(Analysis by Danny Mittal, Benjamin Qi)

The problem reduces to the following: we want to add the minimum number of bits to the Farm ID numbers so that no Farm ID number is a prefix of another. We can solve this problem greedily.

We maintain a partial assignment of Farm ID numbers to new bitstrings with some bits added so that, considering all the bitstrings currently in our assignment, none is a prefix of another.

The current assignment can be thought of as a binary tree where each node represents a bitstring -- the root represents the empty bitstring, and each node's left child represents that bitstring plus '0', while each node's right child represents that bitstring plus '1'.

The nodes that belong to Farm ID numbers in the assignment are leaves, because they can't have any descendants in the assignment (otherwise they would be a prefix of another Farm ID numbers in the assignment), while nodes with children represent bitstrings that can't be used because they are (and so will always be) a prefix of a current Farm ID number. Every node that isn't a leaf has two children, because even if one child isn't in use as a Farm ID number and doesn't have any descendants in use, that node (or its descendants) could potentially be used in the future.

All Farm ID numbers have length exactly $1$

All bitstrings in the input are either "0" or "1"; we cannot make a "0" bitstring the prefix of a "1" bitstring or vice versa by adding bits, so consider them separately. Below, we will instead assume that all bitstrings in the input are "".

The first Farm ID number that we see can just be left as is, and so will become the root of our tree. Each additional Farm ID number needs to be assigned to a new node in such a way that all Farm ID numbers in the assignment remain leaves. A way to do this is to take a current leaf, remove it from the assignment, then make its two children leaves, so that one child represents the bitstring of the old node which we added one bit to, and the other child represents the new bitstring that we were accounting for.

For the second Farm ID number we see, that is the only way to incorporate the Farm ID number, because the only node is the root, so there are no free leaves that we could use instead. Furthermore, because each leaf that becomes a nonleaf now has its two children used, there is never a free leaf to use, so the above way is the only way to incorporate new Farm ID numbers.

If the leaf that we make into a nonleaf was at depth $d$, then we need to add $1$ bit to push down the Farm ID number it represented to one of its children, and $d + 1$ bits to push down the new Farm ID number to its other child, adding $d + 2$ bits in total. This also means that we should greedily push down the nodes at smaller depths first.

The first Farm ID number is free. The second involves pushing down the root, at depth $0$, costing $0 + 2$ bits. After that, at depth $k$ we have $2^k$ nodes that can be pushed down at a cost of $k + 2$ bits. We can go through each depth until we've assigned all the Farm ID numbers, adding up the costs, to get our answer (for each of "0" and "1"). This runs in $O(N)$ (and can be made even faster using formulas).

$N \leq 10^2$ and the total length of Farm ID numbers $\leq 10^3$

The key observation here is that we should incorporate the longer Farm ID numbers first, because they have more limited options to choose from and so should be prioritized.

More rigorously, if we attained some assignment by first incorporating a shorter (at the beginning) Farm ID number, which becomes the bitstring $x$, then a longer (at the beginning) Farm ID number, which becomes the bitstring $y$, swapping the order of those incorporations could change nothing, or it could mean assigning the longer Farm ID number to $x$ and the shorter one to $y$, but that wouldn't change the cost, so we may as well consider the longer ones first.

The algorithm is thus to simply incorporate the Farm ID numbers in the input in order of decreasing length. A Farm ID number that is currently of length $l$ can be incorporated by pushing down an in use leaf at depth $d$ at a cost of $(d - l) + 2$, similarly to what was mentioned before, with the $- l$ accounting for the fact that we need to add less bits if the Farm ID number already has some of them. It can also be incorporated by using up a previously unused leaf at depth $d$ at a cost of $d - l$.

