DSA Notes
HubAlgorithm vs Data Structure — Key Differences, Definitions, and How They Work TogetherApplications of DSA — How Data Structures and Algorithms Power Real-World SoftwareTime and Space Complexity — A Complete BeginnerWhat is DSA? — A Complete BeginnerWhy Learn DSA? — Career Benefits, Real-World Impact, and Long-Term ValueAsymptotic Analysis — Complete Framework with Recurrence Relations and Master TheoremBest, Average, and Worst Case Analysis — A Practical Guide for Every AlgorithmBig O Notation — Complete Guide with Formal Definition, Rules, and ExamplesBig Omega Notation — Lower Bounds, Best Case, and Formal DefinitionBig Theta (Θ) Notation — Tight Bounds and Exact Growth RatesSpace Complexity — Memory Analysis, Auxiliary Space, and Time-Space TradeoffsTime Complexity Examples — 25 Worked Problems from Basic to AdvancedArray Interview Problems — 15 Classic Problems with Complete SolutionsArray Operations — Every Operation Explained with Code, Complexity, and Edge CasesArray Rotation — Algorithms, Techniques, and Interview ProblemsArray Searching — Linear Search, Binary Search, and Advanced TechniquesArrays — The Complete Introduction to the Most Fundamental Data StructureDynamic Arrays — How Python Lists and ArrayLists Work InternallyMultidimensional Arrays — 2D Arrays, Matrices, and Memory LayoutSliding Window Technique — Complete Guide with 10 Classic ProblemsAnagram Problems — Detection, Grouping, and Sliding Window VariationsPalindrome Problems — Check, Longest, Count, and Expand Around CenterPattern Searching — KMP, Rabin-Karp, and Boyer-Moore AlgorithmsString Interview Problems — 10 Classic Problems with Complete SolutionsString Matching — Longest Common Subsequence, Edit Distance, and Z-AlgorithmString Operations — Complete Reference with Complexity and ImplementationStrings — Complete Introduction to String Data Structure and OperationsCircular Linked List — Structure, Operations, and ApplicationsDoubly Linked List — Complete Implementation with Bidirectional OperationsLinked List Interview Problems — 12 Classic Problems with Complete SolutionsLinked Lists — A Complete Introduction with Memory Model and TypesLinked List Operations — Every Technique with Code and ComplexityReverse Linked List — All Approaches from Basic to AdvancedSingly Linked List — Complete Implementation with All OperationsBalanced Parentheses — Stack-Based Validation with All VariantsInfix, Prefix, and Postfix Expressions — Stack-Based Conversion and EvaluationStack Implementation — Three Complete Implementations with Edge CasesStack Interview Problems — 10 Classic Problems with Complete SolutionsStack — Complete Introduction to LIFO Data StructureStack Using Arrays — Memory Layout, Capacity Management, and PerformanceStack Using Linked List — Implementation, Analysis, and When to Use ItCircular Queue — The Ring Buffer Explained with ImplementationDeque — Double-Ended Queue with Complete Implementation and ApplicationsPriority Queue — Complete Guide from Concept to ImplementationQueue Applications — BFS, Level-Order Traversal, 01-BFS, and MoreQueue Implementation — Array, Linked List, and Two-Stack ApproachesQueue Interview Problems — 10 Classic Problems with Complete SolutionsQueue — Complete Introduction to the FIFO Data StructureAVL Tree - Self-Balancing Binary Search TreeBinary Search Tree (BST)Binary Tree - Structure and OperationsFenwick Tree (Binary Indexed Tree)Red-Black Tree - Another Balanced BSTSegment Tree - Range Query Data StructureTree Interview Problems CollectionTree Data Structure FundamentalsTree Traversals - Inorder, Preorder, Postorder, Level OrderHeap — Complete Introduction to the Priority-Based Tree Data StructureHeap Operations — Complete Reference with Complexity AnalysisHeap Problems — 8 Classic Interview Problems with Complete SolutionsMax Heap — Complete Implementation and ApplicationsMin Heap — Complete Implementation with All OperationsPriority Queue Using Heap — Implementation, Applications, and PatternsBellman-Ford Algorithm — Shortest Path with Negative Weights and Cycle DetectionBFS Algorithm — Breadth-First Search with All Applications and VariantsDFS Algorithm — Depth-First Search with Cycle Detection and All ApplicationsDijkstraFloyd-Warshall Algorithm — All-Pairs Shortest Paths with Complete AnalysisGraph Interview Problems — 12 Classic Problems with Complete SolutionsGraph — Complete Introduction to the Most Powerful Data StructureGraph Representation — Adjacency List, Matrix, Edge List with Complete AnalysisKruskalMinimum Spanning Tree — Definition, Properties, and ApplicationsPrimShortest Path Algorithms — Complete Comparison and Selection GuideTopological Sorting — DFS and BFS (KahnUnion-Find (DSU) in Graphs — Connected Components and Dynamic ConnectivitySeparate Chaining — Complete Implementation and AnalysisCollision Handling — Chaining vs Open Addressing Complete