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 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.
Definition
A subsequence is a sequence derived from another sequence by deleting some (or no) elements without changing the order of the remaining elements.
| String | "ABCBDAB" |
| Valid subsequences | "A", "ABC", "BDB", "ABCB", "ABAB", "ABCBDAB", "" |
| NOT a subsequence | "BA" would require reordering |
The Longest Common Subsequence (LCS) of two strings is the longest sequence that is a subsequence of both.
Example
s1 = "ABCBDAB"
s2 = "BDCAB"
Common subsequences: "B", "BC", "BA", "BDA", "BCAB", "BDAB", "BCAB"
LCS: "BCAB" or "BDAB" — both have length 4
Complete Implementation
Example
def lcs_length(s1, s2):
"""
Find length of Longest Common Subsequence.
Time: O(m×n), Space: O(m×n)
"""
m, n = len(s1), len(s2)
dp = [[0]*(n+1) for _ in range(m+1)]
for i in range(1, m+1):
for j in range(1, n+1):
if s1[i-1] == s2[j-1]:
dp[i][j] = dp[i-1][j-1] + 1
else:
dp[i][j] = max(dp[i-1][j], dp[i][j-1])
return dp[m][n]
print(lcs_length("ABCBDAB", "BDCAB")) # 4
print(lcs_length("AGGTAB", "GXTXAYB")) # 4 (GTAB)
DP Table Visualization
For s1="ABCD", s2="ACBD":
| dp[4][4] = 3 | LCS length is 3 |
| dp[4][4]=3: s1[3]='D' ≠ s2[3]='D' | wait, 'D'='D'! → came from dp[3][3]+1=2+1=3 |
| dp[3][3]=2: s1[2]='C' vs s2[2]='B' | not equal → came from dp[2][3]=2 (max of dp[2][3],dp[3][2]) |
| dp[2][3]=2: s1[1]='B' vs s2[2]='B' | EQUAL → came from dp[1][2]+1=1+1=2 |
| dp[1][2]=1: s1[0]='A' vs s2[1]='C' | not equal → came from dp[0][2]=0 or dp[1][1]=1 → dp[1][1] |
| dp[1][1]=1: s1[0]='A' vs s2[0]='A' | EQUAL → came from dp[0][0]+1=0+1=1 |
Reconstruct the Actual LCS String
Example
def lcs_reconstruct(s1, s2):
"""
Find and return the actual LCS string.
Time: O(m×n), Space: O(m×n)
"""
m, n = len(s1), len(s2)
dp = [[0]*(n+1) for _ in range(m+1)]
for i in range(1, m+1):
for j in range(1, n+1):
if s1[i-1] == s2[j-1]:
dp[i][j] = dp[i-1][j-1] + 1
else:
dp[i][j] = max(dp[i-1][j], dp[i][j-1])
# Backtrack to find the string
result = []
i, j = m, n
while i > 0 and j > 0:
if s1[i-1] == s2[j-1]:
result.append(s1[i-1])
i -= 1; j -= 1
elif dp[i-1][j] > dp[i][j-1]:
i -= 1
else:
j -= 1
return ''.join(reversed(result))
print(lcs_reconstruct("ABCBDAB", "BDCAB")) # "BCAB" or "BDAB"
Space Optimization — O(n) Space
Example
def lcs_space_optimized(s1, s2):
"""
Only keep two rows: previous and current.
Time: O(m×n), Space: O(n)
"""
if len(s1) < len(s2):
s1, s2 = s2, s1 # Make s1 the longer string
m, n = len(s1), len(s2)
prev = [0] * (n + 1)
for i in range(1, m+1):
curr = [0] * (n+1)
for j in range(1, n+1):
if s1[i-1] == s2[j-1]:
curr[j] = prev[j-1] + 1
else:
curr[j] = max(prev[j], curr[j-1])
prev = curr
return prev[n]
print(lcs_space_optimized("ABCBDAB", "BDCAB")) # 4
Important LCS Variants
Variant 1 — Longest Common Substring (Contiguous)
Unlike subsequence, substring must be contiguous:
Example
def longest_common_substring(s1, s2):
"""
Longest contiguous substring common to both strings.
dp[i][j] = length of longest common suffix of s1[:i] and s2[:j]
Time: O(m×n), Space: O(m×n) reducible to O(n)
"""
m, n = len(s1), len(s2)
dp = [[0]*(n+1) for _ in range(m+1)]
max_length = 0
end_pos = 0 # End index in s1 of longest common substring
for i in range(1, m+1):
for j in range(1, n+1):
if s1[i-1] == s2[j-1]:
dp[i][j] = dp[i-1][j-1] + 1 # Extend from previous
if dp[i][j] > max_length:
max_length = dp[i][j]
end_pos = i
else:
dp[i][j] = 0 # Reset — must be contiguous
return s1[end_pos-max_length:end_pos], max_length
print(longest_common_substring("ABAB", "BABA")) # ('BAB', 3) or ('ABA', 3)
print(longest_common_substring("GeeksforGeeks", "GeeksQuiz")) # ('Geeks', 5)
Key difference: When characters don't match, LCS variant takes max(left, up), but substring variant resets to 0.
