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Maximum Total Damage With Spell Casting

Medium

Citadel

4 views

Topics:

Dynamic ProgrammingArraysGreedy Algorithms

A magician has various spells.

You are given an array power, where each element represents the damage of a spell. Multiple spells can have the same damage value.

It is a known fact that if a magician decides to cast a spell with a damage of power[i], they cannot cast any spell with a damage of power[i] - 2, power[i] - 1, power[i] + 1, or power[i] + 2.

Each spell can be cast only once.

Return the maximum possible total damage that a magician can cast.

Example 1:

Input: power = [1,1,3,4]

Output: 6

Explanation:

The maximum possible damage of 6 is produced by casting spells 0, 1, 3 with damage 1, 1, 4.

Example 2:

Input: power = [7,1,6,6]

Output: 13

Explanation:

The maximum possible damage of 13 is produced by casting spells 1, 2, 3 with damage 1, 6, 6.

Constraints:

1 <= power.length <= 10⁵
1 <= power[i] <= 10⁹

Try coding it in LeetCode

Maximum Total Damage With Spell Casting

Medium

Citadel

4 views

Topics:

Dynamic ProgrammingArraysGreedy Algorithms

A magician has various spells.

You are given an array power, where each element represents the damage of a spell. Multiple spells can have the same damage value.

Each spell can be cast only once.

Return the maximum possible total damage that a magician can cast.

Example 1:

Input: power = [1,1,3,4]

Output: 6

Explanation:

The maximum possible damage of 6 is produced by casting spells 0, 1, 3 with damage 1, 1, 4.

Example 2:

Input: power = [7,1,6,6]

Output: 13

Explanation:

The maximum possible damage of 13 is produced by casting spells 1, 2, 3 with damage 1, 6, 6.

Constraints:

1 <= power.length <= 10⁵
1 <= power[i] <= 10⁹

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 are the constraints on the number of spells, their damage values, and the duration of the spell effects? Are there maximum or minimum values I should be aware of?
Can a spell's damage or duration be zero or negative?
If multiple combinations of spells yield the same maximum total damage, is any one of them acceptable, or is there a tie-breaking condition?
Is there a limit to the total number of spells I can cast, or can I cast all spells available?
If no spells are cast, should I return 0 or null? Is an empty spell list a possible input?

Brute Force Solution

Approach

The brute force approach to maximizing spell damage involves trying out every possible combination of spells and calculating the total damage for each. It's like testing every possible scenario to see which one gives you the highest damage output. This ensures we don't miss the optimal solution.

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

Consider using only the first spell.
Calculate the total damage if you only use that spell.
Now, consider using the first two spells in every possible combination (first then second, second then first).
Calculate the total damage for each of these combinations.
Continue this process, adding one more spell at a time and exploring all possible orders of the spells you're considering.
For each possible order of spells, calculate the total damage, taking into account any combo bonuses or special effects.
After you have considered every possible combination and order of spells, compare the total damage of each combination.
Select the combination that resulted in the highest total damage. This is your brute force solution.

Code Implementation

def maximum_total_damage_brute_force(spells, combo_bonus):	# Initialize the maximum damage to 0
	maximum_damage = 0

	# Iterate through all possible subsets of spells
	for i in range(1, 1 << len(spells)):
		subset = []
		for j in range(len(spells)):
			# Select spells for the current subset
			if (i >> j) & 1:
				subset.append(spells[j])

		# Generate all permutations of the subset
		import itertools
		for permutation in itertools.permutations(subset):
			current_damage = 0
			previous_spell_type = None

			# Calculate the damage for the current permutation
			for spell in permutation:
				current_damage += spell['damage']

				# Apply combo bonus if applicable
				if previous_spell_type == spell['type']:
					current_damage += combo_bonus
				previous_spell_type = spell['type']

			# Update the maximum damage if the current permutation's damage is higher
			maximum_damage = max(maximum_damage, current_damage)

	return maximum_damage

Big(O) Analysis

Time Complexity

O(n * n!) – The brute force approach involves exploring all possible permutations of spells to maximize total damage. With 'n' spells, there are n! (n factorial) possible orderings to consider. For each of these n! permutations, we need to calculate the total damage, which involves iterating through all 'n' spells in that specific order. Therefore, the time complexity is driven by generating all permutations (n!) and calculating damage for each (n), leading to a total of approximately n * n! operations. This results in a time complexity of O(n * n!).

Space Complexity

O(N!) – The brute force approach described explores all possible permutations of the N spells. While calculating the damage for each permutation, we need to store the current permutation of spells. In the worst case, where we consider all N spells, the space required to store a permutation can be O(N). However, because we explore every possible ordering (N! permutations), the space to hold these orderings while exploring could reach a space complexity related to N!. Specifically, recursive calls would build up on the stack to create these permutations. Thus, auxiliary space is largely dominated by the call stack for the permutations, leading to O(N!) space complexity.

Optimal Solution

Approach

The goal is to maximize the total damage you can inflict with a limited number of spells. We'll achieve this by strategically choosing which spells to cast and in what order, prioritizing the spells that give us the most damage considering their increasing cost as we cast more spells of the same type.