ComparisonHash Functions — Design, Properties, and Common ImplementationsHashing — Complete Introduction to Hash Tables and Hash FunctionsHashing Problems — 10 Classic Interview Problems with Complete SolutionsOpen Addressing — Linear Probing, Quadratic Probing, and Double HashingBubble Sort — Complete Analysis with Optimizations and When to Use ItCounting Sort — O(n+k) Sorting That Breaks the Comparison Lower BoundHeap Sort — Guaranteed O(n log n) with O(1) SpaceInsertion Sort — The O(n) Best Case Sort That Powers PythonMerge Sort — Complete Guide to the Guaranteed O(n log n) Stable SortQuick Sort — The Fastest Practical Sorting Algorithm with Complete AnalysisRadix Sort — O(nk) Digit-by-Digit Sorting for Large Integer RangesSelection Sort — Complete Analysis with Swaps Minimization PropertySorting Algorithm Comparison — The Complete Decision GuideSorting — Complete Introduction to Sorting AlgorithmsBinary Search — Complete Guide with All Variants and Proof of CorrectnessInterpolation Search — O(log log n) Search for Uniformly Distributed DataLinear Search — Complete Guide with All Variants and ApplicationsSearch Algorithm Comparison — Choosing the Right Search for Every SituationSearching Interview Problems — 10 Classic Problems with Complete SolutionsTernary Search — Finding Maximum/Minimum of Unimodal FunctionsBacktracking Introduction — The Recursive Exploration with UndoRecursion Basics — The Complete BeginnerRecursion Problems — 12 Classic Problems from Basic to AdvancedRecursion Tree — Visualizing Recursion for Complexity AnalysisRecursion vs Iteration — When to Use Each and How to ConvertTail Recursion — Definition, Optimization, and Converting to IterationDP Interview Problems — 12 Classic Problems with Pattern Recognition GuideDynamic Programming — Complete Introduction with Properties and Problem IdentificationFibonacci DP — From Exponential to Linear with Memoization and Matrix ExponentiationKnapsack Problem — 0/1, Unbounded, and Fractional with Complete AnalysisLongest Common Subsequence — Complete Guide with DP Table and String VariantsLongest Increasing Subsequence — O(n log n) with Patience SortingMatrix Chain Multiplication — Interval DP with Complete AnalysisMemoization — Top-Down Dynamic Programming with CachingTabulation — Bottom-Up Dynamic Programming with Complete Table BuildingActivity Selection Problem — Greedy Scheduling with Proof of CorrectnessFractional Knapsack — Greedy by Value Density with ProofGreedy Interview Problems — 10 Classic Problems with Complete SolutionsGreedy Algorithms — Complete Introduction with Proof TechniquesHuffman Coding — Optimal Prefix-Free Data CompressionJob Sequencing Problem — Greedy Scheduling with DeadlinesBacktracking — The Complete Guide to Recursive Exploration with UndoBacktracking Problems — 12 Classic Interview Problems with Complete SolutionsN-Queens Problem — Complete Solution with Backtracking and OptimizationRat in a Maze — Backtracking Path Finding with All SolutionsSubset Generation — All Subsets, Combinations, and Partitions with BacktrackingSudoku Solver — Backtracking with Constraint PropagationCodeforces Guide — Complete Strategy to Reach 1800+ RatingCommon CP Techniques — Essential Algorithms Every Competitive Programmer Must KnowContest Strategies — How to Perform Your Best in Competitive Programming ContestsCompetitive Programming — Complete Introduction for BeginnersFast Input/Output — Why I/O Speed Matters and Complete Templates for C++ and PythonLeetCode Guide — Complete Strategy to Crack FAANG InterviewsBit Manipulation — Complete Guide with All Tricks and Interview ProblemsBloom Filter — Space-Efficient Probabilistic Membership TestingComputational Geometry — Points, Lines, Convex Hull, and Geometric AlgorithmsDisjoint Set Union (DSU) — Union-Find with Path Compression and Union by RankRandomized Algorithms — Las Vegas, Monte Carlo, and Probabilistic TechniquesSkip List — The Probabilistic Data Structure that Competes with Balanced BSTsSuffix Array — Construction, LCP Array, and String Search ApplicationsTrie Data Structure — Complete Guide with Implementation and ApplicationsAutocomplete System Using TrieLibrary Management System Using DSARoute Finder Using GraphsMini Search Engine ProjectSocial Network Analysis Using GraphsEssential Coding Patterns for InterviewsDSA Cheatsheet: Quick ReferenceEssential DSA Interview QuestionsFAANG DSA Sheet: Company-Specific Patterns and ProblemsProblem Solving Strategies for Coding InterviewsComplexity CheatsheetImportant Algorithms SummaryBest Practice Platforms for DSARecommended Books for DSA
Master Longest Increasing Subsequence (LIS) completely — the O(n²) DP solution, the O(n log n) patience sorting algorithm, reconstructing the actual LIS, variants including longest non-decreasing subsequence and bitonic subsequence, and applications in scheduling and sequence analysis.
