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
Complete 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.
Introduction
Searching is one of the most common operations in all of computing. Every time a database responds to a query, every time a search engine ranks results, every time your phone looks up a contact — a search algorithm is running.
For arrays specifically, the choice of search algorithm depends on one critical question: Is the array sorted?
- Unsorted array: Linear search (O(n)) — no way to do better without additional structure
- Sorted array: Binary search (O(log n)) — one of the most elegant algorithms in computer science
This lesson covers both in complete depth, along with advanced variants and the interview problems that test them.
Binary Search
Binary search is one of the most important algorithms in computer science. It works by repeatedly halving the search range — eliminating half the remaining candidates with each comparison.
Requirement: The array must be sorted.
Core Algorithm — Iterative
Example
def binary_search(arr, target):
"""
Search for target in sorted array arr.
Returns index if found, -1 if not found.
Invariant: If target exists, it must be in arr[left..right].
"""
left = 0
right = len(arr) - 1
while left <= right:
# Compute midpoint — avoid overflow: (left + right) // 2
# In Python, integers don't overflow, but in C/Java use:
# mid = left + (right - left) // 2
mid = (left + right) // 2
if arr[mid] == target:
return mid # Found
elif arr[mid] < target:
left = mid + 1 # Target is in the RIGHT half
# arr[mid] is too small, so discard left half
else: # arr[mid] > target
right = mid - 1 # Target is in the LEFT half
# arr[mid] is too large, so discard right half
return -1 # left > right: search space empty, target not found
Step-by-step trace:
| arr[5] = 23 = target | found! Return 5 |
| arr[5] = 23 < 38 | left = 6 |
| arr[8] = 56 > 38 | right = 7 |
| arr[6] = 38 = target | found! Return 6 |
| Iteration 1 | mid=5, arr[5]=23 < 30, left=6 |
| Iteration 2 | mid=8, arr[8]=56 > 30, right=7 |
| Iteration 3 | mid=6, arr[6]=38 > 30, right=5 |
| left(6) > right(5) | return -1 |
Recursive Implementation
Example
def binary_search_recursive(arr, target, left=None, right=None):
"""Recursive binary search — O(log n) time, O(log n) space (call stack)."""
if left is None:
left = 0
if right is None:
right = len(arr) - 1
if left > right:
return -1 # Base case: search space empty
mid = (left + right) // 2
if arr[mid] == target:
return mid
elif arr[mid] < target:
return binary_search_recursive(arr, target, mid + 1, right)
else:
return binary_search_recursive(arr, target, left, mid - 1)
Note: The iterative version is preferred in practice because it uses O(1) space vs. O(log n) stack space for the recursive version.
Complexity — Binary Search:
| Case | Comparisons | Notation |
|---|---|---|
| Best | 1 | Ω(1) — target at exact midpoint |
| Average | log n - 1 | Θ(log n) |
| Worst | ⌊log₂ n⌋ + 1 | O(log n) |
| Space (iterative) | 1 | O(1) |
| Space (recursive) | log n | O(log n) |
Power of binary search:
| n | O(n) comparisons | O(log n) comparisons |
|---|---|---|
| 1,000 | 1,000 | 10 |
| 1,000,000 | 1,000,000 | 20 |
| 1,000,000,000 | 1,000,000,000 | 30 |
Binary Search Variants
The standard binary search finds one occurrence. Real problems often need variations.
Find Leftmost (First) Occurrence
Example
def binary_search_leftmost(arr, target):
"""
Find the index of the FIRST occurrence of target.
Returns -1 if target not present.
"""
left, right = 0, len(arr) - 1
result = -1
while left <= right:
mid = (left + right) // 2
if arr[mid] == target:
result = mid # Record this match
right = mid - 1 # But keep searching LEFT for earlier occurrences
elif arr[mid] < target:
left = mid + 1
else:
right = mid - 1
return result
arr = [1, 2, 2, 2, 3, 4, 5]
print(binary_search_leftmost(arr, 2)) # 1 (first occurrence at index 1)
Find Rightmost (Last) Occurrence
Example
def binary_search_rightmost(arr, target):
"""Find the index of the LAST occurrence of target."""
left, right = 0, len(arr) - 1
result = -1
while left <= right:
mid = (left + right) // 2
if arr[mid] == target:
result = mid # Record this match
left = mid + 1 # Keep searching RIGHT for later occurrences
elif arr[mid] < target:
left = mid + 1
else:
right = mid - 1
return result
arr = [1, 2, 2, 2, 3, 4, 5]
print(binary_search_rightmost(arr, 2)) # 3 (last occurrence at index 3)
Count Occurrences
Example
def count_occurrences(arr, target):
"""Count how many times target appears in sorted arr."""
first = binary_search_leftmost(arr, target)
if first == -1:
return 0
last = binary_search_rightmost(arr, target)
return last - first + 1
arr = [1, 2, 2, 2, 3, 4, 5]
print(count_occurrences(arr, 2)) # 3
print(count_occurrences(arr, 6)) # 0
Find Insertion Position (Lower Bound)
Example
def lower_bound(arr, target):
"""
Find the leftmost position where target can be inserted
to keep the array sorted (i.e., first index where arr[i] >= target).