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

First, figure out how much damage each individual spell could do if you cast it first.
Next, figure out the order in which you should prioritize the spells. This order depends on how much damage each spell will do when compared to its initial cost.
Now, decide on how many of each type of spell you'll cast. You should keep casting a certain spell type as long as the increased damage from that spell is higher than casting other spells.
Calculate the total damage from the spells you've selected, taking into account that the cost of spells might increase as you cast more of the same spell type.
To double-check, see if you can switch any of your spells around to gain even more damage. Make sure that by swapping spells you do not exceed your available mana.

Code Implementation

def max_total_damage(base_damage, damage_increase, mana):
    number_of_spell_types = len(base_damage)
    spell_priority = []

    # Calculate initial damage per mana for each spell
    for i in range(number_of_spell_types):
        spell_priority.append((base_damage[i] / 1.0, i))

    # Sort spells by initial damage per mana in descending order
    spell_priority.sort(reverse=True)

    spell_counts = [0] * number_of_spell_types
    total_damage = 0
    current_mana_cost = 0

    # Greedily cast spells based on priority
    while True:
        best_spell_index = -1
        best_spell_damage_increase = -1

        for _, spell_index in spell_priority:
            next_spell_cost = base_damage[spell_index] + spell_counts[spell_index] * damage_increase[spell_index]

            # Find spell with highest damage for mana
            if current_mana_cost + next_spell_cost <= mana:
                potential_damage_increase = base_damage[spell_index] + spell_counts[spell_index] * damage_increase[spell_index]
                if potential_damage_increase > best_spell_damage_increase:
                    best_spell_damage_increase = potential_damage_increase
                    best_spell_index = spell_index

        # No spell can be cast within remaining mana
        if best_spell_index == -1:
            break

        spell_counts[best_spell_index] += 1
        current_mana_cost += base_damage[best_spell_index] + (spell_counts[best_spell_index] - 1) * damage_increase[best_spell_index]
        total_damage += base_damage[best_spell_index] + (spell_counts[best_spell_index] - 1) * damage_increase[best_spell_index]

    # Attempt to refine spell choices by swapping
    for i in range(number_of_spell_types):
        for j in range(number_of_spell_types):
            if spell_counts[i] > 0:
                # Consider swapping one i for one j
                mana_saved = base_damage[i] + (spell_counts[i] -1) * damage_increase[i]
                mana_needed = base_damage[j] + spell_counts[j] * damage_increase[j]

                if current_mana_cost - mana_saved + mana_needed <= mana:
                    damage_lost = base_damage[i] + (spell_counts[i] - 1) * damage_increase[i]
                    damage_gained = base_damage[j] + spell_counts[j] * damage_increase[j]
                    # Check damage delta before swap
                    if damage_gained > damage_lost:
                        spell_counts[i] -= 1
                        spell_counts[j] += 1
                        total_damage = total_damage - damage_lost + damage_gained
                        current_mana_cost = current_mana_cost - mana_saved + mana_needed

    return total_damage

Big(O) Analysis

Time Complexity

O(n log n) – The initial calculation of damage for each spell type involves iterating through the input list of spells, which takes O(n) time where n is the number of spells. Determining the priority order likely involves sorting the spells based on their initial damage-to-cost ratio, leading to O(n log n) complexity. The process of deciding how many of each spell to cast might involve iterating and comparing damage increases, but this is still bounded by n since it is looping through the different spell types. Therefore, the dominant factor becomes the sorting step resulting in O(n log n) time complexity.

Space Complexity

O(N) – The algorithm needs to store the potential damage each spell could do if cast first, implying an auxiliary array to hold these initial damage values. The size of this array is directly proportional to the number of unique spell types which we can denote as N. Therefore, the space complexity is O(N), where N is the number of different spell types.

Edge Cases

Case	How to Handle
Empty spells or damages array	Return 0 immediately since no spells can be cast.
Spells and damages arrays have different lengths	Handle by throwing an IllegalArgumentException or returning an error code, based on the API contract.
All spells have 0 power	Return 0 as the maximum total damage, as no damage amplification is possible.
All spells have the same power and all damages are the same	The sorting will still work correctly; return total sum after applying amplification.
Damages array contains negative numbers	Account for negative damage values by either excluding them or by handling them correctly in the damage calculation.
Large input arrays exceeding available memory	Optimize memory usage, potentially using an iterative approach with O(1) additional space where possible.
Spell powers are very large numbers leading to integer overflow during multiplication	Use long or BigInteger for damage calculations and comparisons to avoid integer overflow.
Zero spell power	Any damage value at this index will be unchanged, resulting in the base damage value.

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 are the constraints on the number of spells, their damage values, and the duration of the spell effects? Are there maximum or minimum values I should be aware of?
Can a spell's damage or duration be zero or negative?
If multiple combinations of spells yield the same maximum total damage, is any one of them acceptable, or is there a tie-breaking condition?
Is there a limit to the total number of spells I can cast, or can I cast all spells available?
If no spells are cast, should I return 0 or null? Is an empty spell list a possible input?

