Euclid's Algorithm in Java:

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public static int gcd(int a, int b) {
    if (b == 0) {
        return a;
    } else {
        return gcd(b, a % b);
    }
}

Sieve of Eratosthenes in Java:

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public static boolean[] sieveOfEratosthenes(int n) {
    boolean[] isPrime = new boolean[n+1];
    Arrays.fill(isPrime, true);
    isPrime[0] = false;
    isPrime[1] = false;
    for (int i = 2; i * i <= n; i++) {
        if (isPrime[i]) {
            for (int j = i * i; j <= n; j += i) {
                isPrime[j] = false;
            }
        }
    }
    return isPrime;
}

Note: The gcd function calculates the GCD of two integers a and b using Euclid's method. The sieveOfEratosthenes function returns a boolean array where isPrime[i] is true if i is prime and false otherwise.

Big O notation is used in computer science to describe the complexity of an algorithm. It tells us how the time and space requirements of an algorithm grow with the input size.

The Big O notation is represented by the letter O followed by parentheses containing an expression that describes the upper bound of the algorithm. Here are some common examples:

• O(1) describes an algorithm that takes constant time to run, regardless of the size of the input. For example, accessing an array element by index is a constant time operation:

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  const arr = [1, 2, 3];
  const element = arr[0]; // takes constant time O(1)
  

• O(n) describes an algorithm that takes time proportional to the size of the input. For example, iterating through an array is a linear time operation:

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  const arr = [1, 2, 3];
  for(let i = 0; i < arr.length; i++) { // takes linear time O(n)
  console.log(arr[i]);
  }
  

• O(n²⁾ describes an algorithm that takes time proportional to the square of the input size. For example, a nested loop over an array is a quadratic time operation:

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  const arr = [1, 2, 3];
  for(let i = 0; i < arr.length; i++) {
  for(let j = 0; j < arr.length; j++){
  console.log(arr[i] + arr[j]); // takes quadratic time O(n^2)
  }
  }
  

These are just a few examples, and there are many different variations of Big O notation that can describe different types of algorithms.

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public int fibo(int number) {
    if (number <= 1) {  // Base cases for fibonacci series
        return 1;
    }

    int prev1 = 1, prev2 = 1, fibo = 1;

    for (int i = 2; i <= number; i++) {  // Loop starts from i = 2
        fibo = prev1 + prev2;  // Calculation of next fibonacci number

        prev2 = prev1;
        prev1 = fibo;
    }

    return fibo;
}

The time complexity of this algorithm is O(n) where n is the value of the parameter passed to the function. This is the optimal time complexity for finding any fibonacci number since we have to go through all the numbers from 0 to n.

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def partition(head, end, new_head, new_end):
    pivot = end
    prev = None
    cur = head
    tail = pivot

    while cur != pivot:
        if cur.data < pivot.data:
            if new_head is None:
                new_head = cur

            prev = cur
            cur = cur.next
        else:
            if prev:
                prev.next = cur.next
            temp = cur.next
            cur.next = None
            tail.next = cur
            tail = cur
            cur = temp

    if new_head is None:
        new_head = pivot

    new_end = tail
    return pivot


def quicksort(head, end=None):
    if not head or head == end:
        return head

    new_head = None
    new_end = None

    pivot = partition(head, end, new_head, new_end)

    if new_head != pivot:
        temp = new_head
        while temp.next != pivot:
            temp = temp.next
        temp.next = None
        new_head = quicksort(new_head, temp)
        temp = get_tail(new_head)
        temp.next = pivot

    pivot.next = quicksort(pivot.next, new_end)

    return new_head


def get_tail(head):
    while head and head.next:
        head = head.next
    return head

Note: The code mentioned is in Python.

Counting sort is a sorting algorithm that works by counting the number of occurrences of each element in the input array and then using arithmetic to calculate the position of each element in the sorted output array. Its basic logic is to determine the number of elements smaller than each element in the array by counting the number of elements between 0 and the maximum value in the array. This information is then used to place each element in the correct position in the sorted order. The advantage of counting sort is that it can sort elements in linear time, making it faster than comparison-based sorting algorithms such as merge sort and quicksort in certain situations. The main disadvantage of counting sort is that it requires extra memory to store the count array.

Counting Sort is a sorting algorithm that operates on an array or list of integers. It works by counting the number of occurrences of each unique element in the list, and then using the counts to reposition the elements in order.

