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 the Activity Selection Problem completely — problem definition, why sorting by end time is the correct greedy criterion, exchange argument proof, implementation variants (maximum activities, weighted interval scheduling, minimum rooms), and interval scheduling interview problems.
Problem Statement
Given n activities, each with a start time s[i] and finish time f[i], select the maximum number of non-overlapping activities that a single machine can perform.
Two activities i and j are compatible if they don't overlap: f[i] ≤ s[j] or f[j] ≤ s[i].
| Activities | [(1,4),(3,5),(0,6),(5,7),(3,9),(5,9),(6,10),(8,11),(8,12),(2,14),(12,16)] |
| Start | 1 3 0 5 3 5 6 8 8 2 12 |
| Finish | 4 5 6 7 9 9 10 11 12 14 16 |
| Optimal selection | 4 non-overlapping activities |
Implementation
Example
def activity_selection(activities):
"""
Select maximum number of non-overlapping activities.
activities: list of (start, finish) tuples
Time: O(n log n) for sorting, O(n) for selection
Space: O(n) for result
"""
# Sort by finish time (greedy criterion)
sorted_activities = sorted(activities, key=lambda x: x[1])
selected = [sorted_activities[0]] # Always take the first (earliest finish)
last_finish = sorted_activities[0][1]
for start, finish in sorted_activities[1:]:
if start >= last_finish: # Compatible with last selected
selected.append((start, finish))
last_finish = finish
return selected
activities = [(1,4),(3,5),(0,6),(5,7),(3,9),(5,9),(6,10),(8,11),(8,12),(2,14),(12,16)]
result = activity_selection(activities)
print(f"Selected {len(result)} activities: {result}")
# 4 activities: (1,4), (5,7), (8,11), (12,16)
Proof of Correctness — Exchange Argument
Theorem: Sorting by earliest finish time and greedily selecting compatible activities yields a maximum-size set.
Proof:
Let G = greedy solution, OPT = any optimal solution.
Order both sets by finish time:
- G: g₁, g₂, ..., g_k (greedy selections)
- OPT: o₁, o₂, ..., o_m (optimal selections)
Claim: k = m (greedy selects as many as optimal).
Proof by induction on i: After i selections, greedy's i-th selected activity finishes no later than optimal's i-th activity.
Base case (i=1): Greedy picks g₁ with the earliest finish. f(g₁) ≤ f(o₁) because g₁ has the globally minimum finish time. ✓
Inductive step: Assume f(gᵢ) ≤ f(oᵢ). Then:
- oᵢ₊₁ starts after oᵢ finishes: s(oᵢ₊₁) ≥ f(oᵢ) ≥ f(gᵢ)
- So oᵢ₊₁ is compatible with gᵢ
- Greedy considers all activities compatible with gᵢ (including oᵢ₊₁)
- Greedy picks the earliest-finishing one: f(gᵢ₊₁) ≤ f(oᵢ₊₁) ✓
Since OPT selects m activities and at each step greedy leaves at least as much room, greedy also selects m activities. ∎
Variant 1 — Minimum Number of Rooms (Interval Partitioning)
Example
import heapq
def min_meeting_rooms(intervals):
"""
Minimum number of rooms to schedule all meetings without overlap.
Greedy: for each new meeting, assign to room that freed up earliest.
Time: O(n log n)
"""
if not intervals: return 0
intervals.sort(key=lambda x: x[0]) # Sort by start time
heap = [] # Min-heap of finish times (one per room in use)
for start, finish in intervals:
if heap and heap[0] <= start:
# Reuse the room that became free earliest
heapq.heapreplace(heap, finish)
else:
# Need a new room
heapq.heappush(heap, finish)
return len(heap)
print(min_meeting_rooms([[0,30],[5,10],[15,20]])) # 2
print(min_meeting_rooms([[7,10],[2,4]])) # 1
print(min_meeting_rooms([[1,5],[2,3],[3,7]])) # 2
Variant 2 — Non-Overlapping Intervals (Minimum Removals)
Example
def erase_overlap_intervals(intervals):
"""
Minimum intervals to remove so remaining are non-overlapping.
Equivalent: n - max_non_overlapping.
Sort by end time, count non-overlapping (greedy keeps them).
