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 all collision resolution strategies for hash tables — separate chaining with linked lists, open addressing (linear probing, quadratic probing, double hashing), comparison of all methods, load factor effects, and when to use each approach.
Why Collisions Are Inevitable
By the pigeonhole principle: if you hash n keys into m slots where n > m, at least one slot must contain multiple keys. Even with n ≤ m, the birthday paradox tells us collisions become likely once n > √m.
For a table of size m = 365, collisions are likely when n > √365 ≈ 19 keys.
The table cannot avoid collisions — it can only handle them.
Strategy 2 — Open Addressing
All elements stored in the table itself — no external chains. When a collision occurs, probe a sequence of slots until an empty one is found.
Linear Probing
Example
class LinearProbingHashTable:
DELETED = object() # Sentinel for deleted slots
def __init__(self, size=16):
self.size = size
self.keys = [None] * size
self.values = [None] * size
self.count = 0
def _hash(self, key):
return hash(key) % self.size
def _probe_sequence(self, key):
"""Generate probe sequence: h(k), h(k)+1, h(k)+2, ..."""
start = self._hash(key)
for i in range(self.size):
yield (start + i) % self.size
def put(self, key, value):
if self.count / self.size > 0.5:
self._resize()
for idx in self._probe_sequence(key):
if self.keys[idx] is None or self.keys[idx] is self.DELETED:
self.keys[idx] = key
self.values[idx] = value
self.count += 1
return
elif self.keys[idx] == key:
self.values[idx] = value # Update
return
def get(self, key, default=None):
for idx in self._probe_sequence(key):
if self.keys[idx] is None:
return default # Empty slot — key not in table
if self.keys[idx] == key:
return self.values[idx]
return default
def delete(self, key):
for idx in self._probe_sequence(key):
if self.keys[idx] is None:
return False
if self.keys[idx] == key:
self.keys[idx] = self.DELETED # Mark as deleted (tombstone)
self.values[idx] = None
self.count -= 1
return True
return False
def _resize(self):
old_keys = self.keys
old_vals = self.values
self.size *= 2
self.keys = [None] * self.size
self.values = [None] * self.size
self.count = 0
for k, v in zip(old_keys, old_vals):
if k is not None and k is not self.DELETED:
self.put(k, v)
Linear probing clustering problem:
| Table | [_, _, A, B, C, _, D, _] (A,B,C,D hashed to same region) |
| idx 2: A ≠ E | try idx 3 |
| idx 3: B ≠ E | try idx 4 |
| idx 4: C ≠ E | try idx 5 |
| idx 5: empty | insert E |
| The cluster [A,B,C,D] grows with each insertion | "primary clustering" |
Quadratic Probing
python example
def _probe_sequence(self, key):
"""Probe: h(k), h(k)+1, h(k)+4, h(k)+9, ... (i²)"""
start = self._hash(key)
for i in range(self.size):
yield (start + i * i) % self.sizeReduces primary clustering but introduces secondary clustering — all keys with the same initial hash follow the same probe sequence.
Double Hashing
Example
def _probe_sequence(self, key):
"""
Probe: h1(k), h1(k)+h2(k), h1(k)+2*h2(k), ...
h2(k) must never be 0 and should be coprime to m.
"""
h1 = hash(key) % self.size
h2 = 1 + (hash(key) % (self.size - 1)) # h2 ∈ [1, m-1]
for i in range(self.size):
yield (h1 + i * h2) % self.size
Double hashing eliminates both primary and secondary clustering. Each key has its own probe sequence. Best performance among open addressing methods.
Tombstone (Lazy Deletion) Problem
With open addressing, you cannot simply mark a deleted slot as empty — this would break lookups for keys that probed past this slot.
| Table after insertions | [_, A, B, C, _, D, _] |
| Delete A: mark as DELETED (tombstone) | [_, DEL, B, C, _, D, _] |
| idx 1: DEL (not empty!) | continue probing |
| idx 2: B == B | found! ✓ |
| idx 1: None | STOP (would incorrectly return "not found"!) |
Tombstones work but accumulate over time. A full rehash (remove tombstones) is needed periodically.
Load Factor Effects on Performance
Chaining
α = 0.5: average chain = 0.5 → fast
α = 1.0: average chain = 1.0 → acceptable
α = 2.0: average chain = 2.0 → slower
No hard limit — chains can grow indefinitely
Open Addressing (Linear Probing)
α = 0.5: avg probes ≈ 1.5 → fast
α = 0.7: avg probes ≈ 2.0 → acceptable
α = 0.9: avg probes ≈ 5.5 → slow
α = 1.0: impossible (table full)
Formula: avg probes ≈ 1/(1-α) for lookup failure
This grows dramatically as α → 1
Typical thresholds
Python dict: rehash when α > 0.67
Java HashMap: rehash when α > 0.75
Open addressing: rehash when α > 0.5
Chaining vs Open Addressing Comparison
| Feature | Chaining | Open Addressing (Double Hash) |
|---|---|---|
| Load factor limit | No limit (just slower) | Must stay < 1 |
| Practical α | ≤ 1 recommended | ≤ 0.5-0.7 |
| Memory | Extra pointers + heap | Compact array |
| Cache performance | Poor (pointer chasing) | Excellent (array-based) |
| Deletion | Simple (remove from list) | Complex (tombstones) |
| Worst case | O(n) one big chain | O(n) full table scan |
| Typical use | Java HashMap | Python dict, open addressing tables |
Python's dict uses open addressing with a variant of linear probing (perturbed probing) for excellent cache performance.
Robin Hood Hashing
A clever improvement to open addressing: when inserting, a key can "steal" a slot from a key that is "richer" (closer to its home slot).
| If new key's displacement > current key's displacement | SWAP |
| (The new key is "poorer" | it needs the slot more) |
| Result | Maximum displacement across all keys is minimized. |
This makes worst-case lookup O(log n) with high probability — much better than standard open addressing.
Summary
| Method | Mechanism | Pros | Cons |
|---|---|---|---|
| Chaining | Linked list per slot | Simple, handles high α, easy delete | Extra memory, poor cache |
| Linear Probing | Step by 1 | Cache friendly | Primary clustering |
| Quadratic Probing | Step by i² | Less clustering | Secondary clustering, misses slots |
| Double Hashing | Two hash functions | No clustering | Slower hash computation |
| Robin Hood | Minimize displacement | Lower variance | More complex |
For most applications: Python's built-in dict (open addressing with smart probing) or Java's HashMap (chaining with TreeMap for long chains) is the right choice.
*Next Lesson: Chaining*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Collision Handling — Chaining vs Open Addressing Complete Comparison.
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, collision, handling
Related Data Structures & Algorithms Topics
Continue learning this concept
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