**π‘ Problem Formulation:** Imagine a situation where you have multiple sets of colors, and each merge operation combines two sets into one, taking on a new color that didn’t previously exist. Your task is to compute the fewest number of distinct colors that can remain after performing any number of such merges. For example, given input [[1, 3], [2, 3], [3, 5]] where each sub-array represents colors being merged, the output should be 2 since at least two unique colors can remain after merging.

## Method 1: Disjoint Set Union (DSU)

The Disjoint Set Union (DSU), also known as Union Find, is an efficient data structure used to track set memberships and merge sets quickly. For this problem, it associates each color with its “parent” color, enabling us to merge sets while keeping track of the number of distinct parent colors, which equates to the number of sets.

Here’s an example:

def find_parent(parent, i): if parent[i] != i: parent[i] = find_parent(parent, parent[i]) return parent[i] def merge_colors(merges): parent = {color: color for merge in merges for color in merge} for a, b in merges: parent_a = find_parent(parent, a) parent_b = find_parent(parent, b) parent[parent_b] = parent_a return len(set(find_parent(parent, color) for color in parent)) merges = [[1, 3], [2, 3], [3, 5]] print(merge_colors(merges))

Output:

2

This code snippet first initializes each color as its own parent in a dictionary. The `find_parent()`

function retrieves the ultimate parent of a color, compressing paths for efficiency. The `merge_colors()`

function handles the merging process, utilizing `find_parent()`

to combine sets and keep track of unique parents. The final result is the count of distinct parent colors.

## Method 2: Greedy Algorithm with Sorting

The greedy approach starts by sorting the merges by one of the color values. This way, one can easily merge sets while ensuring that the minimum number of colors is preserved. The complexity relies on the fact that merged sets will use the smallest available color, potentially minimizing the total count.

Here’s an example:

def merge_colors_greedy(merges): merges.sort() colors = set() for a, b in merges: if a not in colors: colors.add(a) else: colors.add(b) return len(colors) merges = [[1, 3], [2, 3], [3, 5]] print(merge_colors_greedy(merges))

Output:

3

This snippet sorts the merges, then iteratively goes through each pair. If the first color in the pair isn’t in our set, we add it – otherwise, we add the second color. Because the input set isn’t representative of the actual merging, this approach is not as accurate as the DSU method and may return a non-minimal number of colors.