Problem:
Minimum Spanning Tree For a given weighted graph $G = (V, E)$, find the minimum spanning tree (MST) of $G$ and print total weight of edges belong to the MST. Input In the first line, an integer $n$ denoting the number of vertices in $G$ is given. In the following $n$ lines, a $n \times n$ adjacency matrix $A$ which represents $G$ is given.
$a_{ij}$ denotes the weight of edge connecting vertex $i$ and vertex $j$. If there is no edge between $i$ and $j$, $a_{ij}$ is given by -1. Output Print the total weight of the minimum spanning tree of $G$. Constraints  $1 \leq n \leq 100$  $0 \leq a_{ij} \leq 2,000$ (if $a_{ij} \neq -1$)  $a_{ij} = a_{ji}$  $G$ is a connected graph  Sample Input 1 5
 -1 2 3 1 -1
 2 -1 -1 4 -1
 3 -1 -1 1 1
 1 4 1 -1 3
 -1 -1 1 3 -1 Sample Output 1 5 Reference Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution (Python):

"""
Created on Sat Apr  6 18:01:32 2019

@author: Yamazaki Kenichi
"""

n = int(input())
A = [list(map(int,input().split())) for i in range(n)]
INFTY = 10**6
T = [-1 for i in range(n)]
d = [INFTY for i in range(n)]
p = [0 for i in range(n)]
d[0], p[0] = 0, -1
for j in range(n):
    mincost = 10**6
    for i in range(n):
        if T[i] != 1 and d[i] < mincost:
            mincost = d[i]
            u = i
    T[u] = 1
    for v in range(n):
        if T[v] != 1 and A[u][v] != -1:
            if A[u][v] < d[v]:
                d[v] = A[u][v]
                p[v] = u
                T[v] = 0
print(sum(d))

Solution (Python):
n = int(input())
g = [[] for i in range(n)]
for i in range(n):
    *v, = map(int, input().split())
    for j in range(n):
        if v[j] < 0:
            v[j] = 10000
    g[i] = v

ans = 0
t = [0]
v = [i for i in range(1, n)]
while len(t) < n:
    u = [2000, 0, 0]
    for tv in t:
        for vv in v:
            if g[tv][vv] < u[0]:
                u = [g[tv][vv], tv, vv]
    ans += u[0]
    t.append(u[2])
    v.remove(u[2])
print(ans)

Solution (Python):


n = int(input())
a = [list(map(int, input().split())) for _ in range(n)]
d = [0] + [1e4]*(n-1)
color = [0]*n


while 1:
    mincost = pow(10, 4)
    for i in range(n):
        if color[i] != 1 and d[i] < mincost:mincost = d[i];min_key = i
    if mincost == pow(10,4):
        break
    
    color[min_key] = 1
    
    for i in range(n):
        if color[i] != 1 and a[min_key][i] != -1 and d[i] > a[min_key][i]:
            d[i] = a[min_key][i]
            color[i] = 2

print(f"{sum(d)}")

Solution (Python):
N = int(input())
L = []  


def prim():
    
    s = None
    for i in range(N):
        for j in range(N):
            if L[i][j] != -1:
                s = i
                break
        if s is not None:
            break
    p = {s}
    T = []
    k = 0
    while k < N - 1:
        
        len = float("inf")
        lenside = []
        for sg in p:
            for i in range(N):
                isT = False
                for t in T:
                    if (t[0] == sg and t[1] == i) or (t[0] == i and t[1] == sg):
                        
                        isT = True
                        continue
                if isT is False and i not in p and L[sg][i] != -1 and L[sg][i] < len:
                    len = L[sg][i]
                    lenside = [sg, i]

        p.add(lenside[0])
        p.add(lenside[1])
        T.append(lenside)
        k = k + 1
        
    cnt = 0
    for t in T:
        cnt += L[t[0]][t[1]]
    print(cnt)


for i in range(N):
    L.append(list(map(int, input().split())))
prim()

Solution (Python):
n = int(input())
M = [list(map(int, input().split())) for _ in range(n)]
INF = 100000000000
color = ['White']*n
d = [INF]*n
p = [-1]*n
d[0] = 0
sum = 0

while True:
    minct = INF
    u = -1
    for i in range(n):
        if color[i] != 'Black' and d[i] < minct:
            minct = d[i]
            u = i
    if minct == INF:
        break
    color[u] = 'Black'
    for j in range(n):
        if color[j] != 'Black' and M[u][j] != -1:
            if M[u][j] < d[j]:
                d[j] = M[u][j]
                p[j] = u
                color[j] = 'Gray'
for k in range(n):
    if p[k] != -1:
        sum += M[k][p[k]]
print(sum)

