C++ Coding Test Course, Utilizing Time Complexity

One of the most important things when preparing for coding tests is understanding and optimizing problem-solving skills along with time complexity. This article will detail the process of solving problems considering time complexity using C++.

Problem: Longest Increasing Subsequence

This problem involves finding the length of the longest increasing subsequence in a given integer array.

Problem Description

Given an integer array nums, return the length of the longest increasing subsequence. A subsequence is formed by selected elements from the array that maintain their original order but do not need to be contiguous.

Example

    Input: nums = [10, 9, 2, 5, 3, 7, 101, 18]
    Output: 4
    Explanation: The longest increasing subsequence is [2, 3, 7, 101], with a length of 4.
    

Approach

This problem can be solved using a combination of Dynamic Programming and Binary Search. Considering time complexity, the following approaches can be utilized:

1. Approach using Dynamic Programming

First, we record the length of the increasing subsequence that ends with each element. Then, we can iterate through each element and compare it with all previous elements to update the length of the subsequence if possible.

2. Combined with Binary Search

Ultimately, we use an array to record the end elements of subsequences and apply binary search on this array to find the position to insert the current element. This method allows us to achieve more optimized performance.

Implementation

Below is the code to solve this problem using C++:

#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

int lengthOfLIS(vector<int>& nums) {
    if (nums.empty()) return 0;

    vector<int> dp;
    for (int num : nums) {
        auto it = lower_bound(dp.begin(), dp.end(), num);
        if (it == dp.end()) {
            dp.push_back(num);
        } else {
            *it = num;
        }
    }
    return dp.size();
}

int main() {
    vector<int> nums = {10, 9, 2, 5, 3, 7, 101, 18};
    cout << "Length of Longest Increasing Subsequence: " << lengthOfLIS(nums) << endl;
    return 0;
}
    

Time Complexity Analysis

The time complexity of this problem is O(n log n). A binary search is performed for each element to find the appropriate position, and the time complexity of binary search is log n. Therefore, the overall time complexity is O(n log n), and the space complexity is O(n).

Conclusion

For problems like finding the longest increasing subsequence in an array, it is important to combine Dynamic Programming and Binary Search to reduce time complexity. Utilizing useful functions and data structures provided by C++’s STL can help write more efficient code.

This article explained time complexity analysis and algorithm problem-solving methods using C++. I hope it will be helpful in your future coding test preparation!