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 10 classic greedy interview problems — gas station, candy distribution, jump game, task scheduler, assign cookies, lemonade change, boats to save people, partition labels, minimum arrows, and queue reconstruction by height — with proofs and pattern recognition.
Problem 1 — Gas Station
Can you complete a full circle? If yes, find the starting station.
Example
def can_complete_circuit(gas, cost):
"""
Greedy insight: If total gas >= total cost, a solution exists.
If we run out at station i, the start must be after i.
Time: O(n), Space: O(1)
"""
total_gas = sum(gas) - sum(cost)
if total_gas < 0:
return -1 # Impossible
start = 0
tank = 0
for i in range(len(gas)):
tank += gas[i] - cost[i]
if tank < 0:
start = i + 1 # Current start doesn't work, try next
tank = 0
return start
print(can_complete_circuit([1,2,3,4,5],[3,4,5,1,2])) # 3
print(can_complete_circuit([2,3,4],[3,4,3])) # -1
Problem 3 — Jump Game (Can You Reach the End?)
Example
def can_jump(nums):
"""
Track max reachable index as we scan.
Greedy: if i > max_reach, we're stuck.
Time: O(n), Space: O(1)
"""
max_reach = 0
for i, jump in enumerate(nums):
if i > max_reach:
return False # Cannot reach index i
max_reach = max(max_reach, i + jump)
return True
print(can_jump([2,3,1,1,4])) # True
print(can_jump([3,2,1,0,4])) # False
Problem 4 — Task Scheduler
Minimum intervals to execute tasks with n cooldown between same tasks.
Example
from collections import Counter
import heapq
def least_interval(tasks, n):
"""
Greedy: always execute the most frequent remaining task.
Use a cooldown queue to respect the constraint.
Time: O(T log T) where T = number of unique tasks.
"""
freq = Counter(tasks)
max_heap = [-f for f in freq.values()]
heapq.heapify(max_heap)
time = 0
cooldown = [] # (available_time, neg_freq)
while max_heap or cooldown:
time += 1
if max_heap:
freq_left = heapq.heappop(max_heap) + 1 # Execute task
if freq_left < 0: # More instances remain
heapq.heappush(cooldown, (time + n, freq_left))
# Release tasks that have cooled down
if cooldown and cooldown[0][0] <= time:
_, f = heapq.heappop(cooldown)
heapq.heappush(max_heap, f)
return time
# Mathematical approach (faster):
def least_interval_math(tasks, n):
freq = sorted(Counter(tasks).values(), reverse=True)
f_max = freq[0]
idle_slots = (f_max - 1) * n
for f in freq[1:]:
idle_slots -= min(f_max - 1, f)
return len(tasks) + max(0, idle_slots)
print(least_interval(["A","A","A","B","B","B"], 2)) # 8
print(least_interval(["A","A","A","B","B","B"], 0)) # 6
Problem 5 — Assign Cookies
Example
def find_content_children(g, s):
"""
Greedily satisfy smallest greed factor children first.
Give each child the smallest cookie that satisfies them.
Time: O(n log n + m log m)
"""
g.sort(); s.sort()
child = cookie = 0
while child < len(g) and cookie < len(s):
if s[cookie] >= g[child]:
child += 1 # Child satisfied
cookie += 1 # Move to next cookie regardless
return child # Number of content children
print(find_content_children([1,2,3],[1,1])) # 1
print(find_content_children([1,2],[1,2,3])) # 2
Problem 6 — Lemonade Change
Example
def lemonade_change(bills):
"""
Can you make change for every customer?
Greedy: when making $15 change, prefer to give $10+$5 (preserves $5s).
Time: O(n), Space: O(1)
"""
fives = tens = 0
for bill in bills:
if bill == 5:
fives += 1
elif bill == 10:
if fives == 0: return False
fives -= 1; tens += 1
else: # bill == 20
if tens > 0 and fives > 0:
tens -= 1; fives -= 1 # Prefer $10+$5 (saves $5 bills)
elif fives >= 3:
fives -= 3
else:
return False
return True
print(lemonade_change([5,5,5,10,20])) # True
print(lemonade_change([5,5,10,10,20])) # False
Problem 7 — Boats to Save People
Example
def num_rescue_boats(people, limit):
"""
Pair heaviest with lightest person if possible.
Two-pointer greedy.
Time: O(n log n)
"""
people.sort()
left, right = 0, len(people) - 1
boats = 0
while left <= right:
if people[left] + people[right] <= limit:
left += 1 # Lightest can ride with heaviest
right -= 1 # Heaviest always takes one boat
boats += 1
return boats
print(num_rescue_boats([1,2],[3])) # 1
print(num_rescue_boats([3,2,2,1],[3])) # 3
print(num_rescue_boats([3,5,3,4],[5])) # 4
Problem 8 — Partition Labels
Example
def partition_labels(s):
"""
Partition string so each character appears in at most one part.
Greedy: for each position, extend partition to last occurrence of all chars seen.
"""
last = {c: i for i, c in enumerate(s)} # Last index of each char
partitions = []
start = end = 0
for i, c in enumerate(s):
end = max(end, last[c]) # Extend to cover last occurrence of c
if i == end: # Reached end of current partition
partitions.append(end - start + 1)
start = i + 1
return partitions
print(partition_labels("ababcbacadefegdehijhklij")) # [9,7,8]
Problem 9 — Minimum Arrows to Burst Balloons
Example
def find_min_arrow_shots(points):
"""
Minimum arrows to burst all balloons (intervals).
Sort by end, shoot at end of each non-covered balloon.
Same pattern as activity selection!
Time: O(n log n)
"""
points.sort(key=lambda x: x[1])
arrows = 1
arrow_pos = points[0][1] # Shoot at end of first balloon
for start, end in points[1:]:
if start > arrow_pos: # This balloon not hit by current arrow
arrows += 1
arrow_pos = end
return arrows
print(find_min_arrow_shots([[10,16],[2,8],[1,6],[7,12]])) # 2
print(find_min_arrow_shots([[1,2],[3,4],[5,6],[7,8]])) # 4
Problem 10 — Queue Reconstruction by Height
Example
def reconstruct_queue(people):
"""
people[i] = [h, k] where h=height, k=number of taller/equal people in front.
Greedy: sort by height desc (ties by k asc), then insert at position k.
"""
people.sort(key=lambda x: (-x[0], x[1]))
result = []
for person in people:
result.insert(person[1], person) # Insert at position k
return result
print(reconstruct_queue([[7,0],[4,4],[7,1],[5,0],[6,1],[5,2]]))
# [[5,0],[7,0],[5,2],[6,1],[4,4],[7,1]]
Pattern Recognition
| Trigger in Problem | Likely Greedy Strategy |
|---|---|
| "Minimum cost/steps" | Sort by some criterion, process greedily |
| "Maximum compatible items" | Earliest end time (interval scheduling) |
| "Fill remaining with best" | Value density ratio (knapsack) |
| "Always feasible?" | Check total balance (gas station) |
| "Pair items" | Sort, use two pointers from both ends |
| "Partition into sections" | Track maximum reachable endpoint |
| "Satisfy constraints" | Smallest-first or largest-first pairing |
*Greedy Algorithms section complete.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Greedy Interview Problems — 10 Classic Problems with Complete 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, greedy, interview, problems
Related Data Structures & Algorithms Topics
Continue learning this concept
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