[LC][array] Add to List 561. Array Partition I
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一、问题描述
Given an array of 2n integers, your task is to group these integers into n pairs of integer, say (a1, b1), (a2, b2), ..., (an, bn) which makes sum of min(ai, bi) for all i from 1 to n as large as possible.
Example 1:
Input: [1,4,3,2]Output: 4Explanation: n is 2, and the maximum sum of pairs is 4 = min(1, 2) + min(3, 4).
Note:
- n is a positive integer, which is in the range of [1, 10000].
- All the integers in the array will be in the range of [-10000, 10000].
二、我的解法
因为要求min()的sum最大。min取决于最小的那个数字,所以如果一个小的数字拉一个大的数字垫背的话,就会损失很多。最好的方法是让两个相近的数字组合在一起,损失的小。
于是乎,先排序,然后sum的时候步长为2加起来就行了。
class Solution { public int arrayPairSum(int[] nums) { if(nums == null || nums.length == 0){ return 0; } Arrays.sort(nums); int sum = 0; for(int i = 0; i < nums.length; i = i + 2){ sum += nums[i]; } return sum; }}
三、淫奇技巧
已经是目前最优了。但是大神给出了证明……
- Assume in each pair
i
,bi >= ai
. - Denote
Sm = min(a1, b1) + min(a2, b2) + ... + min(an, bn)
. The biggestSm
is the answer of this problem. Given1
,Sm = a1 + a2 + ... + an
. - Denote
Sa = a1 + b1 + a2 + b2 + ... + an + bn
.Sa
is constant for a given input. - Denote
di = |ai - bi|
. Given1
,di = bi - ai
. DenoteSd = d1 + d2 + ... + dn
. - So
Sa = a1 + a1 + d1 + a2 + a2 + d2 + ... + an + an + di = 2Sm + Sd
=>Sm = (Sa - Sd) / 2
. To get the maxSm
, givenSa
is constant, we need to makeSd
as small as possible. - So this problem becomes finding pairs in an array that makes sum of
di
(distance betweenai
andbi
) as small as possible. Apparently, sum of these distances of adjacent elements is the smallest. If that's not intuitive enough, see attached picture. Case 1 has the smallestSd
四、知识点&注意事项
Arrays.sort()这个东西之前不知道。。
Array有关的操作:
http://www.codeceo.com/article/10-java-array-method.html
http://www.jianshu.com/p/bef8c1b93555
五、扩展问题
如果不给定2n会怎样?要注意单数的情况,从1开始取。
如果要求sum最小?排序后加最小的那半。
六、碎碎念
这个题目解得挺快。。。明天继续加油
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