Maximum Number of Weeks for Which You Can Work - Problem

Imagine you're a project manager juggling multiple exciting projects, each with their own milestones to complete! 🎯

You have n projects numbered from 0 to n - 1, and you're given an integer array milestones where each milestones[i] represents the number of milestones the i-th project has.

The Challenge: You want to maximize your productivity while following these strict rules:

  • πŸ”Ή Consistency: Every week, you must finish exactly one milestone from one project
  • πŸ”Ή Variety: You cannot work on the same project for two consecutive weeks

Your goal is to determine the maximum number of weeks you can work before being forced to stop due to these constraints. You might not be able to complete all milestones due to the alternating rule!

Input: An array of integers representing milestone counts for each project

Output: The maximum number of weeks you can work

Input & Output

example_1.py β€” Balanced Projects
$ Input: milestones = [1,2,3]
β€Ί Output: 6
πŸ’‘ Note: We can complete all milestones since no single project dominates. One possible schedule: Project 2 β†’ Project 1 β†’ Project 2 β†’ Project 0 β†’ Project 2 β†’ Project 1 (total: 6 weeks)
example_2.py β€” Dominant Project
$ Input: milestones = [5,2,1]
β€Ί Output: 7
πŸ’‘ Note: Project 0 has 5 milestones while others total 3. We can work 7 weeks: P0β†’P1β†’P0β†’P2β†’P0β†’P1β†’P0. We cannot complete all of project 0's milestones due to alternating constraint.
example_3.py β€” Single Project
$ Input: milestones = [5]
β€Ί Output: 1
πŸ’‘ Note: With only one project, we can work just 1 week since we can't work on the same project for consecutive weeks.

Constraints

  • n == milestones.length
  • 1 ≀ n ≀ 105
  • 1 ≀ milestones[i] ≀ 109
  • The sum of all milestones can be very large (up to 1014)

Visualization

Tap to expand
Project Scheduling Visualizationβœ… Balanced Scenario3432Max: 4, Others: 8Result: All 12 weeks!⚠️ Bottleneck Scenario10221Max: 10, Others: 5Result: 2Γ—5+1 = 11 weeksOptimal Pattern for Bottleneck Case:OtherMaxOtherMaxOtherMaxOtherPattern: Other-Max-Other-Max-...-OtherWe get 2Γ—(others) + 1 total weeksFormula: max > others ? 2Γ—others + 1 : totalTime: O(n) | Space: O(1)
Understanding the Visualization
1
Assess the Situation
Calculate total work and identify the most demanding project
2
Check for Bottleneck
If largest project > sum of all others, it's the limiting factor
3
Apply Strategy
Either complete all work or work optimally within constraints
Key Takeaway
🎯 Key Insight: The problem reduces to comparing the largest project against all others combined. If it's too large, it becomes our bottleneck and limits total achievable weeks!
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