算法来找到最大总和在重叠的时间间隔的序列 [英] Algorithm to find the maximum sum in a sequence of overlapping intervals
问题描述
我试图解决的问题有间隔数轴上,每一个pre定义得分列表。我需要返回最大可能的总成绩。
The problem I am trying to solve has a list of intervals on the number line, each with a pre-defined score. I need to return the maximum possible total score.
美中不足的是,该区间重叠,重叠的区间,我可以只用一个。下面是一个例子。
The catch is that the intervals overlap, and of the overlapping intervals I can use only one. Here is an example.
Intervals - Score
0- 5 - 15
4- 9 - 18
10-15 - 12
8-21 - 19
25-30 - 25
在这里,间隔0-5,4-9和8-21重叠。
该区间10-15和8-21也有重叠。
最大的一笔是55(18 + 12 + 25)。
Here, the intervals 0-5, 4-9 and 8-21 overlap.
The intervals 10-15 and 8-21 also overlap.
The maximum sum would be 55 (18+12+25).
,这里要注意,我们选择了第一批重叠,即使它不具有最高分数的三间隔的间隔4-9是很重要的。
It is important to note here that we select the interval 4-9 of the first batch of overlapping intervals even though it does not have the highest score of the three.
这是因为在选择的时间间隔8-21将prevent我们从使用间隔:10-15以后,从而降低整体总和(在此情况下,整体的总和将是19 + 25 = 44)。
This is because selecting the interval 8-21 would prevent us from using the interval 10-15 later on, thereby reducing the overall sum (In that case, the overall sum would be 19+25=44).
我要寻找一个O(nlogn)或者O(n)的解决这个问题。我认为,动态规划可以用,但我可能是错的。可能有人提出一个解决方案/算法(S),可以做的伎俩吗?
I am looking for an O(nlogn) or O(n) solution to this problem. I think dynamic programming can be used, but I may be wrong. Could someone suggest a solution/algorithm(s) that could do the trick here?
编辑:间隔是没有特定的顺序。
The intervals are in no particular order.
推荐答案
这是间隔调度加权变化;它是可解的 O(N日志N)
与动态规划。
This is a weighted variation of interval scheduling; it's solvable in O(N log N)
with dynamic programming.
让的间隔 G(启动,停止,得分)
,让他们可以通过排序停止
。为简单起见,我们假设现在所有的停止
是独一无二的。
Let an interval be g(start, stop, score)
, and let them be sorted by stop
. For simplicity, let's assume for now that all stop
is unique.
让最好的[I]
是最好的成绩时,我们允许使用,我们可以得到政[1],...,政[I]
。我们不必使用,当然所有这些,以及通常我们不能因为间隔我们使用了子集必须是非重叠的。
Let best[i]
be the best score we can get when we're allowed to use g[1], ..., g[i]
. We don't have to use them all, of course, and generally we can't because the subset of intervals we use must be non-overlapping.
- 很明显
最好的[0] = 0
。也就是说,因为我们不能使用任何的间隔,最好的成绩,我们可以得到的是0。 - 对于任何
1< = K< = N
,我们有:-
最好的[K] = MAX(最好[K-1],最好的研究[J] + G [K] .score)
,其中-
Ĵ
是最大的指数,这样政[J] .stop<政[K]。开始
(Ĵ
可能为零)
- Clearly
best[0] = 0
. That is, since we can't use any interval, the best score we can get is 0. - For any
1 <= k <= N
, we have:best[k] = max( best[k-1], best[j] + g[k].score )
, wherej
is the largest index such thatg[j].stop < g[k].start
(j
may be zero)
这是,因为我们允许使用
政[1],...政[K]
,我们能做的最好的是更好的得分王这两个选项:That is, given that we're allowed to use
g[1], ... g[k]
, the best we can do is the better scoring of these two options:- 我们不包括
政[K]
。因此,此选项的得分最好[K-1]
。- ...因为这是我们可以用
政[1],...政[K-1]
请最好的
- We do not include
g[k]
. Thus, the score of this option isbest[k-1]
.- ... because that's the best we can do with
g[1], ... g[k-1]
(注意动态规划体现在上式的最优子和重叠的子组件)。
(Note the optimal substructure and overlapping subproblems components of dynamic programming embodied in the above equation).
总体问题的答案是
最好[N]
,即最好成绩时,我们可以使用所有的基因,我们可以得到的。哎呀,我说的基因?我的意思是时间间隔。The overall answer to the question is
best[N]
, i.e. the best score we can get when we're allowed to use all the genes. Oops, did I say genes? I mean intervals.这是
O(N日志N)
,因为:- 排序的所有时段采用
O(N日志N)
- 查找
Ĵ
每个K
是O(日志N)
使用二进制搜索
- Sorting all the intervals takes
O(N log N)
- Finding
j
for eachk
isO(log N)
using binary search
如果几个基因可以有相同的
停止
的值,然后什么都没有改变:你还是要寻找最右侧的Ĵ
。在如Python中这是很容易与bisect_right
。在Java中,其中标准库中的二进制搜索不保证其索引的情况下返回的关系,你可以(在许多选项)与线性搜索(对O(N)$ c遵循它$ C>最坏情况的性能),或另一系列二进制搜索以发现最右索引
If several genes can have the same
stop
values, then nothing changed: you still have to search for the rightmostj
. In e.g. Python this is easy withbisect_right
. In Java where the standard library binary search doesn't guarantee which index is returned in case of ties, you can (among many options) follow it with a linear search (forO(N)
worst-case performance), or another series of binary searches to find the right most index.抱歉没我再说一遍基因呢?我的意思是时间间隔。
Oops did I say genes again? I mean intervals.
- <一个href="http://stackoverflow.com/questions/2218931/extension-of-binary-search-algo-to-find-the-first-and-last-index-of-the-key-value">Extension二进制搜索,找到键值 的第一个和最后一个索引
- Extension of binary search to find the first and last index of the key value
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- ... because that's the best we can do with
- ...因为这是我们可以用
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