	
	return backtrack(r + 1, c, m, n) + backtrack(r, c + 1, m, n);
}
int uniquePaths(int m, int n)
{
	return backtrack(0, 0, m, n);
}


//解法二、DP(from bottom to up)
//动态规划状态转移方程：dp[m][n] = dp[m][n-1] + dp[m-1][n]，其中dp[m][n]表示到达网格[m][n]时有多少条唯一的路径
int uniquePaths(int m, int n)
{
	if(m == 0 || n == 0)
		return 0;
	int dp[m][n];
	for(int r = 0; r < m; ++r)
		dp[r][0] = 1;
	for(int c = 0; c < n; ++c)
		dp[0][c] = 1;
	for (int r = 1; r < m; r++)
		for (int c = 1; c < n; c++)
			dp[r][c] = dp[r-1][c] + dp[r][c-1];

	return dp[m - 1][n - 1];
}