Question

1782. Count Pairs Of Nodes

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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] = [ui, vi] indicates that there is an undirected edge between ui and vi. 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 jth 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 _jth_ query_.

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

 

Example 1:

Input: n = 4, edges = [[1,2],[2,4],[1,3],[2,3],[2,1]], queries = [2,3]
Output: [6,5]
Explanation: The calculations for incident(a, b) are shown in the table above.
The answers for each of the queries are as follows:
- answers[0] = 6. All the pairs have an incident(a, b) value greater than 2.
- answers[1] = 5. All the pairs except (3, 4) have an incident(a, b) value greater than 3.

Example 2:

Input: n = 5, edges = [[1,5],[1,5],[3,4],[2,5],[1,3],[5,1],[2,3],[2,5]], queries = [1,2,3,4,5]
Output: [10,10,9,8,6]

 

Constraints:

• 2 <= n <= 2 * 104
• 1 <= edges.length <= 105
• 1 <= ui, vi <= n
• ui != vi
• 1 <= queries.length <= 20
• 0 <= queries[j] < edges.length

Solutions

Solution 1: Hash Table + Sorting + Binary Search

From the problem, we know that the number of edges connected to the point pair $(a, b)$ is equal to the "number of edges connected to $a$" plus the "number of edges connected to $b$", minus the number of edges connected to both $a$ and $b$.

Therefore, we can first use the array $cnt$ to count the number of edges connected to each point, and use the hash table $g$ to count the number of each point pair.

Then, for each query $q$, we can enumerate $a$. For each $a$, we can find the first $b$ that satisfies $cnt[a] + cnt[b] > q$ through binary search, add the number to the current query answer, and then subtract some duplicate edges.

The time complexity is $O(q \times (n \times \log n + m))$, and the space complexity is $O(n + m)$. Where $n$ and $m$ are the number of points and edges respectively, and $q$ is the number of queries.

class Solution:
  def countPairs(
    self, n: int, edges: List[List[int]], queries: List[int]
  ) -> List[int]:
    cnt = [0] * n
    g = defaultdict(int)
    for a, b in edges:
      a, b = a - 1, b - 1
      a, b = min(a, b), max(a, b)
      cnt[a] += 1
      cnt[b] += 1
      g[(a, b)] += 1

    s = sorted(cnt)
    ans = [0] * len(queries)
    for i, t in enumerate(queries):
      for j, x in enumerate(s):
        k = bisect_right(s, t - x, lo=j + 1)
        ans[i] += n - k
      for (a, b), v in g.items():
        if cnt[a] + cnt[b] > t and cnt[a] + cnt[b] - v <= t:
          ans[i] -= 1
    return ans

class Solution {
  public int[] countPairs(int n, int[][] edges, int[] queries) {
    int[] cnt = new int[n];
    Map<Integer, Integer> g = new HashMap<>();
    for (var e : edges) {
      int a = e[0] - 1, b = e[1] - 1;
      ++cnt[a];
      ++cnt[b];
      int k = Math.min(a, b) * n + Math.max(a, b);
      g.merge(k, 1, Integer::sum);
    }
    int[] s = cnt.clone();
    Arrays.sort(s);
    int[] ans = new int[queries.length];
    for (int i = 0; i < queries.length; ++i) {
      int t = queries[i];
      for (int j = 0; j < n; ++j) {
        int x = s[j];
        int k = search(s, t - x, j + 1);
        ans[i] += n - k;
      }
      for (var e : g.entrySet()) {
        int a = e.getKey() / n, b = e.getKey() % n;
        int v = e.getValue();
        if (cnt[a] + cnt[b] > t && cnt[a] + cnt[b] - v <= t) {
          --ans[i];
        }
      }
    }
    return ans;
  }

  private int search(int[] arr, int x, int i) {
    int left = i, right = arr.length;
    while (left < right) {
      int mid = (left + right) >> 1;
      if (arr[mid] > x) {
        right = mid;
      } else {
        left = mid + 1;
      }
    }
    return left;
  }
}

class Solution {
public:
  vector<int> countPairs(int n, vector<vector<int>>& edges, vector<int>& queries) {
    vector<int> cnt(n);
    unordered_map<int, int> g;
    for (auto& e : edges) {
      int a = e[0] - 1, b = e[1] - 1;
      ++cnt[a];
      ++cnt[b];
      int k = min(a, b) * n + max(a, b);
      ++g[k];
    }
    vector<int> s = cnt;
    sort(s.begin(), s.end());
    vector<int> ans(queries.size());
    for (int i = 0; i < queries.size(); ++i) {
      int t = queries[i];
      for (int j = 0; j < n; ++j) {
        int x = s[j];
        int k = upper_bound(s.begin() + j + 1, s.end(), t - x) - s.begin();
        ans[i] += n - k;
      }
      for (auto& [k, v] : g) {
        int a = k / n, b = k % n;
        if (cnt[a] + cnt[b] > t && cnt[a] + cnt[b] - v <= t) {
          --ans[i];
        }
      }
    }
    return ans;
  }
};

func countPairs(n int, edges [][]int, queries []int) []int {
  cnt := make([]int, n)
  g := map[int]int{}
  for _, e := range edges {
    a, b := e[0]-1, e[1]-1
    cnt[a]++
    cnt[b]++
    if a > b {
      a, b = b, a
    }
    g[a*n+b]++
  }
  s := make([]int, n)
  copy(s, cnt)
  sort.Ints(s)
  ans := make([]int, len(queries))
  for i, t := range queries {
    for j, x := range s {
      k := sort.Search(n, func(h int) bool { return s[h] > t-x && h > j })
      ans[i] += n - k
    }
    for k, v := range g {
      a, b := k/n, k%n
      if cnt[a]+cnt[b] > t && cnt[a]+cnt[b]-v <= t {
        ans[i]--
      }
    }
  }
  return ans
}

function countPairs(n: number, edges: number[][], queries: number[]): number[] {
  const cnt: number[] = new Array(n).fill(0);
  const g: Map<number, number> = new Map();
  for (const [a, b] of edges) {
    ++cnt[a - 1];
    ++cnt[b - 1];
    const k = Math.min(a - 1, b - 1) * n + Math.max(a - 1, b - 1);
    g.set(k, (g.get(k) || 0) + 1);
  }
  const s = cnt.slice().sort((a, b) => a - b);
  const search = (nums: number[], x: number, l: number): number => {
    let r = nums.length;
    while (l < r) {
      const mid = (l + r) >> 1;
      if (nums[mid] > x) {
        r = mid;
      } else {
        l = mid + 1;
      }
    }
    return l;
  };
  const ans: number[] = [];
  for (const t of queries) {
    let res = 0;
    for (let j = 0; j < s.length; ++j) {
      const k = search(s, t - s[j], j + 1);
      res += n - k;
    }
    for (const [k, v] of g) {
      const a = Math.floor(k / n);
      const b = k % n;
      if (cnt[a] + cnt[b] > t && cnt[a] + cnt[b] - v <= t) {
        --res;
      }
    }
    ans.push(res);
  }
  return ans;
}