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 searching interview problems — find in rotated sorted array, search in 2D matrix, find peak element, kth smallest in sorted matrix, find duplicate, median of two sorted arrays, capacity to ship packages, aggressive cows, and more.
Problem 1 — Search in Rotated Sorted Array
Example
def search_rotated(nums, target):
"""Binary search in rotated sorted array. O(log n)."""
left, right = 0, len(nums) - 1
while left <= right:
mid = left + (right - left) // 2
if nums[mid] == target: return mid
if nums[left] <= nums[mid]: # Left half sorted
if nums[left] <= target < nums[mid]:
right = mid - 1
else:
left = mid + 1
else: # Right half sorted
if nums[mid] < target <= nums[right]:
left = mid + 1
else:
right = mid - 1
return -1
print(search_rotated([4,5,6,7,0,1,2], 0)) # 4
print(search_rotated([4,5,6,7,0,1,2], 3)) # -1
Problem 3 — Find Peak Element
Example
def find_peak_element(nums):
"""Find any element greater than both neighbors. O(log n)."""
left, right = 0, len(nums) - 1
while left < right:
mid = left + (right - left) // 2
if nums[mid] < nums[mid + 1]:
left = mid + 1 # Ascending slope → peak is to the right
else:
right = mid # Descending slope → peak is at mid or left
return left # left == right at this point
print(find_peak_element([1,2,3,1])) # 2
print(find_peak_element([1,2,1,3,5,6,4])) # 5
Problem 4 — Kth Smallest in Sorted Matrix
Example
import heapq
def kth_smallest(matrix, k):
"""
Find kth smallest in n×n matrix where rows and columns are sorted.
Binary search on value range.
Time: O(n log(max-min)), Space: O(1)
"""
n = len(matrix)
left, right = matrix[0][0], matrix[n-1][n-1]
def count_less_equal(mid):
"""Count elements <= mid using O(n) staircase traversal."""
count = 0
row, col = n - 1, 0
while row >= 0 and col < n:
if matrix[row][col] <= mid:
count += row + 1 # All elements in this column up to row
col += 1
else:
row -= 1
return count
while left < right:
mid = left + (right - left) // 2
if count_less_equal(mid) < k:
left = mid + 1
else:
right = mid
return left
matrix = [[1,5,9],[10,11,13],[12,13,15]]
print(kth_smallest(matrix, 8)) # 13
Problem 5 — Find Duplicate Number (Binary Search Approach)
Example
def find_duplicate(nums):
"""
Array of n+1 integers in [1,n]. Binary search on value range.
Count numbers <= mid: if > mid, duplicate is in [1,mid].
Time: O(n log n), Space: O(1)
"""
left, right = 1, len(nums) - 1
while left < right:
mid = left + (right - left) // 2
count = sum(1 for x in nums if x <= mid)
if count > mid:
right = mid # Duplicate is in [left, mid]
else:
left = mid + 1
return left
print(find_duplicate([1,3,4,2,2])) # 2
print(find_duplicate([3,1,3,4,2])) # 3
Problem 6 — Capacity to Ship Packages
Example
def ship_within_days(weights, days):
"""
Find minimum ship capacity to ship all packages within 'days' days.
Binary search on capacity from max(weights) to sum(weights).
"""
def can_ship(capacity):
day_count, current_load = 1, 0
for w in weights:
if current_load + w > capacity:
day_count += 1
current_load = 0
current_load += w
return day_count <= days
left = max(weights) # Minimum: must fit heaviest package
right = sum(weights) # Maximum: ship everything in one day
while left < right:
mid = left + (right - left) // 2
if can_ship(mid):
right = mid # Try smaller capacity
else:
left = mid + 1 # Need larger capacity
return left
print(ship_within_days([1,2,3,4,5,6,7,8,9,10], 5)) # 15
print(ship_within_days([3,2,2,4,1,4], 3)) # 6
Problem 7 — Aggressive Cows (Classic CP Problem)
Problem: Place k cows in n stalls such that the minimum distance between any two cows is maximized.
Example
def aggressive_cows(stalls, k):
"""
Binary search on the answer (minimum distance).
Check if k cows can be placed with at least 'dist' apart.
"""
stalls.sort()
def can_place(min_dist):
count = 1
last_placed = stalls[0]
for i in range(1, len(stalls)):
if stalls[i] - last_placed >= min_dist:
count += 1
last_placed = stalls[i]
if count == k: return True
return False
left, right = 1, stalls[-1] - stalls[0]
while left < right:
mid = left + (right - left + 1) // 2 # Upper mid to avoid infinite loop
if can_place(mid):
left = mid # Try larger minimum distance
else:
right = mid - 1 # Too large, reduce
return left
print(aggressive_cows([1,2,8,4,9], 3)) # 3
Problem 8 — Median of Two Sorted Arrays
Example
def find_median(nums1, nums2):
"""
Find median of two sorted arrays combined.
Binary search on partition of smaller array.
