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 Job Sequencing Problem — scheduling jobs with deadlines and profits to maximize total profit, greedy strategy of sorting by profit, DSU-based efficient scheduling, complete implementations, proof of correctness, and variants including minimize lateness.
Problem Statement
Given n jobs, each with:
- A deadline d[i] (must finish by this time)
- A profit p[i] (earned if completed on time)
Each job takes exactly 1 unit of time. Find the sequence that maximizes total profit.
| Jobs | (id, deadline, profit) |
| J1 | (1, d=2, p=100) |
| J2 | (2, d=1, p=19) |
| J3 | (3, d=2, p=27) |
| J4 | (4, d=1, p=25) |
| J5 | (5, d=3, p=15) |
| Greedy by profit (descending) | J1, J3, J4, J2, J5 |
| Place J1 (p=100, d=2) | schedule at slot 2. Available: [slot 1, slot 3] |
| Place J3 (p=27, d=2) | slot 2 taken, try slot 1. Available: [slot 3] |
| Place J4 (p=25, d=1) | slot 1 taken, no slot ≤ 1 available. Skip. |
| Place J2 (p=19, d=1) | slot 1 taken, skip. |
| Place J5 (p=15, d=3) | schedule at slot 3. |
| Result | J3 (slot 1), J1 (slot 2), J5 (slot 3), profit = 27+100+15 = 142 |
Efficient Implementation with DSU
The naive O(n²) approach can be improved to O(n log n) using Union-Find to quickly find the latest available slot:
Example
class DSU:
def __init__(self, n):
self.parent = list(range(n+1))
def find(self, x):
if self.parent[x] != x:
self.parent[x] = self.find(self.parent[x])
return self.parent[x]
def union(self, x):
"""Mark slot x as used, point to x-1."""
self.parent[x] = x - 1
def job_sequencing_dsu(jobs):
"""
DSU-optimized job sequencing.
Time: O(n log n) for sort + O(n α(n)) for DSU operations.
"""
jobs = sorted(jobs, key=lambda x: -x[2])
max_d = max(j[1] for j in jobs)
dsu = DSU(max_d + 1)
selected = []
total = 0
for job_id, deadline, profit in jobs:
# Find latest available slot ≤ deadline
available = dsu.find(deadline)
if available > 0:
total += profit
selected.append(job_id)
dsu.union(available) # Mark slot as used, redirect to available-1
return total, selected
jobs = [(1,2,100),(2,1,19),(3,2,27),(4,1,25),(5,3,15)]
profit, schedule = job_sequencing_dsu(jobs)
print(f"Maximum profit: {profit}") # 142
Minimize Total Lateness (Related Problem)
A related greedy problem: Schedule all jobs to minimize total lateness (where lateness = max(0, finish_time - deadline)).
Example
def minimize_lateness(jobs):
"""
Schedule jobs to minimize total lateness.
Greedy: sort by deadline (earliest deadline first — EDF).
Time: O(n log n)
"""
# Sort by deadline (EDF — Earliest Deadline First)
jobs_sorted = sorted(jobs, key=lambda x: x[1])
current_time = 0
total_lateness = 0
schedule = []
for job_id, deadline, duration in jobs_sorted:
start = current_time
finish = current_time + duration
lateness = max(0, finish - deadline)
total_lateness += lateness
schedule.append((job_id, start, finish, lateness))
current_time = finish
return total_lateness, schedule
# Each job: (id, deadline, duration)
jobs = [(1,6,3),(2,8,2),(3,9,1),(4,9,4),(5,14,3),(6,15,2)]
lateness, schedule = minimize_lateness(jobs)
print(f"Total lateness: {lateness}")
for jid, start, finish, late in schedule:
print(f" Job {jid}: [{start},{finish}], lateness={late}")
Why EDF works for minimizing lateness? Exchange argument: If job i with earlier deadline d_i is scheduled AFTER job j with later deadline d_j, swapping them reduces total lateness.
Variants
Minimize Number of Late Jobs
Example
def min_late_jobs(jobs):
"""
Maximize jobs completed on time (equivalently minimize late jobs).
Moore's algorithm: greedy by deadline, remove longest job if late.
Time: O(n log n + n²) or O(n log n) with priority queue.
"""
jobs_sorted = sorted(jobs, key=lambda x: x[1]) # By deadline
import heapq
current_time = 0
scheduled = [] # Max-heap by duration (negate)
for job_id, deadline, duration in jobs_sorted:
heapq.heappush(scheduled, (-duration, job_id, deadline))
current_time += duration
if current_time > deadline:
# Remove the longest job (it's making things late)
longest = heapq.heappop(scheduled)
current_time += longest[0] # longest[0] is negated duration
return len(scheduled), [jid for _, jid, _ in scheduled]
jobs = [(1,3,1),(2,3,1),(3,3,1),(4,3,1),(5,3,1)] # 5 jobs, all due by time 3
print(min_late_jobs(jobs)) # Only 3 jobs can be done on time
Complexity Summary
| Variant | Greedy Criterion | Time | Algorithm |
|---|---|---|---|
| Max profit with deadlines | Sort by profit desc | O(n²) | Linear search |
| Max profit (efficient) | Sort by profit desc | O(n log n) | DSU |
| Minimize total lateness | Sort by deadline (EDF) | O(n log n) | Simple |
| Minimize late jobs | EDF + remove longest | O(n log n) | Priority queue |
Summary
Job Sequencing:
- Greedy choice: Process jobs in decreasing profit order, assign latest available slot ≤ deadline
- Why correct: Taking the most profitable jobs first, using latest available slots preserves flexibility for lower-profit jobs
- DSU optimization:
find(d)gives latest available slot ≤ d in O(α(n)) - Minimize lateness variant: Sort by earliest deadline (EDF) — classic greedy for scheduling
*Next Lesson: Greedy Interview Problems*
Continue the lesson path
Study next in this module
01Greedy 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.02Greedy 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.03Greedy 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.04Greedy 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.