Problem:
 Score : 500 points    Problem Statement There are N towns on a plane. The i -th town is located at the coordinates (x_i,y_i) . There may be more than one town at the same coordinates.  You can build a road between two towns at coordinates (a,b) and (c,d) for a cost of min(|a-c|,|b-d|) yen (the currency of Japan). It is not possible to build other types of roads.  Your objective is to build roads so that it will be possible to travel between every pair of towns by traversing roads. At least how much money is necessary to achieve this?      Constraints  2 ≤ N ≤ 10^5  0 ≤ x_i,y_i ≤ 10^9  All input values are integers.         Input Input is given from Standard Input in the following format:  N  x_1  y_1  x_2  y_2 : x_N  y_N       Output Print the minimum necessary amount of money in order to build roads so that it will be possible to travel between every pair of towns by traversing roads.        Sample Input 1 3
1 5
3 9
7 8      Sample Output 1 3  Build a road between Towns 1 and 2 , and another between Towns 2 and 3 . The total cost is 2+1=3 yen.       Sample Input 2 6
8 3
4 9
12 19
18 1
13 5
7 6      Sample Output 2 8

Solution (Python):


from math import sin, cos, tan, asin, acos, atan, sqrt, floor, ceil, pi, log10, log, gcd, hypot
from itertools import accumulate, permutations, combinations
from collections import deque, defaultdict, Counter
from bisect import bisect_left, bisect_right
from copy import deepcopy
from decimal import Decimal
import heapq
import sys

mod = 10**9 + 7



sys.setrecursionlimit(10**7)
input = sys.stdin.readline


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

    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 + 1) 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())


class BIT():
    
    def __init__(self, n):
        self.n = n
        self.data = [0]*(n+1)

    def to_sum(self, i):
        
        s = 0
        while i > 0:
            s += self.data[i]
            i -= (i & -i)
        return s

    def add(self, i, x):
        
        while i <= self.n:
            self.data[i] += x
            i += (i & -i)

    def get(self, i, j):
        
        return self.to_sum(j)-self.to_sum(i-1)


class SegTree():
    
    def __init__(self, lists, function, basement):
        self.n = len(lists)
        self.K = (self.n-1).bit_length()
        self.f = function
        self.b = basement
        self.seg = [basement]*(2**(self.K+1)+1)
        X = 2**self.K
        for i, v in enumerate(lists):
            self.seg[i+X] = v
        for i in range(X-1, 0, -1):
            self.seg[i] = self.f(self.seg[i << 1], self.seg[i << 1 | 1])

    def update(self, k, value):
        X = 2**self.K
        k += X
        self.seg[k] = value
        while k:
            k = k >> 1
            self.seg[k] = self.f(self.seg[k << 1], self.seg[(k << 1) | 1])

    def query(self, L, R):
        num = 2**self.K
        L += num
        R += num
        vL = self.b
        vR = self.b
        while L < R:
            if L & 1:
                vL = self.f(vL, self.seg[L])
                L += 1
            if R & 1:
                R -= 1
                vR = self.f(self.seg[R], vR)
            L >>= 1
            R >>= 1
        return self.f(vL, vR)

    def find_max_index(self, L, R, X):
        
        return self.fMi(L, R, X, 1, 0, 2**self.K)

    def find_min_index(self, L, R, X):
        
        return self.fmi(L, R, X, 1, 0, 2**self.K)

    def fMi(self, a, b, x, k, l, r):
        if self.seg[k] > x or r <= a or b <= l:
            
            return -1
        else:
            if k >= 2**self.K:
                return k-2**self.K
            else:
                vr = self.fMi(a, b, x, (k << 1) | 1, (l + r) // 2, r)
                if vr != -1:
                    return vr
                return self.fMi(a, b, x, k << 1, l, (l + r) // 2)

    def fmi(self, a, b, x, k, l, r):
        if self.seg[k] > x or r <= a or b <= l:
            
            return -1
        else:
            if k >= 2**self.K:
                return k-2**self.K
            else:
                vl = self.fmi(a, b, x, k << 1, l, (l+r)//2)
                if vl != -1:
                    return vl
                return self.fmi(a, b, x, k << 1 | 1, (l+r)//2, r)


class LCA():

