Problem Description

In this problem, you are provided with two arrays called spells and potions. The spells array contains elements representing the strength of each spell, and similarly, the potions array contains elements representing the strength of each potion. The task is to determine for each spell, how many potions can successfully combine with it to achieve or exceed a given success threshold. A spell and potion pair is deemed successful if the product (multiplication) of their strengths is at least equal to the success value provided. You should return an array where each value corresponds to the number of successful potion combinations for each spell in the spells array.

To clarify with an example, suppose spells = [10, 20], potions = [5, 8, 10], and success = 100. To find successful pairs, you multiply each spell with each potion:

• For the first spell (strength 10), it pairs successfully with two potions (potions of strength 10 and 8) since both products are at least 100.
• For the second spell (strength 20), it pairs successfully with all three potions, as all products will be 100 or greater. The result array will be [2, 3] as the first spell has 2 successful combinations, and the second has 3.

Intuition

The goal is to find an efficient way to pair spells with potions such that the product of their strengths is at least the success target. A brute-force approach would be to try all possible pairs, but this can be inefficient, especially with large arrays.

Instead, by first sorting the potions array, we can take advantage of binary search to quickly find the number of successful potions for each spell. Binary search operates by repeatedly dividing the search interval in half, which is much faster than scanning every element.

Once the potions array is sorted, for each spell, we want to find the position where the potion strength is just enough or more to meet the success target when paired with the spell. We need to find the smallest potion that, when multiplied by the spell's strength, equals or exceeds the success threshold. All potions to the right in the sorted array are guaranteed to also form successful pairs because they are equal or stronger.

The solution leverages the bisect_left function from Python's bisect module, which uses binary search to find the insertion point for a given element to maintain sorted order. In this context, it finds the first index in potions where the potion strength is sufficient for the success criteria with a given spell. We then subtract this index from the total number of potions (m) to get the count of successful potions for that spell.

For each spell in spells, we apply this logic and generate the final result array.

Learn more about Two Pointers, Binary Search and Sorting patterns.

Solution Approach

The solution implements a binary search mechanism to optimize the process of finding successful spell-potion pairs. Here's a step-wise explanation of the approach used in the solution:

1. Sorting the Potions: Begin by sorting the potions array. This is crucial for binary search to work since binary search requires a sorted array to function correctly. Sorting is done in ascending order.

2. Using Binary Search: For each spell's strength value, we apply a binary search to determine the number of potions that, when multiplied by this spell's strength, will at least be equal to the success value. This is done by finding the leftmost (smallest) potion index that can achieve the required target when combined with the current spell.

3. Calculating Successful Pairs:

   • We use the bisect_left function which finds the index at which we could insert the value success / spell_strength into potions to maintain sorted order.
   • The value we are searching for is calculated by success / spell_strength because we want to find the potion strength that, at the minimum, when multiplied by the spell's strength, is equal to success.
   • Since potions is sorted, every potion at and beyond the index returned by bisect_left would result in a successful pair when combined with the current spell.

4. Storing Results:

   • For each spell, we calculate m - index to find out the number of successful potions, where m is the total number of potions and index is the position returned by bisect_left.
   • This operation gives us the count because all elements from the position index to the end of the potions array will be successful pairs due to the sorted property of the array.

5. Generating the Output: Finally, we return a list comprehension that iterates over every spell in spells, applies the above logic using bisect_left, and calculates the number of successful pairs for that spell.

Here's the critical code snippet with explanations for clarity:

potions.sort() # Step 1: [Sorting](/problems/sorting_summary) the 'potions' array.
m = len(potions) # Storing the length of the 'potions' array.
# Step 2, 3, and 4: List comprehension iterating over 'spells'.
return [m - bisect_left(potions, success / v) for v in spells]

Through sorting and binary searching, this algorithm achieves a complexity of O(n log m), where n is the number of spells, and m is the number of potions. Sorting takes O(m log m), and then each binary search operation takes O(log m), with the linear iteration over spells representing the n in the complexity.

Example Walkthrough

Let's walk through a small example based on the solution approach given above.

Consider we have spells = [15, 10], potions = [1, 5, 20, 8], and success = 120. We aim to determine for each spell, how many potions achieve or exceed the success threshold when multiplied by the spell.

1. Sorting the Potions: The first step is sorting the potions array. Before: [1, 5, 20, 8] After sorting: [1, 5, 8, 20]

2. Using Binary Search: We then apply binary search to find the count of potions that can pair with a spell to achieve the success.

   • For spell 15: We search for the smallest potion that when multiplied is at least 120. That potion should be 120 / 15 = 8.
   • For spell 10: We need a potion of at least 120 / 10 = 12.

3. Calculating Successful Pairs:

   • For spell 15: Using bisect_left, we find the index where 8 could fit in the sorted potions array [1, 5, 8, 20], which is index 2.
   • For spell 10: Using bisect_left, we find the index where 12 could fit, which is after 8 and before 20. Hence, the index would be 3.

4. Storing Results:

   • For spell 15: m - index is 4 - 2 = 2. So, there are 2 successful potion combinations for the spell 15.
   • For spell 10: m - index is 4 - 3 = 1. Therefore, there is 1 successful potion combination for the spell 10.

5. Generating the Output: According to the steps described in the solution approach, we create a list with the counts of successful combinations for each spell. The result for our example will be [2, 1].

For better understanding, here's how the binary search part of the code would operate in a step-by-step manner:

• Spell is 15: bisect_left([1, 5, 8, 20], 120/15)

• It finds the position of 8 because 8 * 15 is exactly 120.

• index = 2, total potions m = 4, successful combinations: 4 - 2 = 2.

• Spell is 10: bisect_left([1, 5, 8, 20], 120/10)

• It passes over 8 (since 8 * 10 is less than 120) and settles before 20.

• index = 3, successful combinations: 4 - 3 = 1.

Thus, the final returned value for this particular example with the spells and potions given would be [2, 1]. Each spell has a corresponding count of potions that, when multiplied, meet or exceed the success threshold.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code snippet consists of two parts: sorting the potions list and performing binary searches using the bisect_left function.

1. Sorting: The potions.sort() operation has a time complexity of O(n log n) where n is the number of potions.
2. Binary Search: The binary search inside the list comprehension using bisect_left is performed for each spell. If there are s spells, and the binary search has a time complexity of O(log n) for each spell, the entire list comprehension has a time complexity of O(s log n).

Therefore, combining both operations, the total time complexity is O(n log n + s log n).

Space Complexity

The space complexity is determined by the additional space required beyond the input data.

1. Sorting Space: The sorting is done in place, which does not require additional space, so it's O(1).
2. Output List: There is a list comprehension that generates a list of the same length as the number of spells. Therefore, it has a space complexity of O(s) where s is the number of spells.

Given that O(s) is the larger term between O(1) and O(s), the overall space complexity is O(s).

Learn more about how to find time and space complexity quickly using problem constraints.