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: def main():
      
      mod=1000000007
      # mod=998244353
      # nCr =  make_nCr_mod()
      
      tc=1
      for _ in range(tc):
          n,m1,m2=ria()
          
          # Two Disjoin Sets Mocha and Diana
          # LOGIC - https://www.youtube.com/watch?v=t01t6aLbBXs
          
          Mocha=DisjointSetUnion(n+1)
          Diana=DisjointSetUnion(n+1)
          
          for i in range(m1):
              a,b=ria()
              Mocha.union(a,b)
          
          for i in range(m2):
              a,b=ria()
              Diana.union(a,b)
          
          ans=[]
          
          for i in range(2,n+1):
              if Mocha.get(1)!=Mocha.get(i) and Diana.get(1)!=Diana.get(i):
                  ans.append([i,1])
                  Mocha.union(i,1)
                  Diana.union(i,1)
   
          vis=[0]*(n+1)
          vis[1]=1
          
          ptr1,ptr2=1,1
          
          while ptr1<=n and ptr2<=n:
              
              while ptr1<=n and ((Mocha.get(ptr1)==Mocha.get(1)) or vis[Mocha.get(ptr1)]): ptr1+=1
              while ptr2<=n and ((Diana.get(ptr2)==Diana.get(1)) or vis[Diana.get(ptr2)]): ptr2+=1
              
              if ptr1>n or ptr2>n: break
              
              vis[Mocha.get(ptr1)] = vis[Diana.get(ptr2)] = 1
              
              ans.append([ptr1,ptr2])
              
          wi(len(ans))
          for i in ans:
              wia(i)
          
  
  def rant():
      # RANT BEGINS-
      # 
      # Did you look at the constraints dummy? Read problem again carefully understand it correctly
      # Are base cases correct n=0,n=1,n=N or string is all 0 or all 1?
      # 
      # 1.Greedy? What matters what doesnt? Is my claim correct? 
      #   Does my implementation correctly implements the greedy i thought of? 
      #   Dictionary? keep a count of values if constraint on something is low(like only alphabets(26) count)? 
      # 
      # 2.DP? Think about state's ranges maybe one of the states have lesser possible values than it may look like? 
      #   Obviously think about order of evaluation
      # 
      # 3.Binary Search?(Monotonic? any one directed order in which we have to perform something?)
      # 
      # 4.Graph?DSU?Edge property or Vertex property Read carefully? Bipartitenes Property??? Cycle Property??? 
      #   Hidden but dependency ordering of values(in 1d array or 2d Table/Array) is that of a graph???
      # 
      # 5.Number Theory?(GCD subtraction?)
      # 
      # 6.Bruteforce?(Redundant part of N which may not give answer?Constraints?)
      # 
      # 7.Range Queries?
      # 
      # 8.Any Equivalency?(We have A and B and have to do 
      #   something between them maybe difficult if there was A~C and C~B then A~B
      #   C could be max or min or some other thing)
      # 
      # 9.Reverse Engineering?(From Answer to quesn or last step to first step)
      #
      # 10.Constructive? Mod? Bruteforce for smaller case and analyze patterns?
      #
      # 11.String? Constructive Greedy example- 
      #   Bracket sequence make first part unknown open and second part unknown closed 
      #   Remember always only n//2 open and n//2 closed brackets are there
      #
      # 12.Combinatorics?--> (n to i-1 index ways) * (ith index ways) * Ways (for i-1 till 0 index)?
      #
      # 13.Look for patterns and dependency of the patterns(maybe only first row or column matters to construct other rows and columns)
      #   for smaller cases maybe make bruteforce solution and analyze the patterns
      # 
      # RANT ENDS
      return "AAAAAAAAAAAAAAAAARRRGHHHHHHHHHHHHHHHHHHHHHHHH"
      
  def SieveOfEratosthenes(limit):
      """Returns all primes not greater than limit."""
      isPrime = [True]*(limit+1)
      isPrime[0] = isPrime[1] = False
      primes = []
      for i in range(2, limit+1):
          if not isPrime[i]:continue
          primes += [i]
          for j in range(i*i, limit+1, i):
              isPrime[j] = False
      return primes
  
  def gcd(x, y):
      """greatest common divisor of x and y"""
      while y:
          x, y = y, x % y
      return x
  
  def memodict(f):
      """memoization decorator for a function taking a single argument"""
      class memodict(dict):
          def __missing__(self, key):
              ret = self[key] = f(key)
              return ret
  
      return memodict().__getitem__
  
  def pollard_rho(n):
      """returns a random factor of n"""
      if n & 1 == 0:
          return 2
      if n % 3 == 0:
          return 3
  
