最大嵌套区间集

Question

这是一个基于求最大嵌套区间集的大小的问题。

有许多间隔，每个间隔由包含一个起点和一个结束点(si，ei)的对定义。如果i1和i2完全位于i1内部，则称为嵌套i2。示例：- (2,6)和(3,4)嵌套，因为(3,4)是(2,6)的一部分。类似地，k间隔i1、i2、i3....ik称为嵌套if，i2位于i1中，i3位于i2中，...and等等。从给定的间隔确定最大间隔集的大小，以便该集合中的所有间隔都能产生嵌套。

我是这样想的：-让我们用一个例如- (0,7) (0,5) (1,21) (1,9) (2,8) (2,4) (3,20) (4,16) (5,15) (6,21)排序，使ai-1<=ai & bi-1>=bi，而不是从第一个区间开始，我们开始一个链表。如果下一个间隔在一个区间内，我们沿着节点向下移动并遍历创建的图(主列表除外)，..we在此图中存储最大级别节点的指针，新的间隔可以在该指针上适应。并在链接列表中进一步遍历，以查看当前间隔在谁下面。最后，我们有一个指向节点的指针，该节点必须附加当前间隔。并将此节点的级别与我们已有的最大级别进行比较……最大值的最终值是最大嵌套区间集的大小。

上述解决方案的复杂性可能是：- O(n(k+l) + nlogn)

我想这样很难，但我没有别的选择.如果有人有其他算法来解决它..。请张贴，因为我的算法将需要更长的时间来实现(很多数据结构).谢谢！

Answer 1

编辑:一些问题的解决方案在http://codeforces.com/blog/entry/5541上发布，包括两个自称O(n lg n)的解决方案。然而，我认为O(n lg n)解不起作用。我在那一页上发表了评论，说明了原因。如果有人有O(n lg n)的解决方案，我很想听听。

二次解

这个问题可以用动态规划在O(n^2)时间内解决：

1. 计算每个间隔包含多少间隔(可以用两个嵌套循环来完成)
2. 按包含的间隔数的升序排序间隔。
3. 使用递归MaxNestedIntervals解决问题

*注:步骤1可以在O(n lg n)时间内使用这里的解来完成：计算嵌套间隔的子O(n^2)算法？步骤2可以在O(n lg n)时间内完成，并使用任何基于最优比较的排序算法。也许有一种优化步骤3的方法，但我还没有找到它。

递归

MaxNestedIntervals(i) =
    max {j = 0 to i-1} :
        1 + MaxNestedIntervals(j)    if interval i contains interval j
        0                            if interval i doesn't contain interval j

基案例

MaxNestedIntervals(i) =
    0    if interval i contains 0 intervals
    1    if interval i contains 1 interval

示例代码

import java.util.*;

public class MaxNestedIntervals {
    public static void main(String[] args) {
        Interval[] intervals = new Interval[10];
        intervals[0] = new Interval(0, 7);
        intervals[1] = new Interval(0, 5);
        intervals[2] = new Interval(1, 21);
        intervals[3] = new Interval(1, 9);
        intervals[4] = new Interval(2, 8);
        intervals[5] = new Interval(2, 4);
        intervals[6] = new Interval(3, 20);
        intervals[7] = new Interval(4,16);
        intervals[8] = new Interval(5,15);
        intervals[9] = new Interval(6,21);

        int n = intervals.length;
        AugmentedInterval[] augmentedIntervals = new AugmentedInterval[n];

        for (int i = 0; i < n; i++) {
            augmentedIntervals[i] = new AugmentedInterval(intervals[i]);
        }

        for (int i = 0; i < n; i++) {
            AugmentedInterval outerInterval = augmentedIntervals[i];

            for (int j = 0; j < n; j++) {
                if (i == j) {
                    continue;
                }

                AugmentedInterval innerInterval = augmentedIntervals[j];

                if (outerInterval.contains(innerInterval)) {
                    outerInterval.numContainedIntervals++;

                    if (outerInterval.childInterval == null) {
                        outerInterval.childInterval = innerInterval;
                    }
                }
            }
        }

        Arrays.sort(augmentedIntervals, new Comparator<AugmentedInterval>() {
            public int compare(AugmentedInterval i, AugmentedInterval j) {
                return i.numContainedIntervals - j.numContainedIntervals;
            }
        });

        int maxNestedIntervals = 0;
        AugmentedInterval parentInterval = null;

        for (int i = 0; i < n; i++) {
            AugmentedInterval currentInterval = augmentedIntervals[i];

            if (currentInterval.numContainedIntervals == 0) {
                currentInterval.maxNestedIntervals = 0;
            } else if (currentInterval.numContainedIntervals == 1) {
                currentInterval.maxNestedIntervals = 1;
            } else {
                int maxNestedIntervalsForCurrentInterval = 0;

                for (int j = 0; j < i; j++) {
                    AugmentedInterval candidateNestedInterval = augmentedIntervals[j];
                    int maxNestedIntervalsForCurrentCandidate = candidateNestedInterval.maxNestedIntervals + 1;

                    if (currentInterval.contains(candidateNestedInterval) && maxNestedIntervalsForCurrentCandidate >= maxNestedIntervalsForCurrentInterval) {
                        maxNestedIntervalsForCurrentInterval = maxNestedIntervalsForCurrentCandidate;
                        currentInterval.childInterval = candidateNestedInterval;
                    }
                }

                currentInterval.maxNestedIntervals = maxNestedIntervalsForCurrentInterval;

                if (maxNestedIntervalsForCurrentInterval >= maxNestedIntervals) {
                    maxNestedIntervals = maxNestedIntervalsForCurrentInterval;
                    parentInterval = currentInterval;
                }
            }
        }

        if (n == 0) {
            System.out.println("The largest set of nested intervals is the empty set.");
        } else if (maxNestedIntervals == 0) {
            System.out.println("The largest set of nested intervals has 1 interval.\n");
            System.out.println("That interval is:");
        } else {
            System.out.println("The largest set of nested intervals has " + (maxNestedIntervals + 1) + " intervals.\n");
            System.out.println("Those intervals are:");
        }

        for (AugmentedInterval currentInterval = parentInterval; currentInterval != null; currentInterval = currentInterval.childInterval) {
            System.out.println(currentInterval);
        }

        System.out.println();
    }

    private static class Interval implements Comparable<Interval> {
        public int start = 0;
        public int end = 0;

        public Interval(int start, int end) {
            this.start = start;
            this.end = end;
        }

        public int size() {
            return this.end - this.start;
        }

        public boolean contains(Interval other) {
            return (this.start <= other.start && this.end >= other.end);
        }

        public int compareTo(Interval other) {
            return this.size() - other.size();
        }

        public String toString() {
            return "[" + this.start + ", " + this.end + "]";
        }
    }

    private static class AugmentedInterval extends Interval {
        public int numContainedIntervals = 0;
        public int maxNestedIntervals = 0;
        public AugmentedInterval childInterval = null;

        public AugmentedInterval(Interval interval) {
            super(interval.start, interval.end);
        }
    }
}