description: This is the hard version of the problem. The only difference between the two versions is the constraint on n. You can make hacks only if all versions of the problem are solved.
  
  A forest is an undirected graph without cycles (not necessarily connected).
  
  Mocha and Diana are friends in Zhijiang, both of them have a forest with nodes numbered from 1 to n, and they would like to add edges to their forests such that: 
  
    * After adding edges, both of their graphs are still forests. 
    * They add the same edges. That is, if an edge (u, v) is added to Mocha's forest, then an edge (u, v) is added to Diana's forest, and vice versa. 
  
  
  
  Mocha and Diana want to know the maximum number of edges they can add, and which edges to add.
  
  Input
  
  The first line contains three integers n, m_1 and m_2 (1 ≤ n ≤ 10^5, 0 ≤ m_1, m_2 < n) — the number of nodes and the number of initial edges in Mocha's forest and Diana's forest.
  
  Each of the next m_1 lines contains two integers u and v (1 ≤ u, v ≤ n, u ≠ v) — the edges in Mocha's forest.
  
  Each of the next m_2 lines contains two integers u and v (1 ≤ u, v ≤ n, u ≠ v) — the edges in Diana's forest.
  
  Output
  
  The first line contains only one integer h, the maximum number of edges Mocha and Diana can add.
  
  Each of the next h lines contains two integers u and v (1 ≤ u, v ≤ n, u ≠ v) — the edge you add each time.
  
  If there are multiple correct answers, you can print any one of them.
  
  Examples
  
  Input
  
  
  3 2 2
  1 2
  2 3
  1 2
  1 3
  
  
  Output
  
  
  0
  
  
  Input
  
  
  5 3 2
  5 4
  2 1
  4 3
  4 3
  1 4
  
  
  Output
  
  
  1
  2 4
  
  
  Input
  
  
  8 1 2
  1 7
  2 6
  1 5
  
  
  Output
  
  
  5
  5 2
  2 3
  3 4
  4 7
  6 8
  
  Note
  
  In the first example, we cannot add any edge.
  
  In the second example, the initial forests are as follows.
  
  <image>
  
  We can add an edge (2, 4).
  
  <image>
solution: import java.io.*;
  import java.util.*;
  
  public final class Main {
  
      static PrintWriter out = new PrintWriter(System.out);
      static FastReader in = new FastReader();
      static Pair[] moves = new Pair[]{new Pair(-1, 0), new Pair(1, 0), new Pair(0, -1), new Pair(0, 1)};
  
      public static void main(String[] args) {
          int tt = 1;
          int mod = (int) (1e9 + 7);
  
          loop:
          while (tt-- > 0) {
              int n = i();
              int m1 = i();
              int m2 = i();
              DSU g1 = new DSU(n);
              DSU g2 = new DSU(n);
  
              while (m1-- > 0) {
                  int u = i() - 1;
                  int v = i() - 1;
                  g1.union(u, v);
              }
              while (m2-- > 0) {
                  int u = i() - 1;
                  int v = i() - 1;
                  g2.union(u, v);
              }
  
              List<Pair> ans = new LinkedList<>();
              int i = 0;
              Stack<Integer> v1 = new Stack<>();
              Stack<Integer> v2 = new Stack<>();
  
              for (int j = 1; j < n; j++) {
                  if (!g1.isSameUnion(i, j) && !g2.isSameUnion(i, j)) {
                      g1.union(i, j);
                      g2.union(i, j);
                      ans.add(new Pair(i + 1, j + 1));
                      continue;
                  }
                  if (!g1.isSameUnion(i, j)) {
                      v1.push(j);
                  }
                  if (!g2.isSameUnion(i, j)) {
                      v2.push(j);
                  }
              }
  