    def _bfs(self, node):
        self.visited[node] = True
        for p in tree[node]:
            if self.visited[p] == False:
                self.kprv[0][p] = node
                self.depth[p] = self.depth[node]+1
                self._bfs(p)

    def __init__(self, root, tree):
        self.root = root
        self.n = len(tree)-1
        self.K = (self.n-1).bit_length()
        self.visited = [False for i in range(self.n+1)]
        self.depth = [0 for i in range(self.n+1)]
        self.kprv = [[-1 for i in range(self.n+1)] for j in range(self.K+1)]
        self._bfs(self.root)
        for cnt in range(1, self.K+1):
            for node in range(self.n+1):
                if self.kprv[cnt-1][node] >= 0:
                    self.kprv[cnt][node] = self.kprv[cnt -
                                                     1][self.kprv[cnt-1][node]]
                else:
                    self.kprv[cnt][node] = -1

    def query(self, u, v):
        

        dd = self.depth[v]-self.depth[u]

        if dd < 0:
            u, v = v, u
            dd = -dd

        for i in range(self.K+1):
            if dd & 1:
                v = self.kprv[i][v]
            dd >>= 1
        if u == v:
            return u

        for i in range(self.K-1, -1, -1):
            pu = self.kprv[i][u]
            pv = self.kprv[i][v]
            if pu != pv:
                u = pu
                v = pv
        return self.kprv[0][u]


class Combi():
    def __init__(self, N, mod):
        self.power = [1 for _ in range(N+1)]
        self.rev = [1 for _ in range(N+1)]
        self.mod = mod
        for i in range(2, N+1):
            self.power[i] = (self.power[i-1]*i) % self.mod
        self.rev[N] = pow(self.power[N], self.mod-2, self.mod)
        for j in range(N, 0, -1):
            self.rev[j-1] = (self.rev[j]*j) % self.mod

    def C(self, K, R):
        if K < R:
            return 0
        else:
            return ((self.power[K])*(self.rev[K-R])*(self.rev[R])) % self.mod

    def P(self, K, R):
        if K < R:
            return 0
        else:
            return (self.power[K])*(self.rev[K-R]) % self.mod


class Graph():
    def __init__(self, N, Type):
        self._V = N  
        self._E = 0  
        self.type = Type
        if Type == "D":
            
            self.G = [[] for i in range(self._V+1)]
        elif Type == "W":
            
            self.G = [[float("inf") for i in range(self._V+1)]
                      for j in range(self._V+1)]
        elif Type == "B" or Type == "K":
            
            self.G = []

    def E(self):
        """ 辺数 """
        return self._E

    @property
    def V(self):
        """ 頂点数 """
        return self._V

    def add(self, _from, _to, _cost):
        """ 2頂点と、辺のコストを追加する """
        self._E += 1
        if self.type == "D":
            self.G[_from].append((_to, _cost))

        elif self.type == "W":
            self.G[_from][_to] = _cost

        elif self.type == "B" or self.type == "K":
            self.G.append((_from, _to, _cost))

    def add_both(self, _from, _to, _cost):
        """ 2頂点と、辺のコストを追加する (無向グラフ) """
        self.add(_from, _to, _cost)
        self.add(_to, _from, _cost)


class Dijkstra(Graph):
    def __init__(self, N):
        super().__init__(N, "D")
        self.d = [10**20]*(self._V + 1)
        self.que = []
        heapq.heapify(self.que)

    def shortest_path(self, s):
        for i in range(self._V+1):
            self.d[i] = 10**20
        self.d[s] = 0
        heapq.heappush(self.que, (0, s))
        while self.que:
            cost, v = heapq.heappop(self.que)
            if self.d[v] < cost:
                continue

            for node, cost in self.G[v]:
                if self.d[node] > self.d[v] + cost:
                    self.d[node] = self.d[v] + cost
                    heapq.heappush(self.que, (self.d[node], node))
        return self.d


class Bellman_Ford(Graph):
    def __init__(self, N):
        super().__init__(N, "B")
        self.d = [10**20]*(self._V+1)

    def shortest_path(self, s):
        for i in range(self._V+1):
            self.d[i] = 10**20
        self.d[s] = 0
        for _ in range(self._V):
            flag = False
            for From, To, cost in self.G:
                newlen = self.d[From]+cost
                if newlen < self.d[To]:
                    flag = True
                    self.d[To] = newlen
            if not flag:
                break
        return self.d