Time: O(log(min(m,n))), Space: O(1)
"""
if len(nums1) > len(nums2):
nums1, nums2 = nums2, nums1
m, n = len(nums1), len(nums2)
total = m + n
half = total // 2
left, right = 0, m
while left <= right:
i = left + (right - left) // 2 # Partition in nums1
j = half - i # Partition in nums2
left1 = nums1[i-1] if i > 0 else float('-inf')
right1 = nums1[i] if i < m else float('inf')
left2 = nums2[j-1] if j > 0 else float('-inf')
right2 = nums2[j] if j < n else float('inf')
if left1 <= right2 and left2 <= right1:
if total % 2 == 1:
return min(right1, right2)
return (max(left1, left2) + min(right1, right2)) / 2
elif left1 > right2:
right = i - 1
else:
left = i + 1
print(find_median([1,3], [2])) # 2.0
print(find_median([1,2], [3,4])) # 2.5
Problem 9 — Minimize Maximum Distance to Gas Station
Example
def minimize_max_gas_distance(stations, k):
"""
Add k gas stations to minimize the maximum distance between adjacent stations.
Binary search on the answer (maximum gap).
Time: O(n log(max_gap/precision))
"""
def can_achieve(max_gap):
count = 0
for i in range(1, len(stations)):
count += int((stations[i] - stations[i-1]) / max_gap)
return count <= k
left, right = 0, stations[-1] - stations[0]
for _ in range(100): # 100 iterations → precision ~10^-30
mid = (left + right) / 2
if can_achieve(mid):
right = mid
else:
left = mid
return right
print(minimize_max_gas_distance([1,2,3,4,5,6,7,8,9,10], 9)) # 0.5 (approx)
Problem 10 — Find in Mountain Array
Example
def find_in_mountain(mountain, target):
"""
Search for target in mountain array (first increases then decreases).
Step 1: Find peak using ternary/binary search.
Step 2: Binary search in left half (ascending).
Step 3: Binary search in right half (descending).
Time: O(log n)
"""
n = len(mountain)
# Step 1: Find peak
left, right = 0, n - 1
while left < right:
mid = left + (right - left) // 2
if mountain[mid] < mountain[mid + 1]:
left = mid + 1
else:
right = mid
peak = left
# Step 2: Binary search ascending portion [0, peak]
left, right = 0, peak
while left <= right:
mid = left + (right - left) // 2
if mountain[mid] == target: return mid
elif mountain[mid] < target: left = mid + 1
else: right = mid - 1
# Step 3: Binary search descending portion [peak+1, n-1]
left, right = peak + 1, n - 1
while left <= right:
mid = left + (right - left) // 2
if mountain[mid] == target: return mid
elif mountain[mid] > target: left = mid + 1 # Descending!
else: right = mid - 1
return -1
print(find_in_mountain([1,2,3,4,5,3,1], 3)) # 2 (or 5 — any valid index)
Pattern Summary
| Problem | Binary Search Type | Key Insight |
|---|---|---|
| Rotated Sorted Array | Classic (modified) | Identify which half is sorted |
| 2D Matrix Search | Classic on flattened | Treat 2D as 1D with index math |
| Peak Element | Binary (slope direction) | Move toward higher neighbor |
| Kth Smallest Matrix | Answer space | Count elements ≤ mid |
| Ship Packages | Answer space | Binary search on capacity |
| Aggressive Cows | Answer space | Binary search on minimum gap |
| Median Two Arrays | Partition binary search | Find correct partition point |
| Mountain Array | Binary + Binary | Find peak, then search both sides |
The "binary search on answer" pattern appears in Problems 4-7 and is one of the most powerful interview techniques — when you can verify a candidate answer in O(f(n)), binary search gives O(f(n) × log n) total.
*Searching folder complete. Next: Sorting — Sorting Introduction*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Searching 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, searching, interview, problems
Related Data Structures & Algorithms Topics
Continue learning this concept
SearchingBinary Search — Complete Guide with All Variants and Proof of CorrectnessMaster binary search completely — the core algorithm with loop invariant proof, iterative and recursive implementations, all variants (leftmost, rightmost, lower bound, upper bound), binary search on answer space, and 10 classic interview problems.SearchingInterpolation Search — O(log log n) Search for Uniformly Distributed DataMaster interpolation search — how it improves on binary search using value-based position estimation, when it achieves O(log log n) vs O(n) worst case, the interpolation formula, comparison with binary search, and when to use each.SearchingLinear Search — Complete Guide with All Variants and ApplicationsMaster linear search completely — sequential search, sentinel search, searching with conditions, finding all occurrences, searching in linked lists and 2D arrays, when to use linear search vs alternatives, and the exact time and space complexity analysis.SearchingSearch Algorithm Comparison — Choosing the Right Search for Every SituationA comprehensive comparison of all searching algorithms — linear search, binary search, interpolation search, ternary search, hash-based search, and tree-based search — with performance tables, use cases, and a decision guide for every scenario.SearchingTernary Search — Finding Maximum/Minimum of Unimodal FunctionsMaster ternary search — what it is, why it differs from binary search, how to find the maximum or minimum of a unimodal function in O(log n), ternary search on sorted arrays vs functions, comparison with binary search, and classic applications.ArraysArray Interview Problems — 15 Classic Problems with Complete SolutionsSolve 15 classic array interview problems that appear at top tech companies. Each problem includes the problem statement, multiple approaches from brute force to optimal, step-by-step solution explanation, complete code, and time/space complexity analysis.