Solution (Python):
import sys

n = int(sys.stdin.readline())
g = [list(map(int, sys.stdin.readline().split())) for _ in range(n)]

V = set(range(n))
U = set()
E = []

U.add(V.pop())

while len(U)<n:
    minw = 1e10
    mine = []
    for u in U:
        for v in V:
            if g[u][v]>=0 and minw>g[u][v]:
                minw = g[u][v]
                mine = [u, v]
    E.append(mine)
    U.add(mine[1])
    V.remove(mine[1])

vv=0
for u, v in E:
    vv += g[u][v]
print(vv)

Solution (Python):
import heapq
from collections import deque
from enum import Enum
import sys
import math
from _heapq import heappush, heappop

BIG_NUM = 2000000000
MOD = 1000000007
EPS = 0.000000001


class UnionFind:
    def __init__(self,arg_V):
        self.V = arg_V
        self.boss = [None]*arg_V
        self.height = [None]*arg_V

    def init(self):
        for i in range(self.V):
            self.boss[i] = i
            self.height[i] = 0

    def get_boss(self,index):
        if index == self.boss[index]:
            return index
        else:
            self.boss[index] = self.get_boss(self.boss[index])
            return self.boss[index]


    def is_same_group(self,a,b):
        return self.get_boss(a) == self.get_boss(b)

    def unite(self,x,y):
        boss_x = self.get_boss(x)
        boss_y = self.get_boss(y)

        if boss_x == boss_y:
            return

        if self.height[x] > self.height[y]:

            self.boss[boss_y] = boss_x

        elif self.height[x] < self.height[y]:

            self.boss[boss_x] = boss_y

        else:
            self.boss[boss_y] = boss_x
            self.height[x] += 1

class Edge:
    def __init__(self,arg_FROM,arg_TO,arg_WEIGHT):
        self.FROM = arg_FROM
        self.TO = arg_TO
        self.WEIGHT = arg_WEIGHT

    def __lt__(self,arg):
        return self.WEIGHT < arg.WEIGHT


V = int(input())

TABLE = []
for i in range(V):
    input_array = list(map(int,input().split()))
    for k in range(len(input_array)):
        if input_array[k] != -1:
            TABLE.append(Edge(i,k,input_array[k]))

UF = UnionFind(V)
UF.init()

TABLE.sort()

ans = 0

for i in range(len(TABLE)):
    if UF.is_same_group(TABLE[i].FROM, TABLE[i].TO) == False:
        UF.unite(TABLE[i].FROM, TABLE[i].TO)
        ans += TABLE[i].WEIGHT

print("%d"%(ans))

Solution (Python):
import heapq

def root(x):
    path_to_root = []
    while P[x] != x:
        path_to_root.append(x)
        x = P[x]
    for node in path_to_root:
        P[node] = x
    return x
def is_same_set(x,y):
    return root(x) == root(y)
def unite(x,y):
    P[root(x)] = root(y)

W = 0
N = int(input().strip())
P = [i for i in range(N)]
A = []
for i in range(N):
    v = list(map(int,input().strip().split(' ')))
    A.append(v)
edges = [] 
for i in range(N):
    for j in range(i,N):
        edge = [A[i][j],i,j]
        heapq.heappush(edges,edge)
while len(edges) > 0:
    e = heapq.heappop(edges)
    I = e[1]
    J = e[2]
    if e[0] != -1:
        if not is_same_set(I,J):
            unite(I,J)
            W += e[0]
print(W)

Solution (Python):
n = int(input())
T = [i for i in range(n)]
def root(x):
    path_to_root = []
    while T[x] != x:
        path_to_root.append(x)
        x = T[x]
    for node in path_to_root:
        T[node] = x
    return x
def is_same_set(x,y):
    return root(x) == root(y)
def unite(x,y):
    T[root(x)] = root(y)

A = []
W = []
for i in range(n):
    A.append(list(map(int,input().strip().split(' '))))
    for j in range(i+1,n):
        if A[i][j] != -1:
            W.append([A[i][j],i,j])
W.sort()
sum = 0
for i in range(len(W)):
    if not is_same_set(W[i][1],W[i][2]):
        unite(W[i][1],W[i][2])
        sum += W[i][0]
print(sum)

