Select a subset of k intervals which form maximum length if we take union of these k intervals

Suppose we have $$n$$ intervals given in the form of (startTime,EndTime). I wish to calculate maximum length of the region that is exactly union of $$1<=k<=n$$ intervals.

Note that there are 2 cases for which I am trying to find an algorithm individually

1. Union of selected intervals might not be a single interval
2. Union of selected intervals should be a single interval

For example, if we have intervals $$I_1 = (2,6)\\ I_2 = (4,9)\\ I_3 = (7,11)\\ I_4 = (10,13)\\ I_5 = (8,12)\\$$ and $$k=3$$

If we have the ability to select disjoint intervals (Case 1), one of the solutions would be $$I_1,I_3,I_4$$ whose union would be $$(2,6),(7,12)$$ for total length of 10

If we do not have the ability to select disjoint intervals (Case 2), we would not be able to select $$I_1,I_3,I_4$$ but we could select $$I_1,I_2,I_5$$ spanning $$(2,12)$$ which is also length 10.

For case 1, I came up with a greedy approach where I greedily pick the best length interval and update all other intervals to remove the overlap from the interval I just picked which has the complexity of $$O(n*k)$$, please let me know if I am on the right path.

• Hint: Look for a way to further constrain the problem (i.e., turn the problem into "Find the maximum length of a union of $k$ intervals such that ... also holds) in such a way that the original problem can be decomposed into a multiple instances of this more-constrained problem that can be ordered so that previously computed solutions can be used to efficiently solve later ones. Concretely, the condition here could be "that extends to the left of the $i$th interval endpoint, for $1 \le i \le 2n$" (each interval has 2 endpoints). Mar 18, 2023 at 4:03
• cs.stackexchange.com/q/59964/755
– D.W.
Mar 23, 2023 at 18:54

Case 1: Union of selected intervals might not be a single interval

First of all, sort the intervals in increasing order of $$I_i.start$$

Use the following Dynamic Programming solution:

$$\mathrm dp(i, j, k) = max( \text{extend_union} + dp(i + 1, j’, k - 1), dp(i + 1, j, k) )$$

where \mathrm j' = \begin{cases} i & \text{if } I_i.end > I_j.end \\ % & is your "\tab"-like command (it's a tab alignment character) j & \text{otherwise.} \end{cases} \mathrm extend\_union = \begin{cases} max(0, I_i.end - I_j.end) & \text{if } I_j \text{ intersects }I_i \\ % & is your "\tab"-like command (it's a tab alignment character) I_i.end - I_i.start & \text{otherwise.} \end{cases}

• $$dp(i, j, k)$$ denotes the maximum sum of lengths of the segments in the range the $$[i, N)$$

• $$j$$ is the range $$I_j$$ such that $$I_j.end$$ is the maximum among all the segments that we chose until now

• $$k$$ denotes how many more segments are left to choose.

We have 2 choices at each state

• Choice 1: Consider the interval $$I_i$$ in our subset. In this case, the union length will increase by the amount $$\text{extend_union}$$ as mentioned above

• Choice 2: Don't consider the interval $$I_i$$ in our subset. In this case, the union length will not increase

• If you have reached, with $$i=N$$ and $$k=0$$, return $$0$$ else $$-\infty$$

• Time Complexity: $$O(N^2K)$$

Case 2: Union of selected intervals should be a single interval

First of all, sort the intervals in increasing order of $$I_i.start$$

1. Now, create a graph of $$N$$ nodes such that for every $$i$$, there an edge from $$i$$ to $$j$$ $$(i < j)$$ such that $$j = \underset{j}{\operatorname{argmax}} I_j.end$$ where $$I_i \bigcap I_j \neq \phi$$ and $$I_j.end > I_i.end$$.

Don't add an edge if such $$j$$ doesn't exist

This $$j$$ can be found out using any range querying data structure like segment trees in $$O(log(N))$$ or in $$O(1)$$ with sparse tables and precomputations

This will add $$O(N)$$ edges in the graph

1. The resulting graph will be a DAG, with at most one path between any pairs of nodes $$i$$ to $$j$$
2. It will be similar to this:
3. Now, if we choose any starting node in this graph and traverse $$K$$ nodes to the right, we will end with the most optimal subset of size $$K$$ having a certain union length. We can use this to maximize the answer
4. It may happen that there are $$M < K$$ nodes to traverse, in that case, we can consider this subset only if there are $$K-M$$ other intervals that lie within this union.

Finding these $$K-M$$ other intervals may take $$O(N)$$ time in the worst case

To speed this up, the following observations are helpful:

From the image above, if number of x segments + number of y segments is $$\geq K-M$$, this subset can be consider in our answer

• Let $$l$$ be the index of starting interval of our subset and $$r$$ be the index where we end up in this case
• Notice that number of x segments is simply $$(r - l + 1) - M$$, computing in $$O(1)$$ time To find the number of y segments, we can use binary search to find the rightmost index $$id$$ in the range $$[r+1, N]$$ and $$I_{id}$$ lies within $$I_r$$
• Then the number of y segments is simply $$id - r$$, computed in $$O(log(N))$$ time
• We can use the binary search here because the interval $$I_r$$ can no longer extend further and the ranges after that will lie completely within the interval $$I_r$$ or start anywhere $$> I_r.start$$ If that is the case we can update the answer with the union length of this subset
1. Considering every interval as the starting point of the subset and traversing $$K$$ nodes each time will result in $$O(N*K$$) time
2. To optimize it further, notice that we only require the starting and ending indices and not the entire subset indices. Furthermore, nodes chosen by a subset can be reused by other subsets based on their connectivity in the DAG that we defined above
3. Instead of adding the edges from $$i$$ to $$j$$, we keep them from $$j$$ to $$i$$. This is because the resultant disjoint trees formed will have their roots(nodes that don't go further) to the right. The following diagram depicts what we are doing:
4. DFS on nodes indexed from $$N$$ to $$1$$ and keep a list of nodes that are the nodes traversed from root to this node.

This will help us finding the $$K^{th}$$ node from each node in every DFS call in $$O(1)$$ time

Update the answer for each DFS($$u$$) call considering $$u$$ as the starting interval of the chosen subset

• Pseudo Code
answer = 0
parents = []
visited = [false] * N

# based on the conditions
# mentioned above

function dfs(u):
visited[u] = true
for each neighbor v of u:
dfs(v)
parents.pop(u)

for i = N to 1:
if not visited[i]:
dfs(i)


• Time Complexity: $$O(N*log(N))$$
Unfortunately, you greedy algorithm for case $$1$$ may not yield the wanted maximum length sometimes.
Here is an example. We are given \begin{aligned} I_1 &= (1,6)\\ I_2 &= (3, 9)\\ I_3 &= (5, 10)\\ \end{aligned} and $$k=2$$. The greedy algorithm will pick $$I_2$$ and then $$I_1$$, whose union is length $$8$$. However, the union of $$I_1$$ and $$I_2$$ is length $$9$$.
I would bet no greedy algorithm can always work for case $$1$$.