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 the Rat in a Maze problem — finding all paths from source to destination in a binary grid, backtracking with visited tracking, directional movement (4-directional and 8-directional), finding shortest path, path reconstruction, and variations including multiple exit points.
Problem Statement
A maze is represented as an N×N binary grid where:
- 1 = open cell (can pass through)
- 0 = blocked cell (wall)
A rat starts at the top-left corner (0, 0) and must reach the bottom-right corner (N-1, N-1). Find all possible paths from source to destination.
| Path 1: DDRDRR — (0,0) | (1,0)→(2,0)... wait, (2,0)=0, invalid |
| Let's re-read: (0,0) | (1,0)→(1,1)→(2,1)→(3,1)→(3,2)→(3,3) ✓ |
| Path: D | R → D → D → R → R |
Solution with Coordinate Path
Example
def find_all_paths_with_coords(maze):
"""Returns paths as sequences of (row, col) coordinates."""
n = len(maze)
result = []
visited = [[False] * n for _ in range(n)]
def backtrack(row, col, path):
if row == n-1 and col == n-1:
result.append(list(path))
return
for dr, dc in [(1,0),(-1,0),(0,1),(0,-1)]:
nr, nc = row+dr, col+dc
if 0 <= nr < n and 0 <= nc < n and maze[nr][nc] == 1 and not visited[nr][nc]:
visited[nr][nc] = True
path.append((nr, nc))
backtrack(nr, nc, path)
path.pop()
visited[nr][nc] = False
if maze[0][0] == 1:
visited[0][0] = True
backtrack(0, 0, [(0,0)])
return result
Visualizing a Path
Example
def visualize_path(maze, path_coords):
"""Print the maze with the path marked."""
n = len(maze)
path_set = set(path_coords)
for r in range(n):
row_str = ""
for c in range(n):
if (r, c) in path_set:
row_str += "* " # Path marker
elif maze[r][c] == 1:
row_str += "1 "
else:
row_str += "0 "
print(row_str)
print()
maze = [[1,0,0,0],[1,1,0,1],[0,1,0,0],[1,1,1,1]]
paths = find_all_paths_with_coords(maze)
print(f"Found {len(paths)} path(s)")
if paths:
print("First path:")
visualize_path(maze, paths[0])
Shortest Path in Maze — BFS Approach
Backtracking finds ALL paths but not necessarily the SHORTEST. For shortest path, use BFS:
Example
from collections import deque
def shortest_path_maze(maze):
"""
Find shortest path using BFS (not backtracking).
BFS guarantees shortest path in unweighted grid.
Time: O(n²), Space: O(n²)
"""
n = len(maze)
if not maze[0][0] or not maze[n-1][n-1]:
return []
queue = deque([(0, 0, [(0,0)])]) # (row, col, path_so_far)
visited = [[False]*n for _ in range(n)]
visited[0][0] = True
while queue:
row, col, path = queue.popleft()
if row == n-1 and col == n-1:
return path # First complete path = shortest
for dr, dc in [(1,0),(-1,0),(0,1),(0,-1)]:
nr, nc = row+dr, col+dc
if 0 <= nr < n and 0 <= nc < n and maze[nr][nc] == 1 and not visited[nr][nc]:
visited[nr][nc] = True
queue.append((nr, nc, path + [(nr, nc)]))
return [] # No path exists
maze = [[1,0,0,0],[1,1,0,1],[0,1,0,0],[1,1,1,1]]
short = shortest_path_maze(maze)
print(f"Shortest path length: {len(short)}")
print(f"Path: {short}")
8-Directional Movement
Example
def find_paths_8dir(maze):
"""
Rat can move in all 8 directions (including diagonals).
"""
n = len(maze)
result = []
visited = [[False]*n for _ in range(n)]
# 8 directions: U,D,L,R + 4 diagonals
directions = [
(-1,0,'U'), (1,0,'D'), (0,-1,'L'), (0,1,'R'),
(-1,-1,'UL'), (-1,1,'UR'), (1,-1,'DL'), (1,1,'DR')
]
def backtrack(r, c, path):
if r == n-1 and c == n-1:
result.append(list(path))
return
for dr, dc, name in directions:
nr, nc = r+dr, c+dc
if 0 <= nr < n and 0 <= nc < n and maze[nr][nc] == 1 and not visited[nr][nc]:
visited[nr][nc] = True
path.append(name)
backtrack(nr, nc, path)
path.pop()
visited[nr][nc] = False
if maze[0][0]:
visited[0][0] = True
backtrack(0, 0, [])
return result
Count Paths Only (No Storage)
Example
def count_paths(maze):
"""
Count paths without storing them — saves memory.
"""
n = len(maze)
visited = [[False]*n for _ in range(n)]
def backtrack(r, c):
if r == n-1 and c == n-1:
return 1 # Found one path
count = 0
for dr, dc in [(1,0),(-1,0),(0,1),(0,-1)]:
nr, nc = r+dr, c+dc
if 0 <= nr < n and 0 <= nc < n and maze[nr][nc] == 1 and not visited[nr][nc]:
visited[nr][nc] = True
count += backtrack(nr, nc)
visited[nr][nc] = False
return count
if maze[0][0]:
visited[0][0] = True
return backtrack(0, 0)
return 0
print(count_paths([[1,1,1],[1,0,1],[1,1,1]])) # 2 paths
Complexity Analysis
| Approach | Time | Space | Use When |
|---|---|---|---|
| Backtracking (all paths) | O(4^(n²)) worst | O(n²) | Need all paths |
| BFS (shortest path) | O(n²) | O(n²) | Need shortest path |
| Count only | O(4^(n²)) | O(n²) | Need count, not paths |
| DP (path count) | O(n²) | O(n²) | Count paths in simple grid |
Note: The DP approach works only when there are no cycles (only Down+Right movement). For full 4-directional movement, backtracking or BFS is required.
Common Interview Variations
- Print all paths: Standard backtracking (shown above)
- Count paths: Remove result storage, return counts
- Shortest path: Use BFS instead of DFS/backtracking
- Path with minimum cost: BFS with cost weights (Dijkstra's)
- Multiple starting/ending points: Run BFS from all sources simultaneously
Summary
The Rat in a Maze problem demonstrates core backtracking concepts:
- State: Current position (row, col) + visited cells
- Choice: Which direction to move (D/U/L/R)
- Constraint: Cell must be open (1) and unvisited
- Undo: Unmark the visited cell when backtracking
The visited array is critical — without it, the algorithm would revisit cells infinitely. The backtracking ensures the visited array is always restored to its correct state before trying a different direction.
*Next Lesson: Sudoku Solver*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Rat in a Maze — Backtracking Path Finding with All Solutions.
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, backtracking, rat, maze
Related Data Structures & Algorithms Topics
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