What is the time complexity of Length of Longest Fibonacci Subsequence?

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

What is the space complexity of Length of Longest Fibonacci Subsequence?

The space complexity is O(n^2).

What data structures are used to solve Length of Longest Fibonacci Subsequence?

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

What are the key insights for solving Length of Longest Fibonacci Subsequence?

Insight 1: The array is strictly increasing, which allows for efficient searching and eliminates the need to check for duplicates or maintain order when creating subsequences. Insight 2: Dynamic Programming is suitable because the length of the Fibonacci-like subsequence ending at indices i and j depends on the length of the Fibonacci-like subsequence ending at indices k and i, where arr[k] + arr[i] = arr[j].

Which companies ask LeetCode 873. Length of Longest Fibonacci Subsequence in interviews?

This problem is frequently asked at Baidu.

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Length of Longest Fibonacci Subsequence - LeetCode 873 Solution

Q: How to solve Length of Longest Fibonacci Subsequence?

The solution uses dynamic programming to find the length of the longest Fibonacci-like subsequence. A hash table is used to quickly look up the index of each number in the array. The algorithm iterates through all possible pairs of numbers in the array, and for each pair, it checks if their difference is also in the array and smaller than the first number in the pair. If it is, then a Fibonacci-like subsequence has been found, and the length of the subsequence is updated in the dp table. The maximum length found is then returned.

Length of Longest Fibonacci Subsequence - Complete Solution Guide

Length of Longest Fibonacci Subsequence is LeetCode problem 873, 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

A sequence x 1 , x 2 , ..., x n is Fibonacci-like if: n >= 3 x i + x i+1 == x i+2 for all i + 2 <= n Given a strictly increasing array arr of positive integers forming a sequence, return the length of the longest Fibonacci-like subsequence of arr . If one does not exist, return 0 . A subsequence is derived from another sequence arr by deleting any number of elements (including none) from arr , without changing the order of the remaining elements. For example, [3, 5, 8] is a subsequence of [3, 4,

Detailed Explanation

The problem asks us to find the length of the longest Fibonacci-like subsequence within a given strictly increasing array of positive integers. A Fibonacci-like sequence has at least three elements, and each element (after the first two) is the sum of the two preceding elements. The subsequence doesn't need to be contiguous, meaning we can pick and choose elements from the original array as long as their order is maintained.

Solution Approach

The solution uses dynamic programming to find the length of the longest Fibonacci-like subsequence. A hash table is used to quickly look up the index of each number in the array. The algorithm iterates through all possible pairs of numbers in the array, and for each pair, it checks if their difference is also in the array and smaller than the first number in the pair. If it is, then a Fibonacci-like subsequence has been found, and the length of the subsequence is updated in the dp table. The maximum length found is then returned.

Step-by-Step Algorithm

Step 1: Create a hash table (val_to_index) to store the index of each value in the input array.
Step 2: Initialize a dynamic programming table (dp) to store the lengths of Fibonacci-like subsequences. In Python, Java, and C++, the dp table is a hashmap storing lengths against key (i,j). In C, it's a 2D array dp[i][j].
Step 3: Iterate through all possible pairs of indices (i, j) in the array, where i < j.
Step 4: Calculate the 'target' value: target = arr[j] - arr[i].
Step 5: Check if the target value is less than arr[i] and exists in the val_to_index hash table.
Step 6: If the target value exists, retrieve its index 'k' from the val_to_index table.
Step 7: Update the dp table. In Python, Java, C++, retrieve dp[(k, i)] if available, default to length = 2. Add 1 to the length and set dp[(i, j)] to this new value. In C, length is 2 if dp[k][i] is 0, otherwise it's dp[k][i] + 1, and then dp[i][j] is set to this length.
Step 8: Update the max_len with the maximum length found so far.
Step 9: After iterating through all pairs, return the max_len.

Key Insights

Insight 1: The array is strictly increasing, which allows for efficient searching and eliminates the need to check for duplicates or maintain order when creating subsequences.
Insight 2: Dynamic Programming is suitable because the length of the Fibonacci-like subsequence ending at indices i and j depends on the length of the Fibonacci-like subsequence ending at indices k and i, where arr[k] + arr[i] = arr[j].
Insight 3: Using a hash table (or map) to store the index of each element in the array allows for fast lookups when searching for the third number in the Fibonacci-like sequence.

Complexity Analysis

Time Complexity: O(n^2)

Space Complexity: O(n^2)

Topics

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

Companies

Asked at: Baidu.

