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 all shortest path algorithms — BFS for unweighted graphs, Dijkstra for non-negative weights, Bellman-Ford for negative weights, Floyd-Warshall for all-pairs, DAG shortest paths, A* heuristic search, and a decision guide for choosing the right algorithm for every scenario.
Overview
Finding shortest paths in graphs is one of the most fundamental problems in computer science. The right algorithm depends on:
- Is the graph weighted or unweighted?
- Are there negative edge weights?
- Do you need one source or all pairs?
- Is the graph a DAG?
Algorithm 2 — Dijkstra (Non-Negative Weights)
Example
import heapq
def dijkstra(graph, source):
"""
Dijkstra for non-negative weighted graphs.
Time: O((V+E) log V), Space: O(V)
"""
dist = {v: float('inf') for v in graph}
dist[source] = 0
pq = [(0, source)]
while pq:
d, u = heapq.heappop(pq)
if d > dist[u]: continue
for v, w in graph[u]:
if dist[u] + w < dist[v]:
dist[v] = dist[u] + w
heapq.heappush(pq, (dist[v], v))
return dist
# Correct for non-negative weights
graph = {0: [(1,1),(2,4)], 1: [(2,2),(3,5)], 2: [(3,1)], 3: []}
print(dijkstra(graph, 0)) # {0:0, 1:1, 2:3, 3:4}
Use when: Non-negative edge weights, single source.
Algorithm 3 — Bellman-Ford (Negative Weights)
Example
def bellman_ford(edges, n, source):
"""
Bellman-Ford for graphs with negative weights.
Time: O(V × E), Space: O(V)
Detects negative cycles.
"""
dist = [float('inf')] * n
dist[source] = 0
for _ in range(n - 1):
for u, v, w in edges:
if dist[u] + w < dist[v]:
dist[v] = dist[u] + w
# Negative cycle check
for u, v, w in edges:
if dist[u] + w < dist[v]:
return None # Negative cycle
return dist
# Handles negative edges (Dijkstra would fail here)
edges = [(0,1,4),(0,2,2),(2,1,-3),(1,3,1)]
print(bellman_ford(edges, 4, 0)) # [0, -1, 2, 0]
Use when: Negative edge weights, single source, need negative cycle detection.
Algorithm 4 — Floyd-Warshall (All Pairs)
Example
def floyd_warshall(n, adj):
"""
All-pairs shortest paths.
Time: O(V³), Space: O(V²)
adj[i][j] = weight (inf if no edge)
"""
dist = [row[:] for row in adj]
for k in range(n):
for i in range(n):
for j in range(n):
dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
return dist
INF = float('inf')
adj = [[0,3,INF,7],[8,0,2,INF],[5,INF,0,1],[2,INF,INF,0]]
result = floyd_warshall(4, adj)
for row in result:
print([x if x < INF else '∞' for x in row])
Use when: Need shortest path between ALL pairs of vertices.
Algorithm 5 — DAG Shortest Path (Topological + DP)
When the graph is a DAG, we can find shortest paths in O(V+E) — faster than Dijkstra:
Example
from collections import defaultdict, deque
def dag_shortest_path(graph, n, source):
"""
Shortest path in DAG using topological order.
Time: O(V + E), Space: O(V)
Works with negative weights! (no negative cycles in DAG)
"""
# Topological sort (Kahn's)
in_degree = defaultdict(int)
for u in range(n):
for v, w in graph[u]:
in_degree[v] += 1
queue = deque(i for i in range(n) if in_degree[i] == 0)
topo = []
while queue:
u = queue.popleft(); topo.append(u)
for v, w in graph[u]:
in_degree[v] -= 1
if in_degree[v] == 0: queue.append(v)
# Relax edges in topological order
dist = [float('inf')] * n
dist[source] = 0
for u in topo:
for v, w in graph[u]:
if dist[u] + w < dist[v]:
dist[v] = dist[u] + w
return dist
# DAG with negative edges — Dijkstra would fail, Bellman-Ford would be O(VE)
graph = {0: [(1,2),(2,6)], 1: [(3,7),(4,3)], 2: [(4,-4)], 3: [(5,1)], 4: [(5,5)], 5: []}
print(dag_shortest_path(graph, 6, 0)) # [0, 2, 6, 9, 2, 7]
Use when: Graph is a DAG — handles negative weights in O(V+E).
