What is the brute force approach and complexity for Increasing Triplet Subsequence?

Check every triple of positions i < j < k with time complexity O(N^3) and space complexity O(1).

What is the optimal approach and complexity for Increasing Triplet Subsequence?

Track the smallest value so far, and the smallest middle candidate; a bigger value finishes a triple with optimal time complexity O(N) and space complexity O(1).

How do you identify the pattern for Increasing Triplet Subsequence?

Look for key signals in the prompt: "triple of indices i < j < k" indicates Only the order of the positions matters. The three numbers do not need to touch, so this is a subsequence of length 3.; "nums[i] < nums[j] < nums[k]" indicates A strictly increasing run of length 3. Equal numbers fail the check.; "return true or false" indicates We only need to know if a length-3 chain exists, so we can stop as soon as one is found..

Increasing Triplet Subsequence

Q: What is the optimal approach and complexity for Increasing Triplet Subsequence?

Track the smallest value so far, and the smallest middle candidate; a bigger value finishes a triple with optimal time complexity O(N) and space complexity O(1).

Q: How do you identify the pattern for Increasing Triplet Subsequence?

Look for key signals in the prompt: "triple of indices i < j < k" indicates Only the order of the positions matters. The three numbers do not need to touch, so this is a subsequence of length 3.; "nums[i] < nums[j] < nums[k]" indicates A strictly increasing run of length 3. Equal numbers fail the check.; "return true or false" indicates We only need to know if a length-3 chain exists, so we can stop as soon as one is found..

Medium·LeetCode #334 Pro OnlyBasic Reinforcement

LeetCode

Problem Statement

You are given a list of numbers nums.

Decide if there are three positions i < j < k, in that order, where nums[i] < nums[j] < nums[k]. The three numbers do not need to be next to each other.

Follow-up: can you do it in one pass with only a few variables?

Input

An integer array nums.

Output

true if such a triple exists, otherwise false.

Example 1

Input: nums = [1, 2, 3, 4, 5]

Output: true

Any three positions in order work, for example 1, 2, 3.

Example 2

Input: nums = [5, 4, 3, 2, 1]

Output: false

The numbers only go down, so no triple exists.

Example 3

Input: nums = [2, 1, 5, 0, 4, 6]

Output: true

The numbers 1, 4, 6 are at positions 1, 4, 5 and increase. The numbers 0, 4, 6 at positions 3, 4, 5 also work.

Constraints

• 1 ≤ nums.length ≤ 5 · 10^5
• −2^31 ≤ nums[i] ≤ 2^31 − 1

Pattern Recognition Clues

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

“triple of indices i < j < k”

→ Only the order of the positions matters. The three numbers do not need to touch, so this is a subsequence of length 3.

“nums[i] < nums[j] < nums[k]”

→ A strictly increasing run of length 3. Equal numbers fail the check.

“return true or false”

→ We only need to know if a length-3 chain exists, so we can stop as soon as one is found.

Approach Complexity Ladder

Brute Force — try every tripleO(N^3)O(1)

Check every triple of positions i < j < k

Better — prefix minimum and suffix maximumO(N)O(N)

For each middle item, compare it with the smallest value on its left and the largest on its right

Optimal — two variables: first and secondO(N)O(1)

Track the smallest value so far, and the smallest middle candidate; a bigger value finishes a triple

Brute Force Intuition

Use three nested loops to pick positions i < j < k, and test nums[i] < nums[j] < nums[k]. If any triple passes, the answer is true.

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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