1. Introduction
Coding tests are conducted by many companies to evaluate applicants’ algorithmic thinking and problem-solving skills. Among the various problems with different topics and difficulties, “Finding Binary Numbers” is a good example where the DP (Dynamic Programming) technique can be utilized. In this article, we will explain the problem of finding binary numbers in detail and proceed step by step to solve the problem using C#.
2. Problem Description
A Binary Number is a number made up of 0s and 1s,
which does not have two consecutive 1s.
For example, examples of binary numbers consisting of 0s and 1s are:
0, 1, 010, 001, 100, 0101, 1010, 1001, 0001, etc.
On the other hand, 11, 00000, or 1111 are not binary numbers.
For instance, given N, the problem is to determine the count of binary numbers of length N.
2.1. Input Format
– An integer N is given on the first line. (1 ≤ N ≤ 1,000)
2.2. Output Format
– Output the count of binary numbers of length N.
2.3. Example
Input:
3
Output:
3
In this case, the binary numbers are {001, 010, 100}, totaling 3.
3. Problem Solving Approach
To solve this problem, the following approach is necessary.
3.1. Define DP Array
We can solve this problem using Dynamic Programming.
First, we define a DP array to store the count of binary numbers of length N.
– DP[i] stores the count of binary numbers of length i.
3.2. State Transition Formula
From the current state, we can derive the following state transition formula:
– A binary number of length i can be appended from a binary number of length (i-1) if the last digit is 0,
– or it can be appended from a binary number of length (i-2) if the last digit is 1.
Therefore, it can be expressed as:
– DP[i] = DP[i-1] + DP[i-2]
3.3. Initial Conditions
– DP[1] = 2 (Binary Numbers: 0, 1)
– DP[2] = 3 (Binary Numbers: 00, 01, 10)
4. C# Code Implementation
Based on the above logic, let’s implement the C# code.
using System;
class Program
{
static void Main(string[] args)
{
int N = int.Parse(Console.ReadLine());
long[] dp = new long[N + 1];
// Initial conditions
dp[1] = 2; // Binary Numbers: 0, 1
if (N > 1)
{
dp[2] = 3; // Binary Numbers: 00, 01, 10
}
// Calculate DP array
for (int i = 3; i <= N; i++)
{
dp[i] = dp[i - 1] + dp[i - 2];
}
// Output result
Console.WriteLine(dp[N]);
}
}
5. Code Explanation
In the above code, we read N from the user, initialize the DP array, and then fill the DP array using a loop.
Finally, we can check the count of binary numbers of length N by printing DP[N].
5.1. Time Complexity
The time complexity of this algorithm is O(N). Even when N is at its maximum of 1,000,
it operates quickly to handle large amounts of data.
5.2. Space Complexity
The space complexity is also O(N) because we use the DP array,
which increases memory requirements. However, since N is small,
the memory usage is not a significant issue.
6. Conclusion
In this tutorial, we covered the problem of "Finding Binary Numbers."
We explained the step-by-step process of solving the problem using the techniques of Dynamic Programming
and implemented the code in C#.
Such types of problems are frequently encountered in actual coding tests, so it is advisable to practice and become familiar with them.
While preparing for coding tests, try to solve more problems,
and contemplate various approaches and optimizations for each problem.
Thank you!