Algorithm 6 — A* Algorithm (Heuristic Search)
A* is Dijkstra enhanced with a heuristic that estimates remaining distance to the target:
Example
import heapq
def a_star(graph, source, target, heuristic):
"""
A* search: Dijkstra + heuristic guidance.
f(n) = g(n) + h(n)
g(n) = actual cost from source to n
h(n) = estimated cost from n to target (must be admissible: never overestimate)
Time: O(E log V) but often much faster due to heuristic pruning.
"""
g = {source: 0}
f = {source: heuristic(source, target)}
pq = [(f[source], source)]
parent = {source: None}
while pq:
_, u = heapq.heappop(pq)
if u == target:
path = []
while u is not None:
path.append(u); u = parent[u]
return g[target], list(reversed(path))
for v, w in graph[u]:
new_g = g[u] + w
if v not in g or new_g < g[v]:
g[v] = new_g
f[v] = new_g + heuristic(v, target)
parent[v] = u
heapq.heappush(pq, (f[v], v))
return float('inf'), [] # Target not reachable
# Example: Grid with Manhattan distance heuristic
def manhattan(a, b):
return abs(a[0]-b[0]) + abs(a[1]-b[1])
Use when: Single target, admissible heuristic available (GPS navigation, game AI).
Algorithm Selection Guide
| YES | BFS O(V+E) — simple, optimal |
| YES | DAG Shortest Path O(V+E) — handles negative weights too! |
| YES | Bellman-Ford O(VE) — detects and reports negative cycles |
| NO | Bellman-Ford O(VE) — still correct, just slower |
| YES | Dijkstra O((V+E) log V) — fastest for non-negative weights |
| Small graph (V ≤ 1000)? | Floyd-Warshall O(V³) |
| Large sparse graph? | Dijkstra × V times O(V(V+E) log V) |
| YES | A* — often much faster than Dijkstra in practice |
Comparison Table
| Algorithm | Type | Time | Space | Neg Weights | Neg Cycle | All Pairs |
|---|---|---|---|---|---|---|
| BFS | Unweighted | O(V+E) | O(V) | ❌ | ❌ | ❌ |
| Dijkstra | Non-neg weighted | O((V+E)logV) | O(V) | ❌ | ❌ | ❌ |
| Bellman-Ford | Any weighted | O(VE) | O(V) | ✅ | ✅ | ❌ |
| DAG SP | DAG (any weight) | O(V+E) | O(V) | ✅ | N/A | ❌ |
| Floyd-Warshall | Any weighted | O(V³) | O(V²) | ✅ | ✅ | ✅ |
| A* | Non-neg + heuristic | O(E log V) | O(V) | ❌ | ❌ | ❌ |
Summary
Choosing the right shortest path algorithm is critical for efficiency:
| Scenario | Algorithm |
|---|---|
| Unweighted graph | BFS |
| Non-negative, single source | Dijkstra |
| Negative weights, single source | Bellman-Ford |
| DAG with any weights | Topological + DP |
| All pairs | Floyd-Warshall |
| Single target + heuristic | A* |
*Next Lesson: Union-Find DSU*
Continue the lesson path
Study next in this module
01GraphsPrimMaster Prim02GraphsTopological Sorting — DFS and BFS (KahnMaster topological sorting completely — what it is, why it only works on DAGs, DFS-based algorithm with finish times, Kahn03GraphsMinimum Spanning Tree — Definition, Properties, and ApplicationsMaster Minimum Spanning Tree concepts completely — what an MST is, the cut property and cycle property proofs, why MSTs are optimal, the two fundamental MST algorithms (Kruskal04GraphsUnion-Find (DSU) in Graphs — Connected Components and Dynamic ConnectivityMaster Union-Find (Disjoint Set Union) in graph contexts — finding connected components, dynamic connectivity queries, cycle detection in undirected graphs, Kruskal