765. Couples Holding Hands

Problem Description

In the given problem, we are presented with n couples (making a total of 2n individuals) arranged in a row of 2n seats. Each couple is represented by a pair of consecutive numbers, such as (0, 1), (2, 3), and so on up to (2n - 2, 2n - 1). The arrangement of the individuals is provided in the form of an integer array called row, where row[i] is the ID number of the person sitting in the ith seat.

The goal is to make every couple sit next to each other with the least amount of swapping actions. A swap action is defined as taking any two individuals and having them switch seats with each other.

The challenge is to calculate the minimum number of swaps required to arrange all couples side by side.

Intuition

The intuition behind the solution is to treat the problem as a graph problem where each couple pairing is a connected component. We can build this conceptual graph by considering each couple as a node and each incorrect pair as an edge connecting the nodes. The size of a connected component then represents the number of couples in a mixed group that need to be resolved.

The key insight is to observe that within a mixed group, when you arrange one pair properly, it will automatically reduce the number of incorrect pairs by one since you reduce the group size. It's akin to untangling a set of connected strings; each time you properly untangle a string from the group, the complexity decreases.

We perform union-find operations to group the couples into connected components. The "find" operation is used to determine the representative (or the root) of a given couple, while the "union" (in this case, an assignment in the loop) combines two different couple nodes into the same connected component. This way, each swap connects two previously separate components, thereby reducing the total number of components. The answer to the minimum number of swaps is the initial number of couples n minus the number of connected components after the union-find process.

The reasoning is that for each connected component that is properly arranged, only (size of the component - 1) swaps are needed (consider that when you position the last couple correctly, no swap is needed for them).

The minSwapsCouples function thus iterates over the row array, pairing individuals with their partners by dividing their ID by 2 (since couples have consecutive numbers), and uses union-find to determine the minimum number of swaps required.

Learn more about Greedy, Depth-First Search, Breadth-First Search, Union Find and Graph patterns.

Solution Approach

The implementation of the solution approach for the problem of arranging couples uses the Union-Find algorithm, which is a classic algorithm in computer science used to keep track of elements which are split into one or more disjoint sets. Its primary operations are "find", which finds the representative of a set, and "union", which merges two sets into one.

Here's a step-by-step explanation of how the implementation works:

1. Initialization: First, we initialize a list p which will represent the parent (or root) of each set. Initially, each set has only one element, so the parent of each set is the element itself. In our case, each set is a couple, and there are n couples indexed from 0 to n-1. Hence, p is initialized with range n.

   1n = len(row) >> 1
   2p = list(range(n))

2. Finding the root: The find function is a recursive function that finds the representative of a given element x. The representative (or parent) of a set is the common element shared by all elements in the set. The find function compresses the path, setting the p[x] to the root of x for faster future lookups.

   1def find(x: int) -> int:
   2    if p[x] != x:
   3        p[x] = find(p[x])
   4    return p[x]

3. Merging sets: As we iterate through the row, we take two adjacent people row[i] and row[i + 1]. We find the set representative (root) of their couple indices, which are row[i] >> 1 and row[i + 1] >> 1 (bitwise right shift operator to divide by 2). We then merge these sets by assigning one representative to another. This is an application of the "union" operation.

   1for i in range(0, len(row), 2):
   2    a, b = row[i] >> 1, row[i + 1] >> 1
   3    p[find(a)] = find(b)

4. Count minimum swaps: Finally, the number of swaps needed equals to the total number of couples n minus the number of sets that represent correctly paired couples. This calculation is done by counting the number of times i is the representative of the set it belongs to (i == find(i)). This line of code effectively counts the number of connected components after union-find.

   1return n - sum(i == find(i) for i in range(n))

Mathematically, each set needs size of the set - 1 swaps to arrange the couple properly. The sum hence gives us a total of how many sets were already properly paired initially, by subtracting this sum from n we get the total number of swaps.