Definition
The Longest Increasing Subsequence (LIS) of an array is the longest subsequence where every element is strictly greater than the previous element.
Example
Array: [10, 9, 2, 5, 3, 7, 101, 18]
Some increasing subsequences:
[2, 3, 7, 101] length 4
[2, 5, 7, 101] length 4
[2, 3, 7, 18] length 4
[10, 101] length 2
[9, 101] length 2
LIS length: 4
Solution 2 — O(n log n) with Patience Sorting
The key insight: maintain a tails array where tails[i] = smallest tail element of all increasing subsequences of length i+1.
Example
import bisect
def lis_nlogn(nums):
"""
O(n log n) using binary search.
tails[i] = smallest tail of all IS of length i+1.
For each num: binary search for its position in tails.
"""
tails = []
for num in nums:
# Find leftmost position where tails[pos] >= num
pos = bisect.bisect_left(tails, num)
if pos == len(tails):
tails.append(num) # num extends the longest IS so far
else:
tails[pos] = num # Replace with smaller tail (greedy)
return len(tails)
print(lis_nlogn([10, 9, 2, 5, 3, 7, 101, 18])) # 4
print(lis_nlogn([4, 10, 4, 3, 8, 9])) # 3
Step-by-step trace for [3, 1, 4, 1, 5, 9, 2, 6]:
| Start | tails = [] |
| num=3: tails=[] | append → tails=[3] |
| num=1: bisect_left([3],1)=0 | replace → tails=[1] |
| num=4: bisect_left([1],4)=1 | append → tails=[1,4] |
| num=1: bisect_left([1,4],1)=0 | replace → tails=[1,4] |
| num=5: bisect_left([1,4],5)=2 | append → tails=[1,4,5] |
| num=9: bisect_left([1,4,5],9)=3 | append → tails=[1,4,5,9] |
| num=2: bisect_left([1,4,5,9],2)=1 | replace → tails=[1,2,5,9] |
| num=6: bisect_left([1,2,5,9],6)=3 | replace → tails=[1,2,5,6] |
| len(tails) = 4 | LIS length = 4 ✓ |
| Note | tails is NOT the actual LIS. It's a clever tool for counting. |
Why does this work?
tailsis always sorted (invariant)tails[i]is the optimal (smallest) ending for IS of length i+1- Binary search finds where the current number fits or improves
- Replacing a larger tail with a smaller one maintains the invariant and allows future elements to extend longer sequences
Reconstruct the Actual LIS
Example
import bisect
def lis_reconstruct(nums):
"""
Find the actual LIS, not just its length.
Uses parent tracking alongside the patience sort.
"""
if not nums: return []
n = len(nums)
tails = []
tails_indices = [] # Index in nums of each tail
parent = [-1] * n
for i, num in enumerate(nums):
pos = bisect.bisect_left(tails, num)
if pos == len(tails):
tails.append(num)
tails_indices.append(i)
else:
tails[pos] = num
tails_indices[pos] = i
# Set parent: this element extends the IS ending at tails[pos-1]
parent[i] = tails_indices[pos-1] if pos > 0 else -1
# Reconstruct by following parent pointers from the last tail
result = []
idx = tails_indices[-1]
while idx != -1:
result.append(nums[idx])
idx = parent[idx]
return list(reversed(result))
print(lis_reconstruct([10, 9, 2, 5, 3, 7, 101, 18]))
# [2, 3, 7, 101] or similar valid LIS
LIS Variants
Non-Decreasing (≥ instead of >)
Example
def longest_non_decreasing(nums):
"""LIS where equal elements are allowed."""
tails = []
for num in nums:
pos = bisect.bisect_right(tails, num) # Use bisect_right!
if pos == len(tails): tails.append(num)
else: tails[pos] = num
return len(tails)
print(longest_non_decreasing([1, 3, 3, 5])) # 4 (the whole array)
print(lis_nlogn([1, 3, 3, 5])) # 3 (skip one 3)
Longest Decreasing Subsequence
Example
def longest_decreasing(nums):
"""Reverse the array and find LIS."""
return lis_nlogn(nums[::-1])
print(longest_decreasing([5, 3, 4, 2, 1])) # 4 ([5,3,2,1])
Longest Bitonic Subsequence
A bitonic subsequence first increases then decreases.
