DSA Notes
HubAlgorithm vs Data Structure - Key DifferencesReal-World Applications of DSATime and Space Complexity OverviewWhat is DSA? Data Structures and AlgorithmsWhy Learn Data Structures and AlgorithmsAsymptotic Analysis - Understanding Algorithm GrowthBest, Average, and Worst Case AnalysisBig O Notation - Upper Bound AnalysisBig Omega Notation - Lower Bound AnalysisBig Theta Notation - Tight Bound AnalysisSpace Complexity - Memory AnalysisTime Complexity Examples and AnalysisArray Interview Problems and SolutionsArray Operations - Insert, Delete, SearchArray Rotation TechniquesArray Searching TechniquesArrays - Fundamental Data StructureDynamic Arrays - Flexible SizingMultidimensional Arrays - 2D and BeyondSliding Window Technique for ArraysAnagram Problems - Character RearrangementPalindrome Problems and SolutionsPattern Searching in StringsString Interview Problems CollectionString Matching Algorithms - Pattern SearchingString Operations - Concatenation, Substring, ManipulationString Data Structure - FundamentalsCircular Linked List - Ring StructureDoubly Linked List - Bidirectional TraversalLinked List Interview ProblemsLinked List - Data Structure FundamentalsLinked List Operations - Insert, Delete, ReverseReverse Linked List - Iterative and RecursiveSingly Linked List - Implementation and OperationsBalanced Parentheses - Stack ApplicationInfix, Prefix, Postfix Expressions - Stack ApplicationStack Implementation TechniquesStack Interview Problems and SolutionsStack Data Structure - LIFO PrincipleStack using Arrays - Static ImplementationStack using Linked Lists - Dynamic ImplementationCircular Queue - Efficient Space UsageDeque (Double Ended Queue)Priority Queue - Custom OrderingQueue Applications - Real-World Use CasesQueue Implementation - Array and Linked ListQueue Interview Problems and SolutionsQueue Data Structure - FIFO PrincipleAVL 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 Data StructureHeap OperationsHeap Interview ProblemsMax-HeapMin-Heap Implementation and Use CasesPriority Queue using HeapBellman-Ford AlgorithmBreadth-First Search (BFS) AlgorithmDepth-First Search (DFS) AlgorithmDijkstra\Floyd-Warshall AlgorithmGraph Interview ProblemsGraph Data Structure FundamentalsGraph Representation MethodsKruskal\Minimum Spanning Tree (MST)Prim\Shortest Path Algorithms: Complete ComparisonTopological SortingUnion-Find (Disjoint Set Union)Separate ChainingCollision HandlingHash FunctionsHashing FundamentalsHashing Interview ProblemsOpen AddressingBubble Sort - Simple Sorting AlgorithmCounting Sort - Non-Comparison SortingHeap Sort - Heap Based SortingInsertion Sort - Build Sorted ArrayMerge Sort - Divide and ConquerQuick Sort - Partition Based SortingRadix Sort - Digit Based SortingSelection Sort - Selection Based SortingSorting Algorithms Comparison - Choosing the Right AlgorithmSorting Algorithms - Fundamentals and ConceptsBinary Search - Efficient SearchingInterpolation Search - Intelligent SearchingLinear Search - Sequential Search AlgorithmSearch Algorithms ComparisonSearching Algorithm Interview ProblemsTernary Search AlgorithmBacktracking with RecursionRecursion Fundamentals - Base Case and RecurrenceRecursion Problems and SolutionsRecursion Tree - Visualizing Function CallsRecursion vs Iteration - Comparing ApproachesTail Recursion - Optimization TechniqueDynamic Programming Interview ProblemsDynamic Programming FundamentalsFibonacci with Dynamic Programming0/1 Knapsack ProblemLongest Common Subsequence (LCS)Longest Increasing Subsequence (LIS)Matrix Chain MultiplicationMemoization (Top-Down Dynamic Programming)Tabulation (Bottom-Up Dynamic Programming)Activity Selection ProblemFractional KnapsackGreedy Interview ProblemsGreedy Algorithm FundamentalsHuffman Coding — Data Structures & AlgorithmsJob Sequencing with DeadlinesBacktracking FundamentalsBacktracking Practice ProblemsN-Queens ProblemRat in a MazeSubset GenerationSudoku SolverCodeforces Guide for Competitive ProgrammersCommon Competitive Programming TechniquesContest Strategies for Competitive ProgrammingCompetitive Programming IntroductionFast Input/Output for Competitive ProgrammingLeetCode Guide for Interview PreparationBit Manipulation - Advanced TechniquesBloom Filter - Probabilistic Data StructureComputational Geometry - Algorithms and TechniquesDisjoint Set Union (DSU) - Union FindRandomized Algorithms - Probabilistic ComputingSkip List - Probabilistic Balanced StructureSuffix Array - String IndexingTrie - Prefix Tree for Efficient SearchingAutocomplete 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 the Job Sequencing problem using a greedy approach — maximize profit by sorting by profit and assigning each job to the latest available time slot before its deadline.
