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 Bloom filters — what they are, how multiple hash functions create a probabilistic set, false positive analysis and optimal parameter calculation, complete implementation, applications in databases and networking, and counting Bloom filters for deletion support.
What Is a Bloom Filter?
A Bloom filter is a space-efficient probabilistic data structure used to test whether an element is a member of a set. It can answer with certainty that an element is definitely not in the set, or with some probability that it might be in the set.
The trade-off: Bloom filters never produce false negatives (if it says "not in set," it's definitely not there), but they can produce false positives (saying "in set" when it actually isn't).
Example
Bloom filter query result:
"Definitely NOT in set" → 100% accurate
"Possibly in set" → Might be wrong (false positive)
How It Works
A Bloom filter uses:
- A bit array of size m (all bits start as 0)
- k hash functions that each map an element to a position [0, m-1]
Adding an element:
Run the element through all k hash functions. Set those k bit positions to 1.
Checking membership:
Run through all k hash functions. If ALL k positions are 1 → element might be in set. If ANY position is 0 → element is definitely not in set.
| Bit array | [0,0,0,0,0,0,0,0,0,0] (size m=10) |
| Bit array | [0,0,1,0,0,1,0,0,1,0] |
| Bit array | [0,1,1,0,0,1,0,1,1,0] |
| Positions 2,5,8 | all 1 → "Possibly in set" ✓ |
| h2("cherry") = 3 ← 0 ← ZERO found | "Definitely NOT in set" ✓ |
| h3("grape") = 7 ← 1 | all 1 → "Possibly in set" (FALSE POSITIVE!) |
Complete Implementation
Example
import mmh3 # MurmurHash3 — pip install mmh3
import math
class BloomFilter:
"""
Bloom filter with configurable false positive rate.
Uses MurmurHash3 with different seeds for k hash functions.
"""
def __init__(self, expected_elements: int, false_positive_rate: float = 0.01):
"""
Calculate optimal size m and number of hash functions k.
expected_elements: n — how many items you expect to insert
false_positive_rate: p — acceptable false positive probability (0 to 1)
"""
self.n = expected_elements
self.p = false_positive_rate
# Optimal bit array size: m = -n*ln(p) / (ln(2))²
self.m = self._optimal_size(expected_elements, false_positive_rate)
# Optimal number of hash functions: k = (m/n) * ln(2)
self.k = self._optimal_hashes(self.m, expected_elements)
self.bit_array = [False] * self.m
self.count = 0
print(f"Bloom filter: {self.m} bits, {self.k} hash functions, "
f"expected FP rate: {false_positive_rate:.1%}")
def _optimal_size(self, n: int, p: float) -> int:
"""Optimal bit array size for n elements and false positive rate p."""
m = -(n * math.log(p)) / (math.log(2) ** 2)
return int(m) + 1
def _optimal_hashes(self, m: int, n: int) -> int:
"""Optimal number of hash functions."""
k = (m / n) * math.log(2)
return max(1, int(k))
def _hash_positions(self, item: str) -> list:
"""Get k positions for an item using MurmurHash3 with k different seeds."""
return [mmh3.hash(item, seed) % self.m for seed in range(self.k)]
def add(self, item: str) -> None:
"""Add item to filter. O(k)."""
for pos in self._hash_positions(item):
self.bit_array[pos] = True
self.count += 1
def might_contain(self, item: str) -> bool:
"""
Check if item might be in the set.
Returns:
False → item is DEFINITELY NOT in set
True → item MIGHT be in set (could be false positive)
O(k)
"""
return all(self.bit_array[pos] for pos in self._hash_positions(item))
def current_false_positive_rate(self) -> float:
"""Calculate actual false positive rate based on how full the filter is."""
