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*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Job Sequencing Problem — Greedy Scheduling with Deadlines.
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, greedy, job, sequencing
Related Data Structures & Algorithms Topics
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