Searching Algorithms

Tags: #◇CCDSALG #◇Algorithms #◇Computer Science #◇Searching #◇Data Structures #◇Search Efficiency #◇Sorting #◇Fundamental Data Structures #◇Computational Complexity #◇Analysis of Algorithms #◇Time Complexity #◇Space Complexity

◇Overview

2025-06-05 02:14:45

◇Key Concepts

Best case Worst case Average case Best case Worst case

◇Details / Theory

2025-06-05 02:14:45

Searching and Sorting Arrays

◇What is an Array?

A collection of elements stored in contiguous memory. Indexed data structure (0-based index in most languages). Can be accessed in constant time $O(1)$ using index. Fixed size in many languages (e.g., C, Java).

1. Linear Search

Checks each element sequentially until finding the target or reaching the end.

Characteristics:
Works on both sorted and unsorted arrays. Best case: element is at the start $O(1)$ Worst case: element is not found or at end $O(n)$ Average case: $O(n)$

Code:

int linearSearch(int arr[], int n, int key) {
    for (int i = 0; i < n; i++) {
        if (arr[i] == key) return i;
    }
    return -1;
}

2. Binary Search

Requires a sorted array. Repeatedly divides the search interval in half.

Characteristics:
Best case: element is at the middle $O(1)$ Worst case: element is not found $O(\log n)$ Average case: $O(\log n)$

Iterative Implementation:

int binarySearch(int arr[], int l, int r, int x) {
    while (l <= r) {
        int m = l + (r - l) / 2;
        if (arr[m] == x) return m;
        if (arr[m] < x) l = m + 1;
        else r = m - 1;
    }
    return -1;
}

Recursive Implementation:

int binarySearchRecursive(int arr[], int l, int r, int x) {
    if (r >= l) {
        int mid = l + (r - l) / 2;
        if (arr[mid] == x) return mid;
        if (arr[mid] > x) return binarySearchRecursive(arr, l, mid - 1, x);
        return binarySearchRecursive(arr, mid + 1, r, x);
    }
    return -1;
}

◇Sorting Algorithms

1. Bubble Sort

How it works:
Repeatedly steps through the list. Compares adjacent elements and swaps them if they are in the wrong order.

Characteristics:
Best case: $O(n)$ when array is already sorted Worst case: $O(n^2)$ Average case: $O(n^2)$ Space complexity: $O(1)$ Stable: Yes

Code:

void bubbleSort(int arr[], int n) {
    for (int i = 0; i < n-1; i++) {
        for (int j = 0; j < n-i-1; j++) {
            if (arr[j] > arr[j+1]) {
                int temp = arr[j];
                arr[j] = arr[j+1];
                arr[j+1] = temp;
            }
        }
    }
}

Optimized Bubble Sort:

void optimizedBubbleSort(int arr[], int n) {
    int swapped;
    for (int i = 0; i < n-1; i++) {
        swapped = 0;
        for (int j = 0; j < n-i-1; j++) {
            if (arr[j] > arr[j+1]) {
                int temp = arr[j];
                arr[j] = arr[j+1];
                arr[j+1] = temp;
                swapped = 1;
            }
        }
        if (swapped == 0) break;
    }
}

2. Selection Sort

How it works:
Divides the input list into sorted and unsorted portions. Finds the smallest element in the unsorted portion. Swaps it with the first element of the unsorted portion.

Characteristics:
Best case: $O(n^2)$ Worst case: $O(n^2)$ Average case: $O(n^2)$ Space complexity: $O(1)$ Stable: No

Code:

void selectionSort(int arr[], int n) {
    for (int i = 0; i < n-1; i++) {
        int min_idx = i;
        for (int j = i+1; j < n; j++)
            if (arr[j] < arr[min_idx])
                min_idx = j;

        int temp = arr[i];
        arr[i] = arr[min_idx];
        arr[min_idx] = temp;
    }
}

3. Insertion Sort

How it works:
Builds the sorted array one element at a time. Takes one element from the unsorted portion and places it in the correct position in the sorted portion.