Brute Force Solution

Approach

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

Consider using only the first spell.
Calculate the total damage if you only use that spell.
Now, consider using the first two spells in every possible combination (first then second, second then first).
Calculate the total damage for each of these combinations.
Continue this process, adding one more spell at a time and exploring all possible orders of the spells you're considering.
For each possible order of spells, calculate the total damage, taking into account any combo bonuses or special effects.
After you have considered every possible combination and order of spells, compare the total damage of each combination.
Select the combination that resulted in the highest total damage. This is your brute force solution.

Code Implementation

def maximum_total_damage_brute_force(spells, combo_bonus):	# Initialize the maximum damage to 0
	maximum_damage = 0

	# Iterate through all possible subsets of spells
	for i in range(1, 1 << len(spells)):
		subset = []
		for j in range(len(spells)):
			# Select spells for the current subset
			if (i >> j) & 1:
				subset.append(spells[j])

		# Generate all permutations of the subset
		import itertools
		for permutation in itertools.permutations(subset):
			current_damage = 0
			previous_spell_type = None

			# Calculate the damage for the current permutation
			for spell in permutation:
				current_damage += spell['damage']

				# Apply combo bonus if applicable
				if previous_spell_type == spell['type']:
					current_damage += combo_bonus
				previous_spell_type = spell['type']

			# Update the maximum damage if the current permutation's damage is higher
			maximum_damage = max(maximum_damage, current_damage)

	return maximum_damage

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

First, figure out how much damage each individual spell could do if you cast it first.
Next, figure out the order in which you should prioritize the spells. This order depends on how much damage each spell will do when compared to its initial cost.
Now, decide on how many of each type of spell you'll cast. You should keep casting a certain spell type as long as the increased damage from that spell is higher than casting other spells.
Calculate the total damage from the spells you've selected, taking into account that the cost of spells might increase as you cast more of the same spell type.
To double-check, see if you can switch any of your spells around to gain even more damage. Make sure that by swapping spells you do not exceed your available mana.

Code Implementation

def max_total_damage(base_damage, damage_increase, mana):
    number_of_spell_types = len(base_damage)
    spell_priority = []

    # Calculate initial damage per mana for each spell
    for i in range(number_of_spell_types):
        spell_priority.append((base_damage[i] / 1.0, i))

    # Sort spells by initial damage per mana in descending order
    spell_priority.sort(reverse=True)

    spell_counts = [0] * number_of_spell_types
    total_damage = 0
    current_mana_cost = 0

    # Greedily cast spells based on priority
    while True:
        best_spell_index = -1
        best_spell_damage_increase = -1

        for _, spell_index in spell_priority:
            next_spell_cost = base_damage[spell_index] + spell_counts[spell_index] * damage_increase[spell_index]

            # Find spell with highest damage for mana
            if current_mana_cost + next_spell_cost <= mana:
                potential_damage_increase = base_damage[spell_index] + spell_counts[spell_index] * damage_increase[spell_index]
                if potential_damage_increase > best_spell_damage_increase:
                    best_spell_damage_increase = potential_damage_increase
                    best_spell_index = spell_index

        # No spell can be cast within remaining mana
        if best_spell_index == -1:
            break

        spell_counts[best_spell_index] += 1
        current_mana_cost += base_damage[best_spell_index] + (spell_counts[best_spell_index] - 1) * damage_increase[best_spell_index]
        total_damage += base_damage[best_spell_index] + (spell_counts[best_spell_index] - 1) * damage_increase[best_spell_index]

    # Attempt to refine spell choices by swapping
    for i in range(number_of_spell_types):
        for j in range(number_of_spell_types):
            if spell_counts[i] > 0:
                # Consider swapping one i for one j
                mana_saved = base_damage[i] + (spell_counts[i] -1) * damage_increase[i]
                mana_needed = base_damage[j] + spell_counts[j] * damage_increase[j]

                if current_mana_cost - mana_saved + mana_needed <= mana:
                    damage_lost = base_damage[i] + (spell_counts[i] - 1) * damage_increase[i]
                    damage_gained = base_damage[j] + spell_counts[j] * damage_increase[j]
                    # Check damage delta before swap
                    if damage_gained > damage_lost:
                        spell_counts[i] -= 1
                        spell_counts[j] += 1
                        total_damage = total_damage - damage_lost + damage_gained
                        current_mana_cost = current_mana_cost - mana_saved + mana_needed

    return total_damage

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty spells or damages array	Return 0 immediately since no spells can be cast.
Spells and damages arrays have different lengths	Handle by throwing an IllegalArgumentException or returning an error code, based on the API contract.
All spells have 0 power	Return 0 as the maximum total damage, as no damage amplification is possible.
All spells have the same power and all damages are the same	The sorting will still work correctly; return total sum after applying amplification.
Damages array contains negative numbers	Account for negative damage values by either excluding them or by handling them correctly in the damage calculation.
Large input arrays exceeding available memory	Optimize memory usage, potentially using an iterative approach with O(1) additional space where possible.
Spell powers are very large numbers leading to integer overflow during multiplication	Use long or BigInteger for damage calculations and comparisons to avoid integer overflow.
Zero spell power	Any damage value at this index will be unchanged, resulting in the base damage value.