Binary Search Variations Application

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My first Codeforces blog post! Hooray!

This post was inspired by the problem: E. Iva & Pav of Codeforces round 900 (https://codeforces.me/contest/1878/problem/E).

The Problem

Unfortunately being the newbie that I am, I tried applying basic binary search, lower bound and upper bound but didn't succeed. Here we are required to find the last element which satisfies the given condition thus requiring a slightly different binary search algorithm. This sparked my urge to dive deep into the bowel of Binary Search. So After years (about 0.01 years) of painstaking research I finally compiled a list of five algorithms that will satisfy most problems regarding binary search. It also lead to better my understanding of the binary search algorithm.

The Algorithms

The Basic Binary search algorithm only applies for arrays with distinct elements. If the array has multiple instances of one element we need to apply a different algorithm. For example you could need any of the following elements for your specific problem.

Ex- [1, 2, 3, 5, 6, 8, 9, 10] key = 5 Ex- [1, 1, 1, 2, 2, 3, 4, 4, 4, 7, 7, 9] key = 4

These are the five algorithms for ascending order respectively in the order of the index of required element in array:

Binary Search for strictly increasing Array (Basic Binary Search)

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] == key)
    {
        result = mid;
        break;
    }
    else if(arr[mid] < key)
    {
        l = mid+1;
    }
    else
    {
        r = mid-1;
    }
}

Binary Search for Last element < Key

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] < key)
    {
        result = mid;
        l = mid+1;
    }
    else
    {
        r = mid-1;
    }
}

Binary Search Lower Bound (First Element >= Key)

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] >= key)
    {
        result = mid;
        r = mid-1;
    }
    else
    {
        l = mid+1;
    }
}

Binary Search for Last element <= Key

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] <= key)
    {
        result = mid;
        l = mid+1;
    }
    else
    {
        r = mid-1;
    }
}

Binary Search Upper Bound (First Element > Key)

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] > key)
    {
        result = mid;
        r = mid-1;
    }
    else
    {
        l = mid+1;
    }
}

Their corresponding decreasing array counterparts:

Binary Search for strictly decreasing Array (Basic Binary Search)

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] == key)
    {
        result = mid;
        break;
    }
    else if(arr[mid] > key)
    {
        l = mid+1;
    }
    else
    {
        r = mid-1;
    }
}

Binary Search Last Element > Key

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] > key)
    {
        result = mid;
        l = mid+1;
    }
    else
    {
        r = mid-1;
    }
}

Binary Search Lower Bound (First Element <= Key)

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] <= key)
    {
        result = mid;
        r = mid-1;
    }
    else
    {
        l = mid+1;
    }
}

Binary Search Last Element >= Key

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] >= key)
    {
        result = mid;
        l = mid+1;
    }
    else
    {
        r = mid-1;
    }
}

Binary Search Upper Bound (First Element < Key)

int l = 0;
int r = n-1;
int result = -1;
int mid;

while(l <= r)
{
    mid = l + (r-l)/2;

    if(arr[mid] < key)
    {
        result = mid;
        r = mid-1;
    }
    else
    {
        l = mid+1;
    }
}

Important Takeaways

The main point we can gleam from these algorithms is that when you get an element that satisfies the condition, you immediately set result = mid. Although quite obvious some algorithms I read don't set a different variable for result and only take the value of mid after the loop ends as the answer. This can be quite problematic sometimes. Another important point to note is that the pointer we move (l or r) depends on whether we want the first element or the last element, not whether the array is ascending or descending. Also, I

Rev.	By	When	Δ	Comment
en15	Supersidd	2024-07-11 16:18:19	0	(published)
en14	Supersidd	2024-07-11 15:58:06	2	Tiny change: ' key = 5\nEx- [1, ' -> ' key = 5\n\nEx- [1, '
en13	Supersidd	2024-07-11 15:56:57	14
en12	Supersidd	2024-07-11 15:56:03	178
en11	Supersidd	2024-07-11 15:49:47	636
en10	Supersidd	2024-07-11 15:36:46	1125
en9	Supersidd	2024-07-11 15:17:29	16
en8	Supersidd	2024-07-11 15:15:02	1858	Tiny change: 'oiler>\n\n ' -> 'oiler>\n\n<spoiler summary="">\n```c++\n\n```\n<\spoiler>'
en7	Supersidd	2024-07-11 14:36:13	0	Tiny change: '^ ^ ^ ^\n\nThe' -> '^ ^ ^\n\nThe'
en6	Supersidd	2024-07-11 14:34:45	37	Tiny change: ' ^ ^\n\n \| \| \| \|\n\nThese ' -> ' ^ ^\n\nThese '
en5	Supersidd	2024-07-11 14:34:01	4
en4	Supersidd	2024-07-11 14:33:43	686
en3	Supersidd	2024-07-11 12:57:50	1027
en2	Supersidd	2024-07-11 12:47:17	331
en1	Supersidd	2024-07-11 12:45:27	2257	Initial revision (saved to drafts)

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