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 deque (double-ended queue) — a data structure allowing O(1) insertion and deletion at both ends. Learn complete implementation using doubly linked list and circular array, all operations, monotonic deque applications, and classic sliding window maximum problem.
Introduction
A deque (pronounced "deck") stands for Double-Ended Queue. Unlike a regular queue (enqueue at rear, dequeue at front), a deque allows insertion and deletion at both ends — the front and the rear — all in O(1).
A deque is more general than both a stack and a queue. It can simulate a stack (add/remove from same end) or a queue (add to one end, remove from other end).
Python's collections.deque
Python provides a highly optimized deque in collections.deque, implemented as a doubly-linked list of fixed-size blocks:
Example
from collections import deque
d = deque()
# Add to ends
d.append(3) # Right: [3]
d.appendleft(1) # Left: [1, 3]
d.append(5) # Right: [1, 3, 5]
d.appendleft(0) # Left: [0, 1, 3, 5]
print(d) # deque([0, 1, 3, 5])
# Remove from ends
d.pop() # Remove right: 5, deque = [0, 1, 3]
d.popleft() # Remove left: 0, deque = [1, 3]
# Peek
print(d[0]) # Left end: 1
print(d[-1]) # Right end: 3
# Bounded deque (automatically discards from other end when full)
d_bounded = deque(maxlen=3)
d_bounded.append(1); d_bounded.append(2); d_bounded.append(3)
d_bounded.append(4) # 1 is automatically removed from left
print(d_bounded) # deque([2, 3, 4], maxlen=3)
Custom Deque Implementation — Doubly Linked List
Example
class _DequeNode:
__slots__ = ('data', 'prev', 'next')
def __init__(self, data):
self.data = data
self.prev = None
self.next = None
class Deque:
"""
Double-ended queue using a doubly linked list.
All operations guaranteed O(1).
"""
def __init__(self):
# Sentinel nodes (dummy head and tail) simplify edge cases
self._left = _DequeNode(None) # Dummy left sentinel
self._right = _DequeNode(None) # Dummy right sentinel
self._left.next = self._right
self._right.prev = self._left
self._size = 0
def _insert_between(self, data, before, after):
"""Insert new node between two existing nodes. O(1)."""
node = _DequeNode(data)
node.prev = before
node.next = after
before.next = node
after.prev = node
self._size += 1
return node
def _remove_node(self, node):
"""Remove a specific node. O(1)."""
node.prev.next = node.next
node.next.prev = node.prev
self._size -= 1
return node.data
def append(self, data) -> None:
"""Add to right end. O(1)."""
self._insert_between(data, self._right.prev, self._right)
def appendleft(self, data) -> None:
"""Add to left end. O(1)."""
self._insert_between(data, self._left, self._left.next)
def pop(self):
"""Remove from right end. O(1)."""
if self.is_empty():
raise IndexError("pop from empty deque")
return self._remove_node(self._right.prev)
def popleft(self):
"""Remove from left end. O(1)."""
if self.is_empty():
raise IndexError("popleft from empty deque")
return self._remove_node(self._left.next)
def peek_right(self):
if self.is_empty():
raise IndexError("peek_right from empty deque")
return self._right.prev.data
def peek_left(self):
if self.is_empty():
raise IndexError("peek_left from empty deque")
return self._left.next.data
def is_empty(self) -> bool:
return self._size == 0
def __len__(self) -> int:
return self._size
def __repr__(self) -> str:
elements = []
cur = self._left.next
while cur != self._right:
elements.append(str(cur.data))
cur = cur.next
return "Deque(left→right): [" + ", ".join(elements) + "]"
Deque as Stack and Queue
python example
# Use as Stack (LIFO):
stack = Deque()
stack.append(1) # push
stack.append(2)
print(stack.pop()) # pop from right: 2
# Use as Queue (FIFO):
queue = Deque()
queue.append(1) # enqueue at right
queue.append(2)
print(queue.popleft()) # dequeue from left: 1Monotonic Deque — The Sliding Window Maximum
The most important algorithmic use of a deque: maintaining a monotonic deque for sliding window problems.
Problem: Given an array and window size k, find the maximum in every window of size k.
Example
[1, 3, -1, -3, 5, 3, 6, 7], k=3
Windows: [1,3,-1] [3,-1,-3] [-1,-3,5] [-3,5,3] [5,3,6] [3,6,7]
Maximums: 3 3 5 5 6 7
Naive approach: O(n×k) — recompute max for each window
Optimal with monotonic deque: O(n)
Example
from collections import deque
def sliding_window_maximum(nums, k):
"""
Monotonic decreasing deque: maintains potential maximums.