We simply act greedily and do whatever is cheapest. This works because the options that each choice opens up for later are always at least as expensive: if we use up a free leaf at depth $d$ for a cost of $d - l$, then we now have a used leaf at depth $d$ that will cost $(d - l) + 2$ to push down later on. If we push down a used leaf at depth $d$ at a cost of $(d - l) + 2$, then we now have two used leaves at depth $d + 1$ each at a cost of $(d + 1 - l) + 2$. This implies that what is currently the cheapest option will always be the cheapest option, so there is no reason to not just use it right now.

For each Farm ID number that we incorporate, we can find the cheapest option using BFS, starting from the node that represents the bitstring that the Farm ID is currently equal to, and going through its descendants in depth order. The first in-use leaf that we find, we save for later. If we ever find a free leaf, then we use that; otherwise, if we get to the point where the saved in-use leaf is the best option, which occurs if that leaf was at depth $d$ and we have reached $d + 2$, meaning that either way we pay at least $d - l + 2$, then we just use that.

To bound the runtime of this BFS, we will argue that the size of the tree at any point in time is bounded by the input. To see this, note that the depth of the node that a Farm ID number starts at is equal to its length, and the number of new nodes we add when inserting that Farm ID number initially is at most twice that depth: we at most add each of its ancestors, as well as a sibling for each of its ancestors to make sure that all nonleaves have two children. The only other time we add nodes is when we push down an in-use leaf: in this case we add two new nodes; since we do this at most once per Farm ID number, the number of new nodes is $2N$.

Therefore the size of the tree is at most $2(N + \sum \text{length})$, so each BFS takes $O(\sum \text{length})$ (because the sum of lengths is at least $N$). The overall runtime is therefore $O(N\sum \text{length})$, which is fast enough.

Note that we can analyze the runtime of the BFS in another way to get $O(N\lg N)$, yielding an $O(N^2\lg N)$ runtime.

Full solution

There are many ways to optimize the computation carried out in the previous subtask. Interestingly, we can make one simple change to the BFS to make it efficient.

A bad case for the original BFS is simply the first subtask: if we simply have a lot of "1" bitstrings, then each one (at least, the latter half) will have to BFS through a tree of size $O(N)$.

We can account for this by doing only one BFS for each distinct bitstring in the input. In order to assign nodes to multiple Farm ID numbers, we maintain two queues: one that performs normal BFS, finding leaves, and one that maintains in-use leafs to push down. At each step, if the normal queue is at depth $d_1$ and the second queue is at depth $d_2$, we perform one step of the first queue if $d_1 \leq d_2 + 2$ and of the second queue otherwise. Free leaves that we find in the first queue are used then added to the second queue, in-use leaves that we find in the first queue are just added to the second queue, and in-use leaves from the second queue are pushed down then their children added back.

To analyze this, we consider for each node how many BFSs it could show up in. A node can only be found in a BFS from one of its ancestors, so one bound on the total runtime of the BFSs is the sum of depths of all nodes in the end.

We can do slightly better: say that each Farm ID number in the input has a potential equal to $2l$, where $l$ is its length, that we can spread between nodes, and the goal is to place potential into nodes so that the amount of BFSs which see them is at most the amount of potential they have.

When we assign that Farm ID number, either we place it into a free leaf, in which case we just put $l$ potential into that leaf, or we push down an in-use leaf into two in-use leafs, in which case we place $l$ potential into each leaf. The total number of ancestors of the leaves in both cases could be slightly more than $l$, but because this is the first BFS in which we see that leaf, and we will only BFS from less deep nodes later on, the only nodes whose BFSs could see the new leaf/leaves are their $l$ ancestors that are higher than the node corresponding to the Farm ID number.

The placements of potentials therefore bounds the total number of BFSs that each node can be seen by, and so the total runtime of all BFSs, by $2\sum\text{length}$, and so this algorithm is $O(\sum\text{length})$, meaning that it is in fact linear in the length of the input.

