Count Pairs Of Nodes

Problem Description

You are given an undirected graph defined by an integer n, the number of nodes, and a 2D integer array edges, the edges in the graph, where edges[i] = [u_i, v_i] indicates that there is an undirected edge between u_i and v_i. You are also given an integer array queries.

Let incident(a, b) be defined as the number of edges that are connected to either node a or b. The answer to the j-th query is the number of pairs of nodes (a, b) that satisfy both of the following conditions:

• a < b
• incident(a, b) > queries[j]

Return an array answers such that answers.length == queries.length and answers[j] is the answer of the j-th query.

Note that there can be multiple edges between the same two nodes.

Key Insights

• To determine incident(a, b) efficiently, we need to count the edges connected to each node.
• We can use an adjacency list or degree array to keep track of the number of edges for each node.
• The number of pairs (a, b) can be calculated based on the number of nodes and the values of incident(a, b) compared to the queries.
• Sorting the queries allows for efficient counting of valid pairs using two pointers or binary search techniques.

Space and Time Complexity

Time Complexity: O(n^2 + m log m + q log q) where n is the number of nodes, m is the number of edges, and q is the number of queries. Space Complexity: O(n + m) for storing the graph structure and edge counts.

Solution

To solve this problem, we will:

1. Create a degree array to count the number of edges incident to each node.
2. For each pair of nodes (a, b), calculate incident(a, b) as the sum of their degrees minus the count of edges directly connecting them.
3. Store the results for all pairs in a list.
4. Sort the list of pairs based on their incident values.
5. Use binary search to efficiently answer each query regarding the number of pairs that exceed the query value.

Code Solutions