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 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.
What Is Ternary Search?
Ternary search is a divide-and-conquer algorithm that finds the maximum or minimum of a unimodal function — a function that first increases (or decreases) monotonically, reaches a single peak (or valley), and then decreases (or increases) monotonically.
Unlike binary search which finds a specific value in a sorted array, ternary search finds the *extreme point* of a unimodal function. It divides the search range into three parts at each step.
The Ternary Search Algorithm
Divide the range into three equal parts using two midpoints m1 and m2:
| If f(m1) < f(m2): maximum is in [m1, right] | left = m1 |
| If f(m1) > f(m2): maximum is in [left, m2] | right = m2 |
| If f(m1) == f(m2): maximum is in [m1, m2] | shrink both ends |
Implementation — Find Maximum on Sorted Array
Example
def ternary_search_max(arr):
"""
Find the index of the maximum element in a unimodal array.
Time: O(log n), Space: O(1)
"""
left, right = 0, len(arr) - 1
while right - left > 2:
m1 = left + (right - left) // 3
m2 = right - (right - left) // 3
if arr[m1] < arr[m2]:
left = m1 + 1 # Max is in right portion
elif arr[m1] > arr[m2]:
right = m2 - 1 # Max is in left portion
else:
left = m1 + 1 # Shrink from both sides
right = m2 - 1
# Find max in remaining small range
return max(range(left, right + 1), key=lambda i: arr[i])
arr = [1, 3, 5, 8, 10, 7, 4, 2]
print(ternary_search_max(arr)) # 4 (index of maximum 10)
Implementation — Find Maximum of Continuous Function
Example
def ternary_search_continuous(f, left, right, precision=1e-9):
"""
Find x that maximizes f(x) on [left, right].
f must be unimodal (single peak).
Time: O(log((right-left)/precision)), Space: O(1)
"""
while right - left > precision:
m1 = left + (right - left) / 3
m2 = right - (right - left) / 3
if f(m1) < f(m2):
left = m1
elif f(m1) > f(m2):
right = m2
else:
left = m1
right = m2
return (left + right) / 2
# Find maximum of f(x) = -(x-3)² + 9 (parabola, peak at x=3)
f = lambda x: -(x - 3)**2 + 9
peak_x = ternary_search_continuous(f, 0, 6)
print(f"Peak at x ≈ {peak_x:.4f}, f(x) ≈ {f(peak_x):.4f}")
# Peak at x ≈ 3.0000, f(x) ≈ 9.0000
Ternary Search vs Binary Search
| Feature | Binary Search | Ternary Search |
|---|---|---|
| Purpose | Find specific value | Find maximum/minimum |
| Data requirement | Sorted array | Unimodal function |
| Steps per iteration | 1 comparison | 2 comparisons |
| Reduction per step | Eliminates 1/2 | Eliminates 1/3 |
| Complexity | O(log₂ n) | O(log₁.₅ n) ≈ O(log n) |
| When to use | Known sorted array | Unimodal optimization |
Why not divide into more parts? The more divisions, the more comparisons per step. Binary search with 1 comparison eliminating 1/2 is optimal for sorted array search. Ternary search makes 2 comparisons to eliminate 1/3 — the same total work per "elimination ratio" as binary search.
Classic Application — Optimal Point in Geometry
Example
def closest_to_point_on_line(ax, ay, bx, by, px, py):
"""
Find point on segment AB closest to point P using ternary search.
The distance function d(t) for t in [0,1] is unimodal (convex).
"""
import math
def distance(t):
x = ax + t * (bx - ax)
y = ay + t * (by - ay)
return math.sqrt((x - px)**2 + (y - py)**2)
t = ternary_search_continuous(lambda t: -distance(t), 0, 1)
# We minimize distance = maximize negative distance
x = ax + t * (bx - ax)
y = ay + t * (by - ay)
return x, y
Ternary Search in Competitive Programming
Common CP problems using ternary search:
Example
1. "Find x ∈ [L,R] that minimizes/maximizes f(x)" where f is convex/concave
2. "Find the peak in a bitonic array" (also solvable with binary search)
3. "Optimal position on a line to minimize total distance to N points"
4. "Largest rectangle area in histogram" (ternary not needed, but shows unimodal)
Example
# Competitive programming template for ternary search
def solve():
L, R = 0, 10**18 # Search range (integers)
EPS = 100 # Precision (stop when range <= EPS)
while R - L > EPS:
m1 = L + (R - L) // 3
m2 = R - (R - L) // 3
if f(m1) <= f(m2): # For minimum, use <=
L = m1 + 1
else:
R = m2 - 1
# Check all values in [L, R]
return min(f(x) for x in range(L, R + 1))
Number of Iterations
Ternary search reduces the range to 2/3 each step:
Example
After k iterations: range = (2/3)^k × original_range
To reach precision ε from range R:
(2/3)^k × R = ε
k = log(R/ε) / log(3/2) ≈ 2.41 × log₂(R/ε)
For integers [0, 10^9]:
k ≈ 2.41 × log₂(10⁹) ≈ 2.41 × 30 ≈ 72 iterations
Summary
- Ternary search finds the maximum or minimum of a unimodal function
- Divides range into thirds using two probes m1 and m2
- Eliminates one-third of the range per iteration → O(log n) total
- Requires: Strictly unimodal function on the search range
- Use when: Optimizing over a continuous or discrete domain with a single peak/valley
- For sorted array value search → use binary search instead
*Next Lesson: Search Comparison*
Continue the lesson path
Study next in this module
01SearchingSearching Interview Problems — 10 Classic Problems with Complete SolutionsSolve 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.02SearchingSearch 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.03SearchingLinear 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.04SearchingInterpolation 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.