<p>Winter is coming! During the contest, your first job is to design a standard heater with a fixed warm radius to warm all the houses.</p>

<p>Every house can be warmed, as long as the house is within the heater&#39;s warm radius range.&nbsp;</p>

<p>Given the positions of <code>houses</code> and <code>heaters</code> on a horizontal line, return <em>the minimum radius standard of heaters&nbsp;so that those heaters could cover all houses.</em></p>

<p><strong>Notice</strong> that&nbsp;all the <code>heaters</code> follow your radius standard, and the warm radius will the same.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> houses = [1,2,3], heaters = [2]
<strong>Output:</strong> 1
<strong>Explanation:</strong> The only heater was placed in the position 2, and if we use the radius 1 standard, then all the houses can be warmed.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> houses = [1,2,3,4], heaters = [1,4]
<strong>Output:</strong> 1
<strong>Explanation:</strong> The two heaters were placed at positions 1 and 4. We need to use a radius 1 standard, then all the houses can be warmed.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> houses = [1,5], heaters = [2]
<strong>Output:</strong> 3
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= houses.length, heaters.length &lt;= 3 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= houses[i], heaters[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution {
public:
    bool housesCovered(vector<int> houses, vector<int>& heaters,
                       int radius) {
        int left = heaters[0] - radius;
        int right = heaters[0] + radius;
        int heater_idx = 0, house_idx = 0;
        
        while(house_idx < houses.size()) {
            if(houses[house_idx] >= left && houses[house_idx] <= right) {
                ++house_idx;
            }
            else {
                ++heater_idx;
                if(heater_idx == heaters.size())
                    return false;
                left = heaters[heater_idx] - radius;
                right = heaters[heater_idx] + radius;
            }
        }
        return true;
    }
    
    int findRadius(vector<int>& houses, vector<int>& heaters) {
        if(houses.empty())
            return true;
        if(heaters.empty())
            return false;
        
        sort(houses.begin(), houses.end());
        sort(heaters.begin(), heaters.end());
        
        int low = 0, high = abs(max(houses.back(), heaters.back()) - houses.front());
        
        while(low < high) {
            int radius = low + (high - low) / 2;
            if(housesCovered(houses, heaters, radius))
                high = radius;
            else
                low = radius + 1;
        }
        return high;
    }
};
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