What is the time complexity of Longest Arithmetic Subsequence?

The optimal solution has a time complexity of O(n^2).

What is the space complexity of Longest Arithmetic Subsequence?

The space complexity is O(n^2).

What data structures are used to solve Longest Arithmetic Subsequence?

This problem uses Array, Hash Table, Binary Search, Dynamic Programming. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Longest Arithmetic Subsequence?

Insight 1: The common difference between elements is the key to identifying arithmetic subsequences. Insight 2: Dynamic programming can be used to store and reuse the lengths of arithmetic subsequences ending at each index with a specific common difference.

Which companies ask LeetCode 1027. Longest Arithmetic Subsequence in interviews?

This problem is frequently asked at Snapdeal.

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Longest Arithmetic Subsequence - LeetCode 1027 Solution

Q: How to solve Longest Arithmetic Subsequence?

The provided solution employs a dynamic programming approach. We iterate through the array, and for each element, we consider all previous elements. For each pair of elements (nums[i], nums[j]) where j < i, we calculate the common difference 'diff'. We then use a hash map (dp[i]) to store the length of the longest arithmetic subsequence ending at index 'i' with a common difference of 'diff'. The length is determined by checking the length of the arithmetic subsequence ending at index 'j' with the same common difference and adding 1. We maintain a variable to track the maximum length encountered during the process.

Longest Arithmetic Subsequence - Complete Solution Guide

Longest Arithmetic Subsequence is LeetCode problem 1027, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given an array nums of integers, return the length of the longest arithmetic subsequence in nums . Note that: A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements. A sequence seq is arithmetic if seq[i + 1] - seq[i] are all the same value (for 0 <= i < seq.length - 1 ). Example 1: Input: nums = [3,6,9,12] Output: 4 Explanation: The whole array is an arithmetic sequence with steps of length = 3. Examp

Detailed Explanation

The problem asks us to find the length of the longest arithmetic subsequence within a given array of integers. An arithmetic subsequence is a sequence derived from the original array by deleting some or no elements without changing the order, such that the difference between consecutive elements in the subsequence is constant. For instance, in [9, 4, 7, 2, 10], [4, 7, 10] is an arithmetic subsequence of length 3 with a common difference of 3. The input is an array 'nums' of integers, and the output is the length of the longest arithmetic subsequence.

Solution Approach

The provided solution employs a dynamic programming approach. We iterate through the array, and for each element, we consider all previous elements. For each pair of elements (nums[i], nums[j]) where j < i, we calculate the common difference 'diff'. We then use a hash map (dp[i]) to store the length of the longest arithmetic subsequence ending at index 'i' with a common difference of 'diff'. The length is determined by checking the length of the arithmetic subsequence ending at index 'j' with the same common difference and adding 1. We maintain a variable to track the maximum length encountered during the process.

Step-by-Step Algorithm

Step 1: Initialize a list of hash maps 'dp' where dp[i] will store the length of the longest arithmetic subsequence ending at index 'i' for different common differences.
Step 2: Initialize 'max_len' to 0 to track the maximum length found so far.
Step 3: Iterate through the 'nums' array using nested loops. The outer loop iterates from i = 0 to n-1, and the inner loop iterates from j = 0 to i-1.
Step 4: Inside the inner loop, calculate the common difference 'diff' between nums[i] and nums[j].
Step 5: Check if there is an existing arithmetic subsequence ending at index 'j' with the common difference 'diff'. If so, its length is retrieved from dp[j][diff]; otherwise, a default length of 1 is used (representing a subsequence of length 2 containing nums[j] and nums[i]).
Step 6: The length of the arithmetic subsequence ending at index 'i' with the common difference 'diff' is then set to be length from step 5 + 1. This length is stored in dp[i][diff].
Step 7: Update 'max_len' with the maximum length found so far.
Step 8: After iterating through all pairs of elements, return 'max_len'.

Key Insights

Insight 1: The common difference between elements is the key to identifying arithmetic subsequences.
Insight 2: Dynamic programming can be used to store and reuse the lengths of arithmetic subsequences ending at each index with a specific common difference.
Insight 3: We need to efficiently store and retrieve the length of the arithmetic subsequence ending at each index with a given common difference, so using a hash map (dictionary) is crucial.

Complexity Analysis

Time Complexity: O(n^2)

Space Complexity: O(n^2)

Topics

This problem involves: Array, Hash Table, Binary Search, Dynamic Programming.

Companies

Asked at: Snapdeal.

