Approach 1: Linear Scan

Intuition

Checking every index for target exhausts the search space, so it must work.

Algorithm

First, we do a linear scan of nums from the left, breaking when we find an instance of target. If we never break, then target is not present, so we can return the "error code" of [-1, -1] early. Given that we did find a valid left index, we can do a second linear scan, but this time from the right. In this case, the first instance of target encountered will be the rightmost one (and because a leftmost one exists, there is guaranteed to also be a rightmost one). We then simply return a list containing the two located indices.

Complexity Analysis

  • Time complexity :

    This brute-force approach examines each of the n elements of nums exactly twice, so the overall runtime is linear.

  • Space complexity :

    The linear scan method allocates a fixed-size array and a few integers, so it has a constant-size memory footprint.



Intuition

Because the array is sorted, we can use binary search to locate the left and rightmost indices.

Algorithm

The overall algorithm works fairly similarly to the linear scan approach, except for the subroutine used to find the left and rightmost indices themselves. Here, we use a modified binary search to search a sorted array, with a few minor adjustments. First, because we are locating the leftmost (or rightmost) index containing target (rather than returning true iff we find target), the algorithm does not terminate as soon as we find a match. Instead, we continue to search until lo == hi and they contain some index at which target can be found.

The other change is the introduction of the left parameter, which is a boolean indicating what to do in the event that target == nums[mid]; if left is true, then we "recurse" on the left subarray on ties. Otherwise, we go right. To see why this is correct, consider the situation where we find target at index i. The leftmost target cannot occur at any index greater than i, so we never need to consider the right subarray. The same argument applies to the rightmost index.

The first animation below shows the process for finding the leftmost index, and the second shows the process for finding the index right of the rightmost index.

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Complexity Analysis

  • Time complexity :

    Because binary search cuts the search space roughly in half on each iteration, there can be at most iterations. Binary search is invoked twice, so the overall complexity is logarithmic.

  • Space complexity :

    All work is done in place, so the overall memory usage is constant.