#2662

Medium

LeetCode Problem Workspace

Minimum Cost of a Path With Special Roads

Find the minimum cost path between two points, using special roads or direct moves in a 2D space.

array graph heap priority queue shortest path

Problem Statement

You are given a 2D grid where the start and target points are represented by two arrays, start and target. Each array contains two integers, startX, startY for the start position and targetX, targetY for the target position.

The challenge is to find the minimum cost to go from the start position to the target. You can move directly between positions in the grid with a cost calculated by the Manhattan distance formula: |x2 - x1| + |y2 - y1|. Additionally, you are provided with special roads, each defined by a start and end point with a cost to traverse.

Examples

Example 1

Input: start = [1,1], target = [4,5], specialRoads = [[1,2,3,3,2],[3,4,4,5,1]]

Output: 5

So the total cost is 1 + 2 + 1 + 1 = 5.

Example 2

Input: start = [3,2], target = [5,7], specialRoads = [[5,7,3,2,1],[3,2,3,4,4],[3,3,5,5,5],[3,4,5,6,6]]

Output: 7

It is optimal not to use any special edges and go directly from the starting to the ending position with a cost |5 - 3| + |7 - 2| = 7. Note that the specialRoads[0] is directed from (5,7) to (3,2).

Example 3

Input: start = [1,1], target = [10,4], specialRoads = [[4,2,1,1,3],[1,2,7,4,4],[10,3,6,1,2],[6,1,1,2,3]]

Output: 8

Constraints

start.length == target.length == 2
1 <= startX <= targetX <= 105
1 <= startY <= targetY <= 105
1 <= specialRoads.length <= 200
specialRoads[i].length == 5
startX <= x1i, x2i <= targetX
startY <= y1i, y2i <= targetY
1 <= costi <= 105

Solution Approach

Graph Representation

The grid and special roads can be modeled as a graph where nodes represent positions in the 2D space. Special roads act as directed edges with specific costs. You must consider both the direct distance and the special roads to find the optimal path.

Dijkstra's Algorithm with Priority Queue

Since the problem involves finding the shortest path, you can apply Dijkstra's algorithm with a priority queue to handle the variable costs of moving from one position to another. Each node's cost will be updated based on the direct path or special road with the least cost.

Limitations on Path Selection

It is proven that optimal paths should only involve start, target, or special road endpoints. This drastically reduces the number of possible paths and ensures that we can use Dijkstra’s algorithm efficiently.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the graph representation and the priority queue operations, typically O((V + E) log V), where V is the number of nodes and E is the number of edges (special roads). The space complexity also depends on the graph structure and is O(V + E).

What Interviewers Usually Probe

Evaluating how the candidate handles pathfinding in a graph with variable edge costs.
How the candidate optimizes the graph search based on the start and target points.
Whether the candidate can apply Dijkstra's algorithm correctly with the given constraints.

Common Pitfalls or Variants

Common pitfalls

Not reducing the problem space by only considering key points (start, target, special road endpoints).
Ignoring the optimization of pathfinding using a priority queue, leading to inefficient solutions.
Failing to handle edge cases such as when no special roads are useful or when a direct path is optimal.

Follow-up variants

Consider adding more special roads with varied costs to increase the complexity of the problem.
Modify the problem to allow for multiple target positions and find the shortest path to any of them.
Allow bidirectional special roads, which would require additional logic to handle the reverse direction.

FAQ

What is the Minimum Cost of a Path With Special Roads problem?

The problem asks you to find the minimum cost path between a start and target point in a 2D grid, considering both direct moves and special roads with defined costs.

How does the special road affect the cost calculation?

Special roads provide a cost-effective way to travel between points in the grid. These roads are factored into the minimum cost path calculation alongside the direct distance.

Why should we only consider start, target, and special road endpoints?

It has been proven that optimal paths only require considering these key points, as any other paths would incur higher costs or be redundant.

What algorithm is most suitable for solving this problem?

Dijkstra’s algorithm is most suitable as it efficiently handles the shortest path in graphs with varying edge weights, especially when combined with a priority queue.

How can GhostInterview assist in solving this problem?

GhostInterview provides step-by-step guidance on applying graph algorithms, optimizing your solution, and avoiding common pitfalls like inefficient path selection.

terminal

Solution

Solution 1: Dijkstra

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 $(x_2, y_2)$.

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class Solution:
    def minimumCost(
        self, start: List[int], target: List[int], specialRoads: List[List[int]]
    ) -> int:
        def dist(x1: int, y1: int, x2: int, y2: int) -> int:
            return abs(x1 - x2) + abs(y1 - y2)

        q = [(0, start[0], start[1])]
        vis = set()
        ans = inf
        while q:
            d, x, y = heappop(q)
            if (x, y) in vis:
                continue
            vis.add((x, y))
            ans = min(ans, d + dist(x, y, *target))
            for x1, y1, x2, y2, cost in specialRoads:
                heappush(q, (d + dist(x, y, x1, y1) + cost, x2, y2))
        return ans

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