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 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.
What Is Binary Search?
Binary search finds a target in a sorted array by repeatedly halving the search range. At each step, it compares the target with the middle element and eliminates half the remaining candidates.
It is one of the most important algorithms in computer science — simple in concept, powerful in practice, and subtle in correct implementation.
Loop Invariant — Proof of Correctness
The loop invariant is: "If target exists in the array, it must be in arr[left..right]."
- Initialization: Before the loop, left=0 and right=n-1, so the invariant holds (entire array).
- Maintenance: After each iteration, we narrow the range to the half where target can be, preserving the invariant.
- Termination: When left > right, the search space is empty. By the invariant, target is not in the array → return -1.
This invariant guarantees the algorithm is correct for any sorted input.
Step-by-Step Trace
| Iter 1 | left=0, right=10, mid=5 |
| arr[5]=23 == target | return 5 ✓ |
| Iter 1: left=0, right=10, mid=5 arr[5]=23 < 38 | left=6 |
| Iter 2: left=6, right=10, mid=8 arr[8]=56 > 38 | right=7 |
| Iter 3: left=6, right=7, mid=6 arr[6]=38 == 38 | return 6 ✓ |
| Iter 1: left=0, right=10, mid=5 arr[5]=23 < 30 | left=6 |
| Iter 2: left=6, right=10, mid=8 arr[8]=56 > 30 | right=7 |
| Iter 3: left=6, right=7, 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):
"""O(log n) time, O(log n) space (call stack)."""
if left is None: left, right = 0, len(arr) - 1
if left > right: return -1 # Base case: empty search space
mid = left + (right - left) // 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)
Prefer iterative — avoids O(log n) stack space.
Complexity Analysis
| Initial search space | n elements |
| After 1 comparison | n/2 elements |
| After 2 comparisons | n/4 elements |
| After k comparisons | n/2^k elements |
| n < 2^k | k > log₂(n) |
| Maximum iterations | ⌊log₂(n)⌋ + 1 |
| For n = 1,000,000 | at most 20 comparisons |
| For n = 1,000,000,000 | at most 30 comparisons |
Time: O(log n) | Space: O(1) iterative, O(log n) recursive
All Binary Search Variants
Find Leftmost (First) Occurrence
Example
def leftmost(arr, target):
"""Find index of FIRST occurrence. Returns -1 if not found."""
left, right = 0, len(arr) - 1
result = -1
while left <= right:
mid = left + (right - left) // 2
if arr[mid] == target:
result = mid # Record this position
right = mid - 1 # Keep searching LEFT for earlier occurrence
elif arr[mid] < target:
left = mid + 1
else:
right = mid - 1
return result
arr = [1, 2, 2, 2, 3, 4, 5]
print(leftmost(arr, 2)) # 1 (index of first 2)
Find Rightmost (Last) Occurrence
Example
def rightmost(arr, target):
"""Find index of LAST occurrence. Returns -1 if not found."""
left, right = 0, len(arr) - 1
result = -1
while left <= right:
mid = left + (right - left) // 2
if arr[mid] == target:
result = mid # Record this position
left = mid + 1 # Keep searching RIGHT for later occurrence
elif arr[mid] < target:
left = mid + 1
else:
right = mid - 1
return result
print(rightmost(arr, 2)) # 3 (index of last 2)
Count Occurrences
Example
def count_occurrences(arr, target):
first = leftmost(arr, target)
if first == -1: return 0
last = 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
Lower Bound — First index where arr[i] >= target
Example
def lower_bound(arr, target):
"""
First position where arr[i] >= target.
If all elements < target, returns len(arr) (insert at end).
"""
left, right = 0, len(arr) # Note: right = len(arr), not len(arr)-1
while left < right: # Note: strict <, not <=
mid = left + (right - left) // 2
if arr[mid] < target:
left = mid + 1
else:
right = mid
return left # Index of first element >= target
arr = [1, 3, 5, 7, 9]
print(lower_bound(arr, 5)) # 2 (arr[2]=5 is first >=5)
print(lower_bound(arr, 6)) # 3 (arr[3]=7 is first >=6)
print(lower_bound(arr, 10)) # 5 (insert after all)
print(lower_bound(arr, 0)) # 0 (insert before all)
Upper Bound — First index where arr[i] > target
Example
def upper_bound(arr, target):
"""First position where arr[i] > target."""
left, right = 0, len(arr)
while left < right:
mid = left + (right - left) // 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(lower_bound(arr, 5)) # 2 (first 5)
print(upper_bound(arr, 5)) # 5 (after last 5)
# Count of 5s: upper_bound - lower_bound = 5 - 2 = 3 ✓
Binary Search on Answer Space
A powerful technique: binary search over the *answer* rather than over array indices.
