LeetCode #873 — MEDIUM

Length of Longest Fibonacci Subsequence

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

A sequence x₁, x₂, ..., 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, 5, 6, 7, 8].

Example 1:

Input: arr = [1,2,3,4,5,6,7,8]
Output: 5
Explanation: The longest subsequence that is fibonacci-like: [1,2,3,5,8].

Example 2:

Input: arr = [1,3,7,11,12,14,18]
Output: 3
Explanation: The longest subsequence that is fibonacci-like: [1,11,12], [3,11,14] or [7,11,18].

Constraints:

3 <= arr.length <= 1000
1 <= arr[i] < arr[i + 1] <= 10⁹

Step 01

Brute Force Baseline

Problem summary: A sequence x1, x2, ..., xn is Fibonacci-like if: n >= 3 xi + xi+1 == xi+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, 5, 6, 7, 8].

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Hash Map · Dynamic Programming

Example 1

[1,2,3,4,5,6,7,8]

Example 2

[1,3,7,11,12,14,18]

Core Insight

What unlocks the optimal approach

Can we use dynamic programming here?
Consider a sequence ending at index <code>i</code. The previous two elements must sum to <code>arr[i]</code>, which can be done in linear time per element.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #873: Length of Longest Fibonacci Subsequence
class Solution {
    public int lenLongestFibSubseq(int[] arr) {
        int n = arr.length;
        int[][] f = new int[n][n];
        Map<Integer, Integer> d = new HashMap<>();
        for (int i = 0; i < n; ++i) {
            d.put(arr[i], i);
            for (int j = 0; j < i; ++j) {
                f[i][j] = 2;
            }
        }
        int ans = 0;
        for (int i = 2; i < n; ++i) {
            for (int j = 1; j < i; ++j) {
                int t = arr[i] - arr[j];
                Integer k = d.get(t);
                if (k != null && k < j) {
                    f[i][j] = Math.max(f[i][j], f[j][k] + 1);
                    ans = Math.max(ans, f[i][j]);
                }
            }
        }
        return ans;
    }
}

// Accepted solution for LeetCode #873: Length of Longest Fibonacci Subsequence
func lenLongestFibSubseq(arr []int) (ans int) {
	n := len(arr)
	f := make([][]int, n)
	for i := range f {
		f[i] = make([]int, n)
	}

	d := make(map[int]int)
	for i, x := range arr {
		d[x] = i
		for j := 0; j < i; j++ {
			f[i][j] = 2
		}
	}

	for i := 2; i < n; i++ {
		for j := 1; j < i; j++ {
			t := arr[i] - arr[j]
			if k, ok := d[t]; ok && k < j {
				f[i][j] = max(f[i][j], f[j][k]+1)
				ans = max(ans, f[i][j])
			}
		}
	}

	return
}

# Accepted solution for LeetCode #873: Length of Longest Fibonacci Subsequence
class Solution:
    def lenLongestFibSubseq(self, arr: List[int]) -> int:
        n = len(arr)
        f = [[0] * n for _ in range(n)]
        d = {x: i for i, x in enumerate(arr)}
        for i in range(n):
            for j in range(i):
                f[i][j] = 2
        ans = 0
        for i in range(2, n):
            for j in range(1, i):
                t = arr[i] - arr[j]
                if t in d and (k := d[t]) < j:
                    f[i][j] = max(f[i][j], f[j][k] + 1)
                    ans = max(ans, f[i][j])
        return ans

// Accepted solution for LeetCode #873: Length of Longest Fibonacci Subsequence
use std::collections::HashMap;
impl Solution {
    pub fn len_longest_fib_subseq(arr: Vec<i32>) -> i32 {
        let n = arr.len();
        let mut f = vec![vec![0; n]; n];
        let mut d = HashMap::new();
        for i in 0..n {
            d.insert(arr[i], i);
            for j in 0..i {
                f[i][j] = 2;
            }
        }
        let mut ans = 0;
        for i in 2..n {
            for j in 1..i {
                let t = arr[i] - arr[j];
                if let Some(&k) = d.get(&t) {
                    if k < j {
                        f[i][j] = f[i][j].max(f[j][k] + 1);
                        ans = ans.max(f[i][j]);
                    }
                }
            }
        }
        ans
    }
}

// Accepted solution for LeetCode #873: Length of Longest Fibonacci Subsequence
function lenLongestFibSubseq(arr: number[]): number {
    const n = arr.length;
    const f: number[][] = Array.from({ length: n }, () => Array(n).fill(0));
    const d: Map<number, number> = new Map();
    for (let i = 0; i < n; ++i) {
        d.set(arr[i], i);
        for (let j = 0; j < i; ++j) {
            f[i][j] = 2;
        }
    }
    let ans = 0;
    for (let i = 2; i < n; ++i) {
        for (let j = 1; j < i; ++j) {
            const t = arr[i] - arr[j];
            const k = d.get(t);
            if (k !== undefined && k < j) {
                f[i][j] = Math.max(f[i][j], f[j][k] + 1);
                ans = Math.max(ans, f[i][j]);
            }
        }
    }
    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n^2)

Space

O(n^2)

Approach Breakdown

RECURSIVE

O(2ⁿ) time

O(n) space

Pure recursion explores every possible choice at each step. With two choices per state (take or skip), the decision tree has 2ⁿ leaves. The recursion stack uses O(n) space. Many subproblems are recomputed exponentially many times.

DYNAMIC PROGRAMMING

O(n × m) time

O(n × m) space

Each cell in the DP table is computed exactly once from previously solved subproblems. The table dimensions determine both time and space. Look for the state variables — each unique combination of state values is one cell. Often a rolling array can reduce space by one dimension.

Shortcut: Count your DP state dimensions → that’s your time. Can you drop one? That’s your space optimization.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.