hello-algo/en/docs/chapter_searching/binary_search_edge.md

Finding boundaries with binary search

Finding the left boundary

!!! question

Given a sorted array `nums` of length $n$ that may contain duplicate elements, return the index of the leftmost occurrence of `target`. If the array does not contain this element, return $-1$.

Recall the method for finding an insertion point with binary search: when the search finishes, i points to the leftmost target. Finding the insertion point therefore amounts to finding the index of the leftmost target.

We can use the insertion-point function to find the left boundary. However, target may be absent from the array, leading to either of the following outcomes.

• The insertion index i is out of bounds.
• nums[i] is not equal to target.

In either case, simply return -1. The code is as follows:

[file]{binary_search_edge}-[class]{}-[func]{binary_search_left_edge}

Finding the right boundary

How can we find the rightmost target? The most direct approach is to modify the pointer update for nums[m] == target. We omit the code here; interested readers can implement it themselves.

Here are two shortcuts.

Reusing the left-boundary search

We can use the function for finding the leftmost element to find the rightmost one: turn the search for the rightmost target into a search for the leftmost target + 1.

As shown below, when the search finishes, pointer i points to the leftmost target + 1, if it exists, while j points to the rightmost target. We therefore return $j$.

The returned insertion point is i, so subtract 1 to obtain j:

[file]{binary_search_edge}-[class]{}-[func]{binary_search_right_edge}

Turning a boundary search into an element search

When the array does not contain target, i ultimately points to the first element greater than target, while j points to the rightmost element smaller than target.

As shown below, we can therefore construct a value absent from the array and use it to find the left and right boundaries.

• To find the leftmost target, search for target - 0.5 and return pointer i.
• To find the rightmost target, search for target + 0.5 and return pointer j.

The code is omitted here, but keep two points in mind.

• The given array contains no fractional values, so we do not need to handle equality.
• Since this method introduces fractional values, change the function's target variable to a floating-point type. No change is needed in Python.