How do you solve Minimum Cost of a Path With Special Roads on LeetCode?

Minimum Cost of a Path With Special Roads (LeetCode #2662) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n log n) time complexity and O(n) space complexity. Key insight: Using special roads can significantly reduce travel costs if chosen wisely.

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

The optimal solution for Minimum Cost of a Path With Special Roads runs in O(n log n) time and uses O(n) space. This complexity is due to the priority queue operations and the number of nodes processed, where n is the number of unique positions considered.

What is the brute force solution for Minimum Cost of a Path With Special Roads?

The brute force approach for Minimum Cost of a Path With Special Roads is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach calculates the cost of all possible paths from the starting point to the target. It checks every special road and the direct path to find the minimum cost. The optimal Optimal Solution approach improves this to O(n log n).

What are the interview tips for Minimum Cost of a Path With Special Roads?

Always clarify the problem and constraints before diving into coding. Think about edge cases, such as no special roads or starting at the target. Explain your thought process clearly while coding, as it shows your understanding.

What are common mistakes when solving Minimum Cost of a Path With Special Roads?

Failing to consider the cost of using special roads versus direct paths. Not properly implementing the priority queue for efficient pathfinding.

#2662

Minimum Cost of a Path With Special Roads

Medium

ArrayGraph TheoryHeap (Priority Queue)Shortest PathGraph TraversalDijkstra's Algorithm

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n log n)
Space	O(1)	O(n)

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Intuition

Time O(n log n)Space O(n)

The optimal solution treats the problem as a graph where nodes are the start, target, and endpoints of special roads. We use Dijkstra's algorithm to efficiently find the minimum cost path.

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Algorithm

3 steps

1Step 1: Create a graph with nodes representing the start, target, and endpoints of special roads.
2Step 2: Use a priority queue to explore the minimum cost paths from the start node to all other nodes.
3Step 3: Update costs as shorter paths are found, and stop when the target node is reached.

solution.py30 lines

1# Full working Python code
2import heapq
3
4def minCost(start, target, specialRoads):
5    graph = {}
6    nodes = [tuple(start), tuple(target)]
7    for road in specialRoads:
8        x1, y1, x2, y2, cost = road
9        nodes.append((x1, y1))
10        nodes.append((x2, y2))
11        graph[(x1, y1)] = graph.get((x1, y1), []) + [(cost, (x2, y2))]
12    min_cost = {node: float('inf') for node in nodes}
13    min_cost[tuple(start)] = 0
14    pq = [(0, tuple(start))]
15    while pq:
16        current_cost, current_node = heapq.heappop(pq)
17        if current_node == tuple(target):
18            return current_cost
19        for next_cost, neighbor in graph.get(current_node, []):
20            total_cost = current_cost + next_cost
21            if total_cost < min_cost[neighbor]:
22                min_cost[neighbor] = total_cost
23                heapq.heappush(pq, (total_cost, neighbor))
24        direct_cost = abs(current_node[0] - target[0]) + abs(current_node[1] - target[1])
25        total_cost = current_cost + direct_cost
26        if total_cost < min_cost[tuple(target)]:
27            min_cost[tuple(target)] = total_cost
28            heapq.heappush(pq, (total_cost, tuple(target)))
29    return min_cost[tuple(target)]
30

ℹ

Complexity note: This complexity is due to the priority queue operations and the number of nodes processed, where n is the number of unique positions considered.

1Using special roads can significantly reduce travel costs if chosen wisely.
2Direct paths may sometimes be the best option, especially when special roads are not advantageous.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.