(Analysis by Nathan Pinsker)

This problem steers us towards doing a large number of string comparisons of substrings within a larger string. As a result, it's a good candidate for using a data structure called a suffix array, which is simply an alphabetically-sorted array of all suffixes in the string. Although we have multiple strings, we can concatenate them together into one large string consisting of all $N$ strings concatenated together with a special separator character (such as "?") between each one, so that we can compare suffixes across the $N$ strings that we're given.

Now that we have our sorted suffixes, we can efficiently compare them to one another. To solve this problem, we need to accomplish two things. First, for each suffix $S$, we must compute the longest prefix of $S$ that is shared by another suffix from a different original name, because that prefix doesn't contribute to the uniqueness factor. Second, we must count the number of strings in $S$ that *do* contribute to the uniqueness factor, but without double-counting any strings that occur twice in the same name.

Since our suffixes are alphabetically sorted, consider the suffixes to the direct left and the direct right of $S$. One of these suffixes must be the suffix (or one such suffix, at least) in $S$ that has the longest possible common prefix with $S$, because of the way we've constructed the suffix array. Similarly, consider the closest suffixes to $S$'s left and right which don't originate from the same name as $S$. Again, over the set of suffixes that aren't from $S$'s original name, one of these two strings must have the longest possible common prefix with $S$.

This insight allows us to solve the problem. We iterate over each suffix $S$ in the array, from left to right. Meanwhile, we keep track of suffixes $L$ and $R$, the closest suffixes to $S$'s left and right that don't originate from the same name as $S$. We compute the largest of the longest common prefix between $S$ and $L$, $S$ and $R$, and, if $S$'s immediate left neighbor is from the same string, $S$ and that neighbor (to handle the double-counting problem mentioned above). Finally, we compute the difference between the length of $S$ and this number. Collating all these results and bucketing them by the original names will give us the uniqueness factor for every cow.

This iterative procedure can be done in $O(1)$ per suffix, for a total of $O(|N|)$, although a bit of care is required to keep track of $L$ and $R$. The runtime is instead dominated by the construction of the suffix array, which is $O(|N| \lg |N|)$ at best. Since $|N| \leq 10^5$, this is easily fast enough for full credit.

Here is Mark's code:

#include <iostream>
#include <cstring>
#include <vector>
#include <algorithm>

using namespace std;

struct suffix_array {
suffix_array(const char* S) : N(strlen(S)) {
vector<int> V;
for(int i = 0; i < N; i++) V.push_back(S[i]);
init(V);
}

suffix_array(const vector<int>& VV) : N(VV.size()) {
vector<int> V(VV);
init(V);
}

int N;
vector<int> SA;
vector<int> RA;

void compress(vector<int>& V, vector<int>& C) {
copy(V.begin(), V.end(), C.begin());
sort(C.begin(), C.end());
auto cend = unique(C.begin(), C.end());
for(int i = 0; i < N; i++) {
V[i] = lower_bound(C.begin(), cend, V[i]) - C.begin() + 1;
}
V.push_back(0); C.push_back(0);
}

void compute_sa(vector<int>& V, vector<int>& C) {
vector<int> T(N + 1);
for(int i = 0; i <= N; i++) SA.push_back(i);
for(int ski = 0; V[SA[N]] < N; ski = ski ? ski << 1 : 1) {
fill(C.begin(), C.end(), 0);
for(int i = 0; i < ski; i++) T[i] = N - i;
for(int i = 0, p = ski; i <= N; i++) if(SA[i] >= ski) T[p++] = SA[i] - ski;
for(int i = 0; i <= N; i++) C[V[i]]++;
for(int i = 1; i <= N; i++) C[i] += C[i - 1];
for(int i = N; i >= 0; i--) SA[--C[V[T[i]]]] = T[i];

T[SA] = 0;
for(int j = 1; j <= N; j++) {
int a = SA[j];
int b = SA[j - 1];
T[a] = T[b] + (a + ski >= N || b + ski >= N ||
V[a] != V[b] || V[a + ski] != V[b + ski]);
}
V.swap(T);
}
}

void compute_lcp(const vector<int>& OV) {
LCP = vector<int>(N);
int len = 0;
for(int i = 0; i < N; i++, len = max(0, len - 1)) {
int si = RA[i];
int j = SA[si - 1];
for(; i + len < N && j + len < N && OV[i + len] == OV[j + len]; len++);
LCP[si - 1] = len;
}
}

void init(vector<int>& V) {
vector<int> OV(V);
vector<int> C(N);
compress(V, C);
compute_sa(V, C);
RA.resize(N + 1);
for(int i = 0; i <= N; i++) RA[SA[i]] = i;
compute_lcp(OV);
}

vector<int> LCP;
};

int main() {
ios_base::sync_with_stdio(false);

string S;
vector<pair<int, int> > A;

int N; cin >> N;
for (int i = 0; i < N; i++) {
string T; cin >> T;

S += T;
S += "?";
for (int j = 0; j < T.size(); j++) {
A.push_back(make_pair(i, T.size() - j));
}
A.push_back(make_pair(-1, -1));
}
A.push_back(make_pair(-1, -1));

vector<int> result(N);
suffix_array sa(S.c_str());
sa.LCP.push_back(0);
for (int i = 1; i <= sa.N; ) {
int j = sa.SA[i];
int ind = A[j].first;
if (ind == -1) {
++i;
continue;
}
int sz = 1;
while (i + sz <= sa.N && A[sa.SA[i + sz]].first == ind) {
++sz;
}

int ln = sa.LCP[i - 1];
for (int j = i; j < i + sz; j++) {
result[ind] += max(A[sa.SA[j]].second - max(ln, sa.LCP[j]), 0);
ln = min(ln, sa.LCP[j]);
}
i += sz;
}
for (int x : result) {
cout << x << '\n';
}

return 0;
}