Binary Search
What is Binary Search?
Binary search is an efficient algorithm for finding a target value within a sorted array (or list). It works by repeatedly dividing the search interval in half, allowing you to skip over large portions of the array, making it much faster than a linear search, which checks each element one by one.
How Does Binary Search Work?
Here’s a step-by-step breakdown of how binary search operates:
1. Start with the entire array: Define the leftmost (left) and rightmost (right) boundaries of the array.
2. Find the middle element: Calculate the middle index of the current search range.
3. Compare the middle element with the target:
o If the middle element is equal to the target, the search is complete, and the target is found.
o If the target is smaller than the middle element, focus on the left half of the array (ignore the right half).
o If the target is larger than the middle element, focus on the right half of the array (ignore the left half).
4. Repeat the process: Continue halving the search range until you find the target or the search range becomes empty.
Example: Visualizing Binary Search
Consider the following sorted array:
[10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
Let’s say we are searching for 70.
1. First step: Check the middle element (50).
o 70 is greater than 50, so now search the right half: [60, 70, 80, 90, 100].
2. Second step: Check the middle element of the new range (80).
o 70 is less than 80, so now search the left half: [60, 70].
3. Third step: Check the middle element (70).
o Found it!
Why is Binary Search So Fast?
Binary search works in O(log n) time complexity. This means that as the array size increases, the number of steps required to find the target increases at a much slower rate compared to linear search. In contrast, linear search has a time complexity of O(n), which means it checks each element individually.
Python Code Example for Binary Search:
left = 0
right = len(arr) - 1
while left <= right:
mid = (left + right) // 2
if arr[mid] == target:
return mid # Target found
elif arr[mid] < target:
left = mid + 1 # Focus on the right half
else:
right = mid - 1 # Focus on the left half
return -1 # Target not found
# Example usage:
arr = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
target = 70
result = binary_search(arr, target)
if result != -1:
print(f"Element {target} found at index {result}")
else:
print(f"Element {target} not found")
Real-Time Example: Finding a Name in a Phone Book
Imagine you have a phone book with names sorted alphabetically from A to Z, and you need to find "Durga". Instead of checking each name one by one (a linear search), you can use binary search to find it faster.
Step-by-Step Process:
1. Start with the entire phone book.
o The middle name might be "Nani", for instance.
2. Compare with the middle name.
o If "Durga" comes before "Nani" alphabetically, focus on the left half of the book.
o If "Durga" comes after "Nani", focus on the right half.
3. Narrow down the search.
o Continue halving the search space and comparing the middle name until "Durga" is found.
Why Is Binary Search Faster in This Example?
If the phone book has 1,000 names, a linear search would involve checking up to 1,000 names, one by one. However, a binary search requires only about log₂(1000) ≈ 10 comparisons to find the target. This is a dramatic improvement in speed, especially for large datasets.
Summary of the Phone Book Example:
· Starting with a large set of names (from A to Z), you reduce the number of names to look through by half with each step.
· This makes binary search much more efficient than linear search, which would require examining every name.
Conclusion:
Binary search is an efficient, fast, and essential algorithm for searching through sorted data. Its O(log n) time complexity makes it particularly useful when dealing with large datasets. Understanding binary search is crucial for anyone learning algorithms and data structures.
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