Problem Statement
Given N jobs, each with a deadline and a profit (earned only if the job is completed before or on its deadline), and the constraint that each job takes exactly 1 unit of time, find a sequence that maximizes total profit. Each time slot can hold at most one job.
Greedy Intuition
Since we want to maximize profit, we should prioritize high-profit jobs. But a high-profit job is useless if we cannot schedule it before its deadline. The greedy strategy:
- Sort jobs by profit in descending order (most profitable first).
- For each job, assign it to the latest available slot before its deadline.
Why the latest slot? Because scheduling early wastes precious early slots that might be needed by other jobs with tighter deadlines. Scheduling as late as possible gives maximum flexibility for remaining jobs.
Dry Run Example
| Jobs | (ID, Deadline, Profit) |
| J1 | deadline=4, profit=20 |
| J2 | deadline=1, profit=10 |
| J3 | deadline=1, profit=40 |
| J4 | deadline=1, profit=30 |
| Sorted by profit (descending) | J3(40), J4(30), J1(20), J2(10) |
| Max deadline = 4, so time slots | [_, _, _, _] (slots 1,2,3,4) |
| Latest slot ≤ 1: slot 1 is free | assign J3 to slot 1. |
| Slots | [J3, _, _, _] |
| Latest slot ≤ 1: slot 1 is taken | NO slot available. SKIP. |
| Latest slot ≤ 4: slot 4 is free | assign J1 to slot 4. |
| Slots | [J3, _, _, J1] |
| Latest slot ≤ 1: slot 1 is taken | NO slot available. SKIP. |
| Result | Jobs {J3, J1}, Total profit = 40 + 20 = 60. |
Another Example (More Complex)
| Jobs | (ID, Deadline, Profit) |
| A | deadline=2, profit=100 |
| B | deadline=1, profit=19 |
| C | deadline=2, profit=27 |
| D | deadline=1, profit=25 |
| E | deadline=3, profit=15 |
| Sorted by profit | A(100), C(27), D(25), B(19), E(15) |
| Slots | [_, _, _] (max deadline = 3) |
| A (deadline=2): Latest free ≤ 2 | slot 2. Slots: [_, A, _] |
| C (deadline=2): Latest free ≤ 2 | slot 1. Slots: [C, A, _] |
| D (deadline=1): Latest free ≤ 1 | slot 1 taken. SKIP. |
| B (deadline=1): Latest free ≤ 1 | slot 1 taken. SKIP. |
| E (deadline=3): Latest free ≤ 3 | slot 3. Slots: [C, A, E] |
| Result | {C, A, E}, Profit = 27 + 100 + 15 = 142 |
| Sequence: C(slot 1) | A(slot 2) → E(slot 3) |
Python Implementation
Example
def job_sequencing(jobs):
"""
jobs: list of (job_id, deadline, profit)
Returns: (max_profit, selected_jobs)
"""
# Sort by profit descending
jobs_sorted = sorted(jobs, key=lambda x: x[2], reverse=True)
# Find maximum deadline to determine number of slots
max_deadline = max(job[1] for job in jobs_sorted)
# Slots array: slot[i] = job assigned to slot i (1-indexed)
slots = [None] * (max_deadline + 1) # index 0 unused
total_profit = 0
selected = []
for job_id, deadline, profit in jobs_sorted:
# Find latest available slot <= deadline
for slot in range(min(deadline, max_deadline), 0, -1):
if slots[slot] is None:
slots[slot] = job_id
total_profit += profit
selected.append(job_id)
break
return total_profit, selected
# Test
jobs = [('A', 2, 100), ('B', 1, 19), ('C', 2, 27), ('D', 1, 25), ('E', 3, 15)]
profit, selected = job_sequencing(jobs)
print(f"Max profit: {profit}") # 142
print(f"Selected: {selected}") # ['A', 'C', 'E']
C++ Implementation
Example
#include <vector>
#include <algorithm>
#include <iostream>
using namespace std;
struct Job {
char id;
int deadline, profit;
};
pair<int, vector<char>> jobSequencing(vector<Job>& jobs) {
sort(jobs.begin(), jobs.end(), [](const Job& a, const Job& b) {
return a.profit > b.profit;
});
int maxDeadline = 0;
for (auto& j : jobs) maxDeadline = max(maxDeadline, j.deadline);
vector<int> slot(maxDeadline + 1, -1); // -1 = empty
int totalProfit = 0;
vector<char> selected;
for (auto& job : jobs) {
for (int s = min(job.deadline, maxDeadline); s >= 1; s--) {
if (slot[s] == -1) {
slot[s] = job.id;
totalProfit += job.profit;
selected.push_back(job.id);
break;
}
}
}
return {totalProfit, selected};
}
int main() {
vector<Job> jobs = {{'A',2,100}, {'B',1,19}, {'C',2,27}, {'D',1,25}, {'E',3,15}};
auto [profit, selected] = jobSequencing(jobs);
cout << "Max profit: " << profit << endl;
}
Optimized Approach: Union-Find (Disjoint Set)
