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
Solve 12 classic recursion problems — Tower of Hanoi, string permutations, power set, balanced parentheses, palindrome check, climbing stairs, flood fill, binary search, merge sort, tree problems, and staircase with k steps.
Problem 1 — Tower of Hanoi
Problem: Move n disks from source peg to destination peg using auxiliary peg. Rules: only one disk at a time, never place larger disk on smaller.
python example
def hanoi(n, source, destination, auxiliary):
"""
Move n disks from source to destination using auxiliary.
Time: O(2ⁿ), Space: O(n)
"""
if n == 1:
print(f"Move disk 1 from {source} to {destination}")
return
# Move top n-1 disks from source to auxiliary (using destination)
hanoi(n-1, source, auxiliary, destination)
# Move largest disk from source to destination
print(f"Move disk {n} from {source} to {destination}")
# Move n-1 disks from auxiliary to destination (using source)
hanoi(n-1, auxiliary, destination, source)
hanoi(3, 'A', 'C', 'B')
# Move disk 1 from A to C
# Move disk 2 from A to B
# Move disk 1 from C to B
# Move disk 3 from A to C
# Move disk 1 from B to A
# Move disk 2 from B to C
# Move disk 1 from A to CTotal moves = 2ⁿ - 1
Problem 3 — Power Set (All Subsets)
Example
def power_set(arr, index=0):
"""Return all subsets of arr. 2ⁿ subsets."""
if index == len(arr):
return [[]] # Base case: empty subset
rest = power_set(arr, index + 1) # All subsets without arr[index]
# For each subset in rest, create a new subset with arr[index]
with_current = [[arr[index]] + subset for subset in rest]
return rest + with_current
print(power_set([1, 2, 3]))
# [[], [3], [2], [2,3], [1], [1,3], [1,2], [1,2,3]]
Problem 4 — Generate Valid Parentheses
Example
def generate_parentheses(n):
"""Generate all valid combinations of n pairs of parentheses."""
result = []
def backtrack(s, open_count, close_count):
if len(s) == 2 * n:
result.append(s)
return
if open_count < n:
backtrack(s + '(', open_count + 1, close_count)
if close_count < open_count:
backtrack(s + ')', open_count, close_count + 1)
backtrack("", 0, 0)
return result
print(generate_parentheses(3))
# ['((()))', '(()())', '(())()', '()(())', '()()()']
Problem 5 — Check Palindrome Using Recursion
Example
def is_palindrome(s, left=None, right=None):
if left is None:
left, right = 0, len(s) - 1
if left >= right:
return True # Base case: empty or single character
if s[left] != s[right]:
return False
return is_palindrome(s, left + 1, right - 1)
print(is_palindrome("racecar")) # True
print(is_palindrome("hello")) # False
Problem 6 — Climbing Stairs (k steps at a time)
python example
def climb_stairs(n, k=2):
"""
Count ways to climb n stairs, taking 1 to k steps at a time.
With k=2: standard problem. Time: O(kⁿ) naive, O(kn) with memo.
"""
from functools import lru_cache
@lru_cache(maxsize=None)
def dp(remaining):
if remaining == 0: return 1 # One way: take no step
if remaining < 0: return 0 # Impossible
return sum(dp(remaining - i) for i in range(1, k + 1))
return dp(n)
print(climb_stairs(5, 2)) # 8 ways with max 2 steps
print(climb_stairs(4, 3)) # 7 ways with max 3 stepsProblem 7 — Flood Fill
Example
def flood_fill(image, sr, sc, new_color):
"""
Change color of connected region (DFS/recursion).
