Problem:
 Score : 400 points    Problem Statement There are N persons called Person 1 through Person N .  You are given M facts that "Person A_i and Person B_i are friends." The same fact may be given multiple times.  If X and Y are friends, and Y and Z are friends, then X and Z are also friends. There is no friendship that cannot be derived from the M given facts.  Takahashi the evil wants to divide the N persons into some number of groups so that every person has no friend in his/her group.  At least how many groups does he need to make?      Constraints  2 \leq N \leq 2\times 10^5  0 \leq M \leq 2\times 10^5  1\leq A_i,B_i\leq N  A_i \neq B_i         Input Input is given from Standard Input in the following format:  N  M  A_1  B_1  \vdots  A_M  B_M       Output Print the answer.        Sample Input 1 5 3
1 2
3 4
5 1      Sample Output 1 3  Dividing them into three groups such as \{1,3\} , \{2,4\} , and \{5\} achieves the goal.       Sample Input 2 4 10
1 2
2 1
1 2
2 1
1 2
1 3
1 4
2 3
2 4
3 4      Sample Output 2 4       Sample Input 3 10 4
3 1
4 1
5 9
2 6      Sample Output 3 3

Solution (Python):
class UnionFindVerSize():
    def __init__(self, N):
        self._parent = [n for n in range(0, N)]
        self._size = [1] * N

    def find_root(self, x):
        if self._parent[x] == x: return x
        self._parent[x] = self.find_root(self._parent[x])
        return self._parent[x]

    def unite(self, x, y):
        gx = self.find_root(x)
        gy = self.find_root(y)
        if gx == gy: return

        if self._size[gx] < self._size[gy]:
            self._parent[gx] = gy
            self._size[gy] += self._size[gx]
        else:
            self._parent[gy] = gx
            self._size[gx] += self._size[gy]

    def get_size(self, x):
        return self._size[self.find_root(x)]

    def is_same_group(self, x, y):
        return self.find_root(x) == self.find_root(y)

    def calc_group_num(self):
        N = len(self._parent)
        ans = 0
        for i in range(N):
            if self.find_root(i) == i:
                ans += 1
        return ans

import sys

input=sys.stdin.readline


N,M=map(int,input().split())
uf=UnionFindVerSize(N)
for _ in range(M):
    a,b=map(int,input().split())
    uf.unite(a-1,b-1)

ans=-1
for i in range(N):
    ans=max(ans,uf.get_size(i))

print(ans)

Solution (Python):
import sys
input = sys.stdin.readline

class UnionFind:
    def __init__(self, N):
        self.parent = list(range(N))
        self.rank = [0] * N
        self.size = [1] * N

    def find(self, x):
        r = x
        while self.parent[r] != r:
            r = self.parent[r]
        while self.parent[x] != r:
            x, self.parent[x] = self.parent[x], r
        return r

    def unite(self, x, y):
        x = self.find(x)
        y = self.find(y)
        if x == y:
            return
        if self.rank[x] < self.rank[y]:
            self.parent[x] = y
            self.size[y] += self.size[x]
        else:
            self.parent[y] = x
            self.size[x] += self.size[y]
            if self.rank[x] == self.rank[y]:
                self.rank[x] += 1

    def same(self, x, y):
        return self.find(x) == self.find(y)

    def get_size(self, x):
        return self.size[self.find(x)]

N, M = map(int, input().split())
uf = UnionFind(N+1)
for _ in range(M):
    A, B = map(int, input().split())
    uf.unite(A, B)
ans = max(uf.get_size(i) for i in range(N+1))
print(ans)

Solution (Python):
import sys

sys.setrecursionlimit(500005)
stdin = sys.stdin

ni = lambda: int(ns())
na = lambda: list(map(int, stdin.readline().split()))
ns = lambda: stdin.readline().strip()



n, m = na()

upper = [-1] * n
def root(x):
    if upper[x] < 0:
        return x
    else:
        upper[x] = root(upper[x])
        return upper[x]

def equiv(x, y):
    return root(x) == root(y)

def unite(x, y):
    x, y = root(x), root(y)
    if x != y:
        if upper[y] < upper[x]:
            x, y = y, x
        upper[x] += upper[y]
        upper[y] = x
    return x == y

for i in range(m):
    a, b = na()
    unite(a-1, b-1)

print(-min(upper))