Here's an example of how Counting Sort would work for an array of integers:

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const countingSort = arr => {
  const max = Math.max(...arr);
  const counts = new Array(max + 1).fill(0);
  const sortedArr = [];

  arr.forEach(num => {
    counts[num]++;
  });

  counts.forEach((count, i) => {
    for (let j = 0; j < count; j++) {
      sortedArr.push(i);
    }
  });

  return sortedArr;
};

console.log(countingSort([3, 2, 1, 4])) // Output: [1, 2, 3, 4]

First, we find the maximum value in the array to determine the size of the counting array. Then, we fill the counting array with zeros and iterate through the input array, incrementing the corresponding count for each number. We then iterate through the counting array, adding each number in the right count to the sorted array. Finally, we return the sorted array.

Counting Sort has a time complexity of O(n+k), where n is the number of elements in the input array and k is the range of input. However, it has a space complexity of O(k), which can be prohibitively large for some inputs.

Sure! Here is a simple implementation of Counting Sort Algorithm in JavaScript:

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function countSort(arr, maxVal) {
   var i, z = 0, count = [];

   // Create a counter array to store count of individual elements
   for (i = 0; i <= maxVal; i++) {
       count[i] = 0;
   }

   // Store count of each element
   for (i=0; i < arr.length; i++) {
       count[arr[i]]++;
   }

   // Modify array and store sorted elements
   for (i = 0; i <= maxVal; i++) {
       while (count[i]-- > 0) {
           arr[z++] = i;
       }
   }

   return arr;
}

// Example Usage
var arr = [1, 4, 1, 2, 7, 5, 2];
var sortedArr = countSort(arr, 7); // maxVal = 7 since the maximum value in the array is 7

console.log(sortedArr); // [1, 1, 2, 2, 4, 5, 7]

In the above implementation, arr represents the array to be sorted and maxVal is the maximum value in the array. The countSort function returns the sorted array.

The Counting Sort algorithm works by counting the number of occurrences of each item and using arithmetic to determine the position of each item in the output sequence.

Bubble sort is a sorting algorithm that repeatedly compares adjacent elements in a collection of items and swaps them if they are in the wrong order. It is called "bubble sort" because the smaller elements rise to the top of the collection like bubbles.

In terms of runtime, bubble sort has a worst-case and average-case time complexity of O(n^2), where n is the number of elements in the collection. This means that as the size of the collection increases, the time taken to sort it also increases exponentially.

Here's an example implementation of bubble sort in Python:

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def bubble_sort(arr):
    n = len(arr)

    # Traverse through all elements in array
    for i in range(n):

        # Last i elements are already sorted
        for j in range(0, n-i-1):

            # Swap if the element found is greater than the next element
            if arr[j] > arr[j+1]:
                arr[j], arr[j+1] = arr[j+1], arr[j]

    return arr

In this implementation, we have an outer loop that iterates over all the elements in the array, and an inner loop that iterates over the unsorted elements. Within the inner loop, we compare adjacent elements and swap them if they're in the wrong order. We repeat this process until the array is sorted.

However, as mentioned earlier, bubble sort has a time complexity of O(n^2), which makes it inefficient for larger data sets. There are other sorting algorithms such as merge sort or quicksort with a lower time complexity that are more suitable for large data sets.

This can be done in LMC (Little Man Computer) by following a few steps:

1. Create a loop that continuously runs until all the elements in the data array are sorted.
   
2. Inside the loop, create an inner loop that iterates through the data array and checks the current element with the next one.
3. If the element before is greater than the next element, swap the two elements in the array.
4. When the inner loop is complete, go back to step 1.

Here's an example in LMC code (pseudocode):

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WHILE (Elements are unsorted) 
    LOOP 
        FOR i=0 to count_elements 
        DO 
            IF (data[i] > data[i+1]) 
            THEN 
                Swap data[i], data[i+1] 
            END IF 
        END DO 
    END LOOP 
END WHILE

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def selection_sort(ls):
  for i in range(len(ls)):
      min_index = i
      for j in range(i+1, len(ls)):
          if ls[j] < ls[min_index]:
            min_index = j
      ls[i], ls[min_index] = ls[min_index], ls[i]

# Driver code
ls = [14, 21, 3, 62, 9, 39]
selection_sort(ls)
print(ls)

This program starts a function called selection_sort with a single parameter, ls. Inside the function the outer loop for i in range(len(ls)) loops through from 0 to the length of the list. The inner loop for j in range(i+1, len(ls)) will check for another index which is higher than the outer index i. The program compares every element to every other element and if the condition if ls[j] < ls[min_index]: is met, the program will choose the new index as the min_index and the values in the list will be swapped using ls[i], ls[min_index] = ls[min_index], ls[i].