Characteristics:
Best case: $O(n)$ when array is already sorted Worst case: $O(n^2)$ Average case: $O(n^2)$ Space complexity: $O(1)$ Stable: Yes

Code:

void insertionSort(int arr[], int n) {
    for (int i = 1; i < n; i++) {
        int key = arr[i];
        int j = i - 1;

        while (j >= 0 && arr[j] > key) {
            arr[j+1] = arr[j];
            j--;
        }
        arr[j+1] = key;
    }
}

4. Merge Sort

How it works:
Divide and conquer algorithm. Divides the array into halves, sorts them, and then merges them.

Characteristics:
Best case: $O(n \log n)$ Worst case: $O(n \log n)$ Average case: $O(n \log n)$ Space complexity: $O(n)$ Stable: Yes

Code:

void merge(int arr[], int l, int m, int r) {
    int i, j, k;
    int n1 = m - l + 1;
    int n2 = r - m;

    int L[n1], R[n2];

    for (i = 0; i < n1; i++)
        L[i] = arr[l + i];
    for (j = 0; j < n2; j++)
        R[j] = arr[m + 1 + j];

    i = 0;
    j = 0;
    k = l;
    while (i < n1 && j < n2) {
        if (L[i] <= R[j]) {
            arr[k] = L[i];
            i++;
        } else {
            arr[k] = R[j];
            j++;
        }
        k++;
    }

    while (i < n1) {
        arr[k] = L[i];
        i++;
        k++;
    }

    while (j < n2) {
        arr[k] = R[j];
        j++;
        k++;
    }
}

void mergeSort(int arr[], int l, int r) {
    if (l < r) {
        int m = l + (r - l) / 2;
        mergeSort(arr, l, m);
        mergeSort(arr, m + 1, r);
        merge(arr, l, m, r);
    }
}

5. Quick Sort

How it works:
Divide and conquer algorithm. Selects a 'pivot' element and partitions the array around the pivot.

Characteristics:
Best case: $O(n \log n)$ Worst case: $O(n^2)$ when poorly pivoted Average case: $O(n \log n)$ Space complexity: $O(\log n)$ Stable: No

Code:

int partition(int arr[], int low, int high) {
    int pivot = arr[high];
    int i = (low - 1);

    for (int j = low; j <= high - 1; j++) {
        if (arr[j] < pivot) {
            i++;
            int temp = arr[i];
            arr[i] = arr[j];
            arr[j] = temp;
        }
    }
    int temp = arr[i + 1];
    arr[i + 1] = arr[high];
    arr[high] = temp;
    return (i + 1);
}

void quickSort(int arr[], int low, int high) {
    if (low < high) {
        int pi = partition(arr, low, high);
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

◇Algorithm Comparison

Time Complexity Comparison

◇Common Pitfalls

Using binary search on an unsorted array — binary search requires the array to be sorted first Forgetting that linear search is $O(n)$ in the worst case but can be $O(1)$ in the best case (element at index 0) Off-by-one errors in binary search bounds — use low <= high (not <) and update mid ± 1 (not just mid) Confusing stability of sorting algorithms — bubble sort, insertion sort, and merge sort are stable; selection sort and quick sort are not Assuming bubble sort is always $O(n^2)$ — with an early-termination flag, best case is $O(n)$ on already-sorted input Forgetting that quick sort has $O(n^2)$ worst case (e.g., already-sorted input with naive pivot), despite $O(n \log n)$ average

◇Summary

TL;DR — Searching: linear search is $O(n)$, binary search is $O(\log n)$ but requires sorted input. Sorting: bubble sort $O(n^2)$, selection sort $O(n^2)$, insertion sort $O(n^2)$ worst / $O(n)$ best, merge sort $O(n \log n)$ guaranteed, quick sort $O(n \log n)$ average / $O(n^2)$ worst. Choose based on stability, space, and expected input.

◇Links

Related: Source / Reference:

◇References & Navigation

• Previous Topic: ◇Reccurences
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• Related Notes: ◇Algorithm, ◇Divide and Conquer