Fair Candy Swap

This solution works by finding the difference in total candies between Alice and Bob, then looking for a pair of boxes (one from Alice, one from Bob) such that swapping them would equalize the total candies.

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class Solution:
    def fairCandySwap(self, aliceSizes: List[int], bobSizes: List[int]) -> List[int]:
        diff = (sum(aliceSizes) - sum(bobSizes)) // 2
        aliceSizes = set(aliceSizes)  # Convert aliceSizes to set for faster lookup
        for candy in bobSizes:
            if diff + candy in aliceSizes:
                return [diff + candy, candy]

The function calculates the difference between the sum of Alice’s candies and Bob’s candies and stores it in diff. It then goes through each of Bob’s boxes and checks if Alice has a box such that swapping it with Bob’s box would equalize their total candies.

This solution uses set to store Alice’s candies for faster lookup (O(1) time complexity). This makes the overall solution more efficient, with a time complexity of O(n + m), where n and m are the sizes of aliceSizes and bobSizes respectively. The space complexity is also O(n + m) for storing the arrays and the set.

Choice of Datastructure

The set data type in Python is an unordered collection of unique elements. It is similar to lists and arrays, but with some distinct characteristics that make it suitable for certain scenarios. In this case, the set is used for its advantageous properties regarding membership tests.

When checking for the existence of an element, a set performs this operation in constant time, O(1), regardless of its size. This is because it uses a hash table under the hood, where elements are hashed to a certain position, and checking for existence only requires accessing that position.

On the other hand, if you were to use a list or an array for the same purpose, the time complexity would be O(n), where n is the size of the list or array. This is because, in the worst-case scenario, you would have to traverse the entire list to confirm that an element does not exist.

So, in this solution, converting aliceSizes to a set allows for faster lookups when the algorithm is checking if diff + candy exists in aliceSizes. This helps reduce the overall time complexity of the solution.

Identifying Problem Isomorphism

“Fair Candy Swap” can be mapped to “Two Sum”.

Both problems deal with finding two numbers, one from each of two different lists, that meet a certain condition. In “Fair Candy Swap”, we are looking for two numbers (boxes of candies) such that swapping them equalizes the total amount of candies that Alice and Bob have. In “Two Sum”, we’re looking for two numbers that add up to a specific target.

While the problems are not exactly identical, they share a common core pattern: iterate over elements from one list and look up corresponding elements in the other list.

“Two Sum” is simpler because it doesn’t involve an exchange between two different parties like “Fair Candy Swap”. However, both problems can be solved efficiently using a hash table or set data structure for quick lookups.

Next in the complexity scale is “4Sum”, where you have to find four numbers that add up to a specific target. This problem has additional complexity because of the higher dimensionality and requires more sophisticated techniques to solve efficiently.

Problem Type

The “Fair Candy Swap” is an array processing problem.

Simulation problems typically involve modeling a real-world process or system over time. They often require steps to be followed in a time-dependent sequence, and the state of the simulation at any given moment can depend on its state at previous time steps.

In the “Fair Candy Swap” problem, we’re not modeling a time-dependent process. Instead, we’re performing a mathematical calculation based on the elements of two arrays. We need to find two elements, one from each array, that meet a certain condition (their swap would equalize the sum of elements in both arrays). This is done by analyzing the whole array at once, not by sequentially moving through a series of time-dependent states.

The “Fair Candy Swap” problem is math-oriented. Specifically, it involves understanding and manipulating the sums of elements in two arrays to ensure the sums are equal after a single swap operation. This involves knowledge of arithmetic operations and understanding of how the sum of a set of numbers changes when one number is swapped with another. Furthermore, the usage of a set for lookup also invokes knowledge about data structures and their time complexities. Thus, while the problem involves elements of computer science (like data structures), at its core, it is a problem that can be solved using mathematical reasoning.

Key Takeaway

This problem is a good example of how to express the problem statement in an equation and solve the equation. Translate that into code.

The “Fair Candy Swap” problem is a great example of how to formulate a problem as an equation and then translate it into code.

Here’s how we can express the problem as an equation:

Let Sa be the total number of candies Alice has and Sb be the total number of candies Bob has. We know that Sa ≠ Sb. Let x be the number of candies in the box Alice gives to Bob and y be the number of candies in the box Bob gives to Alice. After the exchange, they should have the same total amount of candy, meaning Sa - x + y = Sb - y + x.

If we simplify the above equation, we get: y = x + (Sa - Sb) / 2

This equation says that for Alice and Bob to end up with the same amount of candies, the number of candies Bob gives to Alice (y) should be the number of candies Alice gives to Bob (x) plus half of the difference between their initial amounts of candies.

Then, the problem translates into finding such x and y that satisfy this equation. This equation guides the implementation of the solution in the code. We use a set to store all the candy boxes Bob has, and for each candy box in Alice’s collection, we check if there’s a box in Bob’s collection that satisfies the equation. If there is, we have found our solution. This formulation of the problem as an equation and its translation into code is a common pattern in solving algorithmic problems.