What is the brute force approach and complexity for Job Sequencing Problem?

Try every subset; keep the most profitable one that can run on time with time complexity O(2^N · N log N) and space complexity O(N).

What is the optimal approach and complexity for Job Sequencing Problem?

Sort by deadline; keep a min-heap of profits; when it grows past the deadline, pop the smallest with optimal time complexity O(N log N) and space complexity O(N).

How do you identify the pattern for Job Sequencing Problem?

Look for key signals in the prompt: "each job takes 1 unit of time" indicates One slot holds one job, so a deadline d means at most d jobs can finish by then.; "profit only if done by its deadline" indicates Jobs can be dropped. This is pick-the-best-set, not 'do them all'.; "maximum total profit" indicates Profits are positive, so a bigger set is always better, as long as it still fits..

Job Sequencing Problem

Medium·GeeksforGeeks Pro OnlySlight Variation

GeeksforGeeks

Problem Statement

Each job takes exactly 1 unit of time, and only one job runs at a time. The time slots are 1, 2, 3, and so on.

A job earns its profit only if it finishes by its deadline. So job i must run in a slot that is at or before deadline[i].

Pick the set of jobs that earns the most profit in total. Return two numbers: how many jobs you picked, and the total profit.

Input

Two arrays deadline and profit of the same length. Job i has deadline deadline[i] and profit profit[i].

Output

A list [count, totalProfit]: the number of jobs picked for the maximum profit, and that maximum profit.

Example 1

Input: deadline = [4, 1, 1, 1], profit = [20, 10, 40, 30]

Output: [2, 60]

Job 1 (deadline 4, profit 20) and job 3 (deadline 1, profit 40) give 20 + 40 = 60. Only one job can use slot 1.

Example 2

Input: deadline = [2, 1, 2, 1, 1], profit = [100, 19, 27, 25, 15]

Output: [2, 127]

Job 1 (profit 100) and job 3 (profit 27) use slots 1 and 2. Total 127.

Example 3

Input: deadline = [3, 1, 2, 2], profit = [50, 10, 20, 30]

Output: [3, 100]

Jobs 1, 3 and 4 (50 + 20 + 30) fit in slots 3, 1 and 2.

Constraints

• The number of jobs is at most 10^5, and `deadline` and `profit` have the same length.
• 1 ≤ deadline[i] ≤ n, where n is the number of jobs
• 1 ≤ profit[i] ≤ 500

Pattern Recognition Clues

Recognising these key signals in the problem description immediately identifies the pattern:

“each job takes 1 unit of time”

→ One slot holds one job, so a deadline d means at most d jobs can finish by then.

“profit only if done by its deadline”

→ Jobs can be dropped. This is pick-the-best-set, not 'do them all'.

“maximum total profit”

→ Profits are positive, so a bigger set is always better, as long as it still fits.

Approach Complexity Ladder

Brute Force — try every subset of jobsO(2^N · N log N)O(N)

Try every subset; keep the most profitable one that can run on time

Better — sort by profit, take the latest free slotO(N · D)O(N)

Sort by profit (highest first); each job takes the latest free slot up to its deadline

Optimal — sort by deadline, keep a min-heap of profitsO(N log N)O(N)

Sort by deadline; keep a min-heap of profits; when it grows past the deadline, pop the smallest

Brute Force Intuition

There are 2^N subsets of jobs. For each subset, check whether it can run on time: sort its deadlines, and the k-th job (1-based) must have deadline ≥ k. Keep the feasible subset with the largest total profit, and report its size.

Recognition rule for this sub-pattern:

Sort by the right key (ratio, deadline, height); argue that swapping any two out-of-order items cannot help.

Remember this · Greedy

Trigger

Maximise count / minimise cost with a natural "best next step", sorting makes the choice obvious, reachability.

Think

Is there a choice I can always make safely without looking ahead?

Key question

Why can't the greedy choice be worse than the optimal one? (Try to break it with a small example.)

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