The driver code creates a list of integers, then calls the selection_sort function with the list as argument. Finally the result is printed to the console.

When completed, the program will output [3, 9, 14, 21, 39, 62] to the console.

There are five basic steps to coding that everyone should know:

1. Understand the problem: Evaluate the problem, determine the inputs and outcomes, and clearly define the scope.
2. Design an algorithm: Break the problem into smaller tasks, plan the structure of the code, and create a step-by-step plan to solve the problem.
3. Implement the algorithm: Write the code out in a programming language.
4. Test your code: Apply different inputs to the code and make sure it produces the correct outputs.
5. Debug: Find and remove any errors in the code.

For more detailed tutorials on how to code, please refer to code-specific tutorials for your language of choice, such as JavaScript tutorials.

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import cv2

# Load the image in greyscale
img = cv2.imread('image.jpg', cv2.IMREAD_GRAYSCALE)

# Blur the image using a Gaussian filter 
img_blur = cv2.GaussianBlur(img, (7, 7), 3)

# Use Otsu segmentation to get a binary mask
ret, mask = cv2.threshold(img_blur, 0, 255, cv2.THRESH_BINARY_INV+cv2.THRESH_OTSU)

# Apply mask to the original image
final_img = cv2.bitwise_and(img, img, mask=mask)

Yes, it is possible to write an algorithm with O(log n) runtime complexity to solve this problem. This can be accomplished using a Binary Search algorithm. The pseudocode to implement a binary search is as follows:

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    // With a sorted array and a target value
    function binarySearch(values, target){ 

        // Set left and right boundaries 
        let left = 0; 
        let right = values.length - 1; 

        // Iterate until left does not 
        // cross right 
        while(left <= right) {

            // Find middle index
            let mid = Math.floor((left + right) / 2); 

            // Check if target is equal to middle element 
            if(target === values[mid]){
                return mid; 
            }

            // Check if target is lower than mid element 
            if(target < values[mid]){
                right = mid - 1; 
            }

            // Check if target is greater than mid element 
            if(target > values[mid]){
                left = mid + 1; 
            } 
        } 
        // Return position to insert: 
            return left; 
    } 

Maintaining a sorted array allows the algorithm to skip elements and search faster, making it possible to achieve O(log n) runtime complexity. Learn more about the binary search algorithm here: https://www.geeksforgeeks.org/binary-search/

A Python Minesweeper Solver is a program that solves a Minesweeper game automatically. Minesweeper is a logic puzzle game where the goal is to reveal all locations on a board that do not contain any mines.

A Python Minesweeper solver can be implemented by using recursive backtracking which is a type of brute-forcing search algorithm. The algorithm will recursively search through all possible solutions, and if it runs into a solution that works, it will keep going until all the mines have been revealed.

The following code is a sample Python Minesweeper Solver that finds all possible solutions to a given Minesweeper board:

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def mine_sweeper_solver(row, col, board):
    if board[row][col] == '*':
        board[row][col] = 'B'
        return True

    if board[row][col] != '.':
        return False

    board[row][col] = 'F'
    for (i, j) in valid_moves(row, col, board):
        if mine_sweeper_solver(i, j, board):
            board[i][j] = '*'           
    return False

def valid_moves(row, col, board):
    moves = []

    for m in [(row - 1, col), (row + 1, col), 
              (row, col - 1), (row, col + 1)]:
        if 0 <= m[0] < len(board) and 0 <= m[1] < len(board[0]):
        moves.append(m)

    return moves

Binary search is an algorithm for finding an element within a sorted array. It works by dividing the array into two halves, checking if the element you are searching for is equal to, or greater than the middle element, and then repeating the process with the appropriate subarray.

In the worst case, the binary search algorithm requires O(log n) comparisons to find an element. Most programming languages provide efficient implementations of binary search in the standard library.

For example, in Python, the bisect.bisect() function provides a binary search of a sorted list.

Below is an example of binary search in Python:

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def binary_search(arr, x): 
    n = len(arr)
    low = 0
    high = n - 1

    while low < high:  
        mid = (low + high) // 2
        if arr[mid] == x: 
            return mid 
        elif arr[mid] < x: 
            low = mid + 1
        else: 
            high = mid - 1
    return -1 #element not found