Kadane's Algorithm is a powerful technique used to solve the Maximum Subarray Problem. This lesson is designed to guide you step-by-step through understanding the problem, exploring different solutions, and finally, mastering Kadane's Algorithm itself. Here's what we'll cover:

1) Kadane’s Algorithm at a Glance (Why this matters)

• What is it? A lightning-fast way to find the most profitable “streak” in an array—i.e., the contiguous subarray with the highest sum.
• Where it shines: Stock gains over days, longest “winning run” in signal processing, anytime you care about the best contiguous chunk.
• Why people love it: It runs in O(n) time with O(1) extra space. Translation: fast and tiny.

2) Meet the Maximum Subarray Problem (with a picture in your head)

• The problem: Given an array (which can include negatives), find the contiguous slice with the biggest sum.
• Mental model: Imagine walking a trail with ups (+) and downs (–). Your goal is to pick one continuous stretch that climbs the most overall.
• Quick example: [-2, 1, -3, 4, -1, 2, 1, -5, 4] → best subarray is [4, -1, 2, 1] with sum 6.

3) Brute Force (the “try everything” approach)

• Idea: Check every possible start and end index, compute each sum, keep the best.
• Pros: Dead simple, great for understanding the problem.
• Cons: Slow when arrays get big.
• Time complexity: O(n²) sums (or O(n³) if you recompute sums every time).

4) Brute Force in Code (get it working first)

• We’ll write the straightforward version so you can see the mechanics: nested loops, track a running best, done.
• This gives you a baseline to compare against Kadane’s speedup later.

5) Kadane’s Algorithm (the elegant upgrade)

• Core idea: As you scan left → right, keep a running sum.

  • If the running sum ever goes negative, drop it and start fresh at the next element.
  • Track the best sum you’ve seen along the way.

• Why it works: A negative running sum can only hurt future totals—so reset early and often.

• Steps you’ll follow:

  1. Initialize best and current with the first element.
  2. For each next number x: current = max(x, current + x) best = max(best, current)
  3. Return best.

• Time complexity: O(n). One pass. No extra storage. Chef’s kiss.

6) Kadane in Code (concise and fast)

• We’ll implement Kadane in a few popular languages so you can copy, paste, and profit (academically).
• You’ll see it’s 3–5 lines for the core logic—clean, tight, and interview-ready.

What you’ll walk away with

By the end, you’ll:

• Understand the Maximum Subarray Problem clearly.
• Have a working brute-force solution (for learning and testing).
• Master Kadane’s Algorithm and know why it’s optimal.
• Be ready to spot and solve “best contiguous streak” problems on sight.