    def have_negative_circle(self):
        for i in range(self._V+1):
            self.d[i] = 10**20
        self.d[1] = 0
        for i in range(1, self._V+1):
            flag = False
            for From, To, cost in self.G:
                newlen = self.d[From]+cost
                if newlen < self.d[To]:
                    flag = True
                    self.d[To] = newlen
            if not flag:
                break
            if i == self._V:
                return True
        return False


class Warshall_Floyd(Graph):

    def __init__(self, N):
        super().__init__(N, "W")

    def shortest_path(self):
        
        _N = self._V
        for i in range(_N+1):
            self.G[i][i] = 0
        for k in range(1, _N+1):
            for i in range(1, _N+1):
                for j in range(1, _N+1):
                    self.G[i][j] = min(self.G[i][j], self.G[i][k]+self.G[k][j])
        return self.G


class Kruskal(Graph):
    def __init__(self, N):
        super().__init__(N, "K")
        self.connected_G = UnionFind(self._V)

    def find_lowest_cost(self):
        res = 0
        self.G.sort(lambda x:x[-1])
        for p, q, cost in self.G:
            if not self.connected_G.same(p, q):
                self.connected_G.union(p, q)
                res += cost
        return res


def union_lcm(lists):
    if len(lists) == 1:
        return lists[0]
    elif len(lists) > 1:
        num = lists[0]
        for i in range(1, len(lists)):
            num = num*lists[i]//gcd(num, lists[i])
        return num


def union_gcd(lists):
    if len(lists) == 1:
        return lists[0]
    elif len(lists) > 1:
        num = lists[0]
        for i in range(1, len(lists)):
            num = gcd(num, lists[i])
        return num


def is_prime(N):
    for i in range(2, N+1):
        if i**2 > N:
            break
        if N % i == 0:
            return False
    return True


class Primes():
    def __init__(self, N):
        self.N = N
        self.prime = {i for i in range(2, self.N+1)}
        for i in range(2, self.N+1):
            if i in self.prime:
                for j in range(i*2, self.N+1, i):
                    if j in self.prime:
                        self.prime.remove(j)

    def show_primes(self):
        return self.prime

    def is_prime(self, X):
        if X < self.N:
            return X in self.prime
        else:
            return "Unknown"


def main():
    
    N = int(input())
    d = []

    for i in range(N):
        x, y = map(int, input().split())
        d.append((i+1, x, y))

    G = Kruskal(N)

    d.sort(key=lambda x: x[1])

    for i in range(N-1):
        a, px, py = d[i]
        b, qx, qy = d[i+1]
        dis = abs(px-qx)
        G.add_both(a, b ,dis)

    d.sort(key=lambda x: x[2])

    for i in range(N-1):
        a, px, py = d[i]
        b, qx, qy = d[i+1]
        dis = abs(py-qy)
        G.add_both(a, b, dis)

    print(G.find_lowest_cost())


if __name__ == "__main__":
    main()

Solution (Python):
class UnionFind():
    
    def __init__(self, n):  
        self.parent = list(range(n)) 
        self.rank = [0] * n   

    def root_of(self, x):
        children = [x]
        while self.parent[x] != x:
            x = self.parent[x]
            children.append(x)

        for ch in children:
            self.parent[ch] = x
        return x

    
    def union(self, x, y):
        rx = self.root_of(x)
        ry = self.root_of(y)
        if rx == ry:
            return
        if self.rank[rx] < self.rank[ry]: 
            self.parent[rx] = ry  
        else:
            self.parent[ry] = rx
            if self.rank[rx] == self.rank[ry]:
                self.rank[rx] += 1

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


N = int(input())
X = [None] * N
Y = [None] * N
for i in range(N):
    x, y = [int(x) for x in input().split()]
    X[i] = (x, i)
    Y[i] = (y, i)

X.sort()
Y.sort()
E = []
xp = X[0][0]
ip = X[0][1]
for (x, i) in X:
    E.append((x - xp, (ip, i)))
    xp = x
    ip = i

yp = Y[0][0]
ip = Y[0][1]
for (y, i) in Y:
    E.append((y - yp, (ip, i)))
    yp = y
    ip = i

