# Subset (POJ 3977)

At first we iterate all subsets of the first half of the array and calculate all the sums and the minimum number of elements to make the sum. Then we put the sums in an array and sort them. Then we iterate the all subsets last half of array and count the number of items. For each sum in from the last half, we build use binary search method to find the neareast value to -sum in the cached array of sums from first half. The two value will make the a least absolute value sum. We use a variable to keep tracking the minimum absolute sum value and minimum number of items to make it and finally return the answer.

import java.io.*;
import java.util.*;

public class Subset {
private static long nearest(long[] array, long value) {
int l = 0, r = array.length - 1;
while (l <= r) {
int mid = (l + r) / 2;
if (array[mid] >= value) {
r = mid - 1;
} else {
l = mid + 1;
}
}
if (r < 0) r = 0;
if (l >= array.length) l = array.length - 1;
if (Math.abs(array[l] - value) > Math.abs(array[r] - value)) return array[r];
else return array[l];
}
private static void solve(long[] array) {
int N = array.length;
int firstHalf = N / 2 + (N & 1);
int lastHalf = N / 2;
HashMap<Long, Integer> map = new HashMap<Long, Integer>(1 << firstHalf);

long min = Long.MAX_VALUE;
int minCount = Integer.MAX_VALUE;
for (int i = 1; i < (1 << firstHalf); i++) {
long sum = 0;
int count = 0;
for (int j = 0; j < firstHalf; j++) {
if (((i >> j) & 1) == 1) {
sum += array[j];
count++;
}
}
if (map.containsKey(sum)) {
map.put(sum, Math.min(count, map.get(sum)));
} else {
map.put(sum, count);
}
sum = Math.abs(sum);
if (min > sum) {
min = sum;
minCount = count;
} else if (min == sum) {
minCount = Math.min(minCount, count);
}
}

long[] firstArray = new long[map.keySet().size()];
int f = 0;
for (long k : map.keySet()) firstArray[f++] = k;

Arrays.sort(firstArray);

for (int i = 1; i < (1 << lastHalf); i++) {
long sum = 0;
int count = 0;
for (int j = 0; j < lastHalf; j++) {
if (((i >> j) & 1) == 1) {
sum += array[firstHalf + j];
count++;
}
}

long near = nearest(firstArray, -sum);
if (Math.abs(sum + near) < Math.abs(sum)) {
sum += near;
count += map.get(near);
}
sum = Math.abs(sum);
if (min > sum) {
min = sum;
minCount = count;
} else if (min == sum) {
minCount = Math.min(minCount, count);
}
}

System.out.printf("%d %d\n", min, minCount);
}
public static void main(String[] args) {
Scanner in = new Scanner(System.in);
while (true) {
int N = in.nextInt();
if (N == 0) return;
long[] array = new long[N];
for (int i = 0; i < N; i++) array[i] = in.nextLong();
solve(array);
}
}
}