Count and say

Problem Overview

The problem of "Count and Say" challenges us to develop an algorithm that generates a sequence based on a specified set of rules. This sequence is notable in computer science for its recursive nature and serves as an introductory problem to help understand the handling of strings and sequences.

Understanding the Problem

The "Count and Say" sequence starts with the term "1" and each subsequent term is derived by describing the previous one. The sequence begins as follows:

• 1 is the initial term.
• The next term is "one 1" or "11".
• The next term after that is "two 1s" or "21".

The aim is to return the nth term of this sequence. This task involves iterating over sequences, counting and grouping identical digits, and representing this description as the next term.

Key DSA Concepts and Theory

String Manipulation

String manipulation involves altering, parsing, and handling string data structures. In this problem, effective string manipulation is essential for iterating through the sequence characters and constructing new terms.

Iteration and Recursion

Recursive patterns are inherent in this problem, where each sequence term is derived from the previous one. While we can solve this iteratively, understanding recursion helps grasp how sequences can be constructed dynamically.

Handling Sequences

Sequences in programming often involve generating the next element based on a straightforward rule or computation derived from previous elements.

Solution Approach

The "Count and Say" problem is best solved using an iterative approach where we start from the base sequence "1" and build out the following terms until we reach the desired nth term.

Step-by-Step Solution

1. Initialize: Start with a base case for n=1, i.e., the sequence starts with "1".

2. Iterate: For each sequence from 2 to n, generate the next term by:

   • Inspecting the current sequence.
   • Counting how many times each digit appears in sequence consecutively.
   • Constructing a new sequence string with counts followed by the digit.

3. Repeat: Repeat the above step until the nth sequence is constructed.

Below are the code examples for this approach in C++ and Java.

C++ Implementation

class Solution {
public:
    string countAndSay(int n) {
        if (n == 1) return "1";
        string result = "1";

        for (int i = 2; i <= n; ++i) {
            string current = "";
            for (int j = 0, count; j < result.size(); j += count) {
                count = 1;
                while (j + count < result.size() && result[j] == result[j + count])
                    ++count;
                current += to_string(count) + result[j];
            }
            result = current;
        }
        return result;
    }
};

Java Implementation

class Solution {
    public String countAndSay(int n) {
        if (n == 1) return "1";
        String result = "1";

        for (int i = 2; i <= n; i++) {
            StringBuilder current = new StringBuilder();
            for (int j = 0, count; j < result.length(); j += count) {
                count = 1;
                while (j + count < result.length() && result.charAt(j) == result.charAt(j + count))
                    ++count;
                current.append(count).append(result.charAt(j));
            }
            result = current.toString();
        }
        return result;
    }
}

Time and Space Complexity Analysis

• Time Complexity: (O(2^n)). Generating each sequence requires processing all previous digits, making the algorithm's complexity double with each additional term.
• Space Complexity: (O(2^n)). Storing sequence strings grows exponentially with the length of the sequence.

Common Mistakes to Avoid

• Off-by-One Errors: When iterating through the string, ensure the loop boundaries are correctly set to avoid out-of-bounds errors.
• Miscounting Consecutive Digits: Carefully count and group consecutive digits to avoid incorrect sequencing.

Similar Problems on Leetcode

• Problem 824. Goat Latin
• Problem 293. Flip Game

Additional Resources and References

• For further exploration of sequence generation problems, consider reviewing recursive algorithms and dynamic programming approaches.
• Official documentation for C++ and Java string manipulation functions can be helpful for developing efficient solutions.

Use this problem as a stepping stone to better understand string operations and sequence generation in programming. Solving such recursive sequence problems helps build a foundational understanding of dynamic patterns in data structures and algorithms.