E.sort()
uf = UnionFind(N)
cost = 0
for c, (e1, e2) in E:
    if uf.same(e1, e2): continue
    cost += c
    uf.union(e1, e2)

print(cost)

Solution (Python):
import sys
from collections import defaultdict
finput=lambda:sys.stdin.readline().strip()

class Unionfind():
    def __init__(self,n):
        self.parents=[-1]*n
        self.ranks=[0]*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.ranks[x]<self.ranks[y]:
            x,y=y,x
        self.parents[x]+=self.parents[y]
        self.parents[y]=x
        if self.ranks[x]==self.ranks[y]:
            self.ranks[x]+=1
    
    def size(self,x):
        return -self.parents[self.find(x)]
    
    def issametree(self,x,y):
        return self.find(x)==self.find(y)

def main():
    n=int(finput())
    coor=[list(map(int,finput().split()))+[i] for i in range(n)]
    coor.sort(key=lambda e:e[0])
    cost = lambda e,f:min(abs(e[0]-f[0]),abs(e[1]-f[1]))
    ew=dict(((coor[i][2],coor[i+1][2]),cost(coor[i][:2],coor[i+1][:2])) for i in range(n-1))
    coor.sort(key=lambda e:e[1])
    ew.update(dict(((coor[i][2],coor[i+1][2]),cost(coor[i][:2],coor[i+1][:2])) for i in range(n-1)))
    edges=sorted(list(iter(ew)),key=lambda e:ew[e])
    v=Unionfind(n)
    totcost=0
    
    
    for edge in edges:
        if not v.issametree(edge[0],edge[1]):
            v.union(edge[0],edge[1])
            totcost+=ew[edge]
    print(totcost)
    
if __name__=='__main__':
    main()

Solution (Python):
import heapq

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 = int(input())
xy = [list(map(int, input().split()))+[i] for i in range(n)]
"""
辺を結びたいがすべての点で結ぶと膨大になる
"""

edge = []
xy.sort()
for i, j in zip(range(n-1), range(1, n)):
    cost = xy[j][0] - xy[i][0]
    heapq.heappush(edge, [cost, xy[i][2], xy[j][2]])

xy.sort(key= lambda x: x[1])
for i, j in zip(range(n-1), range(1, n)):
    cost = xy[j][1] - xy[i][1]
    heapq.heappush(edge, [cost, xy[i][2], xy[j][2]])

uf = UnionFind(n)
ans = 0
while edge:
    c, a, b = heapq.heappop(edge)
    if uf.same(a, b): continue
    uf.union(a, b)
    ans += c

print(ans)

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=int(input())
X=[]
Y=[]
for i in range(n):
    x,y=map(int,input().split())
    X.append([x,i])
    Y.append([y,i])
X.sort()
Y.sort()
E=[]
for i in range(n-1):
    E.append([X[i+1][0]-X[i][0],X[i][1],X[i+1][1]])
    E.append([Y[i+1][0]-Y[i][0],Y[i][1],Y[i+1][1]])
E.sort()
U=UnionFind(n)
ans=0
for i in range(2*n-2):
    d=E[i][0]
    j1=E[i][1]
    j2=E[i][2]
    if not U.same(j1,j2):
        ans+=d
        U.union(j1,j2)
print(ans)

Solution (Python):
import sys
buf = sys.stdin.buffer

class UnionFind:
  def __init__(self, num_of_node):
    self.roots = [-1] * num_of_node
    self.ranks = [1] * num_of_node
    self.groups = num_of_node
 
  def get_root(self, node):
    root = self.roots[node]
    if root == -1:
      return node
    else:
      root = self.get_root(root)
      self.roots[node] = root
      return root

  def is_same_group(self, node1, node2):
    return self.get_root(node1) == self.get_root(node2)

  def unite(self, node1, node2):
    self.groups -= 1
    root1, root2 = self.get_root(node1), self.get_root(node2)
    rank1, rank2 = self.ranks[root1], self.ranks[root2]
    if rank1 < rank2:
      root1, root2 = root2, root1
    self.roots[root2] = root1
    self.ranks[root2] += self.ranks[root1]

def main():
  n = int(buf.readline())
  XY = map(int, buf.read().split())
  Xs = []
  Ys = []
  for i, (x, y) in enumerate(zip(XY, XY)):
    Xs.append((x-1, i))
    Ys.append((y-1, i))
    
