Minimum Cost to travel in a Grid

Let the movement graph choose the algorithm

First clarify whether movement is only right and down, whether four directions are allowed, and whether the cost belongs to entering a cell, leaving it, or traversing an edge. With only right/down moves the graph is acyclic: dp[r][c] = cost(r,c) + min(dp[r-1][c], dp[r][c-1]) is enough. With four-direction nonnegative movement, revisiting is possible, so run Dijkstra from the start over neighboring cells. If costs are only zero or one, 0-1 BFS is a better specialization.

Make conventions explicit

Initialize the start correctly according to whether its cost is paid, reject blocked cells, and use a wide distance type. Do not use a visited boolean in place of Dijkstra relaxation: a cell can first be discovered by a nonoptimal route. The acyclic DP is O(rows*cols) time and can use O(cols) space. Dijkstra is O(rowscols log(rowscols)) time with a priority queue and O(rows*cols) space. Test one cell, blocked start or destination, and a case where going temporarily away from the destination is cheaper.