This approach is efficient because Union-Find has nearly constant time (amortized) complexity for both union and find operations, which allows us to solve the problem in almost linear time relative to the number of people/couples.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have n = 2 couples, so a total of 4 individuals. Our row array representing the seating arrangement is: [1, 0, 3, 2]. This means the first person (1) is paired with the second person (0), and the third person (3) is paired with the fourth person (2). But we want to arrange them such that person 0 is next to 1 and 2 is next to 3.

Using the mentioned approach, let's walk through the steps:

Step 1: Initialization We know n = 2. Initialize parent list p with the indices representing the couples (0-based).

1n = 2
2p = [0, 1]

Step 2: Finding the root The find function will determine the representative of each set. As we haven't made any unions yet, each set's representative is the set itself.

Step 3: Merging sets The iteration goes as follows:

• First pair is (1, 0). Their couple indices are 1 >> 1 = 0 and 0 >> 1 = 0. Since both are the same, no union is needed.
• Second pair is (3, 2). Their couple indices are 3 >> 1 = 1 and 2 >> 1 = 1. Again, no union is needed.

Now comes the merging by using the union operation:

• We set the representative of the first person's couple to be the same as the second person's couple: p[find(0)] = find(1).
• However, both are already correct as p[0] is 0 and find(1) is 1.
• No changes occur to p, it remains [0, 1].

Step 4: Count minimum swaps We have n = 2 couples, we go through the parent list p and count how many times i is equal to find(i).

• For i = 0, find(0) returns 0, so there's 1 correct pair.
• For i = 1, find(1) returns 1, so there's another correct pair.

Thus, the sum of correct pairs is 2. The minimum swaps needed are n - sum which is 2 - 2 = 0. But in the given example, we see that they are not seated correctly. However, according to our merged sets, there should be no swaps because both pairs are already with their respective partners but only in the wrong order, so our process does include the final swap to make the order correct.

We now see there is a discrepancy. The example also illustrates that the merging of the representatives doesn't fully capture the physical seating arrangement - it just ensures elements are in the same set. In this example, a single swap between 0 and 3 is needed to arrange the couples correctly. So, the expected output for swaps is 1, and the final array should look like this [1, 0, 2, 3].

This walkthrough shows that while our parents' array after the find operation accurately identifies the sets, we still need to consider the physical swaps between these sets to achieve the proper arrangement. Thus, the minimum number of swaps calculated will be 1 and we get our desired seating [1, 0, 2, 3].

Solution Implementation

Time and Space Complexity

The given code implements a function to count the minimum number of swaps required to arrange couples in a row. It uses a union-find algorithm to merge the couple pairs. Here's the analysis:

Time Complexity:

The time complexity of this function is dominated by two parts, the union-find operations (find function) and the loop that iterates over the row to perform unions.

• The find function has an amortized time complexity of O(α(n)), where α is the inverse Ackermann function, which grows extremely slowly and is considered practically constant for modest input sizes.

• The loop iterates n/2 times since we're stepping by 2. Inside the loop, we perform at most two find operations and one union operation per iteration. Since the find operation's amortized time complexity is O(α(n)) and the union operation can be considered O(1) in this amortized context, the total time complexity inside the loop is O(n * α(n)).

• After the loop, we have a comprehension that iterates n times to count the number of unique parents, which is O(n).

Combining these, the overall time complexity is O(n * α(n)) + O(n), which simplifies to O(n * α(n)) due to the dominance of the find operation within the loop.

Space Complexity:

The space complexity is determined by the space used to store the parent array p and any auxiliary space used by recursion in the find function.

• The parent array p has a size of n, which is half the number of elements in the initial row because we're considering couples.

• The find function's recursion depth is at most O(α(n)). However, due to the path compression (p[x] = find(p[x])), the depth does not expand with each call; thus, the recursion stack does not significantly increase the space used.

Hence, the space complexity is essentially O(n) for the parent array p, with no significant additional space used for recursive calls due to path compression. Overall the space complexity is O(n).

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