  Xs.sort()
  Ys.sort()
  
  edges = []
  for i, ((x, node), (nx, new_node)) in enumerate(zip(Xs[:-1], Xs[1:])):
    edges.append((nx-x, node, new_node))
 
  for i, ((y, node), (ny, new_node)) in enumerate(zip(Ys[:-1], Ys[1:])):
    edges.append((ny-y, node, new_node))
 
  edges.sort(reverse=True)
 
  ans = 0
  uf = UnionFind(n)
  while uf.groups != 1:
    d, node1, node2 = edges.pop()
    if uf.is_same_group(node1, node2):
      continue
    uf.unite(node1, node2)
    ans += d
  print(ans)

if __name__ == "__main__":
    main()

Solution (Python):

import copy


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

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

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


n = int(input())
rows = [[i] + list(map(int, input().split())) for i in range(n)]


edges = []


rows_x = copy.deepcopy(rows)
rows_x.sort(key=lambda x: x[1])
for i in range(len(rows)-1):
    left = rows_x[i]
    right = rows_x[i+1]
    edges.append((left[0], right[0], right[1] - left[1]))


rows_y = copy.deepcopy(rows)
rows_y.sort(key=lambda x: x[2])
for i in range(len(rows)-1):
    down = rows_y[i]
    up = rows_y[i+1]
    edges.append((down[0], up[0], up[2] - down[2]))



uf = UnionFind(n)

edges.sort(key=lambda edge: edge[2])

result = 0
for edge in edges:
    s, t, c = edge
    unioned = uf.union(s, t)
    if unioned:
        result += c

print(result)

Solution (Python):
ii = lambda : int(input())
mi = lambda : map(int,input().split())
li = lambda : list(map(int,input().split()))

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

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

    def union(self, x, y):
        x = self.find(x)
        y = self.find(y)
        if(x == y):
            return False
        if(self.d[x] > self.d[y]):
            x, y = y, x
        self.d[x] += self.d[y]
        self.d[y] = x
        return True

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

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

n = ii()

uf = UnionFind(n)
path = [] 

xaxis = []
yaxis = []

ans = 0
cnt = 0

for i in range(n):
    x,y = mi()
    xaxis.append([x, i])
    yaxis.append([y, i])

xaxis.sort()
yaxis.sort()

for k,i in enumerate(xaxis[1:]):
    d = i[0] - xaxis[k][0] 
    s = xaxis[k][1]
    t = i[1]
    if d != 0:
        path.append([d,s,t])
    else:
        if uf.same(s, t) == False:
            uf.union(s,t)
            cnt += 1

for k,i in enumerate(yaxis[1:]):
    d = i[0] - yaxis[k][0] 
    s = yaxis[k][1]
    t = i[1]
    if d != 0:
        path.append([d,s,t])
    else:
        if uf.same(s, t) == False:
            uf.union(s,t)
            cnt += 1

path.sort()

for i in path:
    if cnt == n-1:
        break
    d = i[0]
    s = i[1]
    t = i[2]
    if uf.same(s,t) == False:
        uf.union(s,t)
        cnt += 1
        ans += d

print(ans)

Solution (Python):
N = int(input())

def find(x, U, r):
  p = []
  while not x == U[x]:
    p.append(x)
    x = U[x]
  for x_ in p:
    U[x_] == x
  return x
  
def merge(x, y, U, r):
  x = find(x, U, r)
  y = find(y, U, r)
  
  if x == y:
    return False
  
  if r[x] < r[y]:
    x, y = y, x
  U[y] = x
  r[x] = max(r[y]+1, r[x])
  
  return True
  
U = [i for i in range(N)]
r = [0 for _ in range(N)]

B = []
C = []
for i in range(N):
  u, v = map(int, input().split())
  B.append((u, v, i))
  C.append((v, u, i))
B.sort()
C.sort()


