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 linear search completely — sequential search, sentinel search, searching with conditions, finding all occurrences, searching in linked lists and 2D arrays, when to use linear search vs alternatives, and the exact time and space complexity analysis.
What Is Linear Search?
Linear search (also called sequential search) is the simplest search algorithm. It examines every element in a sequence one by one, from start to end, until it finds the target element or exhausts all elements.
No preprocessing is required. The array can be sorted or unsorted, and any data type can be searched as long as equality comparison is possible.
Complete Analysis
Best Case — Ω(1)
Target is the first element:
Example
arr = [42, 7, 19, 3], target = 42
Step 1: arr[0] == 42 → return 0 immediately
Operations: 1 comparison
Worst Case — O(n)
Target is the last element or not present:
Example
arr = [7, 19, 3, 42], target = 99
Compare 7 ≠ 99, 19 ≠ 99, 3 ≠ 99, 42 ≠ 99 → return -1
Operations: n comparisons
Average Case — Θ(n)
Assuming target is equally likely at any position (or not present): Expected comparisons ≈ (n+1)/2 ≈ n/2 = Θ(n)
Variants
Find All Occurrences
Example
def linear_search_all(arr, target):
"""Return list of all indices where target appears."""
indices = []
for i, val in enumerate(arr):
if val == target:
indices.append(i)
return indices # Empty list if not found
arr = [1, 3, 5, 3, 7, 3, 9]
print(linear_search_all(arr, 3)) # [1, 3, 5]
print(linear_search_all(arr, 8)) # []
Search by Condition (Predicate Search)
Example
def find_first(arr, condition):
"""Find first element satisfying a condition. O(n)."""
for i, val in enumerate(arr):
if condition(val):
return i, val
return -1, None
arr = [8, 15, 3, 22, 7, 11, 19]
# Find first element > 10
idx, val = find_first(arr, lambda x: x > 10)
print(f"Index {idx}, Value {val}") # Index 1, Value 15
# Find first even number
idx, val = find_first(arr, lambda x: x % 2 == 0)
print(f"Index {idx}, Value {val}") # Index 0, Value 8
Sentinel Linear Search (Optimization)
Example
def sentinel_search(arr, target):
"""
Place target at the end to eliminate the bounds check inside the loop.
Reduces comparisons from 2n to n+1 in worst case.
Note: modifies arr temporarily.
"""
n = len(arr)
last = arr[n-1] # Save last element
arr[n-1] = target # Place sentinel
i = 0
while arr[i] != target: # Only ONE comparison per iteration instead of two
i += 1
arr[n-1] = last # Restore
if i < n-1 or arr[n-1] == target:
return i
return -1
Why it's faster: The standard loop checks both arr[i] != target AND i < n per iteration. The sentinel eliminates the bounds check by guaranteeing the target will always be found.
Searching in Different Data Structures
Linked List — O(n)
python example
def search_linked_list(head, target):
current = head
index = 0
while current:
if current.data == target:
return index
current = current.next
index += 1
return -12D Array — O(m×n)
Example
def search_2d(matrix, target):
"""Search in unsorted 2D array. O(m×n)."""
for i in range(len(matrix)):
for j in range(len(matrix[0])):
if matrix[i][j] == target:
return (i, j)
return None
# Optimized for row-sorted, column-sorted matrix: O(m+n)
def search_sorted_matrix(matrix, target):
"""
Matrix where rows and columns are sorted.
Start from top-right corner.
"""
if not matrix: return None
row, col = 0, len(matrix[0]) - 1
while row < len(matrix) and col >= 0:
if matrix[row][col] == target:
return (row, col)
elif matrix[row][col] > target:
col -= 1 # Too big, go left
else:
row += 1 # Too small, go down
return None
When to Use Linear Search
| Scenario | Use Linear Search? |
|---|---|
| Unsorted array, one-time search | ✅ Yes — no preprocessing cost |
| Small array (n < 50) | ✅ Yes — overhead of binary search not worth it |
| Array changes frequently | ✅ Yes — maintaining sorted order costs O(n) |
| Searching by complex condition | ✅ Yes — only option |
| Large sorted array, repeated searches | ❌ No — use binary search O(log n) |
| Fast lookup by key | ❌ No — use hash map O(1) |
Performance Comparison
| Linear search (worst) | 1,000,000 comparisons |
| Binary search | 20 comparisons |
| Hash map lookup | 1 comparison (average) |
| Linear search (worst) | 10 comparisons |
| Binary search | 4 comparisons |
| Hash map lookup | 1 comparison |
Java and C++ Equivalents
Example
// Java
int linearSearch(int[] arr, int target) {
for (int i = 0; i < arr.length; i++) {
if (arr[i] == target) return i;
}
return -1;
}
// Java Collections — built-in linear search
List<Integer> list = Arrays.asList(3, 1, 4, 1, 5);
int idx = list.indexOf(5); // Returns 4, or -1 if not found
Example
# Python built-in equivalents (all O(n) linear search internally)
arr = [3, 1, 4, 1, 5, 9, 2, 6]
idx = arr.index(5) # First occurrence, raises ValueError if not found
found = 5 in arr # Boolean membership test
count = arr.count(1) # Count occurrences
Summary
- Definition: Check every element one by one until found or exhausted
- Best: Ω(1) — target at index 0
- Average: Θ(n/2) = Θ(n)
- Worst: O(n) — target absent or at end
- Space: O(1) — no extra memory
- Use when: Array is unsorted, small, or changes frequently
- Optimize with: Sentinel (reduces loop overhead), early termination
*Next Lesson: Binary Search — The O(log n) search algorithm for sorted arrays.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Linear Search — Complete Guide with All Variants and Applications.
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, searching, linear, search
Related Data Structures & Algorithms Topics
Continue learning this concept
SearchingBinary Search — Complete Guide with All Variants and Proof of CorrectnessMaster binary search completely — the core algorithm with loop invariant proof, iterative and recursive implementations, all variants (leftmost, rightmost, lower bound, upper bound), binary search on answer space, and 10 classic interview problems.SearchingInterpolation Search — O(log log n) Search for Uniformly Distributed DataMaster interpolation search — how it improves on binary search using value-based position estimation, when it achieves O(log log n) vs O(n) worst case, the interpolation formula, comparison with binary search, and when to use each.SearchingSearch Algorithm Comparison — Choosing the Right Search for Every SituationA comprehensive comparison of all searching algorithms — linear search, binary search, interpolation search, ternary search, hash-based search, and tree-based search — with performance tables, use cases, and a decision guide for every scenario.SearchingTernary Search — Finding Maximum/Minimum of Unimodal FunctionsMaster ternary search — what it is, why it differs from binary search, how to find the maximum or minimum of a unimodal function in O(log n), ternary search on sorted arrays vs functions, comparison with binary search, and classic applications.SearchingSearching Interview Problems — 10 Classic Problems with Complete SolutionsSolve 10 classic searching interview problems — find in rotated sorted array, search in 2D matrix, find peak element, kth smallest in sorted matrix, find duplicate, median of two sorted arrays, capacity to ship packages, aggressive cows, and more.ArraysArray Searching — Linear Search, Binary Search, and Advanced TechniquesComplete guide to searching arrays — linear search with early termination, binary search with all variants, interpolation search, and exponential search. Includes full implementations, complexity analysis, and classic search-based interview problems.