Finding minimal absolute sum of a subarray

问题

There's an array A containing (positive and negative) integers. Find a (contiguous) subarray whose elements' absolute sum is minimal, e.g.:

A = [2, -4, 6, -3, 9]
|(−4) + 6 + (−3)| = 1 <- minimal absolute sum

I've started by implementing a brute-force algorithm which was O(N^2) or O(N^3), though it produced correct results. But the task specifies:

complexity:
- expected worst-case time complexity is O(N*log(N))
- expected worst-case space complexity is O(N)

After some searching I thought that maybe Kadane's algorithm can be modified to fit this problem but I failed to do it.

My question is - is Kadane's algorithm the right way to go? If not, could you point me in the right direction (or name an algorithm that could help me here)? I don't want a ready-made code, I just need help in finding the right algorithm.

回答1:

If you compute the partial sums

such as

2, 2 +(-4), 2 + (-4) + 6, 2 + (-4) + 6 + (-3)...

Then the sum of any contiguous subarray is the difference of two of the partial sums. So to find the contiguous subarray whose absolute value is minimal, I suggest that you sort the partial sums and then find the two values which are closest together, and use the positions of these two partial sums in the original sequence to find the start and end of the sub-array with smallest absolute value.

The expensive bit here is the sort, so I think this runs in time O(n * log(n)).

回答2:

This is C++ implementation of Saksow's algorithm.

int solution(vector<int> &A) {
    vector<int> P;
    int min = 20000 ;
    int dif = 0 ;
    P.resize(A.size()+1);
    P[0] = 0;
    for(int i = 1 ; i < P.size(); i ++)
    {
        P[i] = P[i-1]+A[i-1];

    }
    sort(P.begin(),P.end());
    for(int i = 1 ; i < P.size(); i++)
    {
         dif = P[i]-P[i-1];
         if(dif<min)
         {
             min = dif;
         }
    }
    return min;
}

回答3:

I was doing this test on Codility and I found mcdowella answer quite helpful, but not enough I have to say: so here is a 2015 answer guys!

We need to build the prefix sums of array A (called P here) like: P[0] = 0, P[1] = P[0] + A[0], P[2] = P[1] + A[1], ..., P[N] = P[N-1] + A[N-1]

The "min abs sum" of A will be the minimum absolute difference between 2 elements in P. So we just have to .sort() P and loop through it taking every time 2 successive elements. This way we have O(N + Nlog(N) + N) which equals to O(Nlog(N)).

That's it!

回答4:

The answer is yes, Kadane's algorithm is definitely the way to go for solving your problem.

http://en.wikipedia.org/wiki/Maximum_subarray_problem

Source - I've closely worked with a PhD student who's entire PhD thesis was devoted to the maximum subarray problem.

回答5:

def min_abs_subarray(a):
    s = [a[0]]
    for e in a[1:]:
        s.append(s[-1] + e)
    s = sorted(s)
    min = abs(s[0])
    t = s[0]
    for x in s[1:]:
        cur = abs(x)
        min = cur if cur < min else min
        cur = abs(t-x)
        min = cur if cur < min else min
        t = x
    return min

回答6:

Short Sweet and work like a charm. JavaScript / NodeJs solution

 function solution(A, i=0, sum =0 ) {
    //Edge case if Array is empty
    if(A.length == 0) return 0;
    // Base case. For last Array element , add and substart from sum
    //  and find min of their absolute value
    if(A.length -1 === i){
        return Math.min( Math.abs(sum + A[i]), Math.abs(sum - A[i])) ;
    }
    // Absolute value by adding the elem with the sum.
    // And recusrively move to next elem
    let plus = Math.abs(solution(A, i+1, sum+A[i]));
    // Absolute value by substracting the elem from the sum
    let minus = Math.abs(solution(A, i+1, sum-A[i]));
    return Math.min(plus, minus);
}

console.log(solution([-100, 3, 2, 4]))

回答7:

Here is an Iterative solution in python. It's 100% correct.

 def solution(A):
    memo = []
    if not len(A):
        return 0

    for ind, val in enumerate(A):
        if ind == 0:
            memo.append([val, -1*val])
        else:
            newElem = []
            for i in memo[ind - 1]:
                newElem.append(i+val)
                newElem.append(i-val)
            memo.append(newElem)
    return min(abs(n) for n in memo.pop())

回答8:

Here is a C solution based on Kadane's algorithm. Hopefully its helpful.

#include <stdio.h>
int min(int a, int b)
{
  return (a >= b)? b: a;
}

int min_slice(int A[], int N) {
if (N==0 || N>1000000) 
return 0;

int minTillHere = A[0];
int minSoFar = A[0];
int i;
for(i = 1; i < N; i++){
    minTillHere = min(A[i], minTillHere + A[i]);
    minSoFar = min(minSoFar, minTillHere);
    }
return minSoFar;
}


int main(){
int A[]={3, 2, -6, 4, 0}, N = 5;
//int A[]={3, 2, 6, 4, 0}, N = 5;
//int A[]={-4, -8, -3, -2, -4, -10}, N = 6;
printf("Minimum slice = %d \n", min_slice(A,N));
return 0;
}

回答9:

public static int solution(int[] A) {
    int minTillHere = A[0];
    int absMinTillHere = A[0];
    int minSoFar = A[0];
    int i;
    for(i = 1; i < A.length; i++){
        absMinTillHere = Math.min(Math.abs(A[i]),Math.abs(minTillHere + A[i]));
        minTillHere = Math.min(A[i], minTillHere + A[i]);
        minSoFar = Math.min(Math.abs(minSoFar), absMinTillHere);
        }
    return minSoFar;
}

回答10:

You can run Kadane's algorithmtwice(or do it in one go) to find minimum and maximum sum where finding minimum works in same way as maximum with reversed signs and then calculate new maximum by comparing their absolute value.

Source-Someone's(dont remember who) comment in this site.

来源：https://stackoverflow.com/questions/25965939/finding-minimal-absolute-sum-of-a-subarray