Question

This problem is a kind of closest pair between two disjoint set.
Upperside picture is expressed this problem. there is two kind of disjoint set, blue dots in -x plane, red dots in +x plane.

I want to calculate minimum distance(distance is |y2-y1| + |x2 – x1|) between one blue dot and one red dot, and I think use binary search for finding distance. How to use binary search this kind of problem?
I struggle on only expressing binary search two disjoint sets. I have already know for one set, but I don’t know in case two disjoint sets.

++ ) it is can in linear time using Delaunay triangulation? (ah, it is only my curiosity, I want to use binary search)

below code which I had already coding one set case(using problem-solving technique, divide and qonquer) and coverting to two disjoint set. I don’t understand how to do in two sets.
Example, Hint. okay.. please someone help me?

#include <iostream>
#include <algorithm>
#include <iomanip>
#include <cmath>

/**
test input
10
-16 -4 
-1 -3 
-9 -1 
-4 -10 
-11 -6 
-20 4 
-13 6 
-3 -10 
-19 -1 
-12 -4
10
8 2
10 3 
10 10 
20 -3 
20 3 
16 2
3 -5 
14 -10
8 -2 
14 0

10
-3 39
-2 -28
-1 20
-3 11
-3 45
-2 -44
-1 -47
-5 -35
-5 -19
-5 -45
10
27 5
28 0
28 5
21 5
2 3
13 -1
16 -2
20 -2
33 -3
27 1
 **/


using namespace std;

const int MAX = 10001;

struct point{
    int x,y;
};

bool xCompare(struct point, struct point);
bool yCompare(struct point, struct point);
int dis(struct point, struct point);

int absd(int);
int trace(int,int,int,int);

point p[MAX], q[MAX], tmp[MAX];

int main(){

    int left;
    int right;

    scanf("%d\n", &left);
    memset(p,0,sizeof(p));
    memset(q,0,sizeof(q));
    memset(tmp,0,sizeof(tmp));


    for(int i=0; i<left; i++){
        cin >> p[i].x >> p[i].y;
    }

    scanf("%d\n", &right);

    for(int j=0; j<right; j++){
        cin >> q[j].x >> q[j].y;
    }

    sort(p, p+left, xCompare);
    sort(q, q+right, xCompare);

    int min = trace(0,0, left-1, right-1);

    printf("%d\n", min);


    /** this is one set case.
    while(true){
        cin >> n;

        if(n == 0)  break;

        memset(p,0,sizeof(p));
        memset(tmp,0,sizeof(tmp));

        for(int i= 0;i<n;i++)
            cin >> p[i].x >> p[i].y;

        sort(p,p+n,xCompare);

        int min = trace(0,n-1);

        if(min < 10000 && n > 1){
            cout << fixed;
            cout << setprecision(4) << min << endl;
        }
        else
            cout << "INFINITY" << endl;
    }
     **/

    return 0;
}

int trace(int low1, int low2, int high1, int high2){

    if(high1 - low1 < 3){
        int value = dis(p[low1],q[low2+1]);
        int nextValue;

        if(high1 - low1 == 2){  
            nextValue = dis(p[low1],q[low2+2]);

            if(value > nextValue)
                value = nextValue;

            nextValue = dis(p[low1+1],q[low2+2]);

            if(value > nextValue)
                value = nextValue;
        }
        return value;
    }
    else{

        /* DIVIDE & QONQUER */

        int mid1 = (low1 + high1) >> 1;
        int mid2 = (low2 + high2) >> 1;
        int cnt = 0;

        int leftValue = trace(low1,low2,mid1,mid2);     // left trace
        int rightValue = trace(mid1+1,mid2+1,high1,high2);  // right trace

        // min value find
        int value = leftValue < rightValue ? leftValue : rightValue;

        /* Middle Condition Check : Y Line */

        // saving left
        for(int i = low1;i<=mid1;i++){
            if(abs(p[i].x - q[mid2].x) <= value)
                tmp[cnt++] = p[i];
        }

        // saving right
        for(int i = mid1+1;i<=high1;i++){
            if(absd(p[i].x - q[mid2+1].x) <= value)
                tmp[cnt++] = p[i];
        }

        sort(tmp,tmp+cnt,yCompare);

        for(int i = 0;i<cnt;i++){
            int count = 0;

            for(int j = i-3;count < 6 && j < cnt;j++){
                if(j >= 0 && i != j){
                    int distance = dis(tmp[i],tmp[j]);

                    if(value > distance)
                        value = distance;

                    count++;
                }
            }
        }
        return value;
    }
}

int absd(int x){
    if( x < 0)
        return -x;
    return x;
}

int dis(struct point a, struct point b){
    return (abs(a.x-b.x) + abs(a.y-b.y));
}

bool xCompare(struct point a, struct point b){
    return a.x < b.x;
}

bool yCompare(struct point a, struct point b){
    return a.y < b.y;
}

Answer 1

This problem is usually called the closest bichromatic pair problem. Here are a couple approaches.

1. Delaunay triangulation. (This certainly works with L₂ (= Euclidean) distances; I think the steps generalize to L₁.) For every Delaunay triangulation (there can be more than one in degenerate cases), there exists a minimum spanning tree whose edges all belong to the triangulation. In turn, this minimum spanning tree contains a shortest edge crossing the cut between the color classes.

2. Nearest neighbor data structures.

3. If it is given that the red points are separated in x from the blue points, then you may be able to adapt the O(n) merge step of the Shamos–Hoey divide-and-conquer algorithm for the closest (monochromatic) pair problem, described here.