Efficient way to find the max number in an array

Question

This is an interview question There is an array of integers. The elements in the array can follow the following patterns.

1. numbers are in ascending order
2. numbers are in descending order
3. numbers increases in the beginning and decreases in the end
4. numbers decreases in the beginning and increases in the end

What is the efficient way to find the max number in the array?

Answer 1

In that case, all you need to do is to determine whether it's (3). If not, the answer is max(first, last).

In the case that all elements are equal, you'll need to exhaustively search the array to show that there's not one high number somewhere in the middle. So I think it's O(n) to determine whether you're in (3).

Answer 2

Well, case by case you have

1. The last number.
2. The first number.
3. Move from beginning to end, stopping at first descent and printing previous number.
4. Compare first and last numbers.

If you don't know which case you're in, then you can test this while finding the max by doing the following (in C-like pseudocode):

for (int i=0; i<end; ++i) {
    if (array[i] < array[i+1]) {
        // CASE 1 or 3
        for (int j=i+1; j<end; ++j) {
            if (array[j] > array[j+1]) {
                // CASE 3
                return array[j];
            }
         }
         // CASE 1
         return array[end];
     }
}
// CASE 2 or 4
return max(array[0],array[end]);

Answer 3

You will be able to determine with type of array it is by inspecting the first two and last two elements

1. It is the last element
2. It is the first element
3. see below
4. It is the larger of the first and last elements

For 3, start by looking at two elements at the middle of the array, if they are still increasing the max is higher in the array, if they are decreasing, the max is lower in the array. Repeat in a binary search fashion

Answer 4

Since cases 1-3 all have one peak (value surrounded on both sides by values lower than itself or the edge of the array), and case 4 has two peaks both on the ends of the array, this problem can be solved rather simply in O(log n) time for all cases:

First, apply the 1D peak finding algorithm to find a peak in the array.

If the peak occurs in the middle of the array (not the first or last position), then this is case #3, and the peak is also the maximum.

If the peak is either the first or last element of the array, then this is one of cases 1, 2, or 4, and the array max is max(first, last).

Python-esque pseudo code:

def find-peak(list):
    mid=len(list)/2
    if (list[mid-1] > list[mid]:
        return find-peak(list[:mid-1])
    else if (list[mid+1] > list[mid]:
        return find-peak(list[mid+1:])
    else:
        return mid

def find-max(list):
    peak = find-peak(list)
    if peak==0 or peak==len(list)-1:
        return max(list[0], list[-1])
    else:
        return list[peak]

Answer 5

1.the last number 2.the first number 3.do binary-like search, pick a pivot,calculate the slope, just to decide next to go left or right 4.first or last number

Answer 6

The way to identify the four cases is straight forward if we assume the sequence do not have repeating number:

case 1: arr[0] < arr[1] && arr[end-1] < arr[end]
case 2: arr[0] > arr[1] && arr[end-1] > arr[end]
case 3: arr[0] < arr[1] && arr[end-1] > arr[end]
case 4: arr[0] > arr[1] && arr[end-1] < arr[end]

As mentioned in other answers, the way to find the max is straight forward too:

case 1: arr[end]
case 2: arr[0]
case 3: binary search, until found n that arr[n-1] < arr[n] > arr[n+1]
case 4: max(arr[0],arr[end])

Answer 7

The answer depends on what is meant by "efficiency." If you want fast code, look at someone else's answer. If you want to be efficient as a programmer you should probably just use a library call (like max_element() in C++.)

Answer 8

This problem reminds me of the Golden section algoritm for finding the minimum of an unimodular (ie.: decreasing then increasing) function. It is kind of a souped-up version of binary search that calculates the value of the function (ie.: inspects the array) in as few points as possible.

All you need to do now is translate it into a discrete version and add nome extra whistles to determine wether the function is concave or convex.