Pattern: When you can verify whether a candidate answer x is feasible/valid in O(f(n)) time, and feasibility is monotone (all values ≤ x are feasible, or all ≥ x are feasible), binary search on x.
Example
def min_speed_to_arrive_on_time(dist, hour):
"""
Find minimum integer speed to complete all train trips by 'hour'.
Each trip takes ceil(dist[i]/speed) hours except the last.
Binary search on speed from 1 to max(dist).
"""
import math
def can_arrive(speed):
total = 0
for i in range(len(dist) - 1):
total += math.ceil(dist[i] / speed) # Round up each trip
total += dist[-1] / speed # Last trip: no rounding
return total <= hour
if hour <= len(dist) - 1:
return -1 # Impossible: need at least 1 hour per trip before last
left, right = 1, max(dist)
while left < right:
mid = left + (right - left) // 2
if can_arrive(mid):
right = mid # Speed mid works, try lower
else:
left = mid + 1 # Speed mid too slow, need faster
return left
Python's bisect Module
Python provides built-in binary search via bisect:
Example
import bisect
arr = [1, 3, 5, 7, 9]
# Lower bound: first index >= target
print(bisect.bisect_left(arr, 5)) # 2 (arr[2]=5)
print(bisect.bisect_left(arr, 6)) # 3 (arr[3]=7, first >=6)
# Upper bound: first index > target
print(bisect.bisect_right(arr, 5)) # 3 (first index after all 5s)
print(bisect.bisect(arr, 5)) # Same as bisect_right
# Insert maintaining sort order:
bisect.insort_left(arr, 4) # arr becomes [1,3,4,5,7,9]
bisect.insort(arr, 6) # arr becomes [1,3,4,5,6,7,9]
Classic Binary Search Interview Problems
1 — Search in Rotated Sorted Array
Example
def search_rotated(arr, target):
"""Binary search in sorted array that was rotated. O(log n)."""
left, right = 0, len(arr) - 1
while left <= right:
mid = left + (right - left) // 2
if arr[mid] == target: return mid
if arr[left] <= arr[mid]: # Left half is sorted
if arr[left] <= target < arr[mid]:
right = mid - 1
else:
left = mid + 1
else: # Right half is sorted
if arr[mid] < target <= arr[right]:
left = mid + 1
else:
right = mid - 1
return -1
print(search_rotated([4,5,6,7,0,1,2], 0)) # 4
2 — Find Peak Element
Example
def find_peak(nums):
"""Find any peak element (greater than both neighbors). O(log n)."""
left, right = 0, len(nums) - 1
while left < right:
mid = left + (right - left) // 2
if nums[mid] < nums[mid+1]:
left = mid + 1 # Peak is to the right
else:
right = mid # Peak is at mid or to the left
return left
print(find_peak([1,2,3,1])) # 2 (nums[2]=3 is a peak)
print(find_peak([1,2,1,3,5,6,4])) # 5 (nums[5]=6 is a peak)
3 — Find Minimum in Rotated Sorted Array
Example
def find_min_rotated(nums):
"""Find minimum element in rotated sorted array. O(log n)."""
left, right = 0, len(nums) - 1
while left < right:
mid = left + (right - left) // 2
if nums[mid] > nums[right]:
left = mid + 1 # Minimum is in right half
else:
right = mid # Minimum is at mid or in left half
return nums[left]
print(find_min_rotated([3,4,5,1,2])) # 1
print(find_min_rotated([4,5,6,7,0,1,2])) # 0
Summary
Binary search: O(log n) on sorted arrays.
Core: Compare with middle, discard half, repeat until found or empty.
Variants (all O(log n)):
- Exact match: standard binary search
- First/last occurrence: adjust to keep searching after finding
- Lower/upper bound: find insertion points
- Answer space: binary search over feasibility
Key implementation details:
- Use
left + (right - left) // 2for mid (overflow-safe) - Distinguish
left <= right(exact match) vsleft < right(bound finding) - Always verify: when does
left = mid + 1vsright = mid - 1vsright = mid?
*Next Lesson: Interpolation Search*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Binary Search — Complete Guide with All Variants and Proof of Correctness.
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, binary, search
Related Data Structures & Algorithms Topics
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