# 1118. Data Structure - Monotonic QueueMonotonic Queue

Implement monotonic queue.

## 1. What is Monotonic Queue?

Monotonic queue is a data structure where the order of the elements is strictly increasing or strictly decreasing. For example a strictly increasing monotonic queue will be able to contain [1, 3, 5, 6, 7], but not [1, 1, 3, 5] and [2, 1, 4, 5] as the elements are not strictly increasing (1, 1 is not strictly increasing, and 2, 1 is decreasing).

There are two types of monotonic queue, increasing or decreasing queue.

• Monotonic increasing queue: to push an element e, starts from the rear element, we pop out element s≥e(violation);
• Monotonic decreasing queue: we pop out element s<=e (violation).
• Sometimes, we can relax the strict monotonic condition, and can allow the stack or queue have duplicate value.

To get the feature of the monotonic queue, with [5, 3, 1, 2, 4] as example.

Index Value Increasing queue Decreasing queue
1 5 [5] [5]
2 3 [3] 3 kick out 5 [5, 3] #3->5
3 1 [1] 1 kick out 3 [5, 3, 1] #1->3
4 2 [1, 2] #2->1 [5, 3, 2] 2 kick out 1 #2->3
5 4 [1, 2, 4] #4->2 [5,4] 4 kick out 2, 3 #4->2

## 2. Question of First Smaller Element

Find the first element smaller than current either in the left or in the right.

For example, given array [5,3,1,2,4]:

• The answer to the first element smaller than current in left is [-1,-1,-1,1,2].
• The answer to the first element smaller than current in right is [3,1,-1,-1,-1].

### 2.1 Naive Solution

We can find the smaller element in brute force way. Use one for loop to point at the current element, and another embedding for loop to look for the first element that is smaller than the current, which gives us $O(n^2)$ time complexity.

// input:  [ 5, 3,  1, 2, 4]
// output: [-1,-1, -1, 1, 2]
// O(n^2)
public int[] firstSmallerLeft(int[] nums) {
int[] ans = new int[nums.length];
for (int i = 0; i < nums.length; i++) {
ans[i] = -1;
int curr = nums[i];
for (int j = i - 1; j >=0; j--) { // scan from head
if (nums[j] < curr) {
ans[i] = nums[j];
break;
}
}
}

return ans;
}

// input:  [5, 3,  1,  2,  4]
// output: [3, 1, -1, -1, -1]
// O(n^2)
public int[] firstSmallerRight(int[] nums) {
int[] ans = new int[nums.length];
for (int i = 0; i < nums.length; i++) {
ans[i] = -1;
int curr = nums[i];
for (int j = i + 1; j < nums.length; j++) { // scan from tail
if (nums[j] < curr) {
ans[i] = nums[j];
break;
}
}
}

return ans;
}


### 2.2 Solution with Stack

We can use stack with better performance, the time complexity is $O(n)$.

// input:  [ 5, 3,  1, 2, 4]
// output: [-1,-1, -1, 1, 2]
// O(n)
public int[] firstSmallerLeft(int[] nums) {
int[] ans = new int[nums.length];
Stack<Integer> stack = new Stack<>();   // increasing stack
for (int i = 0; i < nums.length; i++) { // left to right
while (!stack.isEmpty() && stack.peek() >= nums[i]) {
stack.pop();
}
if (stack.isEmpty()) {
ans[i] = -1;
} else {
ans[i] = stack.peek();
}
stack.push(nums[i]);
}

return ans;
}

// input:  [5, 3,  1,  2,  4]
// output: [3, 1, -1, -1, -1]
// O(n)
public int[] firstSmallerRight(int[] nums) {
int[] ans = new int[nums.length];
Stack<Integer> stack = new Stack<>();        // increasing stack
for (int i = nums.length - 1; i >= 0; i--) { // right to left
while (!stack.isEmpty() && stack.peek() >= nums[i]) {
stack.pop();
}
if (stack.isEmpty()) {
ans[i] = -1;
} else {
ans[i] = stack.peek();
}
stack.push(nums[i]);
}

return ans;
}


### 2.3 Solution with Monotonic Queue

We can use monotonic queue with better performance, the time complexity is $O(n)$. Instead of creating two methods to get smaller left and smaller right separately, we can use one method to get both results at once.

// input:  [5, 3, 1, 2, 4]
// output: [[-1,-1, -1, 1, 2], [3, 1, -1, -1, -1]]
// O(n)
public int[][] firstSmaller(int[] nums) {
int[] leftSmaller = new int[nums.length];
int[] rightSmaller = new int[nums.length];
Arrays.fill(leftSmaller, -1);
Arrays.fill(rightSmaller, -1);
Deque<Integer> queue = new LinkedList<>(); // increasing queue
for (int i = 0; i < nums.length; i++) {    // left to right
while (!queue.isEmpty() && nums[queue.peekLast()] >= nums[i]) {
rightSmaller[queue.pollLast()] = nums[i];
}
if (!queue.isEmpty()) {
leftSmaller[i] = nums[queue.peekLast()];
}
queue.offerLast(i);
}

return new int[][]{leftSmaller, rightSmaller};
}

• Use Deque as monotonic queue.
• The deque stores the index of the element, not the value of it.

Similarly, we can get larger left and larger right at once with monotonic queue.

// input:  [5, 3, 1, 2, 4]
// output: [[-1, 5, 3, 3, 5], [-1, 4, 2, 4, -1]]
// O(n)
public int[][] firstLarger(int[] nums) {
int[] leftLarger = new int[nums.length];
int[] rightLarger = new int[nums.length];
Arrays.fill(leftLarger, -1);
Arrays.fill(rightLarger, -1);
Deque<Integer> queue = new LinkedList<>(); // decreasing queue
for (int i = 0; i < nums.length; i++) {    // left to right
while (!queue.isEmpty() && nums[queue.peekLast()] <= nums[i]) {
rightLarger[queue.pollLast()] = nums[i];
}
if (!queue.isEmpty()) {
leftLarger[i] = nums[queue.peekLast()];
}
queue.offerLast(i);
}

return new int[][]{leftLarger, rightLarger};
}