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 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.
Introduction
The array is the most fundamental data structure in computer science. Almost every other data structure is built on top of arrays, implemented using arrays, or compared against arrays. Without a deep understanding of arrays — not just how to use them, but how they work at the memory level — you cannot truly understand any more advanced data structure.
This lesson covers arrays completely: from the raw memory model to Python's dynamic lists, from basic operations to their precise time complexities, from simple 1D arrays to multi-dimensional matrices. Take your time with each section. This is the foundation everything else is built on.
Memory Model — Why Arrays Are Special
Understanding the memory model of arrays reveals *why* they have O(1) random access — the property that makes them so powerful.
The Address Formula
Because all elements are the same size and stored contiguously, the memory address of any element can be calculated directly:
Example
address(arr[i]) = base_address + i × element_size
Where:
base_address = memory address of arr[0]
i = index of the desired element
element_size = bytes per element (4 for int, 8 for double, etc.)
This formula is computed in constant time (a single multiplication and addition). No searching, no traversal — the CPU jumps directly to the correct memory location.
Example with 64-bit (8-byte) doubles:
| Base address | 2000 |
| arr[0] | address 2000 + 0×8 = 2000 |
| arr[2] | address 2000 + 2×8 = 2016 |
| arr[4] | address 2000 + 4×8 = 2032 |
| Access time | O(1) — always one formula calculation |
Why This Makes Random Access O(1)
Accessing arr[999] takes the same time as accessing arr[0]. There is no need to count through 999 elements to find the 999th one — you just compute 2000 + 999×8 = 9992, and the hardware reads from address 9992 directly.
This is fundamentally different from a linked list, where to access the 1000th element, you must follow 999 pointers from the beginning.
Cache Friendliness
Modern processors have a cache hierarchy — small amounts of very fast memory (L1, L2, L3 cache) that buffer recently accessed memory regions. When you access arr[0], the processor loads not just that single value, but an entire cache line (typically 64 bytes = 16 integers) into the fast cache.
When you then access arr[1], arr[2], ..., arr[15], they are already in cache — these accesses are essentially free. This spatial locality makes sequential array traversal dramatically faster in practice than theoretical analysis suggests.
Linked lists, trees, and hash maps with their scattered memory addresses destroy spatial locality, making them much slower in practice despite sometimes having the same theoretical complexity.
Types of Arrays
1. Static Arrays (Fixed-Size)
A static array is allocated with a fixed size that cannot change after creation.
Example
// C — static array, fixed at 5 elements
int scores[5] = {88, 74, 91, 65, 83};
// Cannot expand: scores[5] = 90; // ERROR — undefined behavior
Example
// Java — static array
int[] scores = new int[5]; // Allocates space for exactly 5 ints
scores[0] = 88;
scores[1] = 74;
// scores[5] = 90; // Throws ArrayIndexOutOfBoundsException
Properties:
- Size must be known at compile time (or at least before allocation)
- Cannot grow or shrink
- Memory allocation is on the stack (for local variables) or heap
- Direct hardware support — the most efficient data structure possible
2. Dynamic Arrays (Automatically Resizing)
A dynamic array grows and shrinks automatically as elements are added or removed. Python's list, Java's ArrayList, and C++'s std::vector are dynamic arrays.
python example
# Python list — dynamic array
scores = [] # Empty, size 0
scores.append(88) # Size becomes 1
scores.append(74) # Size becomes 2
scores.append(91) # Size becomes 3
# Python automatically manages the underlying memoryHow dynamic resizing works (under the hood):
When a Python list runs out of internal capacity:
- A new array with a larger capacity is allocated (typically 1.125× to 2× the current size)
- All existing elements are copied to the new array
- The old array is freed
| Start | capacity = 0 |
| append(88): capacity becomes 4 | arr = [88, _, _, _] |
| append(74) | arr = [88, 74, _, _] |
| append(91) | arr = [88, 74, 91, _] |
| append(65) | arr = [88, 74, 91, 65] (full) |
| append(83): capacity doubles to 8 | copy → arr = [88, 74, 91, 65, 83, _, _, _] |
This copying makes individual appends O(n) sometimes, but amortized over n appends, the average cost per append is O(1).
3. Multi-Dimensional Arrays
Arrays can be extended to multiple dimensions.
