未知值的逐次逼近

Question

I have a url with a page system.

For instance https://myURL?p=50

But I want a script to find the last page available, for instance, let's say p=187

I have a function checkEmpty() that tells me whether the page is empty or not. So for instance:

$myUrl = new URL(50); //https://myURL?p=50
$myUrl->checkEmpty();
//This evaluates to false -> the page exists

$myUrl = new URL(188); //https://myURL?p=188
$myUrl->checkEmpty();
//This evaluates to true -> the page does NOT exist

$myUrl = new URL(187); //https://myURL?p=187
$myUrl->checkEmpty();
//This evaluates to false -> the page exists

I did a naive algorithm, that you might guess it, performs too much requests.

My question is: What would be the algorithm to find the last page with the minimal amount of requests?

EDIT As requested by people in the comment here is the checkEmpty() implementation

<?php
public function checkEmpty() : bool
{
    $criteria = "Aucun contenu disponible";
    if(strstr( $this->replace_carriage_return(" ", $this->getHtml()), $criteria) !== false)
    {
        return true;
    }
    else
    {
        return false;
    }
}

Answer 1

Since the upper bound is not known, exponentially increase the page no by 2 starting from 1. The moment you hit a non-existent page, you can do a binary search from previous existing page + 1 till this new upper bound where the page doesn't exist.

This way, you can get your answer in O(log(n)) attempts asymptotically where n is the no. of existing pages here as the sample space.

<?php

$lowerBound = 1;
$upperBound = 1;

while(true){
    $myUrl = new URL($upperBound);
    if($myUrl->checkEmpty()){
        break;
    }
    $lowerBound = $upperBound + 1;
    $upperBound <<= 1;
}

$ans = $lowerBound;

while($lowerBound <= $upperBound){
    $mid = $lowerBound + (($upperBound - $lowerBound) >> 1);
    $myUrl = new URL($mid);
    if($myUrl->checkEmpty()){
        $upperBound = $mid - 1;
    }else{
        $lowerBound = $mid + 1;
        $ans = $lowerBound;
    }
}

echo $ans;