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Subsets
一道非常有意思的题目。
原题如下:
Given a set of distinct integers, S, return all possible subsets.
Note:
- Elements in a subset must be in non-descending order.
- The solution set must not contain duplicate subsets.
For example,
If S = [1,2,3]
, a solution is:
[ [3], [1], [2], [1,2,3], [1,3], [2,3], [1,2], [] ]只记得可以利用位操作来解决这个问题。但是想了一会儿还是没有想起来,就看了下Discuss里面的内容。其中有一个分析十分给力。如下
This is an amazing solution.Learnt a lot.Let me try to explain this to those who didn‘t get the logic.
Number of subsets for {1 , 2 , 3 } = 2^3 . why ?
case possible outcomes for the set of subsets
1 -> Take or dont take = 2
2 -> Take or dont take = 2
3 -> Take or dont take = 2
therefore , total = 2*2*2 = 2^3 = { { } , {1} , {2} , {3} , {1,2} , {1,3} , {2,3} , {1,2,3} }
Lets assign bits to each outcome -> First bit to 1 , Second bit to 2 and third bit to 3
Take = 1
Dont take = 0
0) 0 0 0 -> Dont take 3 , Dont take 2 , Dont take 1 = { }
1) 0 0 1 -> Dont take 3 , Dont take 2 , take 1 = {1 }
2) 0 1 0 -> Dont take 3 , take 2 , Dont take 1 = { 2 }
3) 0 1 1 -> Dont take 3 , take 2 , take 1 = { 1 , 2 }
4) 1 0 0 -> take 3 , Dont take 2 , Dont take 1 = { 3 }
5) 1 0 1 -> take 3 , Dont take 2 , take 1 = { 1 , 3 }
6) 1 1 0 -> take 3 , take 2 , Dont take 1 = { 2 , 3 }
7) 1 1 1 -> take 3 , take 2 , take 1 = { 1 , 2 , 3 }
In the above logic ,Insert S[i] only if (1<<i)&j ==true { j E { 0,1,2,3,4,5,6,7 } i = ith element in the input arra}
element 1 is inserted only into those places where 1st bit of j is 1 if( 1 << i &j ) ==> for above above eg. this is true for sl.no.( j )= 1 , 3 , 5 , 7
element 2 is inserted only into those places where 2nd bit of j is 1 if(1 << i &j ) == for above above eg. this is true for sl.no.( j ) = 2 , 3 , 6 , 7
element 3 is inserted only into those places where 3rd bit of j is 1 if(1 << i &j ) == for above above eg. this is true for sl.no.( j ) = 4 , 5 , 6 , 7
Time complexity : O(n*2^n) , for every input element loop traverses the whole solution set length i.e. 2^n
感谢 对C=+代码的解释,我对Java代码改动了一下。
如此很容易写出Java代码:
public List<List<Integer>> subsets(int[] S) { Arrays.sort(S); int totalNumber = 1 << S.length;//将1左移S.length位 System.out.println(totalNumber); List<List<Integer>> collection = new ArrayList<List<Integer>>(totalNumber); for (int i=0; i<totalNumber; i++) { List<Integer> set = new LinkedList<Integer>(); for (int j=0; j<S.length; j++) { if ((i & (1<<j)) != 0) { set.add(S[j]); //只添加和i相同的j位置的S[j] } } collection.add(set); } return collection; }
Subsets