      s = ((n - 1) & (1 - n)).bit_length() - 1
      d = n >> s
      for a in [2, 325, 9375, 28178, 450775, 9780504, 1795265022]:
          p = pow(a, d, n)
          if p == 1 or p == n - 1 or a % n == 0:
              continue
          for _ in range(s):
              prev = p
              p = (p * p) % n
              if p == 1:
                  return gcd(prev - 1, n)
              if p == n - 1:
                  break
          else:
              for i in range(2, n):
                  x, y = i, (i * i + 1) % n
                  f = gcd(abs(x - y), n)
                  while f == 1:
                      x, y = (x * x + 1) % n, (y * y + 1) % n
                      y = (y * y + 1) % n
                      f = gcd(abs(x - y), n)
                  if f != n:
                      return f
      return n
  
  @memodict
  def prime_factors(n):
      """returns a Counter of the prime factorization of n"""
      if n <= 1:
          return Counter()
      f = pollard_rho(n)
      return Counter([n]) if f == n else prime_factors(f) + prime_factors(n // f)
  
  def distinct_factors(n):
      """returns a list of all distinct factors of n"""
      factors = [1]
      for p, exp in prime_factors(n).items():
          factors += [p**i * factor for factor in factors for i in range(1, exp + 1)]
      return factors
  
  def all_factors(n):
      """returns a sorted list of all distinct factors of n"""
      small, large = [], []
      for i in range(1, int(n**0.5) + 1, 2 if n & 1 else 1):
          if not n % i:
              small.append(i)
              large.append(n // i)
      if small[-1] == large[-1]:
          large.pop()
      large.reverse()
      small.extend(large)
      return small
  
  def make_nCr_mod(max_n=2 * 10**5, mod=10**9 + 7):
      max_n = min(max_n, mod - 1)
  
      fact, inv_fact = [0] * (max_n + 1), [0] * (max_n + 1)
      fact[0] = 1
      for i in range(max_n):
          fact[i + 1] = fact[i] * (i + 1) % mod
  
      inv_fact[-1] = pow(fact[-1], mod - 2, mod)
      for i in reversed(range(max_n)):
          inv_fact[i] = inv_fact[i + 1] * (i + 1) % mod
  
      def nCr_mod(n, r):
          res = 1
          while n or r:
              a, b = n % mod, r % mod
              if a < b:
                  return 0
              res = res * fact[a] % mod * inv_fact[b] % mod * inv_fact[a - b] % mod
              n //= mod
              r //= mod
          return res
  
      return nCr_mod
  
  class DisjointSetUnion:
      def __init__(self, n):
          self.parent = [*range(n+1)]
          self.size = [1]*(n+1)
          self.min, self.max = [*range(n+1)], [*range(n+1)]
          self.count = n
  
      def get(self, a):
          """Returns the identifier (parent) of the set to which a belongs to!"""
          if self.parent[a] == a:
              return a
          x = a
          while a != self.parent[a]:
              a = self.parent[a]
          while x != self.parent[x]:
              self.parent[x], x = a, self.parent[x]
          return a
  
      def union(self, a, b):
          """Join two sets that contain a and b!"""
          a, b = self.get(a), self.get(b)
          if a != b:
              if self.size[a] > self.size[b]:
                  a, b = b, a
              self.parent[a] = b
              self.size[b] += self.size[a]
              self.min[b] = min(self.min[a], self.min[b])
              self.max[b] = max(self.max[a], self.max[b])
              self.count -= 1
  
      def count_sets(self):
          """Returns the number of disjoint sets!"""
          return self.count
      
  class SegTree:
      
      def __init__(self, n):
          self.N = 1 << n.bit_length()
          self.tree = [0] * (self.N<<1)
      
      def update(self, i, j, v):
          i += self.N
          j += self.N
          while i <= j:
              if i%2==1: self.tree[i] += v
              if j%2==0: self.tree[j] += v
              i, j = (i+1) >> 1, (j-1) >> 1
      
      def query(self, i):
          v = 0
          i += self.N
          while i > 0:
              v += self.tree[i]
              i >>= 1
          return v
                  
  class SortedList:
      def __init__(self, iterable=[], _load=200):
          """Initialize sorted list instance."""
          values = sorted(iterable)
          self._len = _len = len(values)
          self._load = _load
          self._lists = _lists = [values[i:i + _load] for i in range(0, _len, _load)]
          self._list_lens = [len(_list) for _list in _lists]
          self._mins = [_list[0] for _list in _lists]
          self._fen_tree = []
          self._rebuild = True
   
      def _fen_build(self):
          """Build a fenwick tree instance."""
          self._fen_tree[:] = self._list_lens
          _fen_tree = self._fen_tree
          for i in range(len(_fen_tree)):
              if i | i + 1 < len(_fen_tree):
                  _fen_tree[i | i + 1] += _fen_tree[i]
          self._rebuild = False
   
      def _fen_update(self, index, value):
          """Update `fen_tree[index] += value`."""
          if not self._rebuild:
              _fen_tree = self._fen_tree
              while index < len(_fen_tree):
                  _fen_tree[index] += value
                  index |= index + 1
   
      def _fen_query(self, end):
          """Return `sum(_fen_tree[:end])`."""
          if self._rebuild:
              self._fen_build()
   
