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 guide to doubly linked lists — node structure with prev and next pointers, all operations with O(1) and O(n) complexities, comparison with singly linked lists, LRU Cache implementation using doubly linked list and hash map, and interview problems.
Introduction
A doubly linked list is a linked list where each node has two pointers — next (pointing forward) and prev (pointing backward). This bidirectionality removes one of the major limitations of singly linked lists: the inability to efficiently delete a node given only its reference.
In a singly linked list, to delete a node, you need to find the *previous* node first (O(n) traversal). In a doubly linked list, every node already knows its previous node — deletion takes O(1).
This property makes doubly linked lists the foundation of one of the most important real-world data structures: the LRU Cache.
Complete Doubly Linked List Implementation
python example
class DoublyLinkedList:
def __init__(self):
self.head = None
self.tail = None
self._size = 0
def __len__(self):
return self._size
def is_empty(self):
return self.head is None
def __repr__(self):
if self.head is None:
return "None"
nodes = []
cur = self.head
while cur:
nodes.append(str(cur.data))
cur = cur.next
return " ↔ ".join(nodes)Insertion Operations
Insert at Head — O(1)
python example
def insert_at_head(self, data):
new_node = DoublyNode(data)
if self.head is None:
self.head = self.tail = new_node
else:
new_node.next = self.head # New node → old head
self.head.prev = new_node # Old head ← new node
self.head = new_node # Update head
self._size += 1Visualization:
Example
Before: head → [None|10|→] ↔ [←|25|None] tail
Insert 5:
new_node(5).next = head(10)
head(10).prev = new_node(5)
head = new_node(5)
After: head → [None|5|→] ↔ [←|10|→] ↔ [←|25|None] tail
Insert at Tail — O(1)
python example
def insert_at_tail(self, data):
new_node = DoublyNode(data)
if self.tail is None:
self.head = self.tail = new_node
else:
new_node.prev = self.tail # New node ← old tail
self.tail.next = new_node # Old tail → new node
self.tail = new_node # Update tail
self._size += 1Insert After a Given Node — O(1)
python example
def insert_after(self, node, data):
"""
Insert a new node immediately after the given node.
O(1) — no traversal needed.
This is the key advantage of doubly linked lists.
"""
if node is None:
raise ValueError("Node cannot be None")
new_node = DoublyNode(data)
new_node.prev = node # new ← node
new_node.next = node.next # new → node.next
if node.next is not None:
node.next.prev = new_node # node.next ← new
else:
self.tail = new_node # new_node is the new tail
node.next = new_node # node → new
self._size += 1Insert at Position — O(n)
python example
def insert_at_position(self, data, position):
if position < 0 or position > self._size:
raise IndexError(f"Position {position} out of range")
if position == 0:
self.insert_at_head(data)
return
if position == self._size:
self.insert_at_tail(data)
return
# Optimization: if position is in second half, traverse from tail
if position <= self._size // 2:
current = self.head
for _ in range(position - 1):
current = current.next
else:
current = self.tail
for _ in range(self._size - position - 1):
current = current.prev
self.insert_after(current, data)Deletion Operations
Delete a Given Node — O(1)
This is the primary advantage over singly linked lists.
python example
def delete_node(self, node):
"""
Delete a specific node in O(1).
In a singly linked list, this would require O(n) to find the previous node.
Here, node.prev gives us the previous node directly.
"""
if node is None:
return
# Update the previous node's next pointer
if node.prev is not None:
node.prev.next = node.next
else:
self.head = node.next # Deleting head
# Update the next node's prev pointer
if node.next is not None:
node.next.prev = node.prev
else:
self.tail = node.prev # Deleting tail
# Clean up the deleted node's pointers
node.prev = None
node.next = None
self._size -= 1Visualization:
Example
Before: [1] ↔ [2] ↔ [3] ↔ [4]
Delete node(3):
node(2).next = node(4) (skip over 3 in forward direction)
node(4).prev = node(2) (skip over 3 in backward direction)
After: [1] ↔ [2] ↔ [4]
Delete Head — O(1)
python example
def delete_head(self):
if self.head is None:
raise ValueError("List is empty")
removed = self.head.data
self.delete_node(self.head)
return removedDelete Tail — O(1)
python example
def delete_tail(self):
"""
O(1) — tail pointer + prev pointer give direct access.
Compare: singly linked list delete tail is O(n).
