Suppose we have an n n grid. We start in the upper-left corner (position (1, 1)), and we would like to reach the lower-right corner (position (n, n)) by taking single steps down and right. Define a function c(i, j) to be the cost of touching position (i, j), and assume it takes constant time to compute. Note that c(i, j) can be negative. Give an algorithm for computing the minimum cost in the most efficient way. What is the runtime (just give the big-O)? Hint: Define v(i, j) to be the minimum cost to reach (i, j) from (1, 1). Can you find a recurrence for v(i, j)? The naive way of implementing the recurrence will take exponential time due to the repeated computation of the same subproblems. Can you make it more efficient? (with the help of additional memory space, probably) Suppose we have an n n grid. We start in the upper-left corner (position (1, 1)), and we would like to reach the lower-right corner (position (n, n)) by taking single steps down and right. Define a function c(i, j) to be the cost of touching position (i, j), and assume it takes constant time to compute. Note that c(i, j) can be negative. Give an algorithm for computing the minimum cost in the most efficient way. What is the runtime (just give the big-O)? Hint: Define v(i, j) to be the minimum cost to reach (i, j) from (1, 1). Can you find a recurrence for v(i, j)? The naive way of implementing the recurrence will take exponential time due to the repeated computation of the same subproblems. Can you make it more efficient? (with the help of additional memory space, probably)