Maximal Network Rank

Problem Description

There is an infrastructure of n cities with some number of roads connecting these cities. Each roads[i] = [a_i, b_i] indicates that there is a bidirectional road between cities a_i and b_i. The network rank of two different cities is defined as the total number of directly connected roads to either city, counting each road only once. The maximal network rank of the infrastructure is the maximum network rank of all pairs of different cities. Given the integer n and the array roads, return the maximal network rank of the entire infrastructure.

Key Insights

• The network rank is calculated based on the number of roads connected to either of the two cities.
• Duplicate connections (a road connecting both cities) should not be double-counted.
• We need to consider all pairs of cities to find the maximum network rank.
• Efficiently storing and accessing the number of roads connected to each city is crucial.

Space and Time Complexity

Time Complexity: O(n^2) - We need to check every pair of cities, which could be up to n(n-1)/2 pairs in the worst case. Space Complexity: O(n) - We need to store the road counts for each city.

Solution

To solve the problem, we can use an array to count the number of roads connected to each city. We will then iterate through all pairs of cities, calculating their network rank using the counts stored in the array. If a pair of cities has a direct road between them, we subtract one from the total count to avoid double-counting that road.

1. Create an array roadCount of size n initialized to zero to count the number of roads connected to each city.
2. For each road in roads, increment the count for both cities involved in the road.
3. Initialize a variable maxRank to zero.
4. Iterate through all pairs of cities (i, j) and calculate their network rank.
5. Update maxRank if the calculated rank for the pair is greater than the current maxRank.

Code Solutions