Minimum Hours of Training to Win a Competition

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You are entering a competition, and are given two positive integers initialEnergy and initialExperience denoting your initial energy and initial experience respectively.

You are also given two 0-indexed integer arrays energy and experience, both of length n.

You will face n opponents in order. The energy and experience of the i^th opponent is denoted by energy[i] and experience[i] respectively. When you face an opponent, you need to have both strictly greater experience and energy to defeat them and move to the next opponent if available.

Defeating the i^th opponent increases your experience by experience[i], but decreases your energy by energy[i].

Before starting the competition, you can train for some number of hours. After each hour of training, you can either choose to increase your initial experience by one, or increase your initial energy by one.

Return the minimum number of training hours required to defeat all n opponents.

Example 1:

Input: initialEnergy = 5, initialExperience = 3, energy = [1,4,3,2], experience = [2,6,3,1]
Output: 8
Explanation: You can increase your energy to 11 after 6 hours of training, and your experience to 5 after 2 hours of training.
You face the opponents in the following order:
- You have more energy and experience than the 0^th opponent so you win.
  Your energy becomes 11 - 1 = 10, and your experience becomes 5 + 2 = 7.
- You have more energy and experience than the 1^st opponent so you win.
  Your energy becomes 10 - 4 = 6, and your experience becomes 7 + 6 = 13.
- You have more energy and experience than the 2^nd opponent so you win.
  Your energy becomes 6 - 3 = 3, and your experience becomes 13 + 3 = 16.
- You have more energy and experience than the 3^rd opponent so you win.
  Your energy becomes 3 - 2 = 1, and your experience becomes 16 + 1 = 17.
You did a total of 6 + 2 = 8 hours of training before the competition, so we return 8.
It can be proven that no smaller answer exists.

Example 2:

Input: initialEnergy = 2, initialExperience = 4, energy = [1], experience = [3]
Output: 0
Explanation: You do not need any additional energy or experience to win the competition, so we return 0.

Constraints:

n == energy.length == experience.length
1 <= n <= 100
1 <= initialEnergy, initialExperience, energy[i], experience[i] <= 100

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the range of possible values for the hours needed for each skill?
Can the initial skill levels or required skill levels be zero?
Is it possible for the required skill level to be less than the initial skill level, and if so, what should I return?
If it's impossible to win the competition (i.e., no amount of training can meet the required skill levels), what should I return?
Are the input arrays guaranteed to be of the same length?

Brute Force Solution

Approach

The brute force strategy explores every possible training schedule to find the one that requires the fewest hours to win. We will try all combinations of additional training hours for each skill to see if a winning scenario can be reached.

Here's how the algorithm would work step-by-step:

Start by considering no extra training at all. Check if the initial skill levels are already enough to win the competition.
If not, try adding one hour of training to the first skill. Then check if this is enough to win. If not, try two hours, and so on, until you've exhausted a reasonable upper limit of training hours for that skill.
Next, reset the training for the first skill to zero. Now, try adding one hour of training to the second skill, and then check all the possible training hours for the first skill again, like in the first step.
Repeat this process for each skill, increasing its training hours one at a time and checking all combinations of training hours for the skills that came before it.
For every possible combination of training hours across all skills, determine if that specific training schedule leads to winning the competition.
Keep track of the total training hours needed for each winning schedule that you find.
Finally, from all the winning schedules, choose the one that requires the smallest total number of training hours. That's your answer.

Code Implementation

def minimum_hours_of_training(initial_skills, required_skills):
    minimum_training_hours = float('inf')
    number_of_skills = len(initial_skills)

    # Iterate through all possible training hour combinations
    training_hours = [0] * number_of_skills

    while True:
        total_training_hours = sum(training_hours)
        current_skills = [initial_skills[i] + training_hours[i] for i in range(number_of_skills)]

        # Check if current training hours lead to a winning scenario
        if all(current_skills[i] >= required_skills[i] for i in range(number_of_skills)):

            # Update minimum if current combination is better
            minimum_training_hours = min(minimum_training_hours, total_training_hours)

        # Increment training hours for the first skill
        skill_index = 0
        while skill_index < number_of_skills:
            training_hours[skill_index] += 1
            if training_hours[skill_index] <= max(required_skills):
                break
            else:

                # Reset current skill and increment the next
                training_hours[skill_index] = 0
                skill_index += 1

        # All combinations exhausted
        if skill_index == number_of_skills:
            break

    # If no winning schedule is found, return -1
    if minimum_training_hours == float('inf'):
        return -1
    return minimum_training_hours

Big(O) Analysis

Time Complexity

O(m^n) – The brute force solution explores all possible training hour combinations for each skill. Let 'n' be the number of skills and 'm' be the maximum number of training hours considered for each skill. We are essentially iterating through 'm' possibilities for each of the 'n' skills. This results in a time complexity of O(m * m * ... * m) (n times), which is O(m^n). Since the number of combinations grows exponentially with the number of skills, 'n', the solution is highly inefficient.

Space Complexity

O(1) – The provided brute force strategy explores training schedules without explicitly storing all combinations. It iteratively increments training hours for each skill and checks for a winning scenario. The algorithm keeps track of the minimum training hours found so far and the current training hours for each skill, which are constant-size variables. Therefore, the auxiliary space required remains constant, irrespective of the number of skills or maximum training hours considered, resulting in O(1) space complexity.

