Minimum Cost of a Path With Special Roads - Solution & Explanation

This approach uses Dijkstra's Algorithm, which is suitable for finding the shortest path in a graph with non-negative weights. Consider each unique position as a node, with bi-directional edges between any two nodes representing the cost to travel between them, either directly or through special roads.

Start at the initial position and explore the reachable positions using both direct Manhattan distance and special roads. Maintain a priority queue (min-heap) to always extend the least costly path available. The algorithm terminates when the target position is reached with the minimum cost.

This solution initializes a priority queue with the starting position at cost zero and iterates through all paths, either direct or via special roads, continuously extending the shortest available path. Once it dequeues the target position from the queue, the accumulated cost is the answer.

The Bellman-Ford algorithm is another valid approach for solving the shortest path problem in a graph with directional edges and no negative weight cycles. Unlike Dijkstra's, this approach systematically updates all paths for a given number of iterations, ensuring the shortest path puts special roads and direct routes on equal footing.

We'll iterate over each node and relax edges associated with both direct movement and special roads. Given the scope of this problem, Bellman-Ford is computationally heavier but provides valuable comparative insights.

This solution uses the Bellman-Ford technique where edges of both traditional and special paths are added. The edge relaxation process ensures iteratively updating the shortest path until no further improvements can be achieved. The final distance derived for the target position will yield the lowest traversal cost.

We can find that for each coordinate (x, y) we visit, suppose the minimum cost from the start point to (x, y) is d. If we choose to move directly to (targetX, targetY), then the total cost is d + |x - targetX| + |y - targetY|. If we choose to go through a special path (x_1, y_1) \rightarrow (x_2, y_2), then we need to spend |x - x_1| + |y - y_1| + cost to move from (x, y) to (x_2, y_2).

Therefore, we can use Dijkstra algorithm to find the minimum cost from the start point to all points, and then choose the smallest one from them.

We define a priority queue q, each element in the queue is a triple (d, x, y), which means that the minimum cost from the start point to (x, y) is d. Initially, we add (0, startX, startY) to the queue.

In each step, we take out the first element (d, x, y) in the queue, at this time we can update the answer, that is ans = min(ans, d + dist(x, y, targetX, targetY)). Then we enumerate all special paths (x_1, y_1) \rightarrow (x_2, y_2) and add (d + dist(x, y, x_1, y_1) + cost, x_2, y_2) to the queue.

Finally, when the queue is empty, we can get the answer.

The time complexity is O(n^2 times log n), and the space complexity is O(n^2). Where n is the number of special paths.