"""
if self.tail is None:
raise ValueError("List is empty")
removed = self.tail.data
self.delete_node(self.tail)
return removedTraversal — Both Directions
python example
def traverse_forward(self):
"""Print list from head to tail."""
result = []
current = self.head
while current:
result.append(str(current.data))
current = current.next
print(" → ".join(result))
def traverse_backward(self):
"""Print list from tail to head."""
result = []
current = self.tail
while current:
result.append(str(current.data))
current = current.prev
print(" → ".join(result))Singly vs Doubly — Direct Comparison
| Operation | Singly Linked | Doubly Linked | Notes |
|---|---|---|---|
| Insert at head | O(1) | O(1) | Same |
| Insert at tail (with tail ptr) | O(1) | O(1) | Same |
| Insert after given node | O(1) | O(1) | Same |
| Insert before given node | O(n) | O(1) | DLL has prev pointer |
| Delete head | O(1) | O(1) | Same |
| Delete tail | O(n) | O(1) | DLL has prev pointer |
| Delete given node | O(n) | O(1) | DLL's biggest advantage |
| Traverse forward | O(n) | O(n) | Same |
| Traverse backward | O(n) | O(n) | Both O(n), but DLL is simpler |
| Memory per node | data + 1 ptr | data + 2 ptrs | SLL uses less memory |
| Reverse traversal | O(n) complex | O(n) simple | DLL has prev ptr |
LRU Cache — The Classic Application
LRU (Least Recently Used) Cache is one of the most important data structures in systems programming. It evicts the least recently used item when the cache is full.
Requirements:
get(key)→ O(1) averageput(key, value)→ O(1) average- When cache is full, remove the least recently used item
Solution: Doubly linked list (for O(1) move-to-front and O(1) delete) + Hash Map (for O(1) lookup).
Example
class LRUCache:
"""
LRU Cache using Doubly Linked List + Hash Map.
Both get() and put() are O(1).
"""
class _Node:
def __init__(self, key, value):
self.key = key
self.value = value
self.prev = None
self.next = None
def __init__(self, capacity):
self.capacity = capacity
self.cache = {} # key → node (for O(1) lookup)
# Doubly linked list with sentinel head and tail nodes
# Most Recently Used end = after head
# Least Recently Used end = before tail
self.head = self._Node(0, 0) # Dummy head
self.tail = self._Node(0, 0) # Dummy tail
self.head.next = self.tail
self.tail.prev = self.head
def _remove(self, node):
"""Remove a node from the doubly linked list. O(1)."""
node.prev.next = node.next
node.next.prev = node.prev
def _add_to_front(self, node):
"""Add a node right after the dummy head (MRU position). O(1)."""
node.prev = self.head
node.next = self.head.next
self.head.next.prev = node
self.head.next = node
def get(self, key):
"""
Get value for key. O(1).
If found, move to front (most recently used).
"""
if key not in self.cache:
return -1
node = self.cache[key]
self._remove(node) # Remove from current position
self._add_to_front(node) # Move to MRU position
return node.value
def put(self, key, value):
"""
Insert/update key-value pair. O(1).
If key exists, update and move to front.
If cache full, evict LRU (node before dummy tail) first.
"""
if key in self.cache:
node = self.cache[key]
node.value = value
self._remove(node)
self._add_to_front(node)
else:
if len(self.cache) == self.capacity:
# Evict LRU: the node just before dummy tail
lru_node = self.tail.prev
self._remove(lru_node)
del self.cache[lru_node.key]
new_node = self._Node(key, value)
self.cache[key] = new_node
self._add_to_front(new_node)
Testing:
python example
lru = LRUCache(capacity=3)
lru.put(1, "one") # Cache: {1:one}
lru.put(2, "two") # Cache: {1:one, 2:two}
lru.put(3, "three") # Cache: {1:one, 2:two, 3:three}
print(lru.get(1)) # "one" → moves 1 to MRU front
lru.put(4, "four") # Evicts 2 (LRU) → Cache: {1:one, 3:three, 4:four}
print(lru.get(2)) # -1 (evicted)
print(lru.get(3)) # "three"Summary
The doubly linked list's key advantages over singly linked list:
- O(1) delete of any node given its reference — via
prevpointer - O(1) insert before a node — via
prevpointer - O(1) delete tail — via
tail.prevpointer - Bidirectional traversal — both forward and backward
Cost: double the pointer memory per node.
The doubly linked list is the backbone of the LRU Cache, which combines it with a hash map to achieve O(1) for both get and put. This data structure is fundamental to OS memory management, database buffer pools, and web browser caches.
*Next Lesson: Circular Linked List — Circular structure for round-robin scheduling and cyclic sequences.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Doubly Linked List — Complete Implementation with Bidirectional Operations.
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, linked, lists, doubly
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