2D Array (Matrix):
Example
# 3×4 matrix
matrix = [
[1, 2, 3, 4 ],
[5, 6, 7, 8 ],
[9, 10, 11, 12]
]
# Access: matrix[row][column]
print(matrix[1][2]) # 7 (row 1, column 2)
print(matrix[0][3]) # 4 (row 0, column 3)
| Logical | [1][2][3][4] | [5][6][7][8] | [9][10][11][12] |
| In memory | 1 2 3 4 5 6 7 8 9 10 11 12 |
| matrix[1][2] | base + (1×4 + 2) × 4 = base + 24 |
3D Array:
Example
# 3D array — e.g., for a video (frames × height × width × channels)
video = [[[0 for _ in range(3)] for _ in range(4)] for _ in range(5)]
# Access: video[frame][row][col]
Array Properties — Summary
| Property | Description |
|---|---|
| Element type | All elements same type (in typed languages) |
| Memory layout | Contiguous — no gaps |
| Size | Fixed (static array) or resizable (dynamic array) |
| Indexing | Zero-based in Python, Java, C/C++, JavaScript |
| Access | O(1) — direct address calculation |
| Cache behavior | Excellent spatial locality |
| Memory overhead | Minimal — just the elements (+ small header) |
Basic Array Operations
1. Creation and Initialization
Example
# Empty array
arr = []
# Array with initial values
arr = [10, 25, 8, 42, 15]
# Array of specific size filled with zeros
arr = [0] * 10 # [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
# Array from a range
arr = list(range(1, 6)) # [1, 2, 3, 4, 5]
# Array with all same value
arr = [None] * 5 # [None, None, None, None, None]
2. Access — O(1)
Example
arr = [10, 25, 8, 42, 15]
# Access by index — O(1)
first = arr[0] # 10
last = arr[-1] # 15 (Python: negative index from end)
third = arr[2] # 8
# Access any index — always same time regardless of index value
print(arr[0]) # O(1)
print(arr[4]) # O(1) — same time as arr[0]
print(arr[1000000]) # O(1) — if index is valid
3. Update — O(1)
Example
arr = [10, 25, 8, 42, 15]
arr[2] = 99 # Change index 2 from 8 to 99
print(arr) # [10, 25, 99, 42, 15]
# Direct address calculation, no traversal needed — O(1)
4. Traversal — O(n)
Example
arr = [10, 25, 8, 42, 15]
# Forward traversal
for element in arr:
print(element)
# With index
for i, element in enumerate(arr):
print(f"arr[{i}] = {element}")
# Reverse traversal
for element in reversed(arr):
print(element)
# While loop traversal
i = 0
while i < len(arr):
print(arr[i])
i += 1
# All O(n) — must visit each element
5. Search — O(n) for unsorted, O(log n) for sorted
Example
arr = [10, 25, 8, 42, 15]
# Linear search — O(n)
def linear_search(array, target):
for i in range(len(array)):
if array[i] == target:
return i # Found at index i
return -1 # Not found
print(linear_search(arr, 42)) # Output: 3
print(linear_search(arr, 99)) # Output: -1
# Binary search — O(log n), requires sorted array
sorted_arr = [5, 12, 18, 30, 47, 55, 63, 77, 88, 95]
def binary_search(array, target):
left, right = 0, len(array) - 1
while left <= right:
mid = (left + right) // 2
if array[mid] == target:
return mid
elif array[mid] < target:
left = mid + 1
else:
right = mid - 1
return -1
print(binary_search(sorted_arr, 47)) # Output: 4
6. Insertion — O(1) at end, O(n) at position
Example
arr = [10, 25, 8, 42, 15]
# Append at end — O(1) amortized
arr.append(99)
print(arr) # [10, 25, 8, 42, 15, 99]
# Insert at beginning — O(n) — must shift all elements right
arr.insert(0, 5)
print(arr) # [5, 10, 25, 8, 42, 15, 99]
# Insert at position i — O(n) — must shift elements from i to end
arr.insert(3, 77)
print(arr) # [5, 10, 25, 77, 8, 42, 15, 99]
Why insert at position is O(n):
| Before | [10, 25, 8, 42, 15] |
| Step 1 | Shift 15 right: [10, 25, 8, 42, _, 15] |
| Step 2 | Shift 42 right: [10, 25, 8, _, 42, 15] |
| Step 3 | Shift 8 right: [10, 25, _, 8, 42, 15] |
| Step 4 | Place 77: [10, 25, 77, 8, 42, 15] |
| Worst case (insert at beginning): n shifts | O(n) |
7. Deletion — O(1) at end, O(n) at position
Example
arr = [10, 25, 8, 42, 15]