Front of deque = maximum of current window.
Invariant: Deque stores indices in decreasing order of their values.
Elements that can never be the maximum are discarded from the back.
Time: O(n), Space: O(k)
"""
result = []
dq = deque() # Stores INDICES, not values
for i in range(len(nums)):
# Remove indices outside the current window
while dq and dq[0] < i - k + 1:
dq.popleft()
# Remove indices whose values are smaller than nums[i]
# (they can never be the max while nums[i] is in the window)
while dq and nums[dq[-1]] < nums[i]:
dq.pop()
dq.append(i)
# Window has k elements starting from index k-1
if i >= k - 1:
result.append(nums[dq[0]]) # Front = maximum
return result
print(sliding_window_maximum([1,3,-1,-3,5,3,6,7], 3))
# [3, 3, 5, 5, 6, 7]
Trace:
| i=0, nums[0]=1 | dq=[0] |
| i=1, nums[1]=3 | 1<3, pop 0. dq=[1] |
| i=2, nums[2]=-1 | dq=[1,2]. Window=[1,2]: max=nums[1]=3. result=[3] |
| i=3, nums[3]=-3 | dq=[1,2,3]. Window=[2,3]: max=nums[1]=3. result=[3,3] |
| i=4, nums[4]=5 | -3<5 pop 3, -1<5 pop 2, 3<5 pop 1. dq=[4]. Window=[3,4]: max=5. result=[3,3,5] |
| i=5, nums[5]=3 | dq=[4,5]. Window=[4,5]: max=nums[4]=5. result=[3,3,5,5] |
| i=6, nums[6]=6 | 3<6 pop 5. dq=[4,6]. Index 4 still in window (6-3+1=4). max=6. result=[3,3,5,5,6] |
| i=7, nums[7]=7 | 6<7 pop 6, 5<7 pop 4. dq=[7]. max=7. result=[3,3,5,5,6,7] |
First Negative in Every Window
Example
def first_negative_in_window(arr, k):
"""
Find the first negative number in each window of size k.
Use deque to track indices of negative numbers.
Time: O(n), Space: O(k)
"""
result = []
dq = deque() # Indices of negative numbers
for i in range(len(arr)):
if arr[i] < 0:
dq.append(i)
# Remove indices outside window
while dq and dq[0] < i - k + 1:
dq.popleft()
if i >= k - 1:
result.append(arr[dq[0]] if dq else 0)
return result
print(first_negative_in_window([-8, 2, 3, -6, 10], 2)) # [-8, 0, -6, -6]
Deque Applications Summary
| Application | Deque Usage |
|---|---|
| Sliding window maximum | Monotonic decreasing deque |
| Sliding window minimum | Monotonic increasing deque |
| BFS with bidirectional search | Both-end access |
| 0-1 BFS (weighted: 0 or 1) | Prepend weight-0 edges, append weight-1 |
| Palindrome check (iterative) | Compare front and back |
| Undo-redo with history limit | Bounded deque (maxlen) |
| LRU cache (combined with hash map) | Doubly linked list = deque |
| Text editor cursor (left/right) | Two stacks ~ two halves of deque |
Summary
- Deque = double-ended queue: O(1) add/remove from both ends
- More general than stack or queue — can simulate both
- Python's
collections.deque: The standard, optimized implementation — use this by default - Custom implementation: Doubly linked list with sentinel nodes
- Monotonic deque: The key algorithmic pattern — maintains potential extremes for sliding window
maxlendeque: Automatically drops oldest elements — perfect for rolling windows
*Next Lesson: Priority Queue — Elements processed by priority, not arrival order — implemented with heaps.*
Continue the lesson path
Study next in this module
01QueuesCircular Queue — The Ring Buffer Explained with ImplementationA deep dive into circular queues — why they exist, the wasted space problem with linear array queues, how modular arithmetic creates the circular wraparound, full implementation with all edge cases, the distinction between full and empty states, and real-world applications as ring buffers.02QueuesPriority Queue — Complete Guide from Concept to ImplementationMaster priority queues — what they are, how they differ from regular queues, heap-based implementation, Python03QueuesQueue Applications — BFS, Level-Order Traversal, 01-BFS, and MoreReal-world and algorithmic applications of queues — BFS for shortest paths in unweighted graphs, level-order tree traversal, 01-BFS for binary-weighted graphs, topological sort using BFS (Kahn04QueuesQueue Implementation — Array, Linked List, and Two-Stack ApproachesComplete queue implementations using three different underlying structures — circular array (O(1) all operations), singly linked list (O(1) all operations), and two stacks (O(1) amortized). Includes the classic interview question of implementing a queue using two stacks.