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 Floyd-Warshall algorithm completely — the all-pairs shortest path approach, the DP recurrence, step-by-step table filling, path reconstruction, negative cycle detection, transitive closure, and comparison with running Dijkstra V times.
What Is Floyd-Warshall?
Floyd-Warshall computes shortest paths between ALL pairs of vertices in a weighted graph. While Dijkstra finds shortest paths from ONE source, Floyd-Warshall gives you the complete distance matrix for every (source, destination) pair.
- Single-source shortest path: Dijkstra (non-negative), Bellman-Ford (any)
- All-pairs shortest path: Floyd-Warshall (any weights), or run Dijkstra V times
Complete Implementation
Example
def floyd_warshall(n, edges, directed=False):
"""
Floyd-Warshall all-pairs shortest path.
Time: O(V³), Space: O(V²)
Works with negative weights (but detects negative cycles).
"""
INF = float('inf')
# Initialize distance matrix
dist = [[INF]*n for _ in range(n)]
for i in range(n):
dist[i][i] = 0 # Distance from i to itself = 0
# Fill in direct edges
for u, v, w in edges:
dist[u][v] = w
if not directed:
dist[v][u] = w
# Main algorithm: try each vertex k as intermediate
for k in range(n):
for i in range(n):
for j in range(n):
if dist[i][k] + dist[k][j] < dist[i][j]:
dist[i][j] = dist[i][k] + dist[k][j]
# Check for negative cycles: dist[i][i] < 0 means negative cycle through i
for i in range(n):
if dist[i][i] < 0:
print(f"Negative cycle detected involving vertex {i}")
return None
return dist
edges = [(0,1,3),(0,3,7),(1,0,8),(1,2,2),(2,0,5),(2,3,1),(3,0,2)]
dist = floyd_warshall(4, edges, directed=True)
for row in dist:
print(row)
# [0, 3, 5, 6]
# [5, 0, 2, 3]
# [3, 6, 0, 1]
# [2, 5, 7, 0]
Step-by-Step Table Filling
| 0 | 1: 3, 0→3: 7, 1→2: 2, 2→0: 5, 2→3: 1, 3→0: 2 |
| 0 | 0 3 ∞ 7 |
| 1 | 8 0 2 ∞ |
| 2 | 5 ∞ 0 1 |
| 3 | 2 ∞ ∞ 0 |
Path Reconstruction
Example
def floyd_warshall_with_path(n, edges, directed=False):
"""Floyd-Warshall with path reconstruction using 'next' matrix."""
INF = float('inf')
dist = [[INF]*n for _ in range(n)]
next_v = [[None]*n for _ in range(n)] # next_v[i][j] = first step from i to j
for i in range(n):
dist[i][i] = 0
for u, v, w in edges:
dist[u][v] = w
next_v[u][v] = v # To go from u to v, first step is v
if not directed:
dist[v][u] = w
next_v[v][u] = u
for k in range(n):
for i in range(n):
for j in range(n):
if dist[i][k] + dist[k][j] < dist[i][j]:
dist[i][j] = dist[i][k] + dist[k][j]
next_v[i][j] = next_v[i][k] # Path goes through k
def reconstruct_path(u, v):
if dist[u][v] == INF:
return None # No path
path = [u]
while u != v:
u = next_v[u][v]
path.append(u)
return path
return dist, reconstruct_path
edges = [(0,1,3),(0,3,7),(1,2,2),(2,0,5),(2,3,1),(3,0,2)]
dist, get_path = floyd_warshall_with_path(4, edges, directed=True)
print(get_path(0, 2)) # [0, 1, 2]
print(dist[0][2]) # 5 (0→1→2 = 3+2)
Transitive Closure
Floyd-Warshall can also compute whether any path exists between each pair (reachability):
Example
def transitive_closure(n, edges, directed=False):
"""
Compute reachability matrix.
reach[i][j] = True if there's a path from i to j.
"""
reach = [[False]*n for _ in range(n)]
for i in range(n):
reach[i][i] = True
for u, v, _ in edges:
reach[u][v] = True
if not directed:
reach[v][u] = True
for k in range(n):
for i in range(n):
for j in range(n):
reach[i][j] = reach[i][j] or (reach[i][k] and reach[k][j])
return reach
edges = [(0,1,1),(1,2,1),(3,4,1)]
reach = transitive_closure(5, edges, directed=True)
print(reach[0][2]) # True (0→1→2)
print(reach[0][3]) # False (no path from 0 to 3)
Comparison: Floyd-Warshall vs Running Dijkstra V Times
For all-pairs shortest paths, you have two options:
| Approach | Time | Space | Handles Negative Weights |
|---|---|---|---|
| Floyd-Warshall | O(V³) | O(V²) | ✅ Yes |
| Dijkstra × V | O(V(V+E) log V) | O(V+E) | ❌ No |
| Bellman-Ford × V | O(V²E) | O(V) | ✅ Yes |
When Floyd-Warshall wins:
- Dense graphs (E ≈ V²): Dijkstra × V = O(V × V² × log V) = O(V³ log V) vs Floyd-Warshall O(V³)
- Negative weights present
- Simple implementation needed
When Dijkstra × V wins:
- Sparse graphs (E ≈ V): Dijkstra × V = O(V × (V+V) log V) = O(V² log V) < O(V³)
- No negative weights
Floyd-Warshall in Practice
python example
# Network latency between all pairs of data centers
# V = 100 data centers, E = 500 connections
# Floyd-Warshall: 100³ = 10^6 operations — fast!
# Social network "degrees of separation" (unweighted)
# V = 10^6 users — Floyd-Warshall: 10^18 operations — impossible!
# Use BFS instead for large sparse graphs
# Practical limit: Floyd-Warshall viable for V ≤ 500-1000Summary
Floyd-Warshall: three nested loops over all vertex triples.
Example
for k in range(n): # Intermediate vertex
for i in range(n): # Source
for j in range(n): # Destination
dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
- Time: O(V³) — suitable for V ≤ 1000
- Space: O(V²) — always this, regardless of edges
- Handles negative weights: Yes (detects negative cycles via
dist[i][i] < 0) - Use for: All-pairs shortest paths, transitive closure, small dense graphs
*Next Lesson: Minimum Spanning Tree*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Floyd-Warshall Algorithm — All-Pairs Shortest Paths with Complete Analysis.
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, graphs, floyd, warshall
Related Data Structures & Algorithms Topics
Continue learning this concept
GraphsBellman-Ford Algorithm — Shortest Path with Negative Weights and Cycle DetectionMaster Bellman-Ford algorithm completely — why it handles negative weights where Dijkstra fails, the V-1 relaxation rounds, negative cycle detection, SPFA optimization, comparison with Dijkstra, and applications in currency arbitrage and network routing protocols.GraphsBFS Algorithm — Breadth-First Search with All Applications and VariantsMaster Breadth-First Search completely — the level-by-level exploration strategy, complete implementation with queue, BFS tree and shortest path guarantee, detecting bipartite graphs, multi-source BFS, 0-1 BFS, BFS on grids, and all classic BFS interview problems.GraphsDFS Algorithm — Depth-First Search with Cycle Detection and All ApplicationsMaster Depth-First Search completely — recursive and iterative implementations, DFS tree and timestamps, cycle detection in directed and undirected graphs, finding articulation points and bridges, DFS on grids, topological sort foundation, and all classic DFS interview problems.GraphsDijkstraMaster DijkstraGraphsKruskalMaster KruskalGraphsPrimMaster Prim