Maximum Total Importance of Roads

Maximum Total Importance of Roads

1. What is this problem about?

The Maximum Total Importance of Roads problem gives you a city with n cities and a list of roads connecting them. You need to assign an "importance" value from 1 to n to each city, using each value exactly once. The importance of a road is defined as the sum of the importance values of the two cities it connects. Your goal is to assign the values such that the total importance of all roads is maximized.

2. Why is this asked in interviews?

This "Maximum Total Importance of Roads interview question" is a favorite for companies like Microsoft and HRT. It is a brilliant test of basic graph theory and greedy optimization. It evaluates whether a candidate can recognize that cities with more connections (higher degrees) should be given higher importance values to maximize their contribution to the total sum. It’s a very clean, high-signal problem.

3. Algorithmic pattern used

The "Graph, Sorting, Heap (Priority Queue), Greedy interview pattern" is very straightforward here.

1. Calculate the degree of each city (the number of roads connected to it).
2. Sort the cities based on their degrees in ascending order.
3. Assign importance values from 1 to n in the same order (the city with the smallest degree gets 1, the city with the largest degree gets n).
4. The total importance is the sum over all cities of degree[i] * assigned_importance[i].

4. Example explanation

Cities: 3, Roads: [[0,1], [1,2]]

• City 0 degree: 1
• City 1 degree: 2
• City 2 degree: 1 Degrees: [1, 2, 1]. Sorted degrees: [1, 1, 2]. Assign values: City 0 = 1, City 2 = 2, City 1 = 3. Total importance = (1+3) [Road 0-1] + (3+2) [Road 1-2] = 4 + 5 = 9. Calculation via degrees: (11) + (23) + (1*2) = 1 + 6 + 2 = 9. Same result!

5. Common mistakes candidates make

In the "Maximum Total Importance of Roads coding problem," the most common mistake is overthinking the graph structure. Since the total sum is just a weighted sum of the importance values (where weights are city degrees), you don't need to traverse the graph or use complex algorithms like DFS/BFS. Another error is incorrectly calculating the degrees or failing to sort them, which leads to a non-optimal assignment of values.

6. Interview preparation tip

Understand the concept of "degree contribution." In any problem where you sum edge values, each node's value is added to the total exactly as many times as there are edges connected to it. This realization simplifies many graph problems into simple array-sorting problems. Being able to explain this "Road Sum = Sum of (City Value * City Degree)" logic clearly will demonstrate a strong mathematical and algorithmic foundation.