**Objective**: Given an array of integers write an algorithm to find the local minima.

**Local Minima**: An element is considered as local minima if it is less than both of its neighbors (if neighbors exist).

**Example**:

int [] arrA = {11, 4, 2, 5, 11, 13, 5}; Output: Local Minima is: 2 int []arrA = {1,2,3,4}; Output: 1 int []arrA = {3}; Output: 3 int []arrA = {6,4}; Output: 4

**NOTE**: There could be many local minima’s, we need to find anyone.\

**Approach**:

**Naïve**: Use for loop and compare every element with its neighbor.

Time Complexity – O(n)

**Binary Search Approach:**

- Check if the mid element is smaller than its left and right neighbors.
- If the left neighbor is less than the mid element then make a recursive call to the left half of the array. (There will be at least one local minima in the left half, take few examples to check)
- If the right neighbor is less than the mid element then make a recursive call to the right half of the array.

Time Complexity – O(logn)

## Java

public class LocalMinimaBinary { public int find(int [] arrA, int start, int end){ //find the mid int mid = start + (end-start)/2; int size = arrA.length; //check the local minima (element is smaller than its left and right neighbors) //first check if left and right neighbor exists if((mid==0 || arrA[mid-1]>arrA[mid]) && (mid==size-1 || arrA[mid+1]>arrA[mid])) return arrA[mid]; //check if left neighbor is less than mid element, if yes go left else if(mid>0 && arrA[mid]>arrA[mid-1]){ return find(arrA, start, mid); }else { //if(mid<size-1 && arrA[mid]>arrA[mid+1]) return find(arrA, mid+1, end); } } public static void main(String[] args) { LocalMinimaBinary l = new LocalMinimaBinary(); int [] arrA = {11, 4, 2, 5, 11, 13, 5}; System.out.println("Local Minima is: " + l.find(arrA,0,arrA.length)); } }

**Output**:

Local Minima is: 2

Reference:** this**