Example
def longest_bitonic(nums):
"""
For each index i, find:
- LIS ending at i (lis[i])
- LDS starting at i (lds[i]) = LIS of reversed array ending at i
Bitonic through i has length lis[i] + lds[i] - 1
"""
n = len(nums)
# LIS ending at each index
lis = [1] * n
for i in range(1, n):
for j in range(i):
if nums[j] < nums[i]:
lis[i] = max(lis[i], lis[j]+1)
# LDS starting at each index (LIS from the right)
lds = [1] * n
for i in range(n-2, -1, -1):
for j in range(i+1, n):
if nums[j] < nums[i]:
lds[i] = max(lds[i], lds[j]+1)
return max(lis[i] + lds[i] - 1 for i in range(n))
print(longest_bitonic([1, 11, 2, 10, 4, 5, 2, 1])) # 6 ([1,2,10,4,2,1])
Number of LIS
Example
def count_lis(nums):
"""Count how many distinct LIS exist."""
n = len(nums)
length = [1] * n
count = [1] * n
for i in range(1, n):
for j in range(i):
if nums[j] < nums[i]:
if length[j]+1 > length[i]:
length[i] = length[j]+1
count[i] = count[j]
elif length[j]+1 == length[i]:
count[i] += count[j]
max_len = max(length)
return sum(c for l, c in zip(length, count) if l == max_len)
print(count_lis([1, 3, 5, 4, 7])) # 2 ([1,3,5,7] and [1,3,4,7])
print(count_lis([2, 2, 2, 2, 2])) # 5 (five length-1 LIS)
Classic Interview Problem — Russian Dolls / Box Stacking
LIS on pairs (sort by one dimension, find LIS on the other):
Example
def max_envelopes(envelopes):
"""
Each envelope [w,h]. Find max envelopes you can nest (like Russian dolls).
Sort by width ascending, then height DESCENDING.
Then find LIS on heights only.
(Descending height ensures we don't count same-width envelopes in LIS)
"""
envelopes.sort(key=lambda x: (x[0], -x[1]))
heights = [h for _, h in envelopes]
return lis_nlogn(heights)
print(max_envelopes([[5,4],[6,4],[6,7],[2,3]])) # 3 ([2,3],[5,4],[6,7])
Complexity Summary
| Approach | Time | Space | Notes |
|---|---|---|---|
| O(n²) DP | O(n²) | O(n) | Simple, good for n ≤ 1000 |
| O(n log n) patience | O(n log n) | O(n) | For n ≤ 10^5 or more |
| With reconstruction | O(n log n) | O(n) | Needs parent array |
| Bitonic | O(n²) | O(n) | Run LIS/LDS for each index |
Summary
LIS is one of the most important DP problems:
- O(n²) solution:
dp[i] = max(dp[j]+1 for j<i where nums[j]<nums[i]) - O(n log n) solution: Maintain sorted tails array, binary search for position
- Key insight for O(n log n): Replacing a larger tail with a smaller one is always beneficial — it gives future elements more chances to extend the subsequence
Common interview appearance: "Scheduling" problems that reduce to LIS, Russian dolls, envelope nesting, and any problem requiring "select a chain where each element satisfies a condition relative to the previous one."
*Next Lesson: Matrix Chain Multiplication*
Continue the lesson path
Study next in this module
01Dynamic ProgrammingLongest Common Subsequence — Complete Guide with DP Table and String VariantsMaster Longest Common Subsequence (LCS) completely — definition with clear examples, 2D DP table construction, string reconstruction, space optimization, and all important variants including Longest Common Substring, Edit Distance, Shortest Common Supersequence, and real-world applications.02Dynamic ProgrammingMatrix Chain Multiplication — Interval DP with Complete AnalysisMaster matrix chain multiplication as the gateway to interval DP — why multiplication order affects cost, the O(n³) interval DP solution, building the DP table from small intervals to large, reconstructing optimal parenthesization, and other interval DP problems like burst balloons and palindrome partitioning.03Dynamic ProgrammingKnapsack Problem — 0/1, Unbounded, and Fractional with Complete AnalysisMaster all knapsack variants — 0/1 knapsack with 2D and 1D DP tables, unbounded knapsack with reusable items, fractional knapsack with greedy, subset sum, partition equal subset sum, and the general framework for recognizing knapsack problems in interviews.04Dynamic ProgrammingMemoization — Top-Down Dynamic Programming with CachingMaster memoization completely — what it is, how it transforms exponential recursion to polynomial time, implementing memo dictionaries in Python, using lru_cache and functools.cache, multi-dimensional memoization, when memoization is better than tabulation, and common pitfalls.