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 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.
What Is a Greedy Algorithm?
A greedy algorithm solves problems by making the locally optimal choice at each step, with the hope that these local choices lead to a globally optimal solution.
At each decision point, a greedy algorithm:
- Looks at the current state
- Makes the best possible choice for the immediate step (without looking ahead)
- Never reconsiders this choice — the decision is irrevocable
The word "greedy" reflects the strategy: take the best thing available right now, don't think about the future.
The Two Essential Properties
A problem can be solved greedily if and only if it has these two properties:
Property 1 — Greedy Choice Property
A globally optimal solution can be built by making locally optimal choices. When we make the greedy choice, there exists an optimal solution consistent with it.
This means: you never need to undo a greedy choice. The choice you make now will never be "wrong" — there is always an optimal solution that includes it.
Property 2 — Optimal Substructure
After making a greedy choice, the remaining subproblem has the same structure as the original. The optimal solution to the whole problem contains the optimal solution to the remaining subproblem.
(This is the same property required for dynamic programming, but greedy additionally requires the greedy choice property.)
How to Prove a Greedy Algorithm is Correct
Method 1 — Exchange Argument
This is the most common proof technique for greedy algorithms.
Structure: "Suppose an optimal solution does NOT make the greedy choice. Show that swapping the greedy choice in (while swapping out whatever the optimal solution chose) makes the solution no worse."
| Example | Activity Selection Problem |
| Greedy | Always pick the activity that ends earliest. |
| Claim | This greedy choice is in some optimal solution. |
| Case 1 | a = b. The greedy choice is already in OPT. Done. |
| Case 2 | a ≠ b. Since a has the earliest end time, end(a) ≤ end(b). |
Method 2 — Contradiction
Assume the greedy algorithm does NOT produce an optimal solution. Show this leads to a contradiction.
Method 3 — Induction
Base case: greedy is correct for small inputs. Inductive step: if greedy is correct for n-1 elements, it's correct for n elements.
When Greedy Fails — Counter-Examples
Not every problem has the greedy choice property. Learning when greedy FAILS is as important as knowing when it works.
Failure 1 — Coin Change (Non-Standard Denominations)
| Denominations | [1, 3, 4], Target: 6 |
| Greedy | 4 + 1 + 1 = 3 coins ← WRONG! |
| Optimal | 3 + 3 = 2 coins |
| Why it fails | Choosing 4 (locally best) prevents using 3+3 (globally best). |
Failure 2 — 0/1 Knapsack
| Items | [(weight=3, value=4), (weight=4, value=5), (weight=5, value=6)] |
| Capacity | 7 |
| Take Item 1 (w=3) | remaining capacity=4 |
| Take Item 2 (w=4) | remaining capacity=0 |
| Take Item 2 + Item 3 | weight=4+5=9 > 7, doesn't fit |
| Take Item 1 + Item 3 | weight=3+5=8 > 7, doesn't fit |
| Items | weight=[3,4,5], value=[4,5,6], capacity=7 |
| Take items 1+2 | w=3+4=7, value=4+5=9 ← greedy got this! |
| Take items 1+3 | w=3+5=8 > 7, invalid |
| Take items 2+3 | w=4+5=9 > 7, invalid |
| Counter-example | items [(w=10,v=60),(w=20,v=100),(w=30,v=120)], cap=50 |
| Greedy by v/w: v/w = [6, 5, 4], take item 1 then item 2 | v=60+100=160 but w=30>50... |
| Actually w=10+20=30 ≤ 50, also take item 3 | 10+20+30=60>50, take item 3 not item 2: |
| item 1+3 | w=40, v=180, item 2+3: w=50, v=220 ← optimal! |
| Optimal | items 2+3 = value 220 |
| Why it fails | Unlike fractional knapsack, you can't take fractions. |
Greedy vs Dynamic Programming
| Aspect | Greedy | Dynamic Programming |
|---|---|---|
| Decision making | One locally optimal choice | Consider all choices |
| Revisiting decisions | Never | Not needed (memoization handles it) |
| Correctness | Requires greedy choice property | Always correct if substructure holds |
| Speed | Usually O(n log n) | Usually O(n²) or O(n×W) |
| Code complexity | Usually simple | More complex |
| Problems | Activity selection, Huffman, Dijkstra | Knapsack, LCS, edit distance |
Key question: "Does making the locally optimal choice now prevent us from finding a better solution later?"
- If NO → Greedy works
- If YES → Need DP
Classic Greedy Problems
| Problem | Greedy Strategy | Time |
|---|---|---|
| Activity Selection | Pick earliest-ending activity | O(n log n) |
| Fractional Knapsack | Pick by value/weight ratio | O(n log n) |
| Job Sequencing | Sort by deadline | O(n log n) |
| Huffman Coding | Merge two lowest-frequency nodes | O(n log n) |
| Dijkstra's | Process closest unvisited vertex | O((V+E) log V) |
| Kruskal's MST | Add cheapest safe edge | O(E log E) |
| Prim's MST | Add cheapest connecting edge | O(E log V) |
| Interval Scheduling | Minimize late jobs | O(n log n) |
The Greedy Algorithm Template
python example
def greedy_algorithm(items, constraint):
"""
General greedy template:
1. Sort items by greedy criterion
2. Iterate and make greedy choice at each step
3. Accumulate result
"""
# Step 1: Sort by greedy criterion (most important step!)
sorted_items = sorted(items, key=greedy_criterion)
result = initial_state
for item in sorted_items:
if is_feasible(result, item):
result = apply(result, item) # Irrevocable choice
return resultThe sorting step is the key design decision — what criterion to sort by determines whether the greedy approach is correct.
Summary
Greedy algorithms make locally optimal choices at each step:
- When correct: Greedy choice property + optimal substructure
- Proof technique: Exchange argument — swap greedy choice into any optimal solution without making it worse
- Failure conditions: When local optima lead to global suboptimality (use DP instead)
- Efficiency advantage: Usually O(n log n) — much faster than DP when applicable
The art of greedy algorithms is recognizing when the greedy choice property holds — and proving it rigorously when it does.
*Next Lesson: Activity Selection Problem*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Greedy Algorithms — Complete Introduction with Proof 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, greedy, introduction, greedy algorithms — complete introduction with proof techniques
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