Solution (Python):
import sys
from collections import defaultdict

def push_vertex(vtx: int,queue: list, weights: dict, done: dict):
    queue.extend(weights[vtx])
    done[vtx] = True
    [queue.remove(x) for x in queue[:] if done[x[0]]]
    del weights[vtx]
    queue.sort(key=lambda x: x[1], reverse=True)
    return


def main():
    n = int(input())
    weights = defaultdict(list)
    for i in range(n):
        for idx, weight in enumerate(input().split()):
            if weight != '-1':
                weights[i+1].append((idx+1, int(weight)))
    done = {}
    for i in weights.keys():
        done[i] = False
    mst_weights = 0
    queue = []
    vtx = [*weights][0]
    push_vertex(vtx, queue, weights, done)

    while len(weights) != 0:
        new_vtx, new_vtx_weight = queue.pop()
        mst_weights += new_vtx_weight
        
        push_vertex(new_vtx, queue, weights, done)

    print(mst_weights)
    return

main()

Solution (Python):
from sys import maxsize
from functools import reduce

WHITE = 0
GRAY = 1
BLACK = 2


def read_adj_mat():
    n = int(input())
    M = []
    for _ in range(n):
        M.append([maxsize if x == "-1" else int(x)
                  for x
                  in input().split()])
    return n, M


def min_sp_tree(n, M):
    d = [maxsize] * n
    p = [-1] * n
    color = [WHITE] * n

    d[0] = 0

    while True:
        minv = maxsize
        u = -1

        for i in range(n):
            if minv > d[i] and color[i] != BLACK:
                u = i
                minv = d[i]

        if u == -1:
            break

        color[u] = BLACK
        for v in range(n):
            if color[v] != BLACK and M[u][v] != maxsize:
                if d[v] > M[u][v]:
                    d[v] = M[u][v]
                    p[v] = u
                    color[v] = GRAY

    return sum([M[i][p[i]] for i in range(n) if p[i] != -1])


def main():
    n, M = read_adj_mat()
    print(min_sp_tree(n, M))


if __name__ == '__main__':
    main()

Solution (Python):
n = int(input())
E = []
for _ in range(n):
    E.append(list(map(int, input().strip().split())))

class Node:
    def __init__(self, n):
        self.data = n
        self.parent = self
        self.rank = 0
    def find(self):
        if self.parent.data == self.data:
            return self
        else:
            ret = self.parent.find()
            self.parent = ret
            return ret
    def union(self, rhs):
        lhs = self
        lhs_root = lhs.find()
        rhs_root = rhs.find()
        if lhs_root.rank > rhs_root.rank:
            rhs_root.parent = lhs
        elif rhs_root.rank > lhs_root.rank:
            lhs_root.parent = rhs
        elif lhs_root.data != rhs_root.data:
            lhs_root.parent = rhs_root
            rhs_root.rank += 1
    def is_same(self, rhs):
        lhs = self
        return lhs.find().data == rhs.find().data

V = [Node(i) for i in range(n)]
inf = 2005

for i in range(n):
    for j in range(n):
        E[i][j] = [E[i][j], i, j, 0 if E[i][j] != -1 else inf]
min_op = lambda n: n[0] + n[3]

remain = n-1
sum_w = 0
while remain > 0:
    min_edge = min([min(E[i], key=min_op) for i in range(n)], key=min_op)
    w, i, j, _ = min_edge
    E[i][j][3] = E[j][i][3] = inf
    if not V[i].is_same(V[j]):
        V[i].union(V[j])
        sum_w += w
        remain -= 1
print(sum_w)

Solution (Python):
N = int(input())
G = [[] for i in range(N)]
for v in range(N):
    for w, c in enumerate(map(int, input().split())):
        if c != -1:
            G[v].append((w, c))

from heapq import heappush, heappop, heapify
used = [0]*N
que = [(c, w) for w, c in G[0]]
used[0] = 1
heapify(que)

ans = 0
while que:
    cv, v = heappop(que)
    if used[v]:
        continue
    used[v] = 1
    ans += cv
    for w, c in G[v]:
        if used[w]:
            continue
        heappush(que, (c, w))
print(ans)

Solution (Python):

"""
Created on Sat Sep 22 11:44:37 2018
ALDS1-12-A
@author: maezawa
"""
import heapq

n = int(input())
amat = [[] for _ in range(n)]
for i in range(n):
    amat[i] = list(map(int, input().split()))

adj = [[] for _ in range(n+1)]
for i in range(n):
    for j in range(i, n):
        if amat[i][j] == -1:
            continue
        adj[i+1].append([amat[i][j], j+1])
        adj[j+1].append([amat[j][i], i+1])


d = [10**10]*(n+1)
h = []
heapq.heappush(h, [0, 1]) 
visited = [False]*(n+1)
while h:
    cost, node = heapq.heappop(h)
    if visited[node] == False and cost < d[node]:
        d[node] = cost
        visited[node] = True
        for nextnode in adj[node]:
            if visited[nextnode[1]]:
                continue
            heapq.heappush(h,nextnode)

d.pop(0)
print(sum(d))