The naive approach searches for a free slot in O(D) per job, giving O(N×D) total. Using Union-Find, we can find the latest free slot in nearly O(1) amortized:
Example
class DSU:
def __init__(self, n):
self.parent = list(range(n + 1))
def find(self, x):
if self.parent[x] != x:
self.parent[x] = self.find(self.parent[x])
return self.parent[x]
def union(self, x, y):
self.parent[self.find(x)] = self.find(y)
def job_sequencing_optimized(jobs):
jobs_sorted = sorted(jobs, key=lambda x: x[2], reverse=True)
max_deadline = max(j[1] for j in jobs_sorted)
dsu = DSU(max_deadline)
total_profit = 0
selected = []
for job_id, deadline, profit in jobs_sorted:
# Find latest available slot
available = dsu.find(min(deadline, max_deadline))
if available > 0:
dsu.union(available, available - 1) # mark slot as used
total_profit += profit
selected.append(job_id)
return total_profit, selected
Complexity with DSU: O(N log N) for sorting + O(N × α(D)) for scheduling ≈ O(N log N) total.
Complexity Analysis
| Approach | Time | Space |
|---|---|---|
| Naive (search for slot) | O(N × D) | O(D) |
| With Union-Find | O(N log N + N × α(D)) ≈ O(N log N) | O(D) |
Where N = number of jobs, D = maximum deadline.
Correctness Argument
The greedy strategy (highest profit first, latest available slot) is optimal because:
- Higher-profit jobs should always be preferred over lower-profit ones.
- Assigning to the latest slot preserves earlier slots for jobs with tighter deadlines.
- An exchange argument shows: if an optimal solution differs from greedy, swapping any job for a higher-profit one in its slot cannot decrease total profit.
Variations
- Weighted Job Scheduling (variable duration): Jobs have different processing times. Requires DP, not greedy.
- Job Sequencing with Machine Assignment: Multiple machines available. Each machine has independent slot arrays.
- Minimizing Lateness: Instead of profit, minimize maximum lateness. Greedy criterion: earliest deadline first.
Real-World Applications
- Freelance project management: Accept highest-paying projects that meet deadlines
- CPU task scheduling: Real-time OS scheduling with hard deadlines
- Order fulfillment: E-commerce warehouse prioritizing high-value orders
- Publishing deadlines: Editors choosing which articles to include before print deadline
- Ad campaign scheduling: Running most profitable ads within campaign windows
Key Takeaways
Job Sequencing demonstrates greedy scheduling at its best: sort by profit, assign to the latest feasible slot. The Union-Find optimization is a classic data structure application that reduces slot-finding from O(D) to nearly O(1). This problem frequently appears in placement interviews — remember the "latest free slot" insight, it is the key that makes the greedy approach work.
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Job Sequencing with Deadlines.
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, job, sequencing
Related Data Structures & Algorithms Topics
Continue learning this concept
Greedy AlgorithmsActivity Selection ProblemSolve the Activity Selection problem with a greedy approach — sort by finish time, select non-overlapping activities, and prove optimality via exchange argument.Greedy AlgorithmsFractional KnapsackSolve the Fractional Knapsack problem using a greedy approach — sort by value-to-weight ratio, fill optimally, and understand why fractions make greedy work.Greedy AlgorithmsGreedy Interview ProblemsMaster four classic greedy interview problems — Minimum Platforms, Gas Station Circuit, Jump Game, and Candy Distribution with detailed approaches and code.Greedy AlgorithmsGreedy Algorithm FundamentalsUnderstand greedy algorithms — the greedy choice property, optimal substructure, when greedy works vs fails, and techniques to prove correctness.Greedy AlgorithmsHuffman Coding — Data Structures & AlgorithmsBuild Huffman codes from scratch — understand frequency-based prefix codes, priority queue construction, tree building, encoding/decoding, and compression analysis.BacktrackingN-Queens ProblemSolve the classic N-Queens problem using backtracking — learn constraint checking, state-space exploration, and trace through a complete 4-Queens walkthrough.