Binary Search Applications

[!NOTE] This module explores the core principles of Binary Search Applications, deriving solutions from first principles and hardware constraints to build world-class, production-ready expertise.

1. The Hook: The Answer Space

Binary Search isn’t just for finding “7” in a sorted array [1, 3, 5, 7, 9]. It is a universal technique for Optimization Problems.

If you have a function f(x) that is Monotonic (e.g., f(x) is true for all x > k), you can finding the boundary k using Binary Search.

Pattern: “Minimize the Maximum…” or “Maximize the Minimum…” usually screams Binary Search on Answer.

2. The Pattern: Search Space

Instead of searching an array indices 0 to N-1, we search the Answer Range [Low, High].

  1. Define Bounds: What is the minimum possible answer? What is the maximum?
  2. Check Function: Write boolean canDo(x) that checks if x works in O(N) time.
  3. Binary Search:
    • mid = low + (high - low) / 2
    • If canDo(mid) works, maybe we can do better? (Move closer to optimal).
    • Else, mid is impossible.

3. Example: Koko Eating Bananas (LeetCode 875)

Problem: Koko wants to eat all piles within H hours. Return the minimum eating speed k.

  • Monotonicity: If speed k=10 works, then k=11 definitely works. If k=5 fails, k=4 definitely fails.
  • Bounds: Low = 1, High = Max(Piles).
  • Check: Iterate piles, calculate hours needed: Math.ceil(pile / k). If totalHours ≤ H, return true.

This turns an Optimization problem into a Decision problem.

4. Deep Dive Strategy Lab: Binary Search Applications

Intuition Through Analogy

Think of this chapter like ranking candidates quickly. The goal is not to memorize a fixed trick, but to repeatedly answer:

  1. What is the bottleneck?
  2. Which constraint dominates (time, memory, latency, correctness)?
  3. Which representation makes the bottleneck easier to eliminate?