Longest Increasing Subsequence in DSA Typescript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Longest Increasing Subsequence (LIS) in an array?

The Longest Increasing Subsequence is the longest sequence of numbers in the array where each number is strictly greater than the previous one, and the sequence does not need to be contiguous.

intermediate

What is the time complexity of the classic dynamic programming approach to find LIS?

The classic dynamic programming approach to find LIS has a time complexity of O(n²), where n is the length of the array.

beginner

In LIS problem, what does the DP array represent in the dynamic programming solution?

The DP array stores the length of the longest increasing subsequence ending at each index of the input array.

advanced

What is the optimized time complexity to find LIS using binary search?

Using binary search with a patience sorting technique, the LIS can be found in O(n log n) time.

beginner

Why does the LIS not require the subsequence to be contiguous?

Because the problem asks for a subsequence, which means elements can be taken from the array in order but not necessarily next to each other, allowing skipping elements to form a strictly increasing sequence.

What does the Longest Increasing Subsequence represent in an array?

A The longest sequence of equal numbers

B The longest contiguous increasing subarray

C The shortest increasing subsequence

D The longest sequence of strictly increasing numbers, not necessarily contiguous

LIS is about the longest strictly increasing sequence that can skip elements, so it is not necessarily contiguous.

What is the time complexity of the naive dynamic programming solution for LIS?

A O(n log n)

B O(n²)

C O(n)

D O(log n)

The classic DP solution compares each element with all previous ones, resulting in O(n²) time.

Which data structure or technique helps optimize LIS to O(n log n)?

A Binary Search

B Stack

C Queue

D Hash Map

Binary search is used on a helper array to efficiently find positions, reducing complexity to O(n log n).

In the DP approach, what does dp[i] represent?

A Length of LIS ending at index i

B Length of LIS starting at index i

C Value of the element at index i

D Index of the next element in LIS

dp[i] stores the length of the longest increasing subsequence that ends exactly at index i.

Can the LIS include elements that are not next to each other in the original array?

A Only if elements are decreasing

B No, it must be contiguous

C Yes, because it is a subsequence

D Only if elements are equal

A subsequence allows skipping elements, so LIS can include non-adjacent elements.

Explain how the dynamic programming approach finds the Longest Increasing Subsequence.

Think about building solutions for smaller parts and combining them.

Describe the optimization that reduces LIS time complexity from O(n²) to O(n log n).

Consider how binary search helps find positions quickly.