              while (v1.size() > 0 && v2.size() > 0) {
                  if (g1.isSameUnion(v1.peek(), i) && g2.isSameUnion(v1.peek(), i)) {
                      v1.pop();
                      continue;
                  }
                  if (g1.isSameUnion(v2.peek(), i) && g2.isSameUnion(v2.peek(), i)) {
                      v2.pop();
                      continue;
                  }
                  int p1 = v1.pop();
                  int p2 = v2.pop();
  
                  ans.add(new Pair(p1 + 1, p2 + 1));
                  g1.union(p1, p2);
                  g2.union(p1, p2);
              }
  
              out.println(ans.size());
              for (Pair pair : ans) {
                  out.println(pair.i + " " + pair.j);
              }
          }
          out.flush();
      }
  
      static class DSU {
  
          int[] p;
  
          public DSU(int n) {
              p = new int[n];
              for (int i = 0; i < n; i++) {
                  p[i] = i;
              }
          }
  
          public int find(int x) {
              return (p[x] == x) ? (x) : (p[x] = find(p[x]));
          }
  
          public boolean union(int u, int v) {
              u = find(u);
              v = find(v);
              if (u == v) {
                  return false;
              }
              if (u > v) {
                  p[u] = v;
              } else {
                  p[v] = u;
              }
              return true;
          }
  
          public boolean isSameUnion(int u, int v) {
              return find(u) == find(v);
          }
      }
  
      static int[] pre(int[] a) {
          int[] pre = new int[a.length];
          pre[0] = a[0];
          for (int i = 1; i < a.length; i++) {
              pre[i] = pre[i - 1] + a[i];
          }
          return pre;
      }
  
      static int[] preRev(int[] a) {
          int[] pre = new int[a.length];
          pre[a.length - 1] = a[a.length - 1];
          for (int i = a.length - 2; i >= 0; i--) {
              pre[i] = pre[i + 1] + a[i];
          }
          return pre;
      }
  
      static void print(char A[]) {
          for (char c : A) {
              out.print(c);
          }
          out.println();
      }
  
      static void print(boolean A[]) {
          for (boolean c : A) {
              out.print(c + " ");
          }
          out.println();
      }
  
      static void print(int A[]) {
          for (int c : A) {
              out.print(c + " ");
          }
          out.println();
      }
  
      static void print(long A[]) {
          for (long i : A) {
              out.print(i + " ");
          }
          out.println();
      }
  
      static void print(List<Integer> A) {
          for (int a : A) {
              out.print(a + " ");
          }
      }
  
      static void printYes() {
          out.println("YES");
      }
  
      static void printNo() {
          out.println("NO");
      }
  
      static int i() {
          return in.nextInt();
      }
  
      static long l() {
          return in.nextLong();
      }
  
      static String s() {
          return in.nextLine();
      }
  
      static int[][] inputWithIdx(int N) {
          int A[][] = new int[N][2];
          for (int i = 0; i < N; i++) {
              A[i] = new int[]{i, in.nextInt()};
          }
          return A;
      }
  
      static int[] input(int N) {
          int A[] = new int[N];
          for (int i = 0; i < N; i++) {
              A[i] = in.nextInt();
          }
          return A;
      }
  
      static long[] inputLong(int N) {
          long A[] = new long[N];
          for (int i = 0; i < A.length; i++) {
              A[i] = in.nextLong();
          }
          return A;
      }
  
      static int GCD(int a, int b) {
          if (b == 0) {
              return a;
          } else {
              return GCD(b, a % b);
          }
      }
  
      static long GCD(long a, long b) {
          if (b == 0) {
              return a;
          } else {
              return GCD(b, a % b);
          }
      }
  
      static long LCM(int a, int b) {
          return (long) a / GCD(a, b) * b;
      }
  
      static long LCM(long a, long b) {
          return a / GCD(a, b) * b;
      }
  
      static void shuffleAndSort(int[] arr) {
          for (int i = 0; i < arr.length; i++) {
              int rand = (int) (Math.random() * arr.length);
              int temp = arr[rand];
              arr[rand] = arr[i];
              arr[i] = temp;
          }
          Arrays.sort(arr);
      }
  