R = []
for i in range(1,N):
  c = B[i][0] - B[i-1][0]
  p1 = B[i][2]
  p2 = B[i-1][2]
  R.append((c, p1, p2))
  
  c = C[i][0] - C[i-1][0]
  p1 = C[i][2]
  p2 = C[i-1][2]
  R.append((c, p1, p2))
R.sort()


U = [i for i in range(N)]
r = [0 for _ in range(N)]

ans = 0
for road in R:
  c = road[0]
  x = road[1]
  y = road[2]
  if merge(x, y, U, r):
    
    ans += c

print(ans)

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 = int(input())
xy1 = [None]*n
for i in range(n):
    x,y = tuple(map(int, input().split()))
    xy1[i] = (x,y,i)
xy2 = xy1[:]
xy1.sort()
xy2.sort(key=lambda x: x[1])

from collections import defaultdict
ns = defaultdict(set)
for i in range(n):
    x1,y1,i1 = xy1[i]
    x2,y2,i2 = xy2[i]
    if i>0:
        ns[i1].add((abs(x1-xy1[i-1][0]), xy1[i-1][2]))
        ns[i2].add((abs(y2-xy2[i-1][1]), xy2[i-1][2]))
    if i<n-1:
        ns[i1].add((abs(x1-xy1[i+1][0]), xy1[i+1][2]))
        ns[i2].add((abs(y2-xy2[i+1][1]), xy2[i+1][2]))
from collections import defaultdict
from heapq import heappush as hpush, heappop as hpop

def prim(ns):
    
    seen = [False] * n
    seen[0] = True
    q = []
    for d,u in ns[0]:
        hpush(q, (d,u))
    val = 0
    for _ in range(n-1):
        while True:
            d,u = hpop(q)
            if not seen[u]:
                val += d
                seen[u] = True
                for d,v in ns[u]:
                    if not seen[v]:
                        hpush(q, (d,v))
                break
    return val
val = prim(ns)
print(val)

Solution (Python):
n = int(input())
x,y = [],[]
for i in range(n):
    a,b = map(int,input().split())
    x.append([a,i])
    y.append([b,i])
x.sort()
y.sort()
g = [[] for _ in range(n)]
for i in range(n-1):
    g[x[i][1]].append([abs(x[i+1][0]-x[i][0]),x[i+1][1]])
    g[x[i+1][1]].append([abs(x[i+1][0]-x[i][0]),x[i][1]])
    g[y[i][1]].append([abs(y[i+1][0]-y[i][0]),y[i+1][1]])
    g[y[i+1][1]].append([abs(y[i+1][0]-y[i][0]),y[i][1]])
import heapq
hq = []
for d,e in g[0]:
    heapq.heappush(hq,[d,e])
cnt = 1
visit = [False]*n
visit[0] = True
ans = 0
while cnt <n:
    cost, end = heapq.heappop(hq)
    if visit[end]: continue
    ans += cost
    visit[end] = True
    cnt += 1
    for d,e in g[end]:
        if not visit[e]:
            heapq.heappush(hq,[d,e])
print(ans)

Solution (Python):

class union_find:
    
    
    
    def __init__(self, N):
        self.par = [-1]*N

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

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

        if x == y:
            return False
        else:
            
            if self.par[x] > self.par[y]:
                x,y = y,x
            self.par[x] += self.par[y]
            self.par[y] = x
            return True

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

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

N = int(input())
XYI = []
for i in range(N):
    x, y = map(int, input().split())
    XYI.append((x, y, i))

import heapq
edges = []
heapq.heapify(edges)

XYI.sort(key=lambda x: x[0])
for j in range(1, N):
    px, py, pi = XYI[j-1]
    x, y, i = XYI[j]
    d = min(abs(x-px), abs(y-py))
    heapq.heappush(edges, (d, pi, i))

XYI.sort(key=lambda x: x[1])
for j in range(1, N):
    px, py, pi = XYI[j-1]
    x, y, i = XYI[j]
    d = min(abs(x-px), abs(y-py))
    heapq.heappush(edges, (d, pi, i))

uf = union_find(N)
ans = 0
while True:
    d, i, j = heapq.heappop(edges)
    if uf.same(i, j): continue
    uf.unite(i, j)
    ans += d
    if uf.size(i) == N: break

print(ans)