          _fen_tree = self._fen_tree
          x = 0
          while end:
              x += _fen_tree[end - 1]
              end &= end - 1
          return x
   
      def _fen_findkth(self, k):
          """Return a pair of (the largest `idx` such that `sum(_fen_tree[:idx]) <= k`, `k - sum(_fen_tree[:idx])`)."""
          _list_lens = self._list_lens
          if k < _list_lens[0]:
              return 0, k
          if k >= self._len - _list_lens[-1]:
              return len(_list_lens) - 1, k + _list_lens[-1] - self._len
          if self._rebuild:
              self._fen_build()
   
          _fen_tree = self._fen_tree
          idx = -1
          for d in reversed(range(len(_fen_tree).bit_length())):
              right_idx = idx + (1 << d)
              if right_idx < len(_fen_tree) and k >= _fen_tree[right_idx]:
                  idx = right_idx
                  k -= _fen_tree[idx]
          return idx + 1, k
   
      def _delete(self, pos, idx):
          """Delete value at the given `(pos, idx)`."""
          _lists = self._lists
          _mins = self._mins
          _list_lens = self._list_lens
   
          self._len -= 1
          self._fen_update(pos, -1)
          del _lists[pos][idx]
          _list_lens[pos] -= 1
   
          if _list_lens[pos]:
              _mins[pos] = _lists[pos][0]
          else:
              del _lists[pos]
              del _list_lens[pos]
              del _mins[pos]
              self._rebuild = True
   
      def _loc_left(self, value):
          """Return an index pair that corresponds to the first position of `value` in the sorted list."""
          if not self._len:
              return 0, 0
   
          _lists = self._lists
          _mins = self._mins
   
          lo, pos = -1, len(_lists) - 1
          while lo + 1 < pos:
              mi = (lo + pos) >> 1
              if value <= _mins[mi]:
                  pos = mi
              else:
                  lo = mi
   
          if pos and value <= _lists[pos - 1][-1]:
              pos -= 1
   
          _list = _lists[pos]
          lo, idx = -1, len(_list)
          while lo + 1 < idx:
              mi = (lo + idx) >> 1
              if value <= _list[mi]:
                  idx = mi
              else:
                  lo = mi
   
          return pos, idx
   
      def _loc_right(self, value):
          """Return an index pair that corresponds to the last position of `value` in the sorted list."""
          if not self._len:
              return 0, 0
   
          _lists = self._lists
          _mins = self._mins
   
          pos, hi = 0, len(_lists)
          while pos + 1 < hi:
              mi = (pos + hi) >> 1
              if value < _mins[mi]:
                  hi = mi
              else:
                  pos = mi
   
          _list = _lists[pos]
          lo, idx = -1, len(_list)
          while lo + 1 < idx:
              mi = (lo + idx) >> 1
              if value < _list[mi]:
                  idx = mi
              else:
                  lo = mi
   
          return pos, idx
   
      def add(self, value):
          """Add `value` to sorted list."""
          _load = self._load
          _lists = self._lists
          _mins = self._mins
          _list_lens = self._list_lens
   
          self._len += 1
          if _lists:
              pos, idx = self._loc_right(value)
              self._fen_update(pos, 1)
              _list = _lists[pos]
              _list.insert(idx, value)
              _list_lens[pos] += 1
              _mins[pos] = _list[0]
              if _load + _load < len(_list):
                  _lists.insert(pos + 1, _list[_load:])
                  _list_lens.insert(pos + 1, len(_list) - _load)
                  _mins.insert(pos + 1, _list[_load])
                  _list_lens[pos] = _load
                  del _list[_load:]
                  self._rebuild = True
          else:
              _lists.append([value])
              _mins.append(value)
              _list_lens.append(1)
              self._rebuild = True
   
      def discard(self, value):
          """Remove `value` from sorted list if it is a member."""
          _lists = self._lists
          if _lists:
              pos, idx = self._loc_right(value)
              if idx and _lists[pos][idx - 1] == value:
                  self._delete(pos, idx - 1)
   
      def remove(self, value):
          """Remove `value` from sorted list; `value` must be a member."""
          _len = self._len
          self.discard(value)
          if _len == self._len:
              raise ValueError('{0!r} not in list'.format(value))
   
