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
A complete introduction to hashing — what a hash table is, how hash functions work, why hash tables achieve O(1) average operations, the load factor, the difference between hash maps and hash sets, and real-world applications of hashing in databases, security, and caching.
What Is a Hash Table?
A hash table (also called a hash map or hash set) is a data structure that provides O(1) average time for three fundamental operations: insert, delete, and lookup.
This is dramatically better than:
- Arrays: O(n) for unsorted search, O(1) for indexed access only
- Sorted arrays: O(log n) search, O(n) insert/delete
- Balanced BSTs: O(log n) for all three
The secret is a hash function — a mathematical transformation that converts a key into an array index.
Collisions — When Two Keys Map to Same Index
A collision occurs when two different keys hash to the same index. With any hash function and finite table, collisions are inevitable (by the pigeonhole principle).
Example
Table size = 5
hash("Alice") = 3
hash("Charlie") = 3 ← COLLISION! Both want index 3
Two main strategies to handle collisions:
- Chaining — store a linked list at each index
- Open Addressing — find another empty slot in the table
Basic Hash Table Structure
Example
class HashTable:
"""
Simple hash table using chaining for collision resolution.
"""
def __init__(self, size=16):
self.size = size
self.buckets = [[] for _ in range(size)] # Each bucket is a list
self.count = 0
def _hash(self, key):
return hash(key) % self.size
def put(self, key, value):
"""Insert or update key-value pair. O(1) average."""
idx = self._hash(key)
for i, (k, v) in enumerate(self.buckets[idx]):
if k == key:
self.buckets[idx][i] = (key, value) # Update
return
self.buckets[idx].append((key, value)) # Insert
self.count += 1
def get(self, key, default=None):
"""Lookup by key. O(1) average."""
idx = self._hash(key)
for k, v in self.buckets[idx]:
if k == key:
return v
return default
def delete(self, key):
"""Delete by key. O(1) average."""
idx = self._hash(key)
bucket = self.buckets[idx]
for i, (k, v) in enumerate(bucket):
if k == key:
bucket.pop(i)
self.count -= 1
return True
return False
def __contains__(self, key):
return self.get(key) is not None
def __len__(self):
return self.count
Load Factor and Rehashing
The load factor α = n/m where n = number of stored items, m = table size.
Example
When α is too high:
→ Buckets grow longer → O(α) operations instead of O(1)
→ Performance degrades toward O(n)
When α is too low:
→ Wasted memory
→ Many empty buckets
Typical target: α ≈ 0.7-0.75
Rehashing: When α exceeds the threshold, create a new table (usually 2× size) and re-insert all entries:
Example
def _rehash(self):
old_buckets = self.buckets
self.size *= 2
self.buckets = [[] for _ in range(self.size)]
self.count = 0
for bucket in old_buckets:
for key, value in bucket:
self.put(key, value)
Python's Built-in Hash Tables
Example
# dict — hash map (key-value pairs)
d = {}
d["name"] = "Alice" # O(1) insert
d["age"] = 25
print(d["name"]) # O(1) lookup: "Alice"
print("age" in d) # O(1) membership: True
del d["age"] # O(1) delete
# set — hash set (keys only, no values)
s = set()
s.add("apple") # O(1)
s.add("banana")
print("apple" in s) # O(1): True
s.remove("apple") # O(1)
# Counter — frequency hash map
from collections import Counter
freq = Counter(["a","b","a","c","b","a"])
print(freq) # Counter({'a': 3, 'b': 2, 'c': 1})
# defaultdict — hash map with default values
from collections import defaultdict
dd = defaultdict(int) # Default value: 0
dd["count"] += 1 # Works without KeyError
Hash Map vs Other Data Structures
| Operation | Array (unsorted) | Sorted Array | BST (balanced) | Hash Map |
|---|---|---|---|---|
| Search | O(n) | O(log n) | O(log n) | O(1) avg |
| Insert | O(1) append | O(n) shift | O(log n) | O(1) avg |
| Delete | O(n) shift | O(n) shift | O(log n) | O(1) avg |
| Min/Max | O(n) | O(1) | O(log n) | O(n) |
| Range query | O(n) | O(log n + k) | O(log n + k) | O(n) |
| Ordered iteration | O(n log n) | O(n) | O(n) | Not supported |
| Memory | Low | Low | Medium | Medium-High |
Hash maps win for: Lookup-heavy workloads, frequency counting, deduplication BSTs/sorted arrays win for: Range queries, ordered operations, minimum/maximum
Real-World Applications of Hashing
1. Database Indexes (Hash Indexes)
| Query | SELECT * WHERE id = 12345 |
| Hash index: hash(12345) | bucket → O(1) lookup |
| (vs B-Tree index | O(log n) traversal) |
2. Caches (Memoization)
python example
from functools import lru_cache
@lru_cache(maxsize=None)
def fib(n):
if n <= 1: return n
return fib(n-1) + fib(n-2)
# Internally uses a hash map: n → computed fib(n)3. Sets and Deduplication
Example
# Remove duplicates while preserving order
def remove_duplicates(arr):
seen = set()
return [x for x in arr if not (x in seen or seen.add(x))]
4. Two Sum and Similar Problems
Example
def two_sum(nums, target):
seen = {} # value → index
for i, num in enumerate(nums):
complement = target - num
if complement in seen:
return [seen[complement], i]
seen[num] = i
5. Password Hashing (Cryptographic)
Example
User sets password "secret123"
System stores: SHA256("secret123" + salt) = "a3f8b2..."
Never stores the actual password
Summary
Hash tables provide the fastest average-case performance for the core dictionary operations:
- O(1) average insert, delete, lookup
- O(n) worst case (when many collisions degrade to linear search)
- Load factor controls the balance between space and time
- Rehashing maintains O(1) amortized performance
Python's dict and set are hash tables. They are the most-used data structures in Python programs and the first tool to reach for when you need fast lookup.
*Next Lesson: Hash Functions*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Hashing — Complete Introduction to Hash Tables and Hash Functions.
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, hashing, introduction, hashing — complete introduction to hash tables and hash functions
Related Data Structures & Algorithms Topics
Continue learning this concept
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