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 graphs — what a graph is, formal definitions, all types of graphs (directed, undirected, weighted, cyclic, DAG), real-world applications, graph terminology (vertex, edge, degree, path, cycle, connectivity), and the two fundamental representations (adjacency list and matrix).
What Is a Graph?
A graph is a non-linear data structure that models relationships between objects. It consists of:
- Vertices (nodes): The objects being modeled
- Edges: The relationships (connections) between vertices
Formally: G = (V, E) where V is a set of vertices and E is a set of edges.
Graphs are arguably the most powerful and widely applicable data structure in computer science. Every network — social, road, internet, biological — is a graph. Understanding graphs means understanding how the connected world works.
Graph Terminology
Vertex (Node)
A fundamental unit of a graph. Usually labeled with a number or name.
Edge
A connection between two vertices. An edge (u, v) connects vertex u and vertex v.
Adjacent Vertices (Neighbors)
Two vertices are adjacent if there is an edge directly connecting them.
| In graph with edges | {(A,B), (B,C), (A,C)} |
| A's neighbors | B, C |
| B's neighbors | A, C |
Degree
The degree of a vertex is the number of edges connected to it.
- Undirected graph: degree = number of edges
- Directed graph:
- In-degree: number of edges coming IN to the vertex
- Out-degree: number of edges going OUT from the vertex
Example
→ B →
A C
→ D →
In-degree of C: 2 (from B and D)
Out-degree of A: 2 (to B and D)
Path
A sequence of vertices where each consecutive pair has an edge between them.
| Graph | A—B—C—D |
| Path from A to D: A | B → C → D |
| Path length | 3 (number of edges) |
Cycle
A path that starts and ends at the same vertex with no repeated edges.
Example
A—B—C—A is a cycle (A→B→C→A)
Connected Graph
An undirected graph where there is a path between every pair of vertices.
Connected Component
A maximal connected subgraph. An isolated vertex is its own component.
Example
Graph: {A—B, C—D, E}
Connected components: {A,B}, {C,D}, {E} — 3 components
Types of Graphs
1. Undirected Graph
Edges have no direction — if there is an edge between A and B, you can travel both ways.
Example
A — B — C
| |
D ——————
Adjacency: A-B, A-D, B-C, C-D (all bidirectional)
Real world: Facebook friendships (both see each other)
2. Directed Graph (Digraph)
Edges have direction — an edge (A, B) means you can go from A to B but NOT necessarily from B to A.
| A | B → C |
| Edges: A | B, B→C, C→D, D→A (one-way) |
| Real world | Twitter follows (you follow someone; they might not follow back) |
3. Weighted Graph
Each edge has a numerical weight (cost, distance, time, etc.).
Example
5
A ——— B
| |
3 2
| |
D ——— C
4
Real world: Road map where edge weights = distances
4. Unweighted Graph
All edges have equal weight (or weight 1). BFS finds shortest paths in unweighted graphs.
5. Cyclic Graph
Contains at least one cycle (path from a vertex back to itself).
6. Acyclic Graph
Contains no cycles.
7. DAG (Directed Acyclic Graph)
Directed graph with no cycles. Extremely important in computer science:
- Prerequisite structures (course A requires course B)
- Build systems (compile module X before module Y)
- Version control graphs
- Neural network architectures
Example
DAG example (course prerequisites):
Math 101 → Math 201 → Math 301
↑
CS 101 ——→ CS 201
8. Tree
A connected undirected graph with no cycles. A tree with n vertices has exactly n-1 edges.
Example
A
/ \
B C
/ \
D E
9. Complete Graph (K_n)
Every vertex is connected to every other vertex. Has n(n-1)/2 edges for n vertices.
10. Bipartite Graph
Vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other. No edges within a set.
Example
Set 1: {A, B} Set 2: {X, Y, Z}
A—X, A—Y, B—Y, B—Z (edges only cross between sets)
Real world: Users and movies they've rated (Netflix)
11. Sparse vs Dense Graph
- Sparse graph: Few edges relative to vertices. E ≈ O(V)
- Dense graph: Many edges relative to vertices. E ≈ O(V²)
This distinction determines whether adjacency list or matrix is more appropriate.