      static void shuffleAndSort(int[][] arr, Comparator<? super int[]> comparator) {
          for (int i = 0; i < arr.length; i++) {
              int rand = (int) (Math.random() * arr.length);
              int[] temp = arr[rand];
              arr[rand] = arr[i];
              arr[i] = temp;
          }
          Arrays.sort(arr, comparator);
      }
  
      static void shuffleAndSort(long[] arr) {
          for (int i = 0; i < arr.length; i++) {
              int rand = (int) (Math.random() * arr.length);
              long temp = arr[rand];
              arr[rand] = arr[i];
              arr[i] = temp;
          }
          Arrays.sort(arr);
      }
  
      static boolean isPerfectSquare(double number) {
          double sqrt = Math.sqrt(number);
          return ((sqrt - Math.floor(sqrt)) == 0);
      }
  
      static void swap(int A[], int a, int b) {
          int t = A[a];
          A[a] = A[b];
          A[b] = t;
      }
  
      static long pow(long a, long b, int mod) {
          long pow = 1;
          long x = a;
          while (b != 0) {
              if ((b & 1) != 0) {
                  pow = (pow * x) % mod;
              }
              x = (x * x) % mod;
              b /= 2;
          }
          return pow;
      }
  
      static long pow(long a, long b) {
          long pow = 1;
          long x = a;
          while (b != 0) {
              if ((b & 1) != 0) {
                  pow *= x;
              }
              x = x * x;
              b /= 2;
          }
          return pow;
      }
  
      static long modInverse(long x, int mod) {
          return pow(x, mod - 2, mod);
      }
  
      static boolean isPrime(long N) {
          if (N <= 1) {
              return false;
          }
          if (N <= 3) {
              return true;
          }
          if (N % 2 == 0 || N % 3 == 0) {
              return false;
          }
          for (int i = 5; i * i <= N; i = i + 6) {
              if (N % i == 0 || N % (i + 2) == 0) {
                  return false;
              }
          }
          return true;
      }
  
      public static String reverse(String str) {
          if (str == null) {
              return null;
          }
          return new StringBuilder(str).reverse().toString();
      }
  
      public static String repeat(char ch, int repeat) {
          if (repeat <= 0) {
              return "";
          }
          final char[] buf = new char[repeat];
          for (int i = repeat - 1; i >= 0; i--) {
              buf[i] = ch;
          }
          return new String(buf);
      }
  
      public static int[] manacher(String s) {
          char[] chars = s.toCharArray();
          int n = s.length();
          int[] d1 = new int[n];
          for (int i = 0, l = 0, r = -1; i < n; i++) {
              int k = (i > r) ? 1 : Math.min(d1[l + r - i], r - i + 1);
              while (0 <= i - k && i + k < n && chars[i - k] == chars[i + k]) {
                  k++;
              }
              d1[i] = k--;
              if (i + k > r) {
                  l = i - k;
                  r = i + k;
              }
          }
          return d1;
      }
  }
  
  class Pair {
  
      int i, j;
  
      Pair(int i, int j) {
          this.i = i;
          this.j = j;
      }
  }
  
  class FastReader {
  
      BufferedReader br;
      StringTokenizer st;
  
      public FastReader() {
          br = new BufferedReader(new InputStreamReader(System.in));
      }
  
      String next() {
          while (st == null || !st.hasMoreElements()) {
              try {
                  st = new StringTokenizer(br.readLine());
              } catch (IOException e) {
                  e.printStackTrace();
              }
          }
          return st.nextToken();
      }
  
      int nextInt() {
          return Integer.parseInt(next());
      }
  
      long nextLong() {
          return Long.parseLong(next());
      }
  
      double nextDouble() {
          return Double.parseDouble(next());
      }
  
      String nextLine() {
          String str = "";
          try {
              str = br.readLine();
          } catch (IOException e) {
              e.printStackTrace();
          }
          return str;
      }
  }