"""
if not intervals: return 0
intervals.sort(key=lambda x: x[1])
kept = 0
last_end = float('-inf')
for start, end in intervals:
if start >= last_end:
kept += 1
last_end = end
return len(intervals) - kept
print(erase_overlap_intervals([[1,2],[2,3],[3,4],[1,3]])) # 1 (remove [1,3])
print(erase_overlap_intervals([[1,2],[1,2],[1,2]])) # 2
Variant 3 — Weighted Interval Scheduling (Need DP)
When each activity has a value and you want to maximize total value:
Example
def weighted_interval_scheduling(intervals):
"""
Maximize total weight of non-overlapping weighted intervals.
This requires DP — greedy doesn't work for weighted version!
intervals: list of (start, finish, weight)
Time: O(n log n)
"""
from bisect import bisect_right
intervals.sort(key=lambda x: x[1]) # Sort by finish
n = len(intervals)
# p[i] = index of last interval that finishes before interval i starts
starts = [intervals[i][0] for i in range(n)]
finishes = [intervals[i][1] for i in range(n)]
p = []
for i in range(n):
j = bisect_right(finishes, intervals[i][0] - 1) # Find compatible predecessor
p.append(j - 1)
dp = [0] * (n + 1)
for i in range(1, n+1):
_, _, w = intervals[i-1]
include = w + dp[p[i-1] + 1] # Include interval i
exclude = dp[i-1] # Exclude interval i
dp[i] = max(include, exclude)
return dp[n]
intervals = [(1,3,2),(2,5,4),(4,6,3),(3,8,5),(5,10,3),(7,9,4),(6,8,2)]
print(weighted_interval_scheduling(intervals)) # 9
Variant 4 — Merge Overlapping Intervals
Example
def merge_intervals(intervals):
"""
Merge all overlapping intervals.
Sort by start, merge when current start ≤ previous finish.
"""
if not intervals: return []
intervals.sort(key=lambda x: x[0])
merged = [intervals[0]]
for start, end in intervals[1:]:
if start <= merged[-1][1]:
merged[-1][1] = max(merged[-1][1], end)
else:
merged.append([start, end])
return merged
print(merge_intervals([[1,3],[2,6],[8,10],[15,18]]))
# [[1,6],[8,10],[15,18]]
Complexity Summary
| Problem | Greedy Criterion | Time | Space |
|---|---|---|---|
| Max activities | Earliest finish first | O(n log n) | O(n) |
| Min rooms | Earliest start, reuse earliest-free | O(n log n) | O(n) |
| Min removals | Earliest finish first | O(n log n) | O(1) |
| Weighted (max) | DP required | O(n log n) | O(n) |
| Merge intervals | Sort by start | O(n log n) | O(n) |
Summary
Activity Selection is the canonical greedy problem:
- Greedy choice: Always pick the activity that finishes earliest
- Why correct: Leaves maximum remaining time for other activities
- Proof: Exchange argument — earliest-finishing activity can always replace any other first choice without reducing the total count
- Weighted variant: Greedy fails — use dynamic programming with binary search
*Next Lesson: Fractional Knapsack*
Continue the lesson path
Study next in this module
01Greedy AlgorithmsFractional Knapsack — Greedy by Value Density with ProofMaster the Fractional Knapsack problem — why greedy by value/weight ratio works, complete implementation with fractional items, proof using exchange argument, comparison with 0/1 knapsack, and real-world applications in resource allocation.02Greedy AlgorithmsGreedy Interview Problems — 10 Classic Problems with Complete SolutionsSolve 10 classic greedy interview problems — gas station, candy distribution, jump game, task scheduler, assign cookies, lemonade change, boats to save people, partition labels, minimum arrows, and queue reconstruction by height — with proofs and pattern recognition.03Greedy AlgorithmsGreedy Algorithms — Complete Introduction with Proof TechniquesMaster greedy algorithms from the ground up — what makes an algorithm greedy, the greedy choice property, optimal substructure, exchange argument proof technique, when greedy works vs when it fails, comparison with DP, and examples of correct and incorrect greedy choices.04Greedy AlgorithmsHuffman Coding — Optimal Prefix-Free Data CompressionMaster Huffman coding completely — what prefix-free codes are, why Huffman coding is optimal, building the Huffman tree with a priority queue, encoding and decoding, calculating compression ratio, and applications in ZIP, JPEG, and MP3 compression.