Java code:

import java.io.BufferedReader;
import java.io.IOException;
import java.util.*;

public class PrefixFreeCodeBFS {

public static void main(String[] args) throws IOException {
int n = Integer.parseInt(in.readLine());
Trie root = new Trie(0);
List<Trie> need = new ArrayList<>();
for (int i = 0; i < n; i++) {
String s = in.readLine();
Trie node = root;
for (char bit : s.toCharArray()) {
node.status = Status.BLOCKED;
node = node.get(bit - '0');
}
if (node.needToAssign == 0) {
}
node.needToAssign++;
}

need.sort(Comparator.comparingInt(node -> -node.depth));
int answer = 0;
for (Trie source : need) {
while (source.needToAssign > 0) {
if (!expandable.isEmpty() && (q.isEmpty() || expandable.peek().depth + 2 <= q.peek().depth)) {
Trie node = expandable.remove();
source.needToAssign--;
node.status = Status.BLOCKED;
answer += node.depth - source.depth + 2;
for (int bit = 0; bit <= 1; bit++) {
node.get(bit).status = Status.IN_USE;
}
} else {
Trie node = q.remove();
if (node.status == Status.FREE) {
source.needToAssign--;
node.status = Status.IN_USE;
answer += node.depth - source.depth;
}
if (node.status == Status.BLOCKED) {
for (int bit = 0; bit <= 1; bit++) {
}
} else {
}
}
}
}
}

enum Status {
FREE, IN_USE, BLOCKED
}

static class Trie {
private Trie[] children;
public int depth;
public Status status;
public int needToAssign;

public Trie(int depth) {
this.children = new Trie[2];
this.depth = depth;
this.status = Status.FREE;
this.needToAssign = 0;
}

public Trie get(int bit) {
if (children[bit] == null) {
children[bit] = new Trie(depth + 1);
}
return children[bit];
}
}

}


C++ code:

//
// Created by Danny Mittal on 3/9/24.
//

#include <iostream>
#include <vector>
#include <queue>
#include <algorithm>

enum Status {
FREE, IN_USE, BLOCKED
};

class Trie {
public:
int depth;
Trie* children[2];
Status status;
int needToAssign;

Trie(int depth) : depth(depth), status(FREE), needToAssign(0) {
children[0] = nullptr;
children[1] = nullptr;
}

Trie* get(int bit) {
if (children[bit] == nullptr) {
children[bit] = new Trie(depth + 1);
}
return children[bit];
}
};

int main() {
int n;
std::cin >> n;
Trie root(0);
std::vector<Trie*> need;
for (int i = 0; i < n; ++i) {
std::string s;
std::cin >> s;
Trie* node = &root;
for (char bitChar : s) {
int bit = bitChar - '0';
node->status = BLOCKED;
node = node->get(bit);
}
if (node->needToAssign == 0) {
need.push_back(node);
}
node->needToAssign++;
}
int answer = 0;
std::sort(need.begin(), need.end(), [](Trie* a, Trie* b) {
return a->depth > b->depth;
});
for (Trie* source : need) {
std::queue<Trie*> q;
std::queue<Trie*> expandable;
q.push(source);
while (source->needToAssign > 0) {
if (!expandable.empty() && (q.empty() || expandable.front()->depth + 2 <= q.front()->depth)) {
Trie* node = expandable.front();
expandable.pop();
source->needToAssign--;
node->status = BLOCKED;
answer += node->depth - source->depth + 2;
for (int bit = 0; bit <= 1; ++bit) {
node->get(bit)->status = IN_USE;
expandable.push(node->get(bit));
}
} else {
Trie* node = q.front();
q.pop();
if (node->status == FREE) {
source->needToAssign--;
node->status = IN_USE;
answer += node->depth - source->depth;
}
if (node->status == BLOCKED) {
for (int bit = 0; bit <= 1; ++bit) {
q.push(node->get(bit));
}
} else {
expandable.push(node);
}
}
}
}
std::cout << answer << std::endl;
return 0;
}


Alternate Full Solution:

Another approach is to use dynamic programming on subtrees. For each prefix $p$ of any input string and integer $x\in [0,N]$, store the minimum time $dp[p][x]$ to solve the problem if we restrict consideration to strings starting with $p$, along with $x$ additional occurrences of $p$. To compute the answer for $p$ we can combine the answers for $p+0'$ and $p+1'$ in $O(N^2)$ time, giving an $O(\sum\text{length}\cdot N^2)$ solution.

