# Introduction to loop

To solve a problem, sometimes it is necessary to repeat a particular code statement several times (possibly doing some key operations at each repetition) until a specific condition is satisfied. It is known as iteration, which allows us to “write code once” and “execute many times.”

The idea of a loop helps us provide the flexibility of code reusability to simplifies complex problem solving, i.e., instead of writing the same code, again and again, we can execute the same code a finite number of times. Think! There are two types of loops mostly used in programming:

## for loop

We use “for” loop when we know how many times the loop will execute.

`for (loop initialisation; loop condition; loop update){    loop body}`

## while loop

It is used when we want to continue until a specific condition is false. We apply this loop idea when we don’t know how many times the loop will execute.

`loop initialisationwhile (loop condition){     loop body    loop update}`

# How to design a correct loop?

We should consider these critical steps during the design and ensure the correctness of the loop.

Let’s try to understand the above idea via examples.

## Example 1: Finding the sum of all integers from 1to n

Pre-condition: we need to define two variables: a variable i that acts as a loop counter and a variable sum to store the sum of all integers. We would like to do a sum from 0 to n, so at the start, we initialize both variables to 0, i.e., sum = 0, i = 1

Post-condition: After the loop termination, the value of the sum must be equals to the sum of all integers from 1 to n.

Loop variant: The loop should terminate when we have added all integers from 1 to n ,i.e, i<= n. In other words, the loop should not terminate until we have added n to the sum.

Loop invariant: Now, we need to set the loop invariant so that when the loop terminates, we will have the correct output. As discussed above, the loop invariant must be true before and after each loop iteration.

Solution Pseudo Code

`int findSum(int n){    int i = 1    int sum = 0    while (i <= n)     {        sum = sum + i        i = i + 1    }    return sum}`

## Example 2: Finding the maxi element in an array of integers

Pre-condition: we need to define two variables: a loop variable i that acts as a loop counter and a variable max to store the maximum of all integers. Before starting the loop to find the max value from X to X[n-1], we initialize max = X and start the loop with i = 1. This pre-condition holds true when we enter the first iteration of the loop.

Post-condition: After the loop termination, the max value must store the max of all values from X to X[n-1].

Loop variant: The loop must terminate after finding the max of all integers from X to X[n-1]. In other words, the loop should not terminate until we have compared X[n-1] to the max i.e. i <= n-1 or i < n.

Loop invariant

Let’s assume invariant is true after (i-1)th iteration, i.e., max store the maximum of all integers from X to X[i-1]. We need to design instruction so that the invariant must be true after ith iterations of the loop — the variable max must be equal to the max of all integers from X to X[i]. Here are the steps of the ith iteration:

Solution Pseudo Code

`int findMax(int X[], int n){    int max = X    for (int i = 1; i < n; i = i + 1)    {        if (X[i] > max)            max = X[i]    }      return max}`

# Common errors in writing loops

## Infinite loops

An infinite loop occurs when a loop condition continuously evaluates to true or not making progress towards loop termination. It appears most of the time due to the incorrect update of the loop variables or some error in the loop condition. Usually, this is an error, but it’s common for infinite loops to occur accidentally.

For example, we want to display the sum of all numbers from 5 to 10 via the following code and ended up with an infinite loop because we did not increment the loop variable. The loop variable remains the same through each iteration, and progress is not made towards loop termination.

`int i = 5int sum = 0while (i <= 10)   sum = sum + i`

Here is the correct version of the code:

`int i = 5int sum = 0while (i <= 10) {   sum = sum + i   i = i + 1}`

## Other examples of the Infinite loop

`//Example 1//For i < 0, this goes into an infinite loop!void infinteLoop(int i){    while (i != 0)    {        i = i - 1     }}//Example 2//loop condition is "1" which is always true.int i = 0  while(1)  {      i =  i + 1       print(i)}`

But in some situations, an infinite loop can be used on purpose. For example, we use an infinite loop for the applications that continuously accept the user input and constantly generate the output until the user comes out of the application manually.

`read the first element while (element is valid ) {     process the element     read the following element }`
`while (true){   // Read request   // Process request}Another popular way is:for ( ; ; ){   // Read request   // Process request}`

## Off-By-One Error

It is an error involving the boundary condition of the loop: the loop variable’s initial value or the loop’s end condition. This problem could arise when a programmer makes mistakes such as:

Off-By-One Example 1

`int Xfor (int i = 0; i <= 5; i = i + 1)    print(X[i])`

The above program will result in an array out of bounds exception because we will try to display the result for X and the upper bound of the array is only 4. This is because the index for the array starts at 0 instead of 1 in most programming languages. The correct code is displayed below.

