Merge Similar Pixels: Comprehensive Background and Solution Guide

1. Background Knowledge

This problem models image segmentation as union-find clustering on a graph where pixels are nodes and edges represent adjacency. Key concepts:

• Graph Representation: Pixels as vertices, adjacency list as edges
• Similarity Metric: $|pixels[i] - pixels[j]| \leq threshold$ defines mergeable regions
• Connected Components: Final count of distinct groups after transitive merging (if A↔B and B↔C, then A,B,C form one group)
• Applications: Region growing, superpixel generation, color quantization

Prerequisites: Graph theory, Union-Find (Disjoint Set Union - DSU) data structure.

2. Algorithm Approach

Union-Find (DSU) with Path Compression and Union by Rank is optimal:

Core Operations:
1. find(x): Return root parent of x (with path compression)
2. union(x, y): Merge sets containing x and y (by rank for balance)
3. count_components(): Number of unique roots

Why DSU? Handles transitive closure efficiently (A connects B, B connects C → A,B,C connected automatically).

Alternatives (less efficient here):

• DFS/BFS traversal: $O(V+E)$ but requires explicit graph traversal
• Kruskal's MST: Similar but more complex

3. Step-by-Step Strategy

class UnionFind:
 def __init__(self, n):
 self.parent = list(range(n))
 self.rank = * n
 self.components = n

 def find(self, x):
 if self.parent[x] != x:
 self.parent[x] = self.find(self.parent[x]) # Path compression
 return self.parent[x]