# Remove last element — O(1)
last = arr.pop()
print(last) # 15
print(arr) # [10, 25, 8, 42]
# Remove from position — O(n) — must shift elements left
arr.pop(1) # Remove element at index 1 (25)
print(arr) # [10, 8, 42]
Why delete from position is O(n):
| Before | [10, 25, 8, 42, 15] |
| Step 1 | Move 42 left: [10, 25, 42, _, 15] |
| Step 2 | Move 15 left: [10, 25, 42, 15, _] |
| Step 3 | Reduce size: [10, 25, 42, 15] |
| Worst case (delete first element): n-1 shifts | O(n) |
Operations Complexity Summary
| Operation | Time Complexity | Space | Notes |
|---|---|---|---|
| Access arr[i] | O(1) | O(1) | Direct address formula |
| Update arr[i] = v | O(1) | O(1) | Direct address formula |
| Append at end | O(1)* | O(1) | *Amortized (O(n) rarely) |
| Insert at position i | O(n) | O(1) | Shifts n-i elements right |
| Delete from end | O(1) | O(1) | |
| Delete from position i | O(n) | O(1) | Shifts n-i-1 elements left |
| Search (unsorted) | O(n) | O(1) | Linear scan |
| Search (sorted) | O(log n) | O(1) | Binary search |
| Sort | O(n log n) | O(log n) | Typical efficient sort |
| Traverse (all elements) | O(n) | O(1) | Must visit each once |
Common Array Operations in Python
Example
arr = [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
# Length — O(1) (stored, not computed)
print(len(arr)) # 10
# Minimum and maximum — O(n)
print(min(arr)) # 1
print(max(arr)) # 9
# Sum — O(n)
print(sum(arr)) # 39
# Sort (in-place) — O(n log n)
arr.sort()
print(arr) # [1, 1, 2, 3, 3, 4, 5, 5, 6, 9]
# Sort (new array) — O(n log n)
arr = [3, 1, 4, 1, 5, 9, 2, 6]
sorted_arr = sorted(arr)
print(sorted_arr) # [1, 1, 2, 3, 4, 5, 6, 9]
print(arr) # [3, 1, 4, 1, 5, 9, 2, 6] (unchanged)
# Reverse (in-place) — O(n)
arr.reverse()
print(arr) # [6, 2, 9, 5, 1, 4, 1, 3]
# Count occurrences — O(n)
arr = [1, 2, 3, 2, 4, 2, 5]
print(arr.count(2)) # 3
# Index of first occurrence — O(n)
print(arr.index(3)) # 2
# Slice (creates new array) — O(k) where k = slice size
arr = [10, 20, 30, 40, 50]
print(arr[1:4]) # [20, 30, 40]
print(arr[:3]) # [10, 20, 30]
print(arr[::2]) # [10, 30, 50] (every other element)
print(arr[::-1]) # [50, 40, 30, 20, 10] (reversed)
# Copy — O(n)
arr_copy = arr.copy() # Shallow copy
arr_copy = arr[:] # Same thing with slice
Arrays vs. Other Data Structures
| Feature | Array | Linked List | Hash Map | BST |
|---|---|---|---|---|
| Access by index | O(1) ✅ | O(n) ❌ | N/A | N/A |
| Access by key | O(n) | O(n) | O(1) ✅ | O(log n) ✅ |
| Insert at end | O(1)* ✅ | O(1) ✅ | O(1)* ✅ | O(log n) |
| Insert at front | O(n) ❌ | O(1) ✅ | N/A | O(log n) |
| Insert at position | O(n) | O(1)† ✅ | N/A | O(log n) |
| Delete from end | O(1) ✅ | O(n) ❌ (or O(1) with tail ptr) | — | — |
| Delete arbitrary | O(n) | O(1)† ✅ | O(1)* ✅ | O(log n) |
| Memory overhead | Low ✅ | High (pointers) | Medium | Medium |
| Cache performance | Excellent ✅ | Poor ❌ | Poor ❌ | Poor ❌ |
| Maintains order | Yes | Yes | No | Yes (sorted) |
*Amortized †Given a pointer to the node
Choose arrays when:
- You need fast random access by index
- You traverse elements sequentially
- Memory efficiency is important
- The number of elements is roughly known in advance
- Cache performance matters
Choose linked lists when:
- You frequently insert/delete at arbitrary positions
- The size varies widely and unpredictably
- You do not need random access
Real-World Applications of Arrays
1. Image Processing
Example
# An image is a 2D array of pixels
# Each pixel is typically a tuple (R, G, B) or a single integer (grayscale)
image = [[255, 200, 180, 160], # Row 0: 4 pixels
[240, 210, 195, 170], # Row 1
[230, 200, 185, 165]] # Row 2
# Access pixel at row 1, column 2
pixel = image[1][2] # 195
# Convert to grayscale: average R, G, B channels
gray_image = [[(r + g + b) // 3 for (r, g, b) in row] for row in color_image]
2. Implementing Other Data Structures
