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
Understand dynamic arrays from the ground up — how automatic resizing works, why append is O(1) amortized, how Python lists and Java ArrayLists are implemented, amortized analysis with the potential method, and how to implement your own dynamic array from scratch.
Introduction
Every time you use a Python list or a Java ArrayList, you are using a dynamic array — a data structure that looks like a simple array but automatically resizes itself as you add elements.
The fact that append() in Python is O(1) amortized despite occasionally requiring O(n) copying might seem like magic. It is not magic — it is a beautifully designed algorithm based on the doubling strategy, and understanding it gives you a deep intuition for amortized analysis that applies far beyond just arrays.
This lesson dissects dynamic arrays completely: the problem they solve, the doubling solution, the formal amortized analysis, their implementation in major languages, and how to build one yourself.
The Doubling Strategy
Rule: When the internal array is full, allocate a new array with twice the current capacity.
| Start | capacity=1, size=0, array=[_] |
| append(1) | size=1, capacity=1, array=[1] |
| append(2): FULL | allocate capacity=2, copy [1], add 2 → [1,2] |
| append(3): FULL | allocate capacity=4, copy [1,2], add 3 → [1,2,3,_] |
| append(4) | size=3, capacity=4, array=[1,2,3,4] |
| append(5): FULL | allocate capacity=8, copy [1,2,3,4], add 5 → [1,2,3,4,5,_,_,_] |
| append(6) | size=5, [1,2,3,4,5,6,_,_] |
| append(7) | size=6, [1,2,3,4,5,6,7,_] |
| append(8) | size=7, [1,2,3,4,5,6,7,8] |
| append(9): FULL | allocate capacity=16, copy 8 elements, add 9 → [1..9,_,_,_,_,_,_,_] |
Resizing (expensive copies) happens at sizes 1, 2, 4, 8, 16, 32, ... (powers of 2).
Amortized Analysis — Why Append is O(1)
Aggregate Method
Total cost of n appends = (cost of actual appends) + (total copy cost during all resizes)
Copy costs occur at: sizes 1, 2, 4, 8, 16, ..., n/2 (last resize before n elements)
Example
Total copies = 1 + 2 + 4 + 8 + ... + n/2
This is a geometric series: Σ 2^i from i=0 to log₂(n)-1
Sum = 2^(log₂(n)) - 1 = n - 1
Total cost of n appends:
- n regular appends: O(n)
- Total copies across all resizes: O(n)
- Grand total: O(2n) = O(n)
Average cost per append: O(n) / n = O(1)
This is the amortized O(1) — averaged over a sequence of operations, each append costs O(1).
Potential Method (More Rigorous)
Define the potential function Φ = 2 × size - capacity
Example
Φ represents "credits stored" — work we have pre-paid for future expensive operations.
When we insert into a non-full array:
Actual cost = 1
ΔΦ = 2×(size+1) - capacity - (2×size - capacity) = 2
Amortized cost = actual + ΔΦ = 1 + 2 = 3 = O(1)
When we insert and resize (array was full, capacity = size):
Before: Φ_before = 2×n - n = n (capacity and size were both n)
After: new capacity = 2n, new size = n+1
Φ_after = 2×(n+1) - 2n = 2
Actual cost = n+1 (copy n elements + 1 append)
ΔΦ = 2 - n
Amortized cost = (n+1) + (2-n) = 3 = O(1)
Every operation has amortized cost O(1), confirming that the total cost over n operations is O(n).
Python List Internals
Python's list is a dynamic array implemented in C (CPython). Key details:
Growth Factor
Python does not exactly double. It uses the formula:
Example
new_capacity = old_capacity + old_capacity // 8 + (6 if old_capacity < 9 else 3)
This gives a growth factor slightly less than double (approximately 1.125×) to avoid over-allocation. The sequence of capacities for a Python list starting empty:
Example
0, 4, 8, 16, 25, 35, 46, 58, 72, 88, 106, ...
Time Complexities
Example
lst = []
# append() — O(1) amortized
lst.append(x)
# pop() from end — O(1)
lst.pop()
# pop(i) from position i — O(n-i)
lst.pop(0) # O(n) — worst case
# insert(i, x) — O(n-i)
lst.insert(0, x) # O(n)
lst.insert(len(lst), x) # O(1) amortized (same as append)
# len() — O(1) — stored as attribute
len(lst)
# in operator (list) — O(n)
x in lst # O(n) — linear search
# indexing — O(1)
lst[i]
# slicing — O(k) where k = slice size
lst[a:b] # O(b-a)
Shrinking Behavior
Python also shrinks the array when elements are removed, but uses a more conservative threshold (shrinks when size drops below capacity/4 or similar) to avoid thrashing.
Java ArrayList and C++ std::vector
Java ArrayList
java example
import java.util.ArrayList;
ArrayList<Integer> list = new ArrayList<>(); // Initial capacity = 10
list.add(5); // O(1) amortized
list.add(0, 5); // O(n) — insert at front
list.remove(0); // O(n) — remove from front
list.get(2); // O(1) — random access
list.size(); // O(1)
// Initial capacity hint (avoid early resizes)
ArrayList<Integer> list = new ArrayList<>(1000);Java's ArrayList uses growth factor 1.5× (adds 50% each resize): new_capacity = old_capacity + (old_capacity >> 1) = old_capacity * 1.5
C++ std::vector
Example
#include <vector>
std::vector<int> v;
v.push_back(5); // O(1) amortized — like append
v.pop_back(); // O(1)
v[2]; // O(1) random access
v.size(); // O(1)
v.capacity(); // Check current allocated capacity
// Reserve to avoid resizes
v.reserve(1000); // Pre-allocate for 1000 elements
// Shrink to fit
v.shrink_to_fit(); // Release excess memory
C++ std::vector uses growth factor 2× (doubles each resize) per most implementations.