Longest Arithmetic Subsequence | LeetCode 1027

Medium

Array Hash Table Binary Search Dynamic Programming

Time: O(n^2)Space: O(n^2)

Solution Code

Available in:

import java.util.HashMap;
import java.util.List;
import java.util.ArrayList;
import java.util.Map;

class Solution {
    public int longestArithSeqLength(int[] nums) {
        int n = nums.length;
        List<Map<Integer, Integer>> dp = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            dp.add(new HashMap<>());
        }
        int max_len = 0;

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < i; j++) {
                int diff = nums[i] - nums[j];
                int length = dp.get(j).getOrDefault(diff, 1) + 1;
                dp.get(i).put(diff, length);
                max_len = Math.max(max_len, length);
            }
        }

        return max_len;
    }
}

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Problem Info

Description

Problem 1027: Longest Arithmetic Subsequence - Detailed explanation and solution approaches.

Topics

Array Hash Table Binary Search Dynamic Programming

Companies

Snapdeal

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:1027

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

8 steps

1
Step 1: Initialize a list of hash maps 'dp' where dp[i] will store the length of the longest arithmetic subsequence ending at index 'i' for different common differences.
2
Step 2: Initialize 'max_len' to 0 to track the maximum length found so far.
3
Step 3: Iterate through the 'nums' array using nested loops. The outer loop iterates from i = 0 to n-1, and the inner loop iterates from j = 0 to i-1.
4
Step 4: Inside the inner loop, calculate the common difference 'diff' between nums[i] and nums[j].
5
Step 5: Check if there is an existing arithmetic subsequence ending at index 'j' with the common difference 'diff'. If so, its length is retrieved from dp[j][diff]; otherwise, a default length of 1 is used (representing a subsequence of length 2 containing nums[j] and nums[i]).
6
Step 6: The length of the arithmetic subsequence ending at index 'i' with the common difference 'diff' is then set to be length from step 5 + 1. This length is stored in dp[i][diff].
7
Step 7: Update 'max_len' with the maximum length found so far.
8
Step 8: After iterating through all pairs of elements, return 'max_len'.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n^2)

🐢 Quadratic - Gets slow with large inputs. Consider optimization!

💾

Space Complexity

O(n^2)

📚 Quadratic space - Uses lots of memory!

📝

Why?

The nested loops contribute to the O(n^2) time complexity. The storage of subsequence lengths in the hash maps (dp) contributes to the O(n^2) space complexity.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Binary Search Approach

Instead of using dynamic programming, for each possible common difference, one can iterate through the sorted array and for each element, try to find the next element that forms the arithmetic progression using binary search. While the implementation is more complex, it is a valid approach.

Time: O(n^2 log n)Space: O(1) or O(n) depending on whether in-place sorting is used.

Trade-offs: This approach might be faster for certain types of input where the number of distinct common differences is small compared to the size of the input array.

✨

Key Insights

1Insight 1: The common difference between elements is the key to identifying arithmetic subsequences.
2Insight 2: Dynamic programming can be used to store and reuse the lengths of arithmetic subsequences ending at each index with a specific common difference.
3Insight 3: We need to efficiently store and retrieve the length of the arithmetic subsequence ending at each index with a given common difference, so using a hash map (dictionary) is crucial.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Binary Search

Halving the search space each step. Like finding a word in a dictionary!

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: Input array with all elements being the same. The solution will correctly identify the entire array as an arithmetic sequence, and return the length of the array.
Edge case 2: Input array with only two elements. The solution will correctly identify the two elements as an arithmetic sequence and return 2.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not handling the case where the common difference has not been seen before for a particular index. In such a case, the length of the subsequence should be initialized to 1 (or 2, considering the current pair of elements).

✗Common mistake 2: Forgetting to update the maximum length found so far. The 'max_len' variable must be updated whenever a new, longer arithmetic subsequence is discovered.

✗Common mistake 3: Failing to consider subsequences and instead only considering contiguous subarrays.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Stock market analysis: Identifying trends in stock prices can be seen as finding arithmetic sequences or subsequences in the stock price data.

🏦Real-world scenario 2: Data compression: Identifying patterns such as arithmetic sequences can be used to compress data by storing only the first element and the common difference.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 6, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 13

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 6

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJun 29

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

How do I choose between a hash map and a hash set?

Use a hash set when you only need to track presence/absence of elements (like detecting duplicates). Use a hash map when you need to associate values with keys (like counting frequencies or storing indices). Hash sets use less memory since they only store keys.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Longest Arithmetic Subsequence"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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