Finding Boundary with Binary Search
Prereq: Binary Search Introduction
An array of boolean values is divided into two sections, the left section consists of all false
and the right section consists of all true
. Find the boundary of the right section, i.e. index of the first true
element. If there is no true
element, return 1.
Input: arr = [false, false, true, true, true]
Output: 2
Explanation: first true
's index is 2.
Try it yourself
Explanation
The binary decision we have to make when we look at an element is
 if the element is
false
, we discard everything to the left and the current element itself.  if the element is
true
, the current element could be the firsttrue
although they may be othertrue
to the left. We discard everything to the right but what about the current element?
We can either keep the current element in the range or record it somewhere and then discard it. Here we choose the latter. We'll discuss the other approach in the alternative solution section.
We keep a variable boundary_index
that represents the leftmost true
's index we currently have. If the current element is true
, then we update boundary_index
with its index and discard everything to the right including the current element itself since its index has been recorded by the variable.
Implemetation
1  1 
 
2  2  
3  3 
 
4   
 
4  + 
 
5  + 
 
6  + 
 
7  + 
 
8  + 
 
9  + 
 
10  + 
 
11  + 
 
12  + 
 
13  +  
14  + 
 
15  +  
5  16 
 
6  17 
 
7  18 
 
8  19 

The good thing with this approach is we don't have to modify the while loop logic in vanilla binary search from the last module besides introducing a variable.
Alternative approach
Another approach to handle case 2 above is to keep the current element in the search range instead of discarding it, i.e. if arr[mid]: right = mid
instead of right = mid  1
. However, doing this without modifying the while
condition will result in an infinite loop. This is because when left == right
, right = mid
will not modify right
and thus not shrink search range and we are stuck in the while loop forever. To make this work we have to remove the equality in the while
condition. In addition, as mentioned in the last module, while
loop without equality will miss the singleelement edge case so we have to add an additional check after the loop to handle this case. Overall, we have to make three modifications to the vanilla binary search to make it work.
Side note: how to not stuck in an infinite loop
 make progress in each step
 have an exit strategy
Summary
This problem is major 🔑 in solving future binary searchrelated problems. As we will see in the following modules, many problems boil down to finding the boundary in a boolean array.