# Fraction of bits set to 1
bits_set = sum(self.bit_array)
fill_rate = bits_set / self.m
return fill_rate ** self.k
def __len__(self):
return self.count
If mmh3 is not available, use hashlib:
python example
import hashlib
def _hash_positions_hashlib(self, item: str) -> list:
"""Alternative using hashlib (no external dependency)."""
positions = []
for seed in range(self.k):
h = hashlib.md5(f"{item}{seed}".encode()).hexdigest()
positions.append(int(h, 16) % self.m)
return positionsFalse Positive Analysis
The false positive probability after inserting n elements into a filter with m bits and k hash functions:
Example
P(false positive) = (1 - e^(-kn/m))^k
Optimal k = (m/n) × ln(2)
With optimal k:
P(false positive) ≈ (0.6185)^(m/n)
Practical table for p=1% false positive rate:
| Elements (n) | Bits per element (m/n) | Memory | Hash functions (k) |
|---|---|---|---|
| 1,000 | 9.59 | ~1.2 KB | 7 |
| 100,000 | 9.59 | ~120 KB | 7 |
| 10,000,000 | 9.59 | ~12 MB | 7 |
| 1,000,000,000 | 9.59 | ~1.2 GB | 7 |
Compare: storing 1 billion 50-byte strings = 50 GB. Bloom filter = 1.2 GB for 1% FP rate, or 2.4 GB for 0.1%.
Testing
Example
# Requires: pip install mmh3
bf = BloomFilter(expected_elements=1000, false_positive_rate=0.01)
# Output: Bloom filter: 9586 bits, 7 hash functions, expected FP rate: 1.0%
# Add elements
for word in ["apple", "banana", "cherry", "date", "elderberry"]:
bf.add(word)
# Check elements
print(bf.might_contain("apple")) # True — was added
print(bf.might_contain("banana")) # True — was added
print(bf.might_contain("grape")) # False — definitely not in set
print(bf.might_contain("kiwi")) # False — definitely not in set
# Check false positive rate
print(f"Current FP rate: {bf.current_false_positive_rate():.4f}")
Applications
1. Database Query Optimization
| 2. If "definitely not" | skip disk read (avoid expensive I/O) |
| 3. If "possibly yes" | do disk read to confirm |
| Used by | HBase, Cassandra, RocksDB, BigTable |
2. Web Crawler URL Deduplication
python example
class WebCrawler:
def __init__(self):
self.visited = BloomFilter(expected_elements=10**9, false_positive_rate=0.001)
def should_visit(self, url: str) -> bool:
if self.visited.might_contain(url):
return False # Probably already visited (occasional false skip is OK)
self.visited.add(url)
return True3. Chrome Safe Browsing
Google Chrome downloads a Bloom filter of malicious URLs. For each URL visited:
- Check Bloom filter: "Possibly dangerous" → query Google's servers
- "Definitely safe" → no server check needed
This saves bandwidth — the filter handles 99.9%+ of cases locally.
Counting Bloom Filter (Supports Deletion)
Standard Bloom filters cannot support deletion (unsetting a bit might affect other elements). Counting Bloom filters use a counter per position instead of a single bit:
Example
class CountingBloomFilter:
"""
Supports deletion by using counters instead of bits.
Space: 4x larger than standard (4-bit counter vs 1-bit)
"""
def __init__(self, expected_elements, false_positive_rate=0.01):
m = int(-(expected_elements * math.log(false_positive_rate)) / (math.log(2)**2)) + 1
k = max(1, int((m / expected_elements) * math.log(2)))
self.m = m
self.k = k
self.counters = [0] * m # Count instead of bit
def _positions(self, item):
return [mmh3.hash(item, seed) % self.m for seed in range(self.k)]
def add(self, item):
for pos in self._positions(item):
self.counters[pos] += 1
def remove(self, item):
"""Delete item (only call if item was previously added!)."""
if self.might_contain(item):
for pos in self._positions(item):
self.counters[pos] -= 1
def might_contain(self, item):
return all(self.counters[pos] > 0 for pos in self._positions(item))
Summary
| Property | Value |
|---|---|
| False negatives | Never — guaranteed |
| False positives | Possible — controlled by m and k |
| Space | O(m) bits — ~10 bits/element for 1% FP rate |
| Add | O(k) — k hash computations |
| Query | O(k) — k hash computations |
| Delete | Not supported (use counting variant) |
| Best used for | Membership testing where false positives are acceptable |
Bloom filters are one of the most practical and widely deployed probabilistic data structures. When you need to answer "have I seen this before?" at massive scale without massive memory, a Bloom filter is the answer.
*Next: Randomized Algorithms*
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