Minimum Fuel Cost to Report to the Capital

Minimum Fuel Cost to Report to the Capital

1. What is this problem about?

The Minimum Fuel Cost to Report to the Capital is a tree-based optimization problem. There is a tree representing cities and roads, with node 0 being the Capital. Every city has one representative who needs to travel to the Capital. Each car has a fixed number of seats (capacity). Representatives can share cars if they meet at a city. Each liter of fuel allows one car to travel one road. The goal is to find the minimum fuel consumed to get everyone to node 0.

2. Why is this asked in interviews?

This problem is asked by companies like Microsoft and Meta because it tests whether a candidate can think about "flow" in a tree structure. The Minimum Fuel Cost to Report to the Capital interview question requires realizing that the fuel cost for a road is determined by the number of people who must cross it. It evaluates your ability to use Depth-First Search (DFS) to aggregate counts from subtrees.

3. Algorithmic pattern used

The pattern is Depth-First Search (DFS) on a Tree. We start a DFS from the Capital. For any road connecting a subtree to its parent (closer to the Capital), we count how many representatives are in that subtree. Let this count be $P$. The number of cars needed to move $P$ people across that single road is $\lceil P / seats \rceil$. The total fuel is the sum of these values across all roads. This "Tree, Graph, DFS interview pattern" effectively maps local subtree information to global fuel costs.

4. Example explanation

Tree: $1-0, 2-0, 3-0$. Seats: 2.

• Subtree 1: 1 person. Crossing road (1,0) needs 1 car. Fuel += 1.
• Subtree 2: 1 person. Crossing road (2,0) needs 1 car. Fuel += 1.
• Subtree 3: 1 person. Crossing road (3,0) needs 1 car. Fuel += 1. Total fuel: 3. If 1, 2, 3 were connected as $3 \to 2 \to 1 \to 0$:
• Road (3,2): 1 person, 1 car, Fuel += 1.
• Road (2,1): 2 people (from 3 and 2), 1 car (seats=2), Fuel += 1.
• Road (1,0): 3 people, 2 cars, Fuel += 2. Total fuel: 4.

5. Common mistakes candidates make

In the Minimum Fuel Cost to Report to the Capital coding problem, a common mistake is trying to model the cars explicitly, which is too complex. Another error is using BFS; while possible, DFS is much more natural for subtree counting. Some candidates also forget to use the correct ceiling division formula or fail to handle large inputs, which can result in total fuel costs exceeding the 32-bit integer limit.

6. Interview preparation tip

Whenever you need to calculate something based on "moving items to a root" in a tree, always think about the "edge contribution." Calculate how many items must pass through each edge. This is a recurring "Tree interview pattern" that simplifies many transport and flow problems. Practice implementing ceiling division as (people + seats - 1) / seats.