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 string matching and similarity algorithms — Longest Common Subsequence (LCS) with DP, Edit Distance (Levenshtein), Longest Common Substring, Z-algorithm for pattern matching, and their applications in spell checking, DNA analysis, and diff utilities.
Longest Common Subsequence (LCS)
Definition: The longest sequence of characters that appears in both strings (not necessarily contiguous).
Example
s1 = "ABCBDAB"
s2 = "BDCABA"
LCS = "BCAB" or "BCBA" — length 4
DP Solution: O(m×n)
Example
def lcs_length(s1, s2):
"""
Find length of LCS using DP table.
dp[i][j] = LCS of s1[:i] and s2[:j]
"""
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 # Characters match
else:
dp[i][j] = max(dp[i-1][j], dp[i][j-1]) # Take best of skipping either
return dp[m][n]
def lcs_string(s1, s2):
"""Reconstruct the actual LCS string."""
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 sequence
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_length("ABCBDAB", "BDCABA")) # 4
print(lcs_string("ABCBDAB", "BDCABA")) # "BCAB"
Time: O(m×n), Space: O(m×n)
Space-optimized to O(min(m,n)):
Example
def lcs_space_optimized(s1, s2):
if len(s1) < len(s2): s1, s2 = s2, s1 # s1 is longer
prev = [0] * (len(s2) + 1)
for char in s1:
curr = [0] * (len(s2) + 1)
for j, c2 in enumerate(s2, 1):
curr[j] = prev[j-1] + 1 if char == c2 else max(prev[j], curr[j-1])
prev = curr
return prev[-1]
Longest Common Substring
Note: Different from LCS — substring must be contiguous.
Example
s1 = "ABCBDAB"
s2 = "BDCABA"
LCS (subsequence) = "BCAB" (length 4)
Longest Common Substring = "AB" or "BD" or "BA" (length 2)
Example
def longest_common_substring(s1, s2):
"""
Find length and value of longest common contiguous substring.
dp[i][j] = length of longest common substring ending at s1[i-1] and s2[j-1]
"""
m, n = len(s1), len(s2)
dp = [[0] * (n+1) for _ in range(m+1)]
max_len = 0
end_pos = 0 # End index in s1
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
if dp[i][j] > max_len:
max_len = dp[i][j]
end_pos = i
else:
dp[i][j] = 0 # Contiguous — reset on mismatch
return s1[end_pos - max_len : end_pos], max_len
print(longest_common_substring("ABCBDAB", "BDCABA")) # ('AB', 2) or similar
print(longest_common_substring("ABABC", "BABCAB")) # ('BABC', 4)
Z-Algorithm: O(n+m)
Builds a Z-array where Z[i] = length of longest substring starting at position i that is also a prefix of the string.
Example
def z_function(s):
"""
Compute Z-array for string s.
Z[i] = length of longest substring starting at s[i] matching prefix of s.
Z[0] = len(s) by convention.
Time: O(n), Space: O(n)
"""
n = len(s)
z = [0] * n
l, r = 0, 0
for i in range(1, n):
if i < r:
z[i] = min(r - i, z[i - l])
while i + z[i] < n and s[z[i]] == s[i + z[i]]:
z[i] += 1
if i + z[i] > r:
l, r = i, i + z[i]
z[0] = n # Convention
return z
def z_search(text, pattern):
"""
Search using Z-algorithm.
Concatenate: combined = pattern + "$" + text
Z values equal to m at indices > m indicate a match.
"""
m = len(pattern)
combined = pattern + "$" + text
z = z_function(combined)
matches = []
for i in range(m + 1, len(combined)):
if z[i] == m:
matches.append(i - m - 1) # Position in text
return matches
print(z_search("aabxaabxcaabxaabxay", "aabx")) # [0, 4, 10, 14]
Comparison Summary
| Problem | Algorithm | Time | Space |
|---|---|---|---|
| Pattern search | KMP, Z-algo | O(n+m) | O(m) |
| String similarity | Edit distance | O(mn) | O(mn) |
| Common subsequence | LCS DP | O(mn) | O(mn) |
| Common substring | DP | O(mn) | O(mn) |
| Fuzzy match | Edit distance | O(mn) | O(mn) |
Applications
| Spell checker | Edit distance between typed word and dictionary |
| DNA alignment | LCS of DNA sequences |
| Git diff | LCS-based to find changed lines |
| Autocomplete | Z-algorithm or KMP for prefix matching |
| Plagiarism | Longest common substring detection |
| Search engines | Multiple pattern matching (Aho-Corasick) |
*Next: Palindrome Problems*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for String Matching — Longest Common Subsequence, Edit Distance, and Z-Algorithm.
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, strings, string, matching
Related Data Structures & Algorithms Topics
Continue learning this concept
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