The objective is to assemble the longest possible concatenated string from a provided vocabulary list, beginning with a specified initial character. Each vocabulary entry may be utilized a maximum of two times during chain construction. When connecting two words, the trailing substring of the first term must perfectly align with the leading substring of the second. This shared segment is merged rather than duplicated. Adjacent entries must not exhibit a containment relationship, meaning the overlapping portion must be strictly shorter than both participating words. Given a dataset size capped at twenty entries, an exhaustive depth-first search (DFS) with state restoration is the optimal approach.

Overlap Determination

The core mechanism relies on identifying the shortest valid intersection between two strings. A helper routine scans possible overlap lengths starting from one character up to the minimum length of both terms. For each candidate length, it verifies character-by-character equality between the suffix of the current term and the prefix of the candidate next term. The loop terminates immediately upon finding the first match, ensuring the minimal shared segment is selected to maximize total concatenation length. Overlaps equal to the full length of either string are explicitly rejected to prevent containment.

Backtracking Strategy

A recursive traversal explores all valid permutations. The algorithm maintains a frequency tracker for each vocabulary entry. Upon entering a recursive state, the current accumulated length is compared against a global maximum. The traversal then iterates through the entire word list, skipping entries that have already been utilized twice. For each eligible candidate, the overlap calculation is performed. If a valid intersection exists, the candidate's usage count increments, and the function recurses with the updated chain length. Following the recursive return, the usage counter decrements to restore the previous state, enabling exploration of alternative branching paths.

Implementation Details

The initialization phase reads the vocabulary size, populates the storage array, and captures the mandatory starting character. Instead of injecting a dummy node, the solver direct iterates through the dataset to locate all entries commencing with the specified character. Each valid starting point triggers the primary recursive routine, after which the global maximum length is printed. The use of standard library utilities for I/O synchronization ensures efficient execution within the constrained limits.

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>

using namespace std;

// Global state management
int vocabulary_size;
vector<string> lexicon;
vector<int> usage_counter;
int maximal_length = 0;

// Computes the shortest valid suffix-prefix intersection
// Returns 0 if no valid overlap exists or if inclusion occurs
int compute_overlap(const string& base, const string& candidate) {
    int boundary = min(base.length(), candidate.length());
    for (int len = 1; len < boundary; ++len) {
        bool is_match = true;
        for (int offset = 0; offset < len; ++offset) {
            if (base[base.length() - len + offset] != candidate[offset]) {
                is_match = false;
                break;
            }
        }
        if (is_match) {
            return len;
        }
    }
    return 0;
}

// Depth-first traversal to explore all valid concatenation paths
void explore_chain(const string& current_term, int current_total_len) {
    if (current_total_len > maximal_length) {
        maximal_length = current_total_len;
    }

    for (int idx = 0; idx < vocabulary_size; ++idx) {
        if (usage_counter[idx] >= 2) {
            continue;
        }

        int shared_segment = compute_overlap(current_term, lexicon[idx]);
        if (shared_segment > 0) {
            usage_counter[idx]++;
            explore_chain(lexicon[idx], current_total_len + lexicon[idx].length() - shared_segment);
            usage_counter[idx]--; // Backtrack state restoration
        }
    }
}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    if (!(cin >> vocabulary_size)) return 0;
    
    lexicon.resize(vocabulary_size);
    usage_counter.assign(vocabulary_size, 0);

    for (int i = 0; i < vocabulary_size; ++i) {
        cin >> lexicon[i];
    }

    char starting_char;
    cin >> starting_char;

    // Initiate search from all words beginning with the required character
    for (int i = 0; i < vocabulary_size; ++i) {
        if (lexicon[i][0] == starting_char) {
            usage_counter[i] = 1;
            explore_chain(lexicon[i], lexicon[i].length());
            usage_counter[i] = 0;
        }
    }

    cout << maximal_length << '\n';
    return 0;
}

Complexity Considerations

The time complexity hinges on the branching factor of valid overlaps and the maximum depth of the recursion tree. With a vocabulary limit of twenty and a strict usage cap of two per entry, the effective search space remains tractable despite the exponential nature of the traversal. The overlap verification operates in linear time relative to the string lengths, which are typically small in competitive programing contexts. Space complexity is linear, dominated by the recursion stack depth and storage for the input lexicon.