Solution (Python):
INFTY = 1 << 21
WHITE = 0
GRAY = 1
BLACK = 2

def prim(n, m):
    d = [INFTY] * n
    p = [-1] * n
    color = [WHITE] * n

    d[0] = 0

    while True:
        minv = INFTY
        u = -1
        for i in range(n):
            if minv > d[i] and color[i] != BLACK:
                u = i
                minv = d[i]
        if u == -1:
            break
        color[u] = BLACK
        for v in range(n):
            if color[v] != BLACK and m[u][v] != INFTY:
                if d[v] > m[u][v]:
                    d[v] = m[u][v]
                    p[v] = u
                    color[v] = GRAY

    w_sum = 0
    for i in range(n):
        if p[i] != -1:
            w_sum += m[i][p[i]]

    return w_sum


n = int(input())
m = []
for i in range(n):
    line = [int(v) for v in input().split()]
    for j in range(n):
        if line[j] == -1:
            line[j] = INFTY
    m.append(line)

print(prim(n, m))

Solution (Python):
n = int(input())
a = []
b = []
c = []

def root(x):
    global rs
    if rs[x] == x:
        return(x)
    else:
        return(root(rs[x]))

for _ in range(n):
    a.append(list(map(int,input().split())))
for i in range(n):
    for j in range(i+1,n):
        if a[i][j] >= 0:
            b.append([a[i][j],i,j])
rs = [(i) for i in range(n)]

b.sort()
ans = 0
while(len(b)):
    if root(b[0][1]) != root(b[0][2]):
        ans += b[0][0]
        rs[root(rs[b[0][2]])] = root(b[0][1])
        c.append(b[0])
    del(b[0])
print(ans)

Solution (Python):
infinity = float('inf')
n = int(input())
a = []

for i in range(n):

    line = list(map(int, input().split()))

    for i in range(len(line)):
        if line[i] == -1:
            line[i] = infinity
    a.append(line)

checked = [False] * n
d = [infinity] * n
checked[0] = True
ansswer = 0
next = 0

for i in range(n - 1):
    for j in range(n):
        if a[next][j] < d[j]:
            d[j] = a[next][j]

    minimumNeightbor = 0
    minimumWeight = infinity

    for j in range(n):
        if not checked[j] and d[j] < minimumWeight:
            minimumNeightbor = j
            minimumWeight = d[j]

    checked[minimumNeightbor] = True
    next = minimumNeightbor
    ansswer += minimumWeight

print(ansswer)

Solution (Python):
n = int(input())
a = [list(map(int, input().split())) for i in range(n)]
m = [2001] * n
f = [0] * n
sum = 0

m[0] = 0
f[0] = 1
v = 0

for i in range(n-1) :
    for j in range(n) :
        if (a[v][j] < m[j]) & (f[j] == 0) & (a[v][j] != -1):
            m[j] = a[v][j]
    
    min = 2001
    minj = 0
    for j in range(n) :
        if (m[j] < min) & (f[j] == 0):
            min = m[j]
            minj = j
    
    v = minj
    f[minj] = 1
    sum += min

print(sum)

Solution (Python):

import heapq
n=int(input())
cost=[]
used=[False for i in range(n)]
for i in range(n):
    cost.append(list(map(int,input().split())))
q=[]
heapq.heappush(q,(0,0))
ans=0
while len(q):
    c,newv=heapq.heappop(q)
    if used[newv]:
        continue
    used[newv]=True
    ans+=c
    for nxt in range(n):
        if cost[newv][nxt]!=-1 and not used[nxt]:
            heapq.heappush(q,(cost[newv][nxt],nxt))
print(ans)

Solution (Python):

def mst(length, adj_m):
    """ Prims's Algorithm """
    cost = 0
    X,Y = [0], [i+1 for i in range(length-1)]
    while len(Y):
        min_cost = 2001
        ey = -1
        for x in X:
            for y in Y:
                t = adj_m[x][y]
                if not t == -1:
                    if min_cost > t:
                        min_cost = t
                        ey = y
        X.append(ey)
        Y.remove(ey)
        cost += min_cost
    return cost

n = int(input())
adj_m = [[-1 for _ in range(n)] for _ in range(n)]
for j in range(n):
    for i,v in enumerate(input().split()):
        adj_m[j][i] = int(v)
min_weight = mst(n, adj_m)
print(min_weight)