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codility MinAbsSum
For a given array A of N integers and a sequence S of N integers from the set {−1, 1}, we define val(A, S) as follows:
val(A, S) = |sum{ A[i]*S[i] for i = 0..N−1 }|
(Assume that the sum of zero elements equals zero.)
For a given array A, we are looking for such a sequence S that minimizes val(A,S).
Write a function:
int solution(int A[], int N);
that, given an array A of N integers, computes the minimum value of val(A,S) from all possible values of val(A,S) for all possible sequences S of N integers from the set {−1, 1}.
For example, given array:
A[0] = 1 A[1] = 5 A[2] = 2 A[3] = -2
your function should return 0, since for S = [−1, 1, −1, 1], val(A, S) = 0, which is the minimum possible value.
Assume that:
- N is an integer within the range [0..20,000];
- each element of array A is an integer within the range [−100..100].
Complexity:
- expected worst-case time complexity is O(N*max(abs(A))2);
- expected worst-case space complexity is O(N+sum(abs(A))), beyond input storage (not counting the storage required for input arguments).
Elements of input arrays can be modified.
题目大意:给n个数,每个数的范围在-100到100之间,然后对于每个数,我们可以选着乘上-1或1,然后让你求出这些数的和的绝对值的最小值。
这题,可以转化成背包问题。
很明显,我们对这些数求和得sum ,用多重背包求得接近<=sum/2的最大值ans,那么sum-ans-ans就是结果了....这里不好语言描述,不过想一想还是很正确的.
把多重背包转换成01背包求。时间复杂度:O(V*Σlog n[i]) 这里的V=sum/2
// you can use includes, for example:// #include <algorithm>// you can write to stdout for debugging purposes, e.g.// cout << "this is a debug message" << endl;int dp[1000010];int V;inline void dp1(int cost,int weight){ for(int v=V;v>=cost;v--) dp[v]=max(dp[v],dp[v-cost]+weight);}inline void dp2(int cost,int weight){ for(int v=cost;v<=V;v++){ dp[v]=max(dp[v],dp[v-cost]+weight); }}inline void dp3(int cost,int weight,int num){ if(cost*num>=V){ dp2(cost,weight); return ; } int k=1; while(k<num){ dp1(k*cost,k*weight); num-=k; k*=2; } dp1(num*cost,num*weight);}int cnt[102];int solution(vector<int> &A) { // write your code in C++11 (g++ 4.8.2) int n=A.size(); int sum,m; sum=m=0; dp[0]=0; for(int i=0;i<n;i++){ int x=(A[i]>=0)?A[i]:(-A[i]); sum+=x; cnt[x]++; if(x>m)m=x; } if(m==0)return 0; V=(sum>>1); for(int i=1;i<=m;i++){ if(cnt[i])dp3(i,i,cnt[i]); } return (sum-(dp[V]<<1)); }
more about 多重背包:http://love-oriented.com/pack/P03.html
codility MinAbsSum