Variant 2 — Edit Distance (Levenshtein)
Minimum operations (insert, delete, substitute) to transform s1 into s2:
Example
def edit_distance(s1, s2):
"""
dp[i][j] = min edits to transform s1[:i] into s2[:j]
Recurrence:
If s1[i-1] == s2[j-1]: dp[i][j] = dp[i-1][j-1] (no edit needed)
Else: dp[i][j] = 1 + min(
dp[i-1][j], # Delete from s1
dp[i][j-1], # Insert into s1
dp[i-1][j-1] # Substitute
)
"""
m, n = len(s1), len(s2)
dp = [[0]*(n+1) for _ in range(m+1)]
for i in range(m+1): dp[i][0] = i # Delete all i chars
for j in range(n+1): dp[0][j] = j # Insert all j chars
for i in range(1, m+1):
for j in range(1, n+1):
if s1[i-1] == s2[j-1]:
dp[i][j] = dp[i-1][j-1]
else:
dp[i][j] = 1 + min(dp[i-1][j], dp[i][j-1], dp[i-1][j-1])
return dp[m][n]
print(edit_distance("kitten", "sitting")) # 3
print(edit_distance("sunday", "saturday")) # 3
print(edit_distance("", "abc")) # 3
Variant 3 — Shortest Common Supersequence
Find the shortest string that has both s1 and s2 as subsequences:
Example
def shortest_common_supersequence(s1, s2):
"""
SCS length = len(s1) + len(s2) - LCS(s1, s2)
Because: include all of both, but LCS is counted only once.
"""
m, n = len(s1), len(s2)
dp = [[0]*(n+1) for _ in range(m+1)]
for i in range(1,m+1):
for j in range(1,n+1):
if s1[i-1]==s2[j-1]: dp[i][j]=dp[i-1][j-1]+1
else: dp[i][j]=max(dp[i-1][j],dp[i][j-1])
lcs_len = dp[m][n]
scs_len = m + n - lcs_len
# Reconstruct the actual SCS
result = []
i, j = m, n
while i > 0 and j > 0:
if s1[i-1] == s2[j-1]:
result.append(s1[i-1])
i -= 1; j -= 1
elif dp[i-1][j] > dp[i][j-1]:
result.append(s1[i-1]); i -= 1
else:
result.append(s2[j-1]); j -= 1
while i > 0: result.append(s1[i-1]); i -= 1
while j > 0: result.append(s2[j-1]); j -= 1
return ''.join(reversed(result)), scs_len
scs, length = shortest_common_supersequence("AGGTAB", "GXTXAYB")
print(scs, length) # Length = 9
Variant 4 — Count LCS (Number of LCS of Maximum Length)
Example
def count_lcs(s1, s2):
"""Count how many distinct LCS strings exist."""
m, n = len(s1), len(s2)
dp = [[0]*(n+1) for _ in range(m+1)]
count = [[0]*(n+1) for _ in range(m+1)]
for i in range(m+1): count[i][0] = 1
for j in range(n+1): count[0][j] = 1
for i in range(1, m+1):
for j in range(1, n+1):
if s1[i-1] == s2[j-1]:
dp[i][j] = dp[i-1][j-1] + 1
count[i][j] = count[i-1][j-1]
elif dp[i-1][j] > dp[i][j-1]:
dp[i][j] = dp[i-1][j]
count[i][j] = count[i-1][j]
elif dp[i-1][j] < dp[i][j-1]:
dp[i][j] = dp[i][j-1]
count[i][j] = count[i][j-1]
else:
dp[i][j] = dp[i-1][j]
count[i][j] = count[i-1][j] + count[i][j-1]
# Subtract overlap if dp[i-1][j-1] == dp[i][j]
if dp[i-1][j-1] == dp[i][j]:
count[i][j] -= count[i-1][j-1]
return dp[m][n], count[m][n]
Real-World Applications of LCS
| 1. Git diff | Shows lines added/removed between two versions of a file |
| 2. DNA Sequence Alignment | Compare two DNA sequences to find shared genes |
| 3. Spell Checkers | Compute edit distance between typed word and dictionary |
| 4. Plagiarism Detection | Find longest common sections in two documents |
| 5. File Comparison Tools | diff, merge, patch utilities |
Complexity Summary
| Problem | Time | Space |
|---|---|---|
| LCS Length | O(m×n) | O(m×n) |
| LCS with reconstruction | O(m×n) | O(m×n) |
| LCS space-optimized | O(m×n) | O(n) |
| Longest Common Substring | O(m×n) | O(m×n) |
| Edit Distance | O(m×n) | O(m×n) |
| Shortest Common Supersequence | O(m×n) | O(m×n) |
Summary
LCS is the foundation of all string DP problems:
- LCS:
dp[i][j] = dp[i-1][j-1]+1if match, elsemax(left, up) - Substring:
dp[i][j] = dp[i-1][j-1]+1if match, else 0 (no max — reset!) - Edit Distance: 3 choices: delete, insert, substitute — take min
- SCS: length = m + n - LCS (combine but share the LCS)
The 2D DP table structure is identical for all these — only the recurrence changes.
*Next Lesson: Longest Increasing Subsequence*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Longest Common Subsequence — Complete Guide with DP Table and String Variants.
Interview Use
Prepare one clear explanation, one practical example, and one common mistake for this Data Structures & Algorithms topic.
Search Terms
data-structures-algorithms, data structures & algorithms, data, structures, algorithms, dynamic, programming, longest
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