Time: O(m×n), Space: O(m×n) stack
"""
old_color = image[sr][sc]
if old_color == new_color:
return image
def fill(r, c):
if r < 0 or r >= len(image): return
if c < 0 or c >= len(image[0]): return
if image[r][c] != old_color: return
image[r][c] = new_color
fill(r+1, c); fill(r-1, c)
fill(r, c+1); fill(r, c-1)
fill(sr, sc)
return image
image = [[1,1,1],[1,1,0],[1,0,1]]
print(flood_fill(image, 1, 1, 2))
# [[2,2,2],[2,2,0],[2,0,1]]
Problem 8 — Binary Search (Recursive)
Example
def binary_search(arr, target, left=None, right=None):
if left is None: left, right = 0, len(arr) - 1
if left > right: return -1
mid = (left + right) // 2
if arr[mid] == target: return mid
elif arr[mid] < target: return binary_search(arr, target, mid+1, right)
else: return binary_search(arr, target, left, mid-1)
print(binary_search([1,3,5,7,9,11], 7)) # 3
Problem 9 — Merge Sort
Example
def merge_sort(arr):
if len(arr) <= 1: return arr
mid = len(arr) // 2
left = merge_sort(arr[:mid])
right = merge_sort(arr[mid:])
return merge(left, right)
def merge(l, r):
result, i, j = [], 0, 0
while i < len(l) and j < len(r):
if l[i] <= r[j]: result.append(l[i]); i += 1
else: result.append(r[j]); j += 1
return result + l[i:] + r[j:]
print(merge_sort([64,34,25,12,22,11,90])) # [11,12,22,25,34,64,90]
Problem 10 — Tree: Maximum Path Sum
Example
def max_path_sum(root):
"""Maximum sum path in a binary tree (any node to any node)."""
max_sum = [float('-inf')]
def helper(node):
if not node: return 0
left = max(helper(node.left), 0) # Ignore negative paths
right = max(helper(node.right), 0)
max_sum[0] = max(max_sum[0], node.val + left + right)
return node.val + max(left, right) # Return max single path
helper(root)
return max_sum[0]
Problem 11 — Flatten Nested List
Example
def flatten(nested):
"""Recursively flatten a nested list."""
result = []
for item in nested:
if isinstance(item, list):
result.extend(flatten(item)) # Recurse on sublists
else:
result.append(item)
return result
print(flatten([1, [2, [3, 4], 5], [6, 7]])) # [1,2,3,4,5,6,7]
Problem 12 — Count Ways to Reach Score
Example
def count_score_ways(n, scores):
"""
Count ways to reach score n using given point values.
Example: n=4, scores=[1,2,3] → how many ways?
"""
from functools import lru_cache
@lru_cache(maxsize=None)
def dp(remaining):
if remaining == 0: return 1 # Exactly reached target
if remaining < 0: return 0
return sum(dp(remaining - s) for s in scores)
return dp(n)
print(count_score_ways(4, (1, 2, 3))) # 7 ways
Complexity Summary
| Problem | Time | Space | Key Technique |
|---|---|---|---|
| Tower of Hanoi | O(2ⁿ) | O(n) | Reduce-to-subproblem |
| Permutations | O(n!×n) | O(n!) | Choose + recurse on rest |
| Power Set | O(2ⁿ×n) | O(2ⁿ) | Include/exclude |
| Valid Parentheses | O(2ⁿ/√n) | O(n) | Constrained backtracking |
| Palindrome Check | O(n) | O(n) | Two pointers inward |
| Climbing Stairs | O(kn) | O(n) | Memoized recursion |
| Flood Fill | O(m×n) | O(m×n) | 4-directional DFS |
| Binary Search | O(log n) | O(log n) | Halving |
| Merge Sort | O(n log n) | O(n) | Divide-and-conquer |
| Max Path Sum | O(n) | O(h) | Post-order DFS |
| Flatten List | O(n) | O(depth) | Recursive flatten |
| Count Ways | O(n×k) | O(n) | Memoized DP |
*Next Lesson: Backtracking Introduction — Using recursion with "undo" to explore all possibilities.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Recursion Problems — 12 Classic Problems from Basic to Advanced.
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, recursion, problems, recursion problems — 12 classic problems from basic to advanced
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