Building Your Own Dynamic Array
Example
class DynamicArray:
"""
Implementation of a dynamic array from scratch.
Demonstrates the doubling strategy and all core operations.
"""
def __init__(self, initial_capacity=1):
self._capacity = initial_capacity
self._size = 0
self._data = [None] * self._capacity
def __len__(self):
return self._size
def __getitem__(self, index):
if not (-self._size <= index < self._size):
raise IndexError(f"Index {index} out of range")
if index < 0:
index += self._size
return self._data[index]
def __setitem__(self, index, value):
if not (-self._size <= index < self._size):
raise IndexError(f"Index {index} out of range")
if index < 0:
index += self._size
self._data[index] = value
def append(self, value):
"""Add element at end. O(1) amortized."""
if self._size == self._capacity:
self._resize(2 * self._capacity) # Double capacity
self._data[self._size] = value
self._size += 1
def insert(self, index, value):
"""Insert at position. O(n)."""
if self._size == self._capacity:
self._resize(2 * self._capacity)
# Shift elements right from index to size-1
for i in range(self._size, index, -1):
self._data[i] = self._data[i - 1]
self._data[index] = value
self._size += 1
def remove_at(self, index):
"""Remove element at index. O(n)."""
if not (0 <= index < self._size):
raise IndexError(f"Index {index} out of range")
removed = self._data[index]
# Shift elements left
for i in range(index, self._size - 1):
self._data[i] = self._data[i + 1]
self._data[self._size - 1] = None # Help garbage collection
self._size -= 1
# Shrink if size drops below 1/4 of capacity (avoid thrashing)
if self._size < self._capacity // 4 and self._capacity > 1:
self._resize(max(1, self._capacity // 2))
return removed
def _resize(self, new_capacity):
"""Allocate new array and copy elements. O(n)."""
new_data = [None] * new_capacity
for i in range(self._size):
new_data[i] = self._data[i]
self._data = new_data
self._capacity = new_capacity
print(f" [Resize: capacity → {new_capacity}]") # For demonstration
def __str__(self):
return "[" + ", ".join(str(self._data[i]) for i in range(self._size)) + "]"
def __repr__(self):
return f"DynamicArray(size={self._size}, capacity={self._capacity})"
Testing our implementation:
Example
da = DynamicArray(initial_capacity=2)
print("Appending elements:")
for i in range(1, 9):
da.append(i)
print(f" append({i}): {da}, capacity={da._capacity}")
# Output:
# append(1): [1], capacity=2
# append(2): [1, 2], capacity=2
# [Resize: capacity → 4]
# append(3): [1, 2, 3], capacity=4
# append(4): [1, 2, 3, 4], capacity=4
# [Resize: capacity → 8]
# append(5): [1, 2, 3, 4, 5], capacity=8
# ...
print("\nAccess:", da[3]) # 4
print("Update:", end=" ")
da[3] = 99
print(da) # [1, 2, 3, 99, 5, 6, 7, 8]
print("Remove at 3:", da.remove_at(3)) # 99
print("After removal:", da) # [1, 2, 3, 5, 6, 7, 8]
Memory Usage Analysis
Example
import sys
# Python list memory usage
lst = []
for n in [0, 10, 100, 1000]:
lst = list(range(n))
print(f"n={n}: {sys.getsizeof(lst)} bytes")
# Output (approximate, depends on Python version):
# n=0: 56 bytes (empty list header)
# n=10: 184 bytes (~12 bytes/element overhead + 56 header)
# n=100: 920 bytes
# n=1000: 8056 bytes
# This includes capacity > size due to doubling strategy
# The "wasted" space is part of the amortized O(1) append guarantee
Summary
- Static arrays have fixed size — resizing requires manual allocation and copying
- Dynamic arrays resize automatically using the doubling strategy
- Doubling: When full, allocate 2× capacity, copy all elements
- Amortized analysis: Total copy cost over n appends = O(n), so average per append = O(1)
- Python list: Growth factor ≈ 1.125, O(1) append/pop from end, O(n) insert/remove from middle
- Java ArrayList: Growth factor 1.5×
- C++ std::vector: Growth factor 2×
- The same amortized O(1) trick appears in: hash table resizing, stack using array, queue using circular buffer
*Next Lesson: Multidimensional Arrays — 2D and 3D arrays, matrix operations, and the row-major vs. column-major memory layout.*
Continue the lesson path
Study next in this module
01ArraysArrays — The Complete Introduction to the Most Fundamental Data StructureA thorough introduction to arrays — what they are, how they work in memory, their properties, all basic operations with time and space complexity, comparison with other structures, and when to use them. The foundation of all data structure knowledge.02ArraysMultidimensional Arrays — 2D Arrays, Matrices, and Memory LayoutA complete guide to multidimensional arrays — how 2D and 3D arrays are stored in memory, row-major vs column-major order, all matrix operations with code and complexity, common matrix algorithms, and interview problems involving matrices.03ArraysArray Searching — Linear Search, Binary Search, and Advanced TechniquesComplete guide to searching arrays — linear search with early termination, binary search with all variants, interpolation search, and exponential search. Includes full implementations, complexity analysis, and classic search-based interview problems.04ArraysSliding Window Technique — Complete Guide with 10 Classic ProblemsMaster the sliding window technique — one of the most powerful array problem-solving patterns. Learn fixed-size and variable-size windows, when to apply this technique, and solve 10 classic interview problems including maximum sum subarray, longest substring without repeats, minimum window substring, and more.