Solution (Python):
class UnionFind:
    def __init__(self, node):
        self.parent = [-1 for _ in range(node)]

    def find(self, target):
        if self.parent[target] < 0:
            return target
        else:
            self.parent[target] = self.find(self.parent[target])
            return self.parent[target]

    def is_same(self, x, y):
        return self.find(x) == self.find(y)

    def union(self, x, y):
        root_x = self.find(x)
        root_y = self.find(y)
        if root_x == root_y:
            return
        if self.parent[root_x] > self.parent[root_y]:
            root_x, root_y = root_y, root_x
        self.parent[root_x] += self.parent[root_y]
        self.parent[root_y] = root_x

    def get_size(self, x):
        return -self.parent[self.find(x)]

    def get_root(self):
        return [i for i, root in enumerate(self.parent) if root < 0]


def main():
    n, m = map(int, input().split())
    uf = UnionFind(n)
    for _ in range(m):
        a, b = map(lambda x: int(x) - 1, input().split())
        uf.union(a, b)
    ans = 0
    for i in range(n):
        ans = max(ans, uf.get_size(i))
    print(ans)


if __name__ == '__main__':
    main()

Solution (Python):
class UnionFind:
    
    
    __slots__ = ["_size", "_first_idx", "_parents"]
    def __init__(self, size: int, first_index: int = 0) -> None:
        self._size = size
        self._first_idx = first_index
        self._parents = [-1] * (size + first_index)
    def find(self, x: int) -> int:
        if self._parents[x] < 0:
            return x
        self._parents[x] = self.find(self._parents[x])
        return self._parents[x]
    def same(self, x: int, y: int) -> bool:
        return self.find(x) == self.find(y)
    def unite(self, x: int, y: int) -> bool:
        x, y = self.find(x), self.find(y)
        if x == y:
            return False
        if self._parents[x] > self._parents[y]:
            x, y = y, x
        self._parents[x] += self._parents[y]
        self._parents[y] = x
        return True
    def size(self, x: int) -> int:
        return -self._parents[self.find(x)]
    def group_count(self) ->int:
        return sum(1 for i in  self._parents if i<0)-self._first_idx
    def connected(self) ->bool:
        return self._parents[self.find(self._first_idx)]==-self._size
n,m=map(int,input().split())

uf=UnionFind(n,1)
for i in range(m):
    uf.unite(*map(int,input().split()))
print(-min(uf._parents))

Solution (Python):
class UnionFind():
    def __init__(self, n):
        self.n = n
        self.parents = [-1] * n

    def find(self, x):
        if self.parents[x] < 0:
            return x
        else:
            self.parents[x] = self.find(self.parents[x])
            return self.parents[x]

    def union(self, x, y):
        x = self.find(x)
        y = self.find(y)

        if x == y:
            return

        if self.parents[x] > self.parents[y]:
            x, y = y, x

        self.parents[x] += self.parents[y]
        self.parents[y] = x

    def size(self, x):
        return -self.parents[self.find(x)]

    def same(self, x, y):
        return self.find(x) == self.find(y)

    def members(self, x):
        root = self.find(x)
        return [i for i in range(self.n) if self.find(i) == root]

    def roots(self):
        return [i for i, x in enumerate(self.parents) if x < 0]

    def group_count(self):
        return len(self.roots())

    def all_group_members(self):
        return {r: self.members(r) for r in self.roots()}

    def __str__(self):
        return '\n'.join('{}: {}'.format(r, self.members(r)) for r in self.roots())

n,m=map(int,input().split())
uf=UnionFind(n)
for i in range(m):
  a,b=map(int,input().split())
  uf.union(a-1,b-1)
ans=1
for i in range(n):
  c=uf.size(i)
  if c>ans:
    ans=c
print(ans)