Solution (Python):
import queue

N = int(input())

pos = []
for _ in range(N):
    a, b = map(int, input().split())
    pos.append((a, b))

visited = {v: False for v in pos}
edges = {v: [] for v in pos}
x_sorted = sorted([(p[0], i) for i, p in enumerate(pos)])
for i in range(len(x_sorted) - 1):
    j = x_sorted[i][1]
    l = x_sorted[i+1][1]
    edges[pos[j]].append((pos[l], abs(pos[l][0] - pos[j][0])))
    edges[pos[l]].append((pos[j], abs(pos[l][0] - pos[j][0])))
y_sorted = sorted([(p[1], i) for i, p in enumerate(pos)])
for i in range(len(y_sorted) - 1):
    j = y_sorted[i][1]
    l = y_sorted[i+1][1]
    edges[pos[j]].append((pos[l], abs(pos[l][1] - pos[j][1])))
    edges[pos[l]].append((pos[j], abs(pos[l][1] - pos[j][1])))

que = queue.PriorityQueue()
que.put((0, pos[0]))

res = 0
while not que.empty():
    cost, vertex = que.get()
    if visited[vertex]:
        continue
    visited[vertex] = True
    res += cost

    for to, next_cost in edges[vertex]:
        if not visited[to]:
            que.put((next_cost, to))

print(res)

Solution (Python):
import sys

sys.setrecursionlimit(10 ** 7)
input = sys.stdin.readline
f_inf = float('inf')
mod = 10 ** 9 + 7

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

    def find(self, x):
        """
        親が同じか判別する
        """
        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):
        """
        yをxの根に繋ぐ
        """
        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 same(self, x, y):
        """
        xとyが同じ連結成分か判別する
        """
        return self.find(x) == self.find(y)

    def size(self, x):
        """
        xの連結成分の大きさを返す
        """
        return -self.parents[self.find(x)]

    def kruskal(self, edge):
        """
        :param edge: edge = [(コスト, 頂点1, 頂点2),...]の形で重み付き隣接リストを渡して下さい
        :return: 最小全域木のコストの和
        """
        edge.sort()
        cost_sum = 0
        for cost, node1, node2 in edge:
            if not self.same(node1, node2):
                cost_sum += cost
                self.union(node1, node2)
        return cost_sum


def resolve():
    n = int(input())
    X, Y = [], []
    for i in range(n):
        x, y = map(int, input().split())
        X.append([x, i])
        Y.append([y, i])
    X.sort()
    Y.sort()
    edge = []
    for i in range(1, n):
        cost1 = X[i][0] - X[i - 1][0]
        edge.append([cost1, X[i - 1][1], X[i][1]])
        cost2 = Y[i][0] - Y[i - 1][0]
        edge.append([cost2, Y[i - 1][1], Y[i][1]])

    uf = UnionFind(n)
    res = uf.kruskal(edge)
    print(res)


if __name__ == '__main__':
    resolve()

Solution (Python):
class UF():
  def __init__(self, N):
    self.par = [-1]*N
 
  def find(self, x):
    if self.par[x] < 0:
      return x
    else:
      self.par[x] = self.find(self.par[x])
      return self.par[x]
 
  def union(self, x,y):
    p1,p2 = self.find(x),self.find(y)
    if p1 == p2: return
    if p1 > p2:
      p1,p2 = p2,p1
    self.par[p1] += self.par[p2]
    self.par[p2] = p1
 
  def same(self, x,y):
    return self.find(x) == self.find(y)

n = int(input())
coords = []
for i in range(n):
  x,y = map(int,input().split())
  coords.append((i,x,y))

edges = []

sorted_x = sorted(coords, key=lambda x: x[1])
for i in range(n-1):
  c = abs(sorted_x[i][1] - sorted_x[i+1][1])
  u,v = sorted_x[i][0], sorted_x[i+1][0]
  edges.append((c,u,v))

sorted_y = sorted(coords, key=lambda x: x[2])
for i in range(n-1):
  c = abs(sorted_y[i][2] - sorted_y[i+1][2])
  u,v = sorted_y[i][0], sorted_y[i+1][0]
  edges.append((c,u,v))

edges.sort()

uf = UF(n)
res = 0
for c,u,v in edges:
  if not uf.same(u,v):
    uf.union(u,v)
    res += c
 
print(res)