"""
left, right = 0, len(arr) # right = len(arr) allows inserting at end
while left < right: # Note: strict less than (no =)
mid = (left + right) // 2
if arr[mid] < target:
left = mid + 1
else:
right = mid
return left
arr = [1, 3, 5, 7, 9]
print(lower_bound(arr, 5)) # 2 (arr[2] = 5, first position >= 5)
print(lower_bound(arr, 6)) # 3 (arr[3] = 7, first position >= 6)
print(lower_bound(arr, 0)) # 0 (insert before everything)
print(lower_bound(arr, 10)) # 5 (insert after everything)
Find Upper Bound
Example
def upper_bound(arr, target):
"""
Find the rightmost position where target can be inserted
(first index where arr[i] > target).
"""
left, right = 0, len(arr)
while left < right:
mid = (left + right) // 2
if arr[mid] <= target: # Note: <= instead of <
left = mid + 1
else:
right = mid
return left
arr = [1, 3, 5, 5, 5, 7, 9]
print(upper_bound(arr, 5)) # 5 (first index after all 5s)
print(lower_bound(arr, 5)) # 2 (first index of 5)
# Count of 5s: upper_bound - lower_bound = 5 - 2 = 3 ✓
Binary Search on Answer
A powerful interview technique: binary search not on array indices, but on the *answer space*.
Pattern: If you can verify whether a given answer is feasible in O(f(n)) time, and the feasibility changes monotonically with the answer value, you can binary search on the answer space.
Example
def minimum_days_to_make_bouquets(bloom_day, m, k):
"""
Given bloom_day[i] = day flower i blooms,
find minimum days to make m bouquets of k consecutive flowers.
Binary search on the answer (number of days).
"""
def can_make(days):
"""Can we make m bouquets if we wait 'days' days?"""
bouquets = consecutive = 0
for bloom in bloom_day:
if bloom <= days:
consecutive += 1
if consecutive == k:
bouquets += 1
consecutive = 0
else:
consecutive = 0
return bouquets >= m
if m * k > len(bloom_day):
return -1 # Impossible
left, right = min(bloom_day), max(bloom_day)
while left < right:
mid = (left + right) // 2
if can_make(mid):
right = mid # Try to do it in fewer days
else:
left = mid + 1 # Need more days
return left
print(minimum_days_to_make_bouquets([1,10,3,10,2], 3, 1)) # 3
print(minimum_days_to_make_bouquets([1,10,3,10,2], 3, 2)) # -1
Interpolation Search
Idea: Instead of always going to the middle, estimate where the target might be based on its value relative to the range.
Example
def interpolation_search(arr, target):
"""
Like binary search but estimates the target's position using interpolation.
Best for uniformly distributed sorted arrays.
Average: O(log log n), Worst: O(n)
"""
left, right = 0, len(arr) - 1
while left <= right and arr[left] <= target <= arr[right]:
if left == right:
return left if arr[left] == target else -1
# Interpolation formula: estimate where target is
pos = left + ((target - arr[left]) * (right - left)
// (arr[right] - arr[left]))
if arr[pos] == target:
return pos
elif arr[pos] < target:
left = pos + 1
else:
right = pos - 1
return -1
arr = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
print(interpolation_search(arr, 70)) # 6
print(interpolation_search(arr, 25)) # -1
When it beats binary search: Uniformly distributed data (like sorted phone numbers, timestamps, or sequential IDs). On uniform data, interpolation search achieves O(log log n) — much faster than O(log n).
When it is worse: Highly non-uniform data (like exponentially distributed data) can degrade to O(n).
Exponential Search
Idea: First find a range [2^(i-1), 2^i] where the target lies (by doubling), then binary search within that range.
Example
def exponential_search(arr, target):
"""
Useful for unbounded/infinite arrays or when the target is near the beginning.
Time: O(log n), Space: O(1)
"""
n = len(arr)
if arr[0] == target:
return 0
# Find the range for binary search
i = 1
while i < n and arr[i] <= target:
i *= 2 # Double i each step
# Binary search in [i//2, min(i, n-1)]
return binary_search_range(arr, target, i // 2, min(i, n - 1))
def binary_search_range(arr, target, left, right):
while left <= right:
mid = (left + right) // 2
if arr[mid] == target: return mid
elif arr[mid] < target: left = mid + 1
else: right = mid - 1
return -1
arr = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21]
print(exponential_search(arr, 13)) # 6
Comparison of Search Algorithms
| Algorithm | Time (Average) | Time (Worst) | Space | Requires Sorted |
|---|---|---|---|---|
| Linear Search | O(n) | O(n) | O(1) | No |
| Binary Search | O(log n) | O(log n) | O(1) | Yes |
| Interpolation | O(log log n) | O(n) | O(1) | Yes + Uniform |
| Exponential | O(log n) | O(log n) | O(1) | Yes |
Summary
- Linear search: O(n), works on any array, use for small or unsorted arrays
- Binary search: O(log n), requires sorted array, the standard for sorted data
- Binary search variants: leftmost/rightmost occurrence, count, lower bound, upper bound
- Binary search on answer: Search over feasible values, not array indices — powerful interview technique
- Interpolation search: O(log log n) average for uniform data
- Exponential search: Good when target is near beginning or array is unbounded
- Always normalize with
k = k % nfor rotation problems
*Next Lesson: Dynamic Arrays — How Python lists and Java ArrayLists work internally, amortized analysis of resize operations, and building your own dynamic array.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Array Searching — Linear Search, Binary Search, and Advanced Techniques.
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, arrays, array, searching
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