`int Xfor (int i = 0; i < 5; i = i + 1)    print(X[i])`

Off-By-One Example 2

In the first case, the loop will be executed (n-1) times and in the second case (n + 1) times, giving error off-by-one. The loop can be written correctly as: for (int i = 0; i < n; i = i + 1) { … }

# Examples of various loops Patterns

## Increasing loop by 1

Finding the nth fibonacci: Bottom up approach of DP

`for(int i = 2; i <= n; i = i + 1)    Fib[i] = Fib[i - 1] + Fib[i - 2]`

Kadane algorithm: finding maximum subarray sum

`for (i = 1; i < n; i = i + 1){    curr_maxSum = max (curr_maxSum + X[i], X[i])    if(maxSum < curr_maxSum)        maxSum = curr_maxSum}`

Boyer–Moore Voting Algorithm: finding majority element in an array

`for(int i = 0; i < n; i = i + 1) {    if(count == 0)         majorityCandidate = X[i]    if(X[i] == majorityCandidate)         count = count + 1    else         count = count - 1}`

## Nested loops

Bubble sort algorithm

`for(int i = 0; i < n; i = i + 1){       for(int j = 0; j < n - i - 1; j = j + 1)    {        if( X[j] > X[j+1])        {            temp = A[j]            X[j] = X[j+1]            X[j+1] = temp        }     }}`

Insertion sort algorithm

`for(int i = 1; i < n - 1; i = i + 1) {      int key = X[i]    int j = i - 1      while (j >= 0 && X[j] > key)     {          X[j + 1] = X[j]        j = j - 1     }    X[j + 1] = key}`

Transpose of a square matrix

`for (int i = 0; i < n; i = i + 1)    for (int j = i + 1; j < n; j + 1)        swap(X[i][j], X[j][i])`

## Increasing loop by 2

Finding max and min in an array

`while (i < n){    if (X[i] < X[i+1])    {        if (X[i] < min)            min = X[i]        if (X[i+1] > max)            max = X[i+1]    }    else    {        if (X[i] > max)            max = X[i]        if (X[i+1] < min)            min = X[i+1]     }    i = i + 2}`

Sorting an array in a wave form

`for (int i = 0; i < n; i = i + 2){    if (i > 0 && X[i-1] > X[i])        swap(X[i], X[i-1])     if (i < n-1 && X[i] < X[i+1])        swap(X[i], X[i + 1])}`

## Increasing loop by power of 2

Exponential search in an unbounded array

`while (i < n && X[i] <= key)    i = i*2`

## Loop decreasing by factor of 2

Iterative Binary Search

`while (left <= right) {    int mid = left + (right - left)/2    if (X[mid] == key)        return mid           if (X[mid] < key)        left = mid + 1          else        right = mid - 1}`

Searching iteratively in a BST

`while (root != NULL) {    if (key < root->data)        root = root->left    else if (key > root->data)        root = root->right    else        return true}`

## Two pointers moving in the opposite direction

Reverse an array

`while (left < right){    int temp = X[left]    X[left] = X[right]    X[right] = temp    left = left + 1    right = right - 1}`

Finding pair sum in a sorted array

`while (l < r){    if(X[l] + X[r] == k)        return 1    else if(X[l] + X[r] < k)        l = l + 1    else        r = r - 1}`

## Two pointers moving in the same direction

Merging algorithm in merge sort

`while (i < n1 && j < n2) {    if (X[i] <= Y[j])     {        A[k] = X[i]        i = i + 1    }    else     {        A[k] = Y[j]        j = j + 1    }    k = k + 1}`

Partition algorithm in quicksort

`for (int j = left; j < right; j = j + 1){    if (x[j] < pivot)    {        i = i + 1        swap(X[i], X[j])    }}`

Floyd algorithm: finding loop in a linked list

`while (slow && fast && fast->next) {    slow = slow->next    fast = fast->next->next    if (slow == fast)        return 1}`

## Loop on a data structure

BFS Traversal of binary tree using queue

`while (Q.empty() == false){    TreeNode temp = Q.dequeue()    print(temp->data)            if (temp->left != NULL)        Q.enqueue(temp->left)     if (temp->right != NULL)        Q.enqueue(temp->right)}`

Pre-order traversal in binary search using Stack

`while (S.empty() == false){    Treenode temp = S.pop()    printf(temp->data)    if (temp->right)        S.push(temp->right)    if (temp->left)        S.push(node->left)}`

# Application of loop in Problem Solving

## Critical Ideas to think in the loop!

`int w// we can use x or w inside ()for(int x = 0; x < 5; x = x + 1) {    int y    // we can use x, y, w here}int z // we can only use w, z here; x and y are only visible in the scope of the loop.`

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