## Tuesday, February 3, 2015

### Codeforces Round #290 (Div. 1 A / Div. 2 C) - Fox and Names

Problem Statement:
512A - Fox and Names

Solution:
The idea to solve this problem is pretty intuitive, but I make no pretence that implementing it is easy,  as I had a lot of trouble trying to write a correct one.

First of all we represent each letter as a node in a directed graph, where a directed edge from u to v means that u has higher "rank" than v. In this problem we are required to build such graph using the strings provided. There are a lot of ways to do this, and some are more clever than the other. My one is particularly messy, but it does the job after some debugging and patience.

After we have the directed graph, we just need to do a topological sort on those nodes, and output one of the feasible linear arrangement. Here is a pseudo-code for doing a topological sort, for future references:

let S and T be stacks.
let G(V,E) be a directed graph.

topological_sort():
push all v in V that has 0 in-degree.
while S is not empty:
u = S.pop()
T.push(u)
for v adjacent to u in G(V,E):
decrease in-degree of v by 1.
if in-degree of v is 0:
S.push(v)
// T will be topologically sorted


Implementation:

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <string>
#include <vector>
#include <utility>
#include <queue>
#include <cmath>
#include <cassert>
#include <map>
#include <set>
using namespace std;

int s[103][103];
int g[103][103];
int indeg[30], vis[30];
int N;
vector<int> S, T;
int $= 26; int main(){ string t; scanf("%d",&N); int maxlen = 0; for(int i=0;i<N;++i){ cin >> t; for(int j=0;j<100;++j){ if(j<t.size()){ s[i][j] = t[j]-'a'; } else { s[i][j] =$;
}
}
maxlen = max(maxlen, (int) t.size());
}
adj = vector<vector<int> > (30);
for(int i=0;i<26;++i){
indeg[i]++;
}
for(int j=0;j<maxlen;++j){
int k = 0;
int prev = $; for(int i=0;i<N;++i){ if(prev != s[i][j]) { ++k; prev = s[i][j]; } else if(j != 0 && i != 0 && g[i-1][j-1] != g[i][j-1]) { ++k; } g[i][j] = k; } for(int i=1;i<N;++i){ if(g[i][j] != g[i-1][j]) { if(j==0 || g[i][j-1] == g[i-1][j-1]) { adj[s[i-1][j]].push_back(s[i][j]); //printf("%c => %c\n",s[i-1][j]+'a',s[i][j]+'a'); indeg[s[i][j]]++; } } } } bool not_empty = false; for(int i=0;i<=$;++i){
not_empty = true;
if(indeg[i] == 0) S.push_back(i);
}
if(S.empty() && not_empty) {
printf("Impossible\n");
return 0;
}
while(!S.empty()){
int u = S.back();
vis[u] = 1;
S.pop_back();
T.push_back(u);
int v = adj[u][i];
//printf("%c->%c\n",u+'a',v+'a');
indeg[v]--;
if(indeg[v]==0) S.push_back(v);
}
}
for(int i=0;i<=$;++i){ if(indeg[i]!=0){ //printf("%c\n",i+'a'); printf("Impossible\n"); return 0; } if(vis[i]==0) T.push_back(i); } for(int i=0;i<T.size();++i){ if(T[i]==$)continue;
printf("%c", T[i]+'a');
}
printf("\n");
return 0;
}