To speed this up, we use the observation that $dp[p][x+2]-dp[p][x+1]\ge dp[p][x+1]-dp[p][x]$ to store $dp[p]$ in a compressed format: store only $dp[p][0]$, and for each $k\in [0, O(\log N))$, the number of $x$ such that $dp[p][x]-dp[p][x-1]=k$. Here, the upper bound on $k$ comes from the fact that we can always reach a leaf after adding $O(\log N)$ bits to $p$. We can merge the compressed forms for $dp[p + 0']$ and $dp[p+1']$ in $O(\log N)$ time.

The overall time and memory complexities are both $O(\sum\text{length}\cdot \log N)$. Depending on your implementation, you may run into the memory limit. Some ways to improve the memory usage are to:

1. Use fixed-length arrays instead of vectors.
2. Avoid using recursion.

Benjamin Qi's code:

#include <algorithm>
#include <array>
#include <cassert>
#include <iostream>
#include <numeric>
#include <vector>
using namespace std;

using ll = long long;
using vi = vector<int>;
int N;

vi operator+(const vi &l, const vi &r) {
vi res = l;
if (size(l) < size(r)) res.resize(size(r));
for (int i = 0; i < size(r); ++i) res.at(i) += r.at(i);
res.insert(begin(res), 0);
int sum = accumulate(begin(res), end(res), 0);
while (true) {
assert(size(res));
if (sum - res.back() < N) break;
sum -= res.back();
res.pop_back();
}
return res;
}

struct DP {
int init;
vi dp;
void advance(int x) {  // push x strings into subtree
for (int i = 0; i < size(dp); ++i) {
int cnt = min(x, dp[i]);
init += cnt * i;
x -= cnt;
dp[i] -= cnt;
}
assert(x == 0);
}
};

DP empty_dp() {  // if no strings starting with p
vi ans{1, 0};
int s = 1;
for (int i = 2; s < N; ++i) {
int x = 1 << (i - 2);
ans.push_back(x);
s += x;
}
return DP{0, ans};
}

DP operator+(const DP &l, const DP &r) {
return DP{l.init + r.init, l.dp + r.dp};
}

struct TrieNode {
int cnt;
array<int, 2> child;
DP dp;
};

vector<TrieNode> nodes(2);
int nxt = 1;

void fill(int idx) {
if (nodes.at(idx).child.at(0) == 0 && nodes.at(idx).child.at(1) == 0) {
nodes.at(idx).dp = nodes.at(0).dp;
} else {
nodes.at(idx).dp = nodes.at(nodes.at(idx).child.at(0)).dp +
nodes.at(nodes.at(idx).child.at(1)).dp;
}
}

int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
cin >> N;
for (int i = 0; i < N; ++i) {
string s;
cin >> s;
int cur = 1;
for (char c : s) {
if (!nodes.at(cur).child.at(c - '0')) {
nodes.at(cur).child.at(c - '0') = ++nxt;
nodes.emplace_back();
}
cur = nodes.at(cur).child.at(c - '0');
}
++nodes.at(cur).cnt;
}
nodes.at(0).dp = empty_dp();
for (int i = nxt; i >= 1; --i) fill(i);
cout << nodes.at(1).dp.init << "\n";
}


Note: This solution can be modified to solve the following extension:

For each Farm ID you are also provided with the number of times it occurs. The sum of the occurrence counts does not exceed $10^{15}$.