Optimal Solution

Approach

The goal is to figure out the fewest hours needed to reach specific skill levels by training and competing. We can efficiently determine this by first sorting the training options and then checking if using the best training options helps us achieve our goal. We repeat this check until we find the absolute minimum training hours that work.

Here's how the algorithm would work step-by-step:

First, sort the training options from easiest to hardest (least to most hours).
Then, calculate the total skill we need to gain to reach the required level.
Now, start with the easiest training option and see if using only it can get us to the required skill level within the allowed hours.
If not, try combining the easiest option with the next easiest, and so on, always aiming to use the most efficient training options first.
Keep adding training options in increasing order of difficulty until you find a combination that allows you to reach the required skill level within the available hours.
Also, consider using competitions to increase skill, prioritizing those that give the highest skill increase. Integrate this into the process above to further reduce the training hours needed.
To find the absolute minimum hours, you might need to make small adjustments by swapping slightly less efficient training for more competition wins, or vice versa, to achieve a lower total time, while still meeting the skill requirement.

Code Implementation

def min_training_hours(initial_skill, required_skill, training_hours, skill_increase, competition_skill):
    training_options = sorted(zip(training_hours, skill_increase))
    number_of_training_options = len(training_options)
    total_skill_needed = required_skill - initial_skill

    if total_skill_needed <= 0:
        return 0

    min_hours = float('inf')

    for i in range(1 << number_of_training_options):
        current_skill = 0
        current_hours = 0
        training_indices = []

        for j in range(number_of_training_options):
            if (i >> j) & 1:
                training_indices.append(j)

        for index in training_indices:
            current_hours += training_options[index][0]
            current_skill += training_options[index][1]

        #Calculate competitions needed after training
        if current_skill < total_skill_needed:
            competitions_needed = (total_skill_needed - current_skill + competition_skill - 1) // competition_skill
        else:
            competitions_needed = 0

        #Here, ensure training+comp skill meets reqs
        if current_skill + competitions_needed * competition_skill >= total_skill_needed:

            # Now we calculate total time including competitions
            total_time = current_hours + competitions_needed
            min_hours = min(min_hours, total_time)

    if min_hours == float('inf'):
        return -1
    else:
        return min_hours

Big(O) Analysis

Time Complexity

O(n log n) – The dominant factor in the time complexity stems from sorting the training options in step 1, which uses a comparison-based sorting algorithm like merge sort or quicksort, contributing O(n log n) where n is the number of training options. Steps 2-7 iterate through the training options to find the optimal combination, but the sorting dominates. While there might be some nested iterations within step 7 to fine-tune the solution, those iterations are bounded by n and do not increase the overall complexity beyond O(n log n) since sorting is performed first.

Space Complexity

O(N) – The provided solution involves sorting the training options which typically requires O(N) auxiliary space, where N is the number of training options, depending on the sorting algorithm used (e.g., merge sort). Furthermore, the process of combining training options and competitions might involve creating temporary lists to store combinations, potentially scaling linearly with N in the worst-case scenario where N represents both the number of training options and competitions. Therefore, the dominant factor for space complexity is likely the temporary data structures needed for the sorting or combination steps. Thus, the space complexity is O(N).

Edge Cases

Case	How to Handle
Empty skills array or empty required skills array	Return 0 immediately, as no training is needed with no skills to improve or requirements to meet.
Skills array contains only 0s and required skills contains only 0s	Return 0, as the participant already meets the trivial requirements.
Skills array is significantly smaller than the required skills array, requiring large training hours	The algorithm should handle potentially large differences between initial and required skill levels without integer overflow.
Required skills are all much higher than the initial skills, approaching integer limits	Handle integer overflow issues when calculating training hours, potentially using larger data types.
Skills array contains negative values (invalid input)	Throw an IllegalArgumentException or return -1 to indicate an invalid input state.
Required skills array contains negative values (invalid input)	Throw an IllegalArgumentException or return -1 to indicate an invalid input state.
Skills array values are very large numbers, close to the maximum integer value	Ensure calculations of training hours don't cause integer overflow, possibly using long data type.
No training is needed as initial skills already meet or exceed required skills	Return 0, as the athlete is already prepared.

Empty skills array or empty required skills array

How to Handle:

Return 0 immediately, as no training is needed with no skills to improve or requirements to meet.

Skills array contains only 0s and required skills contains only 0s

How to Handle:

Return 0, as the participant already meets the trivial requirements.

Skills array is significantly smaller than the required skills array, requiring large training hours

How to Handle:

The algorithm should handle potentially large differences between initial and required skill levels without integer overflow.

Required skills are all much higher than the initial skills, approaching integer limits

How to Handle:

Handle integer overflow issues when calculating training hours, potentially using larger data types.

Skills array contains negative values (invalid input)

How to Handle:

Throw an IllegalArgumentException or return -1 to indicate an invalid input state.

Required skills array contains negative values (invalid input)

How to Handle:

Throw an IllegalArgumentException or return -1 to indicate an invalid input state.

Skills array values are very large numbers, close to the maximum integer value

How to Handle:

Ensure calculations of training hours don't cause integer overflow, possibly using long data type.

No training is needed as initial skills already meet or exceed required skills

How to Handle:

Return 0, as the athlete is already prepared.