Example
# Stack using array
class Stack:
def __init__(self):
self._data = [] # Array as underlying storage
def push(self, item):
self._data.append(item) # O(1) amortized
def pop(self):
return self._data.pop() # O(1)
def peek(self):
return self._data[-1] # O(1)
# Queue using array (inefficient — deque is better)
# Hash table using array of buckets
3. Numerical Computation
Example
import numpy as np
# NumPy arrays — highly optimized C arrays for numerical computation
scores = np.array([88, 74, 91, 65, 83])
# Vectorized operations (no Python loop needed — runs in optimized C)
mean = np.mean(scores) # O(n) but very fast
doubled = scores * 2 # [176, 148, 182, 130, 166]
above_80 = scores[scores > 80] # Boolean indexing
4. Sorting and Searching
Arrays are the natural structure for:
- Sorting algorithms (Merge Sort, Quick Sort, Heap Sort all operate on arrays)
- Binary search (requires sorted array with O(1) random access)
- Sliding window technique (fast subarray queries)
- Two-pointer technique (fast pair finding)
- Prefix sum arrays (fast range sum queries)
Common Mistakes and Pitfalls
Off-by-One Errors
Example
arr = [10, 25, 8, 42, 15]
# WRONG: accessing index n
for i in range(len(arr) + 1): # IndexError when i = len(arr)!
print(arr[i])
# CORRECT: loop from 0 to n-1
for i in range(len(arr)): # i = 0, 1, 2, 3, 4
print(arr[i])
# Last valid index is len(arr) - 1, not len(arr)
last = arr[len(arr) - 1] # arr[4] = 15 ✓
# NOT: arr[len(arr)] # IndexError ✗
Modifying Array While Iterating
Example
arr = [1, 2, 3, 4, 5]
# WRONG: Modifying while iterating can skip elements
for i, x in enumerate(arr):
if x % 2 == 0:
arr.remove(x) # Shifts elements, causes iteration to skip
print(arr) # [1, 3, 5] — seems right, but breaks for consecutive evens
# CORRECT: Collect items to remove, then remove them
to_remove = [x for x in arr if x % 2 == 0]
for x in to_remove:
arr.remove(x)
# OR: Create a new filtered list
arr = [x for x in arr if x % 2 != 0]
Shallow Copy vs. Deep Copy
Example
arr = [[1, 2, 3], [4, 5, 6]]
# Shallow copy — creates new outer list, but inner lists are still shared
shallow = arr.copy() # or arr[:]
shallow[0].append(99) # This ALSO modifies arr[0]!
print(arr[0]) # [1, 2, 3, 99] — arr was modified!
# Deep copy — creates fully independent copy
import copy
deep = copy.deepcopy(arr)
deep[0].append(99) # Only modifies deep, not arr
print(arr[0]) # [1, 2, 3] — arr is unchanged
Summary
Arrays are the bedrock of data structures:
- Definition: Collection of same-type elements in contiguous memory, accessed by index
- Memory model: Direct address formula → O(1) random access → the most important property
- Cache friendly: Spatial locality makes sequential access very fast in practice
- Types: Static (fixed size), Dynamic (resizable — Python list), Multi-dimensional
- Key complexities: Access O(1), search O(n)/O(log n), append O(1), insert/delete at position O(n)
- Best for: Random access, sequential traversal, implementing stacks/queues, numerical computation
- Not best for: Frequent insertions/deletions in the middle (use linked list), fast arbitrary search (use hash map)
Every data structure you study next — linked lists, stacks, queues, heaps, hash tables — will either use arrays internally or will be compared against arrays. The understanding you build here carries forward through your entire DSA journey.
*Next Lesson: Array Operations — Deep dive into every array operation with detailed implementation, edge cases, and optimization techniques.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Arrays — The Complete Introduction to the Most Fundamental Data Structure.
Interview Use
Prepare one clear explanation, one practical example, and one common mistake for this Data Structures & Algorithms topic.
Search Terms
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