Longest Increasing Subsequence (LIS) — Dynamic Programming (DP) Patterns

The Longest Increasing Subsequence (LIS) pattern applies when you need the Longest Increasing Subsequence (LIS) technique within the Dynamic Programming (DP) Patterns pattern. Its time complexity is O(n²) with the classic DP, O(n log n) with binary search (patience sorting) and space complexity O(n). It is used in 4 problems on Thita, including Longest Increasing Subsequence, Longest Increasing Subsequence II and Minimum Number of Removals to Make Mountain Array. Common variations are 1D Array (Word Break Style) and 2D Array (Edit Distance / Levenshtein Distance).

Find LIS using DP and binary search optimization. O(n log n) solution.

Longest Increasing Subsequence (LIS) is one of the 12 subpatterns of the Dynamic Programming (DP) Patterns pattern, which covers fibonacci, Kadane, knapsack, LCS, LIS, edit distance, grid paths, and word break. The whole pattern is about 5 hours of study. This subpattern is an important variation worth recognising on sight.

What Longest Increasing Subsequence (LIS) covers

Master DP patterns including Fibonacci, Kadane, knapsack, LCS, LIS, edit distance, and grid paths. Solve optimization problems with overlapping subproblems. Problems in this subpattern are usually searched for as longest increasing subsequence, LIS, patience sorting, binary search LIS, leetcode 300.

How to practise Longest Increasing Subsequence (LIS) on Thita.ai

Read the theory for Longest Increasing Subsequence (LIS), then work the problems attached to it in the browser editor. Your solution runs against the problem's test cases, and the AI coach offers a hint about the technique you are missing rather than a finished solution. Progress is tracked per subpattern, so the Dynamic Programming (DP) Patterns tracker shows this one as covered once you have solved its problems.

Other subpatterns in Dynamic Programming (DP) Patterns

• 1D Array (Fibonacci Style) — Solve problems with Fibonacci-like recurrence: climbing stairs, house robber, decode ways.
• 1D Array (Kadane's Algorithm for Max/Min Subarray) — Master Kadane's algorithm for finding maximum (or minimum) subarray sum in O(n) time.
• 1D Array (0/1 Knapsack Subset Sum Style) — Solve 0/1 knapsack, partition equal subset sum, and target sum problems.
• 1D Array (Coin Change / Unbounded Knapsack Style) — Solve coin change and unbounded knapsack problems where items can be reused.
• 1D Array (Word Break Style) — Determine if string can be segmented into dictionary words using DP.
• 2D Array (Unique Paths on Grid) — Count paths and find optimal paths in grids using 2D dynamic programming.
• 2D Array (Longest Common Subsequence - LCS) — Find longest common subsequence between two strings using 2D DP.
• 2D Array (Edit Distance / Levenshtein Distance) — Calculate minimum edit operations to transform one string to another using 2D DP.
• Stock problems — Solve stock trading problems with k transactions, cooldown, and transaction fees using DP.
• Catalan Numbers — Solve problems involving Catalan numbers: unique BSTs, parentheses combinations, polygon triangulation.
• Interval DP — Solve problems on intervals like burst balloons and matrix chain multiplication using interval DP.

Related DSA patterns

• Greedy Patterns — Interval scheduling, jump games, stock trading, and task scheduling.
• Backtracking Patterns — Subsets, permutations, combinations, N-Queens, and word search.
• Graph Traversal Patterns (DFS & BFS) — DFS/BFS traversal, topological sort, shortest paths, Union-Find, and MST.
• Bit Manipulation Patterns — XOR tricks, counting bits, power-of-two checks, and Hamming weight.

Where to go next

Longest Increasing Subsequence (LIS) is one lesson in a 16-pattern DSA course. If you are preparing end to end, work the interview-critical patterns first and use the pattern sheet as the checklist; if you are here for one technique, the Dynamic Programming (DP) Patterns guide is the shortest path back to the rest of it.

• The Dynamic Programming (DP) Patterns pattern guide
• All DSA patterns
• The DSA patterns sheet
• Coding practice