What is the brute force approach and complexity for Number of Longest Increasing Subsequence?

Try all 2^N subsets of positions; keep the increasing ones and count the longest with time complexity O(N · 2^N) and space complexity O(1).

What is the optimal approach and complexity for Number of Longest Increasing Subsequence?

For each end, keep its longest length and how many chains reach it; O(N²) time with optimal time complexity O(N^2) and space complexity O(N).

Number of Longest Increasing Subsequence

Q: How do you identify the pattern for Number of Longest Increasing Subsequence?

Look for key signals in the prompt: "number of the longest" indicates We count chains, not just find the length. Two subsequences that pick different positions count as different, even with equal values.; "strictly increasing" indicates Equal values cannot both be in the chain, so the check is <, not <=.; "subsequences" indicates Chains can skip numbers, so each end position starts its own count of chains..

Medium·LeetCode #673 Pro OnlyMedium

LeetCode

Problem Statement

You are given a list of whole numbers nums.

Count the longest strictly increasing subsequences. Two subsequences are different when they pick different positions, even if their values match.

Input

An integer array nums.

Output

One number: how many longest strictly increasing subsequences there are.

Example 1

Input: nums = [1, 3, 5, 4, 7]

Output: 2

The longest length is 4. Two chains reach it: 1, 3, 4, 7 and 1, 3, 5, 7.

Example 2

Input: nums = [2, 2, 2, 2, 2]

Output: 5

The longest length is 1. Each single 2 is one chain, so there are five chains.

Constraints

• 1 ≤ nums.length ≤ 2000
• −10^6 ≤ nums[i] ≤ 10^6
• The answer is guaranteed to fit in a 32-bit signed integer.

Pattern Recognition Clues

Recognising these key signals in the problem description immediately identifies the pattern:

“number of the longest”

→ We count chains, not just find the length. Two subsequences that pick different positions count as different, even with equal values.

“strictly increasing”

→ Equal values cannot both be in the chain, so the check is <, not <=.

“subsequences”

→ Chains can skip numbers, so each end position starts its own count of chains.

Approach Complexity Ladder

Brute Force — list every subset of positionsO(N · 2^N)O(1)

Try all 2^N subsets of positions; keep the increasing ones and count the longest

Optimal — len and cnt for each endO(N^2)O(N)

For each end, keep its longest length and how many chains reach it; O(N²) time

Brute Force Intuition

Each subset of positions is one candidate subsequence. Check each subset: if its values are strictly increasing, record its length. Then count the subsets that have the biggest length.

Recognition rule for this sub-pattern:

O(n²): try every earlier j. O(n log n): binary search the tails array.

Remember this · DP — Longest Increasing Subsequence

Trigger

Longest chain where each item must be bigger/fit inside the previous one (numbers, pairs, envelopes, words).

Think

Does each element extend the best chain ending at some smaller earlier element?

Key question

Sort first? Then what does "increasing" mean for these items?

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