Graph Representation
Representation 1 — Adjacency List
Example
# Each vertex stores a list of its neighbors
graph = {
'A': ['B', 'C'],
'B': ['A', 'D'],
'C': ['A', 'D'],
'D': ['B', 'C']
}
# For weighted graphs: (neighbor, weight) pairs
weighted_graph = {
'A': [('B', 5), ('C', 3)],
'B': [('A', 5), ('D', 2)],
'C': [('A', 3), ('D', 4)],
'D': [('B', 2), ('C', 4)]
}
Space: O(V + E) — ideal for sparse graphs
Representation 2 — Adjacency Matrix
Example
# adj[i][j] = 1 if edge exists, 0 otherwise
# For n vertices, use n×n matrix
n = 4 # Vertices: 0, 1, 2, 3
adj = [[0]*n for _ in range(n)]
adj[0][1] = adj[1][0] = 1 # Edge 0-1 (undirected)
adj[1][2] = adj[2][1] = 1 # Edge 1-2
adj[0][3] = adj[3][0] = 1 # Edge 0-3
# Weighted: store weight instead of 1
INF = float('inf')
weighted_adj = [[INF]*n for _ in range(n)]
for i in range(n): weighted_adj[i][i] = 0 # Self-loop = 0
weighted_adj[0][1] = weighted_adj[1][0] = 5 # Edge with weight 5
Space: O(V²) — practical only for dense graphs
Comparison: List vs Matrix
| Feature | Adjacency List | Adjacency Matrix |
|---|---|---|
| Space | O(V + E) | O(V²) |
| Check if edge (u,v) exists | O(degree) | O(1) |
| Find all neighbors of u | O(degree) | O(V) |
| Add edge | O(1) | O(1) |
| Remove edge | O(degree) | O(1) |
| Best for | Sparse graphs | Dense graphs |
| Used in | BFS, DFS, Dijkstra | Floyd-Warshall |
Edge List Representation
Example
# Simply a list of all edges
edges = [(0,1), (1,2), (0,3), (2,3)] # (source, dest)
weighted_edges = [(0,1,5), (1,2,2), (0,3,3)] # (src, dst, weight)
# Used in: Kruskal's MST algorithm
Building Graphs from Input
Example
from collections import defaultdict
def build_graph(n, edges, directed=False):
"""Build adjacency list from edge list."""
graph = defaultdict(list)
for u, v in edges:
graph[u].append(v)
if not directed:
graph[v].append(u)
return dict(graph)
def build_weighted_graph(n, edges, directed=False):
"""Build weighted adjacency list."""
graph = defaultdict(list)
for u, v, w in edges:
graph[u].append((v, w))
if not directed:
graph[v].append((u, w))
return dict(graph)
# Example: 4-node graph
n = 4
edges = [(0,1), (1,2), (2,3), (0,3)]
g = build_graph(n, edges)
print(g) # {0: [1,3], 1: [0,2], 2: [1,3], 3: [2,0]}
Graph Properties
python example
def graph_properties(graph):
"""Calculate basic graph properties."""
V = len(graph)
E = sum(len(neighbors) for neighbors in graph.values()) // 2 # Undirected
# Degree of each vertex
degrees = {v: len(neighbors) for v, neighbors in graph.items()}
max_degree = max(degrees.values())
avg_degree = sum(degrees.values()) / V
return {
'vertices': V,
'edges': E,
'max_degree': max_degree,
'avg_degree': avg_degree,
'density': 2 * E / (V * (V-1)) if V > 1 else 0
}Summary
Graphs model any network of relationships:
- Vertices = entities, Edges = relationships
- Undirected = bidirectional, Directed = one-way
- Weighted = edges have costs, Unweighted = edges equal
- Cyclic = has cycles, Acyclic = no cycles
- DAG = directed acyclic graph — extremely common in CS
Two representations:
- Adjacency List: O(V+E) space — use for sparse graphs (most real-world graphs)
- Adjacency Matrix: O(V²) space — use for dense graphs or when O(1) edge lookup is needed
*Next Lesson: Graph Representation*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Graph — Complete Introduction to the Most Powerful 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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