使用Binary Search将Interval插入不相交的间隔中

Question

Given

class Interval{
   int start;
   int end;
}

Task is to insert an Interval into a disjoint set or list of Intervals. So for example

<4,8> into <3,7><10,13><20,21><30,31><40,45> gives <3,8><10,13><20,21><30,31><40,45>

<1,30> into <3,7><10,13><20,21><30,31><40,45> gives <1,30><40,45>

and etc.

I know we should use 2 binary searches for the most efficient solution and that we should be comparing the inserted interval's start with the list intervals' end and vice versa. How exactly do we handle binary search when we can't find exactly what we're looking for?

Answer 1

Count the number of nodes. Go back to the centre, insert if applicable. Otherwise decide which half of the list is of interest. Go back to it's centre, insert etc. You need to handle the exception <4 9> into <2 5> <8 12>.

Answer 2

Assuming C++, I'd use an std::map from the interval end to the interval start. For searching, use std::upper_bound() to find the first overlapping interval and then advance the iterator to find all overlapping intervals. Only one binary search is required.

#include <map>
#include <stdio.h>

typedef std::map<int, int> IntervalMap;

struct Interval {
    int start;
    int end;
};

int main()
{
    IntervalMap imap;
    Interval query;

    imap[7] = 3;    // <3,7>
    imap[13] = 10;  // <10,13>

    // Insert: <4,8>
    query.start = 4;
    query.end = 8;

    // Find the first overlapping interval
    auto it = imap.upper_bound(query.start);
    if (it != imap.end() && it->second < query.end)
    {
        // There is one or more overlapping interval
        // Update lower bound for the new interval
        if (it->second < query.start)
            query.start = it->second;

        while (it != imap.end() && it->second < query.end)
        {
            // Update upper bound for the new interval
            if (it->first > query.end)
                query.end = it->first;

            auto tmp = it;
            ++it;
            imap.erase(tmp);
        }
    }

    // Insert the new interval
    imap[query.end] = query.start;

    for (auto it = imap.begin(); it != imap.end(); ++it)
    {
        fprintf(stderr, "<%d,%d>\n", it->second, it->first);
    }

    return 0;
}