Length of Longest Fibonacci Subsequence | LeetCode 873

Medium

Array Hash Table Dynamic Programming

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

Solution Code

Available in:

import java.util.HashMap;
import java.util.Map;

class Solution {
    public int lenLongestFibSubseq(int[] arr) {
        int n = arr.length;
        Map<Integer, Integer> valToIndex = new HashMap<>();
        for (int i = 0; i < n; i++) {
            valToIndex.put(arr[i], i);
        }

        Map<String, Integer> dp = new HashMap<>();
        int maxLen = 0;

        for (int j = 0; j < n; j++) {
            for (int i = 0; i < j; i++) {
                int target = arr[j] - arr[i];

                if (target < arr[i] && valToIndex.containsKey(target)) {
                    int k = valToIndex.get(target);
                    String key = k + "," + i;
                    int length = dp.getOrDefault(key, 2) + 1;
                    dp.put(i + "," + j, length);
                    maxLen = Math.max(maxLen, length);
                }
            }
        }

        return maxLen;
    }
}

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Problem Info

Description

Problem 873: Length of Longest Fibonacci Subsequence - Detailed explanation and solution approaches.

Topics

Array Hash Table Dynamic Programming

Companies

Baidu

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:873

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

9 steps

1
Step 1: Create a hash table (val_to_index) to store the index of each value in the input array.
2
Step 2: Initialize a dynamic programming table (dp) to store the lengths of Fibonacci-like subsequences. In Python, Java, and C++, the dp table is a hashmap storing lengths against key (i,j). In C, it's a 2D array dp[i][j].
3
Step 3: Iterate through all possible pairs of indices (i, j) in the array, where i < j.
4
Step 4: Calculate the 'target' value: target = arr[j] - arr[i].
5
Step 5: Check if the target value is less than arr[i] and exists in the val_to_index hash table.
6
Step 6: If the target value exists, retrieve its index 'k' from the val_to_index table.
7
Step 7: Update the dp table. In Python, Java, C++, retrieve dp[(k, i)] if available, default to length = 2. Add 1 to the length and set dp[(i, j)] to this new value. In C, length is 2 if dp[k][i] is 0, otherwise it's dp[k][i] + 1, and then dp[i][j] is set to this length.
8
Step 8: Update the max_len with the maximum length found so far.
9
Step 9: After iterating through all pairs, return the 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 dominant operations are the nested loops in the algorithm, contributing to O(n^2) time complexity. The space complexity is dominated by the DP table which stores a value for each pair of indices

📖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

Brute Force

Generate all possible subsequences of length at least 3 and check if each subsequence is Fibonacci-like. This approach would involve generating 2^n subsequences, where n is the length of the array.

Time: O(2^n * n), where n is the length of the array.Space: O(n), for storing the subsequence

Trade-offs: The brute-force approach is much less efficient than the dynamic programming approach and should not be used for larger input arrays. It is simple to understand but computationally expensive.

✨

Key Insights

1Insight 1: The array is strictly increasing, which allows for efficient searching and eliminates the need to check for duplicates or maintain order when creating subsequences.
2Insight 2: Dynamic Programming is suitable because the length of the Fibonacci-like subsequence ending at indices i and j depends on the length of the Fibonacci-like subsequence ending at indices k and i, where arr[k] + arr[i] = arr[j].
3Insight 3: Using a hash table (or map) to store the index of each element in the array allows for fast lookups when searching for the third number in the Fibonacci-like sequence.

📚

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!

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: An array with no Fibonacci-like subsequence. The code correctly returns 0 in this case, as max_len will remain 0 if no valid subsequence is found.
Edge case 2: An array with only two elements. The loops won't execute since n >= 3 constraint, so max_len will remain 0. The code handles this correctly.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not considering the constraint that the array is strictly increasing. This leads to unnecessary checks or incorrect logic.

✗Common mistake 2: Forgetting the base case for the dp table. The length of a subsequence formed by the first two numbers is always at least 2, so the dp table needs to be initialized accordingly. Also, not handling case where dp[k][i] is not yet populated in C implementation.

✅

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: Financial analysis - Identifying trends that follow a Fibonacci-like pattern in stock prices or other financial data.

🏦Real-world scenario 2: Data compression - Identifying patterns in data that can be compressed using Fibonacci-like sequences.

💡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

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ 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: November 1, 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

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

@CodeNinjaJul 13

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

@TechLeadJul 6

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jun 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 "Length of Longest Fibonacci 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.

🤖 Need help with this problem?