Welcome, fellow coding enthusiasts! 🚀 In this comprehensive guide, we’ll explore how to tackle the Lexicographically Smallest Equivalent String problem (LeetCode 1061) – an excellent exercise that combines string manipulation with graph theory concepts. Perfect for beginners and intermediate coders alike, we’ll walk through multiple solution approaches with detailed implementations in C++, JavaScript, and Python.
## 🔍 Understanding the Problem Statement
The challenge presents us with:
1. Two strings `s1` and `s2` of equal length
2. A third string called `baseStr`
Each character pair `(s1[i], s2[i])` establishes an equivalence relationship with these properties:
– **Reflexive**: Every character is equivalent to itself
– **Symmetric**: If ‘a’ equals ‘b’, then ‘b’ equals ‘a’
– **Transitive**: If ‘a’ equals ‘b’ and ‘b’ equals ‘c’, then ‘a’ equals ‘c’
Your mission? Transform each character in `baseStr` to its lexicographically smallest equivalent character based on the relationships defined by `s1` and `s2`.
## 🧠 Problem-Solving Approach and Intuition
Visualizing this challenge as a graph problem provides our breakthrough insight:
1. **Graph Representation**: Each character becomes a node
2. **Edges**: Each `(s1[i], s2[i])` pair forms an undirected edge connecting nodes
3. **Connected Components**: Characters in the same equivalence class form connected components
4. **Optimization Goal**: For each component, we want to identify and use the smallest character
The **Disjoint Set Union (DSU)** data structure, also known as Union-Find, emerges as the perfect tool for this scenario with its efficient union and find operations.
## ⚡ Key DSU Operations for This Problem
1. **Find Operation**: Determines the root representative of a character (with path compression)
2. **Union Operation**: Connects two characters while maintaining the lexicographically smallest parent
## 👨💻 Solution Implementations
### C++ Implementation (With Optimized DSU)
“`cpp
#include
#include
using namespace std;
class Solution {
public:
class DSU {
vector parent;
public:
DSU(int n) {
parent.resize(n);
for (int i = 0; i < n; i++) parent[i] = i;
}
int find(int x) {
if (parent[x] != x) parent[x] = find(parent[x]);
return parent[x];
}
void unionByLex(int x, int y) {
int rootX = find(x);
int rootY = find(y);
if (rootX < rootY) parent[rootY] = rootX;
else if (rootY < rootX) parent[rootX] = rootY;
}
};
string smallestEquivalentString(string s1, string s2, string baseStr) {
DSU dsu(26);
for (int i = 0; i i);
}
find(x) {
if (this.parent[x] !== x) {
this.parent[x] = this.find(this.parent[x]);
}
return this.parent[x];
}
unionByLex(x, y) {
const rootX = this.find(x);
const rootY = this.find(y);
if (rootX < rootY) this.parent[rootY] = rootX;
else if (rootY < rootX) this.parent[rootX] = rootY;
}
}
function smallestEquivalentString(s1, s2, baseStr) {
const dsu = new DSU(26);
for (let i = 0; i < s1.length; i++) {
dsu.unionByLex(
s1.charCodeAt(i) – 97,
s2.charCodeAt(i) – 97
);
}
let result = [];
for (const c of baseStr) {
const charCode = dsu.find(c.charCodeAt(0) – 97);
result.push(String.fromCharCode(charCode + 97));
}
return result.join('');
}
“`
### Python Implementation
“`python
class DSU:
def __init__(self, n):
self.parent = list(range(n))
def find(self, x):
if self.parent[x] != x:
self.parent[x] = self.find(self.parent[x])
return self.parent[x]
def union(self, x, y):
rootX = self.find(x)
rootY = self.find(y)
if rootX < rootY:
self.parent[rootY] = rootX
elif rootY str:
dsu = DSU(26)
for a, b in zip(s1, s2):
dsu.union(ord(a) – ord(‘a’), ord(b) – ord(‘a’))
result = []
for c in baseStr:
smallest_char = chr(dsu.find(ord(c) – ord(‘a’)) + ord(‘a’))
result.append(smallest_char)
return ”.join(result)
“`
## 💡 Pro Tips for Mastering Similar Problems
1. Always analyze the problem for underlying graph structures
2. Consider equivalence relations carefully—they often suggest union-find solutions
3. Practice implementing DSU with various optimizations (path compression, union by rank/size)
4. For lexicographical problems, maintain ordering during union operations
5. Don’t forget to convert between character representations and numerical indices
## 🔗 Additional Resources for Problem Solving
Looking to enhance your problem-solving skills further? Consider exploring:
– More graph and DSU problems on LeetCode
– Algorithm visualization tools to better understand union-find
– Competitive programming resources that cover data structures in depth
– Pair programming sessions to discuss different approaches to these problems
Happy coding! Remember, the journey of a thousand algorithms begins with a single for-loop. 🏆
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