<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 {

    bool canBeHeatedWithRadiusR(vector<int> &houses, vector<int> &heaters, int radius) {
        //cout << "radius = " << radius; 
        vector<int> heated(houses.size(), 0);
        for(auto x: heaters) {
            int l = lower_bound(houses.begin(), houses.end(), x-radius) - houses.begin();
            int r = upper_bound(houses.begin(), houses.end(), x+radius) - houses.begin();
            //cout << " l = " << l << " r = " << r << endl;
            if(l< houses.size()) {
                ++heated[l];
            }
            if(r<houses.size()) {
                --heated[r];
            }
        }
        for(int i = 1; i < houses.size(); i++) {
            heated[i] += heated[i-1];
            if(heated[i] < 1) {
                //cout << " not possible " << endl;
                return false;
            }
        }
        //cout << " possible " << endl;
        return heated[0] > 0;
    }

public:
    int findRadius(vector<int>& houses, vector<int>& heaters) {
        sort(houses.begin(), houses.end());
        sort(heaters.begin(), heaters.end());
        int mxhouse = *max_element(houses.begin(), houses.end());
        int minhouse = *min_element(houses.begin(), houses.end());
        int mxHeat = *max_element(heaters.begin(), heaters.end());
        int minHeat = *min_element(heaters.begin(), heaters.end());
        int l = 0, r = max(abs(mxhouse-minHeat), abs(minhouse - mxHeat));
        while(l<=r) {
            int m = l + (r-l)/2;
            if(canBeHeatedWithRadiusR(houses, heaters, m)) {
                r = m - 1;
            } else {
                l = m + 1;
            }
        }
        return l;
    }
};
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