Solution (Python):
belong=list()
group=list()

def init(X):
  global belong
  global group
  belong=list(range(X))
  group=[[i] for i in range(X)]

def unite(x,y):
  if len(group[belong[x]])<len(group[belong[y]]):
    x,y=y,x
  if belong[x]==belong[y]:
    return True
  z=belong[y]
  for i in range(len(group[z])):
    belong[group[z][-1]]=belong[x]
    group[belong[x]].append(group[z][-1])
    del group[z][-1]
  return False

N,M=map(int,input().split())
init(N)
a,b=0,0
for i in range(M):
  a,b=map(int,input().split())
  unite(a-1,b-1)
P=0
for i in range(N):
  P=max(P,len(group[i]))
print(P)

Solution (Python):
FAST_IO = 0
if FAST_IO:
    import io, sys, atexit
    rr = iter(sys.stdin.read().splitlines()).next
    sys.stdout = _OUTPUT_BUFFER = io.BytesIO()

    @atexit.register
    def write():
        sys.__stdout__.write(_OUTPUT_BUFFER.getvalue())
else:
    rr = raw_input
rri = lambda: int(rr())
rrm = lambda: map(int, rr().split())
rrmm = lambda n: [rrm() for _ in xrange(n)]


from collections import defaultdict as ddic, Counter, deque
import heapq, bisect, itertools
MOD = 10 ** 9 + 7

class DSU:
    def __init__(self, N):
        self.par = list(range(N))
        self.sz = [1] * N

    def find(self, x):
        if self.par[x] != x:
            self.par[x] = self.find(self.par[x])
        return self.par[x]

    def union(self, x, y):
        xr, yr = self.find(x), self.find(y)
        if xr == yr:
            return False
        if self.sz[xr] < self.sz[yr]:
            xr, yr = yr, xr
        self.par[yr] = xr
        self.sz[xr] += self.sz[yr]
        self.sz[yr] = self.sz[xr]
        return True

    def size(self, x):
        return self.sz[self.find(x)]
N,M=rrm()
A= [ rrm() for _ in xrange(M)]

dsu = DSU(N)
for u,v in A:
    u-=1;v-=1
    dsu.union(u, v)

from collections import Counter
count = Counter(dsu.find(u) for u in xrange(N))
print max(count.values())

Solution (Python):
import sys
input = lambda : sys.stdin.readline().rstrip()
sys.setrecursionlimit(max(1000, 10**9))
write = lambda x: sys.stdout.write(x+"\n")


n,m = map(int, input().split())
ns = [[] for _ in range(n)]
for _ in range(m):
    u,v = map(int, input().split())
    u -= 1
    v -= 1
    ns[u].append(v)
    ns[v].append(u)
seen = [None]*n
ans = -1
for start in range(n):
    if seen[start]:
        continue
    s = [start]
    seen[start] = True
    val = 0
    while s:
        u = s.pop()
        val += 1
        for v in ns[u]:
            if not seen[v]:
                s.append(v)
                seen[v] = True
    ans = max(ans, val)
print(ans)

Solution (Python):
import sys
input = sys.stdin.readline
sys.setrecursionlimit(10**6)

class union_find():
    
    def __init__(self,n):
        self.n=n
        self.root=[-1]*(n+1)
        self.rank=[0]*(n+1)

    def find_root(self,x):
        if self.root[x]<0:
            return x
        else:
            self.root[x]=self.find_root(self.root[x])
            return self.root[x]

    def unite(self,x,y):
        x=self.find_root(x)
        y=self.find_root(y)

        if x==y:
            return 
        elif self.rank[x]>self.rank[y]:
            self.root[x]+=self.root[y]
            self.root[y]=x
        else:
            self.root[y]+=self.root[x]
            self.root[x]=y

            if self.rank[x]==self.rank[y]:
                self.rank[y]+=1

    def same(self,x,y):
        return self.find_root(x)==self.find_root(y)

    def cnt(self,x):
        return -self.root[self.find_root(x)]

n,m=map(int,input().split())
g=union_find(n)
ans=0
for _ in range(m):
    a,b=map(int,input().split())
    g.unite(a,b)


for i in range(1,n+1):
    ans=max(ans,g.cnt(i))
print(ans)

