Divide Players Into Teams of Equal Skill

class Solution {
public:
    long long splitTeams(vector<int>& abilities) {
        int count = abilities.size();
        int totalAbilities = 0;
        unordered_map<int, int> freqMap;

        // Accumulate skills and build frequency dictionary
        for (int ability : abilities) {
            totalAbilities += ability;
            freqMap[ability]++;
        }

        // Verify if abilities can be divided evenly between two teams
        if (totalAbilities % (count / 2) != 0) {
            return -1;
        }

        int requiredSkill = totalAbilities / (count / 2);
        long long combinedSkill = 0;

        // Loop through each unique skill entry
        for (auto& [skill, freq] : freqMap) {
            int neededSkill = requiredSkill - skill;

            // Ensure there's a matching skill with the correct frequency
            if (freqMap.find(neededSkill) == freqMap.end() || freqMap[neededSkill] != freq) {
                return -1;
            }

            // Sum up the product of skill pairs and their frequencies
            combinedSkill += (long long)skill * (long long)neededSkill * (long long)freq;
        }

        // Result must be halved as each combination is considered twice
        return combinedSkill / 2;
    }
};

The solution for splitting players into teams with equal skill levels involves calculating the total abilities of all players using a frequency map to track the number of times each skill level appears. Follow this step-by-step breakdown:

1. First, calculate the total sum of abilities and the frequency of each ability.

2. Check if the total abilities can be evenly divided between two teams. If not, return -1 indicating it's not possible.

3. Determine the required skill per team by dividing the total abilities by half the number of players.

4. Loop through each unique skill value:

   • Calculate the needed skill to match the current skill to form a balanced pair.
   • Verify if the corresponding skill exists with the correct frequency. If there's a mismatch, return -1 as an equal split is not possible.
   • Calculate the combined skill product, adjusting for frequency.

5. Return half the computed combined skill value to account for duplicate combinations considered in the calculations.

This approach allows you to determine if an even distribution of players based on their abilities is feasible, and if so, calculates the possible combinations. This C++ solution efficiently manages the skill distributions using map operations and ensures that the entire array of player skills is used optimally.

class Solution {

    public long teamDivision(int[] abilities) {
        int count = abilities.length;
        int sumOfAbilities = 0;
        Map<Integer, Integer> abilityFrequency = new HashMap<>();

        // Sum abilities and map their frequencies
        for (int ability : abilities) {
            sumOfAbilities += ability;
            abilityFrequency.put(ability, abilityFrequency.getOrDefault(ability, 0) + 1);
        }

        // Check for even distribution of abilities
        if (sumOfAbilities % (count / 2) != 0) {
            return -1;
        }

        int neededAbility = sumOfAbilities / (count / 2);
        long totalTeamValue = 0;

        // Process each unique ability
        for (int thisAbility : abilityFrequency.keySet()) {
            int thisFreq = abilityFrequency.get(thisAbility);
            int requiredComplement = neededAbility - thisAbility;

            // Verify that the complement ability exists and has the correct frequency
            if (!abilityFrequency.containsKey(requiredComplement) ||
                thisFreq != abilityFrequency.get(requiredComplement)) {
                return -1;
            }

            // Assign team value based on present abilities
            totalTeamValue += (long) thisAbility * (long) requiredComplement * (long) thisFreq;
        }

        // Each pair is considered twice, so divide by 2
        return totalTeamValue / 2;
    }
}

The given Java program implements a method, teamDivision, which takes an array of integers representing abilities. The goal is to distribute players into teams where the total abilities are balanced.

Step-by-step explanation of the code:

1. Initialize variables for counting total abilities and mapping each ability's frequency using a HashMap.

2. Loop over each ability to calculate the sum and update the frequency map.

3. Check if the sum of abilities can be evenly divided among half the players. If not, return -1 indicating that an even distribution is not possible.

4. Define the neededAbility as the target sum for each half team.

5. Iterate through each unique ability:

   • Fetch the frequency of the current ability.
   • Calculate the required complement ability to achieve the target sum with the current ability.
   • Validate the existence and correct frequency of the complement ability. If the check fails, return -1.

6. If a valid complement is found, compute the cumulative product of the current ability, its complement, and its frequency to get the total team contribution for these abilities.

7. After processing all abilities, adjust for double-counting by dividing the total team value by 2.

Finally, the method returns the total team value, or -1 if teams of equal skill cannot be formed. This solution efficiently combines hashing and arithmetic operations to balance out the team abilities.

class Solution:
    def separatePlayers(self, abilities: List[int]) -> int:
        total_players = len(abilities)
        total_abilities = sum(abilities)

        # Validate if abilities are evenly distributable
        if total_abilities % (total_players // 2) != 0:
            return -1

        desired_team_ability = total_abilities // (total_players // 2)
        ability_counts = Counter(abilities)
        total_teamwork = 0

        # Traverse through each unique ability
        for ability, frequency in ability_counts.items():
            required_partner = desired_team_ability - ability

            # Ensure the pair ability exists and frequencies match
            if (
                required_partner not in ability_counts
                or frequency != ability_counts[required_partner]
            ):
                return -1

            # Sum up contributions from all valid pairs
            total_teamwork += ability * required_partner * frequency

        # Since each pair is considered twice, divide the result by two
        return total_teamwork // 2

This solution provides a method to divide players into teams based on their skills, ensuring the teams are as equally skilled as possible. The Python function separatePlayers receives a list of player abilities and aims to distribute them into two teams with equal total skills.

Follow these steps in the function:

1. Calculate the total_players, the number of entries in the abilities list, and total_abilities, the sum of all skill levels.
2. Check if total_abilities is divisible by half the number of players to ensure even distribution. If not, return -1, indicating it's not possible to form the teams fairly.
3. Assuming distribution is possible, compute the desired_team_ability, which is the total abilities divided by half the number of players. This value represents the skill sum each team should achieve.
4. Utilize a Counter to track the frequency of each unique ability in the list.
5. Initialize a counter (total_teamwork) for the skill pairings within teams.
6. Loop through each unique ability in the Counter dictionary:
   • Determine the required_partner skill to satisfy the balance for a given skill level.
   • Check if a skill pair exists in the ability_counts with the needed frequency. If a suitable skill pair is absent, return -1.
   • Update total_teamwork to include the product of the ability, its required partner, and their frequency.
7. Return total_teamwork divided by 2 to avoid double-counting pairs, as the accumulation happens twice for each valid pairing.

This approach ensures that it attempts to verify each potential skill pairing and calculates the teamwork score only when the pairings correctly align across the list, thus reflecting accurate and fair team distribution based on the provided abilities.