Backwards Binary Search

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Unit 7 Session 1 (Click for link to problem statements)

Problem Highlights

• 💡 Difficulty: Medium
• ⏰ Time to complete: 15 mins
• 🛠️ Topics: Binary Search, Arrays, Iterative Algorithms

1: U-nderstand

Understand what the interviewer is asking for by using test cases and questions about the problem.

• Established a set (2-3) of test cases to verify their own solution later.
• Established a set (1-2) of edge cases to verify their solution handles complexities.
• Have fully understood the problem and have no clarifying questions.
• Have you verified any Time/Space Constraints for this problem?

• Q: What should the function do if the target appears multiple times in the list?
  • A: The function should return the index of the last occurrence of the target.

HAPPY CASE
Input: lst = [1, 2, 3, 3, 3, 4, 5], target = 3
Output: 4
Explanation: The last occurrence of the target 3 is at index 4.

EDGE CASE
Input: lst = [1, 2, 3, 4, 5], target = 6
Output: -1
Explanation: The target value 6 is not in the list.

2: M-atch

Match what this problem looks like to known categories of problems, e.g. Linked List or Dynamic Programming, and strategies or patterns in those categories.

This problem is a variation of the binary search algorithm, requiring:

• Modifying the classic binary search to continue searching even after finding the target to ensure it is the last occurrence.

3: P-lan

Plan the solution with appropriate visualizations and pseudocode.

General Idea: Adapt the binary search to locate the last occurrence of a target value within a sorted array.

1) Initialize pointers for the left and right bounds of the list.
2) Initialize a variable `last_occurrence` to store the index of the last found target.
3) While the left pointer is not greater than the right:
   - Calculate the middle index.
   - If the middle element is the target, update `last_occurrence` and move the left pointer to continue searching on the right for possible later occurrences.
   - If the target is less than the middle element, adjust the right pointer to narrow the search to the left half.
   - If the target is greater than the middle element, adjust the left pointer to narrow the search to the right half.
4) Return `last_occurrence` if found; otherwise, return -1.

⚠️ Common Mistakes

• Stopping the search upon finding the first instance of the target without ensuring it is the last occurrence.

4: I-mplement

Implement the code to solve the algorithm.

def find_last(lst, target):
    left, right = 0, len(lst) - 1
    last_occurrence = -1
    
    while left <= right:
        mid = (left + right) // 2
        
        if lst[mid] == target:
            last_occurrence = mid
            left = mid + 1  # Continue searching in the right half
        elif lst[mid] < target:
            left = mid + 1
        else:
            right = mid - 1
    
    return last_occurrence

5: R-eview

Review the code by running specific example(s) and recording values (watchlist) of your code's variables along the way.

• Trace through your code with a list [1, 2, 3, 3, 3, 4, 5] and a target of 3 to ensure it correctly identifies the last occurrence at index 4.
• Validate with a target not in the list (e.g., 6) to check that it returns -1.

6: E-valuate

Evaluate the performance of your algorithm and state any strong/weak or future potential work.

• Time Complexity: O(log n) because the approach still fundamentally relies on binary search, halving the search space with each iteration.
• Space Complexity: O(1) because it uses a few variables for pointers and does not require additional space based on input size.