Solution (Python):
class UnionFind:
    def __init__(self, n):
        self.parent = [-1] * n
        self.rank = [1] * n
        self.count = [1] * n

    def get_root(self, i):
        while self.parent[i] != -1:
            i = self.parent[i]
        return i

    def is_same_tree(self, i, j):
        return self.get_root(i) == self.get_root(j)

    def merge(self, i, j):
        i = self.get_root(i)
        j = self.get_root(j)
        if i == j:
            return

        if self.rank[i] > self.rank[j]:
            self.parent[j] = i
            self.rank[i] = max(self.rank[i], self.rank[j] + 1)
            self.count[i] += self.count[j]
        else:
            self.parent[i] = j
            self.rank[j] = max(self.rank[j], self.rank[i] + 1)
            self.count[j] += self.count[i]

n,m = map(int, input().split())
uf = UnionFind(n)
for _ in range(m):
    a,b = [int(i)-1 for i in input().split()]
    uf.merge(a,b)
print(max(uf.count))

Solution (Python):
class UnionFindNode:
    def __init__(self, group_id, parent=None, value=None):
        self.group_id_ = group_id
        self.parent_ = parent
        self.value = value
        self.rank_ = 1
        self.member_num_ = 1

    def is_root(self):
        return not self.parent_

    def root(self):
        parent = self
        while not parent.is_root():
            parent = parent.parent_
            self.parent_ = parent
        return parent

    def find(self):
        root = self.root()
        return root.group_id_

    def rank(self):
        root = self.root()
        return root.rank_

    def unite(self, unite_node):
        root = self.root()
        unite_root = unite_node.root()

        if root.group_id_ != unite_root.group_id_:
            if root.rank() > unite_root.rank():
                n_root, child = root, unite_root
            else:
                n_root, child = unite_root, root
            child.parent_ = n_root
            n_root.rank_ = max(n_root.rank_, child.rank_ + 1)
            n_root.member_num_ = n_root.member_num_ + child.member_num_


N, M = map(int, input().split())
nodes = [UnionFindNode(i) for i in range(N)]
for _ in range(M):
    a, b = map(int, input().split())
    nodes[a-1].unite(nodes[b-1])
R = 0
for i in range(N):
    R = max(R, nodes[i].root().member_num_)
print(R)

Solution (Python):
import sys
from collections import deque
import bisect
import copy
import heapq
import itertools
import math
input = sys.stdin.readline
sys.setrecursionlimit(1000000)
mod = 10 ** 9 + 7

def read_values(): return map(int, input().split())
def read_index(): return map(lambda x: int(x) - 1, input().split())
def read_list(): return list(read_values())
def read_lists(N): return [read_list() for n in range(N)]


class UF:
    def __init__(self, N):
        self.state = [-1] * N
        self.rank = [0] * N
        self.num_group = N
    
    def get_parent(self, a):
        p = self.state[a]
        if p < 0:
            return a
        
        q = self.get_parent(p)
        self.state[a] = q
        return q

    def make_pair(self, a, b):
        pa = self.get_parent(a)
        pb = self.get_parent(b)
        if pa == pb:
            return

        if self.rank[pa] > self.rank[pb]:
            pa, pb = pb, pa
            a, b = b, a
        elif self.rank[pa] == self.rank[pb]:
            self.rank[pb] += 1

        self.state[pb] += self.state[pa]
        self.state[pa] = pb
        self.state[a] = pb
        self.num_group -= 1
    
    def is_pair(self, a, b):
        return self.get_parent(a) == self.get_parent(b)

    def get_size(self, a):
        return -self.state[self.get_parent(a)]


def main():
    N, M = read_values()
    uf = UF(N)
    for _ in range(M):
        A, B = read_index()
        uf.make_pair(A, B)

    res = 0
    for n in range(N):
        res = max(res, uf.get_size(n))

    print(res)


if __name__ == "__main__":
    main()