Median of Two Sorted Arrays

hardPython

Lesson

Binary Search for Optimization Problems

Binary search isn't just for finding elements in sorted arrays—it's a powerful technique for solving optimization problems where you need to find the "best" partition or cut point in a search space.

The key insight is recognizing when a problem can be transformed into a binary search. This happens when:

  1. You can define a search space with clear boundaries
  2. The search space has a monotonic property (if condition X is true at position i, it remains true for all positions beyond i)
  3. You can efficiently test whether a candidate solution is "too small," "too large," or "just right"

For array partition problems, binary search helps you find the optimal way to divide arrays. Instead of checking every possible partition (which would be O(n)), you can eliminate half the search space at each step.

The process follows this pattern:

  1. Define boundaries: Set left and right bounds for your search space
  2. Choose a partition point: Pick a midpoint to test
  3. Evaluate the partition: Check if this partition satisfies your constraints
  4. Adjust search space: Based on the evaluation, eliminate half the remaining possibilities
  5. Repeat: Continue until you find the optimal partition

This approach transforms many O(n) or O(n²) problems into O(log n) solutions. The challenge lies in identifying the right property to search on and designing the evaluation function that can quickly determine which direction to search next.

Example
1def find_partition_point(arr, target_sum): 2 """Find partition point where left sum >= target_sum using binary search""" 3 left, right = 0, len(arr) 4 5 while left < right: 6 mid = (left + right) // 2 7 left_sum = sum(arr[:mid]) # Sum of left partition 8 9 if left_sum >= target_sum: 10 right = mid # Try smaller partition 11 else: 12 left = mid + 1 # Need larger partition 13 14 return left
L5Binary search on partition positions, not array values
L7Evaluate current partition by checking our constraint
L9Monotonic property: if sum is too big here, it's too big for larger partitions

Key Takeaways

  • •Binary search works on any monotonic search space, not just sorted arrays
  • •Transform optimization problems by searching on partition points or constraint boundaries
  • •The key is identifying what property to search on and how to evaluate candidate solutions
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