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 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.
Problem Statement
Given items with weights and values, and a knapsack with capacity W, maximize the total value. You can take fractions of items (unlike 0/1 knapsack where each item is all-or-nothing).
| Items | (weight, value) |
| Item 1: (10 kg, $60) | value density = $6/kg |
| Item 2: (20 kg, $100) | value density = $5/kg |
| Item 3: (30 kg, $120) | value density = $4/kg |
| Knapsack capacity | 50 kg |
| Take all of Item 1 | 10 kg, $60 (remaining: 40 kg) |
| Take all of Item 2 | 20 kg, $100 (remaining: 20 kg) |
| Take 2/3 of Item 3 | 20 kg, $80 (remaining: 0 kg) |
| Total | $240 |
Implementation
Example
def fractional_knapsack(weights, values, capacity):
"""
Fractional Knapsack using greedy by value density.
Time: O(n log n) for sorting, O(n) for selection.
Space: O(n)
"""
n = len(weights)
# Create (value_density, weight, value) triples
items = sorted(zip(values, weights), key=lambda x: x[0]/x[1], reverse=True)
total_value = 0.0
remaining = capacity
selected = []
for value, weight in items:
if remaining <= 0:
break
if weight <= remaining:
# Take entire item
total_value += value
remaining -= weight
selected.append((weight, value, 1.0)) # fraction = 1
else:
# Take fraction of item
fraction = remaining / weight
total_value += value * fraction
selected.append((weight, value, fraction))
remaining = 0
return total_value, selected
weights = [10, 20, 30]
values = [60, 100, 120]
total, selection = fractional_knapsack(weights, values, 50)
print(f"Maximum value: ${total:.2f}") # $240.00
for w, v, f in selection:
if f == 1.0:
print(f" Take all: weight={w}, value={v}")
else:
print(f" Take {f:.1%}: weight={w*f:.1f}, value={v*f:.2f}")
Proof of Correctness
Theorem: Greedy by value density yields the optimal fractional knapsack solution.
Proof (exchange argument):
Sort items by value density: v₁/w₁ ≥ v₂/w₂ ≥ ... ≥ vₙ/wₙ.
Suppose OPT is any optimal solution. Let x₁, x₂, ..., xₙ be the fractions taken (xᵢ ∈ [0,1]).
Greedy takes fractions g₁, g₂, ..., gₙ where:
- gᵢ = 1 for items until capacity is reached
- One item is partially taken
- gᵢ = 0 for remaining items
Claim: Greedy value ≥ OPT value.
If OPT ≠ Greedy, there exist i < j such that OPT takes less of i than greedy and more of j.
- Since i has higher density: vᵢ/wᵢ ≥ vⱼ/wⱼ
- Shifting ε weight from j to i (in OPT): gain ε·(vᵢ/wᵢ) - lose ε·(vⱼ/wⱼ) ≥ 0
- Value doesn't decrease after the swap
We can keep swapping until OPT matches Greedy without decreasing value. So Greedy value ≥ OPT value, and since OPT is optimal, they're equal. ∎
Fractional vs 0/1 Knapsack
| Feature | Fractional | 0/1 |
|---|---|---|
| Items | Can be split | All-or-nothing |
| Algorithm | Greedy O(n log n) | DP O(n×W) |
| Optimal by greedy | ✅ Yes | ❌ No |
| Value density heuristic | ✅ Correct | ❌ Can fail |
| Practical example | Filling bag with sand | Packing indivisible boxes |
Why 0/1 knapsack cannot be solved greedily:
| Items | [(w=3,v=4),(w=4,v=5),(w=5,v=6)], capacity=7 |
| Densities | 4/3≈1.33, 5/4=1.25, 6/5=1.2 |
| Total | $9 |
| Actually | take only item 2: v=5, or item 3: v=6, or items 1+2: v=9 |
| Wait, items 1+2 | w=3+4=7≤7, v=4+5=9 ← same as greedy here |
| Better example | items=[(2,3),(3,4),(4,5)], cap=5 |
| Densities | 1.5, 1.33, 1.25 |
| Greedy | take item1(2kg,3$) then item2(3kg,4$) but 2+3=5≤5, total=7 |
| Optimal | take item1+item2: 7$ ← same as greedy in this case! |
| True failure | items=[(1,1),(2,6),(3,10),(5,16)], cap=7 |
| Densities | 1, 3, 3.33, 3.2 |
| Greedy | item3(3,10), item2(2,6), item1(1,1)... wait 3+2+1=6≤7, value=17 |
| Actually take item4(5,16) | remaining=2, take item2(2,6): value=22 |
| Greedy with correct density | item3 density=3.33 first: w=3, value=10 |
| remaining=4 | item4 density=3.2: w=5>4, take 4/5 fraction: value=16×(4/5)=12.8 |
| Fractional total | 22.8 |
| For 0/1 | item4+item2=5+2=7, value=22 (optimal for 0/1) |
| Greedy 0/1 | take item3(3,10), item2(2,6), item1(1,1): total=17 ← suboptimal! |
Continuous Knapsack Applications
Example
def resource_allocation(tasks, time_budget):
"""
Given tasks with time requirements and rewards,
maximize total reward within time budget.
Same as fractional knapsack.
"""
items = sorted(zip([t['reward'] for t in tasks],
[t['time'] for t in tasks]),
key=lambda x: x[0]/x[1], reverse=True)
total_reward = 0
remaining_time = time_budget
for reward, time in items:
if remaining_time <= 0: break
fraction = min(1.0, remaining_time / time)
total_reward += reward * fraction
remaining_time -= time * fraction
return total_reward
tasks = [
{'name': 'Task A', 'time': 5, 'reward': 15}, # 3/hr
{'name': 'Task B', 'time': 3, 'reward': 12}, # 4/hr
{'name': 'Task C', 'time': 8, 'reward': 24}, # 3/hr
]
print(f"Max reward: {resource_allocation(tasks, 10)}") # 34.0
Summary
Fractional Knapsack:
- Greedy choice: Sort by value/weight ratio (value density), take items greedily
- Time: O(n log n) for sorting + O(n) for selection
- Correct: Exchange argument proves optimality
- Key difference from 0/1: Fractions allowed → greedy works
The ability to take fractions is what makes greedy correct. Without fractions (0/1 knapsack), greedy fails — use dynamic programming instead.
*Next Lesson: Huffman Coding*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Fractional Knapsack — Greedy by Value Density with Proof.
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, fractional, knapsack
Related Data Structures & Algorithms Topics
Continue learning this concept
Greedy AlgorithmsActivity Selection Problem — Greedy Scheduling with Proof of CorrectnessMaster 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.Greedy 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.Greedy 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.Greedy 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.Greedy AlgorithmsJob Sequencing Problem — Greedy Scheduling with DeadlinesMaster 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.Dynamic ProgrammingKnapsack Problem — 0/1, Unbounded, and Fractional with Complete AnalysisMaster all knapsack variants — 0/1 knapsack with 2D and 1D DP tables, unbounded knapsack with reusable items, fractional knapsack with greedy, subset sum, partition equal subset sum, and the general framework for recognizing knapsack problems in interviews.