      def pop(self, index=-1):
          """Remove and return value at `index` in sorted list."""
          pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
          value = self._lists[pos][idx]
          self._delete(pos, idx)
          return value
   
      def bisect_left(self, value):
          """Return the first index to insert `value` in the sorted list."""
          pos, idx = self._loc_left(value)
          return self._fen_query(pos) + idx
   
      def bisect_right(self, value):
          """Return the last index to insert `value` in the sorted list."""
          pos, idx = self._loc_right(value)
          return self._fen_query(pos) + idx
   
      def count(self, value):
          """Return number of occurrences of `value` in the sorted list."""
          return self.bisect_right(value) - self.bisect_left(value)
   
      def __len__(self):
          """Return the size of the sorted list."""
          return self._len
   
      def __getitem__(self, index):
          """Lookup value at `index` in sorted list."""
          pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
          return self._lists[pos][idx]
   
      def __delitem__(self, index):
          """Remove value at `index` from sorted list."""
          pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
          self._delete(pos, idx)
   
      def __contains__(self, value):
          """Return true if `value` is an element of the sorted list."""
          _lists = self._lists
          if _lists:
              pos, idx = self._loc_left(value)
              return idx < len(_lists[pos]) and _lists[pos][idx] == value
          return False
   
      def __iter__(self):
          """Return an iterator over the sorted list."""
          return (value for _list in self._lists for value in _list)
   
      def __reversed__(self):
          """Return a reverse iterator over the sorted list."""
          return (value for _list in reversed(self._lists) for value in reversed(_list))
   
      def __repr__(self):
          """Return string representation of sorted list."""
          return 'SortedList({0})'.format(list(self))
  
  import sys, os, io
  def rs(): return sys.stdin.readline().rstrip()
  def ri(): return int(sys.stdin.readline())
  def ria(): return list(map(int, sys.stdin.readline().split()))
  def ws(s): sys.stdout.write(s + '\n')
  def wi(n): sys.stdout.write(str(n) + '\n')
  def wia(a): sys.stdout.write(' '.join([str(x) for x in a]) + '\n')
  import math,datetime,functools,itertools,operator,bisect,fractions,statistics
  from collections import deque,defaultdict,OrderedDict,Counter
  from fractions import Fraction
  from decimal import Decimal
  from sys import stdout
  from heapq import heappush, heappop, heapify ,_heapify_max,_heappop_max,nsmallest,nlargest
  # sys.setrecursionlimit(111111) 
  INF=999999999999999999999999
  alphabets="abcdefghijklmnopqrstuvwxyz"
  
  class FastReader(io.IOBase):
      newlines = 0
  
      def __init__(self, fd, chunk_size=1024 * 8):
          self._fd = fd
          self._chunk_size = chunk_size
          self.buffer = io.BytesIO()
  
      def read(self):
          while True:
              b = os.read(self._fd, max(os.fstat(self._fd).st_size, self._chunk_size))
              if not b:
                  break
              ptr = self.buffer.tell()
              self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
          self.newlines = 0
          return self.buffer.read()
  
      def readline(self, size=-1):
          while self.newlines == 0:
              b = os.read(self._fd, max(os.fstat(self._fd).st_size, self._chunk_size if size == -1 else size))
              self.newlines = b.count(b"\n") + (not b)
              ptr = self.buffer.tell()
              self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
          self.newlines -= 1
          return self.buffer.readline()
  
  class FastWriter(io.IOBase):
  
      def __init__(self, fd):
          self._fd = fd
          self.buffer = io.BytesIO()
          self.write = self.buffer.write
  
      def flush(self):
          os.write(self._fd, self.buffer.getvalue())
          self.buffer.truncate(0), self.buffer.seek(0)
  
  class FastStdin(io.IOBase):
      def __init__(self, fd=0):
          self.buffer = FastReader(fd)
          self.read = lambda: self.buffer.read().decode("ascii")
          self.readline = lambda: self.buffer.readline().decode("ascii")
  
  class FastStdout(io.IOBase):
      def __init__(self, fd=1):
          self.buffer = FastWriter(fd)
          self.write = lambda s: self.buffer.write(s.encode("ascii"))
          self.flush = self.buffer.flush
  
  if __name__ == '__main__':
      sys.stdin = FastStdin()
      sys.stdout = FastStdout()
      starttime=datetime.datetime.now()
      if(os.path.exists('input.txt')):
          sys.stdin = open("input.txt","r")
          sys.stdout = open("output.txt","w")
      main()
      endtime=datetime.datetime.now()
      time=(endtime-starttime).total_seconds()*1000
      if(os.path.exists('input.txt')):
          print("Time:",time,"ms")  
      