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*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Ternary Search — Finding Maximum/Minimum of Unimodal Functions.
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, ternary, search
Related Data Structures & Algorithms Topics
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