### Problem: Write a function which returns the square root of an integer. Return an integer which is the floor of the integer closest to the value. Implement an algorithm with efficiency better than O(sqrt(n))

int sqrtFlr(int x) {...}

In the last post, we implemented a solution to this problem, which was O(log(n)).

The solution was:

1. Create a ceiling and floor, where the floor is 0 and the ceiling is the number.

2. Try the square of halfway between the ceiling and floor.

3. Adjust the ceiling or floor based up on the result.

4. Continue 2 and 3 and return the floor when the ceiling is one more than the floor.

Now, let's implement this is code:

int sqrtFlr(int x) {

int floor = 0;

int ceiling = x;

int halfway;

while (floor != (ceiling - 1)) {

halfway = floor + (ceiling - floor) / 2);

if ((halfway * halfway) > x) { // halfway is the ceiling!

ceiling = halfway;

} else if ((halfway * halfway < x)) {//halfway is the floor!

floor = halfway;

else { //we have a perfect square

return halfway;

}

return floor;

}

Let's try an example now, say 284.

When we start:

floor = 0

celing = 284.

we enter the while loop.

Now, halfway is 142.

142^2 is much bigger than 284, so we get:

floor = 0;

ceiling = 142

continuing through the while loop:

floor = 0;

ceiling = 142;

halfway = 71;

71^2 is still much bigger than 284, so we get:

floor = 0;

ceiling = 71;

staying in the while loop, we continue:

floor = 0;

ceiling = 71;

halfway = 35; (35^2 is much bigger still)

Now, let's implement this is code:

int sqrtFlr(int x) {

int floor = 0;

int ceiling = x;

int halfway;

while (floor != (ceiling - 1)) {

halfway = floor + (ceiling - floor) / 2);

if ((halfway * halfway) > x) { // halfway is the ceiling!

ceiling = halfway;

} else if ((halfway * halfway < x)) {//halfway is the floor!

floor = halfway;

else { //we have a perfect square

return halfway;

}

return floor;

}

Let's try an example now, say 284.

When we start:

floor = 0

celing = 284.

we enter the while loop.

Now, halfway is 142.

142^2 is much bigger than 284, so we get:

floor = 0;

ceiling = 142

continuing through the while loop:

floor = 0;

ceiling = 142;

halfway = 71;

71^2 is still much bigger than 284, so we get:

floor = 0;

ceiling = 71;

staying in the while loop, we continue:

floor = 0;

ceiling = 71;

halfway = 35; (35^2 is much bigger still)

staying in the while loop, we continue:

floor = 0;

ceiling = 35;

halfway = 17; (17^2 is much bigger still)

staying in the while loop, we continue:

floor = 0;

ceiling = 17;

halfway = 8; (8^2 = 64, which is smaller)

staying in the while loop, we continue:

floor = 8;

ceiling = 17;

halfway = 12; (12^2 = 144, which is smaller)

staying in the while loop, we continue:

floor = 15;

ceiling = 17;

halfway = 16; (16^2 = 256, which is smaller) and we set floor to 16.

Now, when we go to the while loop, it fails and we fall out, and return 16.

16 is the correct answer here.

So, this solution seems to work.

Let's think of edge cases here. Try a perfect square, say 16.

floor = 0;

ceiling = 16;

halfway = 8; (8^2 = 64, which is larger).

Continuing through the for loop:

floor = 0;

ceiling = 8;

halfway = 4; (4^2 = 16, which is equal, trigger the else in the if statement, and returns true, so this works).

And there is is. An O(log n) solution to returning square root floor!

floor = 0;

ceiling = 35;

halfway = 17; (17^2 is much bigger still)

staying in the while loop, we continue:

floor = 0;

ceiling = 17;

halfway = 8; (8^2 = 64, which is smaller)

staying in the while loop, we continue:

floor = 8;

ceiling = 17;

halfway = 12; (12^2 = 144, which is smaller)

staying in the while loop, we continue:

floor = 12;

ceiling = 17;

halfway = 14; (14^2 = 196, which is smaller)

floor = 12;

ceiling = 17;

halfway = 14; (14^2 = 196, which is smaller)

staying in the while loop, we continue:

floor = 14;

ceiling = 17;

halfway = 15; (15^2 = 225, which is smaller)

floor = 14;

ceiling = 17;

halfway = 15; (15^2 = 225, which is smaller)

floor = 15;

ceiling = 17;

halfway = 16; (16^2 = 256, which is smaller) and we set floor to 16.

Now, when we go to the while loop, it fails and we fall out, and return 16.

16 is the correct answer here.

So, this solution seems to work.

Let's think of edge cases here. Try a perfect square, say 16.

floor = 0;

ceiling = 16;

halfway = 8; (8^2 = 64, which is larger).

Continuing through the for loop:

floor = 0;

ceiling = 8;

halfway = 4; (4^2 = 16, which is equal, trigger the else in the if statement, and returns true, so this works).

Ok-- any edge cases. Well, we could try 0 and 1, and this works. How about negative numbers. Well, let's return -1 in this case, and then we're done.

int sqrtFlr(int x) {

int floor = 0;

int ceiling = x;

int halfway;

if (x < 0) return -1; //error case.

while (floor != (ceiling - 1)) {

halfway = floor + (ceiling - floor) / 2);

if ((halfway * halfway) > x) { // halfway is the ceiling!

ceiling = halfway;

} else if ((halfway * halfway < x)) {//halfway is the floor!

floor = halfway;

else { //we have a perfect square

return halfway;

}

return floor;

}

int floor = 0;

int ceiling = x;

int halfway;

if (x < 0) return -1; //error case.

while (floor != (ceiling - 1)) {

halfway = floor + (ceiling - floor) / 2);

if ((halfway * halfway) > x) { // halfway is the ceiling!

ceiling = halfway;

} else if ((halfway * halfway < x)) {//halfway is the floor!

floor = halfway;

else { //we have a perfect square

return halfway;

}

return floor;

}

And there is is. An O(log n) solution to returning square root floor!

/* Please send along any bugs or optimizations! */