Ancient Language (Codeforces 1424M)
Tags: Others, graphs

While exploring the old caves, researchers found a book, or more precisely, a stash of mixed pages from a book. Luckily, all of the original pages are present and each page contains its number. Therefore, the researchers can reconstruct the book.After taking a deeper look into the contents of these pages, linguists think that this may be some kind of dictionary. What's interesting is that this ancient civilization used an alphabet which is a subset of the English alphabet, however, the order of these letters in the alphabet is not like the one in the English language.Given the contents of pages that researchers have found, your task is to reconstruct the alphabet of this ancient civilization using the provided pages from the dictionary.

Input: First-line contains two integers: $$$n$$$ and $$$k$$$ ($$$1 \le n, k \le 10^3$$$) — the number of pages that scientists have found and the number of words present at each page. Following $$$n$$$ groups contain a line with a single integer $$$p_i$$$ ($$$0 \le n \lt 10^3$$$) — the number of $$$i$$$-th page, as well as $$$k$$$ lines, each line containing one of the strings (up to $$$100$$$ characters) written on the page numbered $$$p_i$$$.

Output: Output a string representing the reconstructed alphabet of this ancient civilization. If the book found is not a dictionary, output "IMPOSSIBLE" without quotes. In case there are multiple solutions, output any of them.


Sample Inputs:
3 3
2
b
b
bbac
0
a
aca
acba
1
ab
c
ccb

Sample Outputs:
acb

Solution (Python):
import io
import os
import sys
from io import BytesIO, IOBase
BUFSIZE = 8192

class FastIO(IOBase):
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = 'x' in file.mode or 'r' not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            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):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            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()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            (self.buffer.truncate(0), self.buffer.seek(0))

class IOWrapper(IOBase):

    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode('ascii'))
        self.read = lambda : self.buffer.read().decode('ascii')
        self.readline = lambda : self.buffer.readline().decode('ascii')
(sys.stdin, sys.stdout) = (IOWrapper(sys.stdin), IOWrapper(sys.stdout))

def input():
    return sys.stdin.readline().rstrip('\r\n')
(n, m) = map(int, input().split())
inc = {}
l = [None for i in range(n * m)]
for i in range(n):
    t = int(input())
    for j in range(m):
        ipt = input()
        l[t * m + j] = ipt
        if len(inc) == 26:
            continue
        for q in ipt:
            inc.setdefault(q, 0)
d = {}

def ae(u, v):
    t = d.setdefault(u, set())
    if v not in t:
        t.add(v)
        inc[v] = inc.get(v, 0) + 1
for (p, i) in enumerate(l[1:]):
    for j in range(min(len(l[p]), len(i))):
        if l[p][j] != i[j]:
            ae(i[j], l[p][j])
            break
        if len(i) - 1 == j and len(l[p]) - 1 > j:
            print('IMPOSSIBLE')
            exit(0)
ans = []
dq = []
ptr = 0
ninc = {}
for (k, v) in inc.items():
    if v == 0:
        dq.append(k)
    else:
        ninc[k] = v
inc = ninc
while ptr != len(dq):
    fst = dq[ptr]
    ans.append(fst)
    for i in d.get(fst, []):
        inc[i] -= 1
        if inc[i] == 0:
            dq.append(i)
            inc.pop(i)
    ptr += 1
if len(inc):
    print('IMPOSSIBLE')
    exit(0)
print(*reversed(ans), sep='')

Solution (Python):
from bisect import bisect_left as bl
from bisect import bisect_right as br
from heapq import heappush, heappop
import math
from collections import *
from functools import reduce, cmp_to_key, lru_cache
import io, os
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
M = mod = 10 ** 9 + 7

def factors(n):
    return sorted(set(reduce(list.__add__, ([i, n // i] for i in range(1, int(n ** 0.5) + 1) if n % i == 0))))

def inv_mod(n):
    return pow(n, mod - 2, mod)

def li():
    return [int(i) for i in input().rstrip().split()]

def st():
    return str(input().rstrip())[2:-1]

def val():
    return int(input().rstrip())

def li2():
    return [str(i)[2:-1] for i in input().rstrip().split()]

def li3():
    return [int(i) for i in st()]
graph = defaultdict(set)
(n, k) = li()
l = []
allwords = set()
for i in range(n):
    p = val()
    l1 = []
    for j in range(k):
        l1.append(st())
    allwords |= set(l1[-1])
    l.append([p, l1[:]])
l.sort(key=lambda x: x[0])
l = [i[1] for i in l]
if n == k == 1:
    print(''.join(set(l[0][0])))
    exit()
ingraph = defaultdict(int)

def match(a, b):
    for j in range(min(len(a), len(b))):
        if a[j] == b[j]:
            continue
        elif b[j] in graph[a[j]]:
            return
        else:
            graph[a[j]].add(b[j])
            ingraph[b[j]] += 1
            return
    if len(a) > len(b):
        print('IMPOSSIBLE')
        exit()
finl = []
for i in l:
    finl.extend(i)
l = finl
for i in range(1, len(l)):
    match(l[i - 1], l[i])
if min([ingraph[j] for j in graph]) != 0:
    print('IMPOSSIBLE')
    exit()
ans = ''
d = deque()
for j in graph:
    if ingraph[j] == 0:
        d.append(j)
while d:
    node = d.popleft()
    ans += node
    for j in graph[node]:
        ingraph[j] -= 1
        if not ingraph[j]:
            d.append(j)
if len(ans) != len(allwords):
    print('IMPOSSIBLE')
    exit()
print(ans)

Solution (Python):
from bisect import bisect_left as bl
from bisect import bisect_right as br
from heapq import heappush, heappop
import math
from collections import *
from functools import reduce, cmp_to_key, lru_cache
import io, os
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
M = mod = 10 ** 9 + 7

def factors(n):
    return sorted(set(reduce(list.__add__, ([i, n // i] for i in range(1, int(n ** 0.5) + 1) if n % i == 0))))

def inv_mod(n):
    return pow(n, mod - 2, mod)

def li():
    return [int(i) for i in input().rstrip().split()]

def st():
    return str(input().rstrip())[2:-1]

def val():
    return int(input().rstrip())

def li2():
    return [str(i)[2:-1] for i in input().rstrip().split()]

def li3():
    return [int(i) for i in st()]
graph = defaultdict(set)
(n, k) = li()
l = []
allwords = set()
for i in range(n):
    p = val()
    l1 = []
    for j in range(k):
        l1.append(st())
    allwords |= set(l1[-1])
    l.append([p, l1[:]])
l.sort(key=lambda x: x[0])
l = [i[1] for i in l]
if n == k == 1:
    print(''.join(set(l[0][0])))
    exit()
ingraph = defaultdict(int)

def match(a, b):
    for j in range(min(len(a), len(b))):
        if a[j] == b[j]:
            continue
        elif b[j] in graph[a[j]]:
            return
        else:
            graph[a[j]].add(b[j])
            ingraph[b[j]] += 1
            return
    if len(a) > len(b):
        print('IMPOSSIBLE')
        exit()
finl = []
for i in l:
    finl.extend(i)
l = finl
for i in range(1, len(l)):
    match(l[i - 1], l[i])
if min([ingraph[j] for j in graph]) != 0:
    print('IMPOSSIBLE')
    exit()
ans = ''
d = deque()
for j in graph:
    if ingraph[j] == 0:
        d.append(j)
while d:
    node = d.popleft()
    ans += node
    for j in graph[node]:
        ingraph[j] -= 1
        if not ingraph[j]:
            d.append(j)
if len(ans) != len(allwords):
    print('IMPOSSIBLE')
    exit()
print(ans)