Choose n unique elements from n lists - algorithm

Question

I have been thinking about a programing problem. If we have n lists, we want to output n diffrent elements (each one from a different list). I suspect this can be solved with some kind of a backtrack algorithm but I don't see how to correctly implement it.

Answer 1

Though you could solve this with backtracking as suggested in a comment, a more efficient solution would be to use a max-flow algorithm.
Model it as a graph. A Source, a Sink a node for each distinct element and a node for each list.
You have the source connected to each distinct element. Each element connected to every list it is in. and the lists connected to a sink node. Each edge with capacity 1.
The maximum flow is the maximum number of distinct elements from distinct lists you can select.
https://en.wikipedia.org/wiki/Maximum_flow_problem
https://en.wikipedia.org/wiki/Edmonds%E2%80%93Karp_algorithm

Answer 2

I'm not exactly sure that this has no bugs and solves what is being asked; and I hope I have understood the question right this time :)
//test
int[][] arr = new int[][] {
new int[] { 1, 1, 1, 1, 6 },
new int[] { 1, 1, 1, 1, 5 },
new int[] { 1, 1, 1, 4, 9 },
new int[] { 1, 1, 3, 8, 1 },
new int[] { 1, 2, 7, 1, 1 },
new int[] { 1, 1, 1, 1, 1 } };
int[] res = getItems(arr).ToArray(); //6,5,4,3,2,1
private static IEnumerable<T> getItems<T>(T[][] array)
{
int N = array.GetLength(0);
item<T>[] items = new item<T>[N];
HashSet<T> hs = new HashSet<T>();
for (int i = 0; i < N; i++)
{
bool ok = false;
T[] arr = array[i];
for (int j = items[i].Index; j < arr.Length; j++)
{
T val = arr[j];
if (hs.Add(val))
{
items[i] = new item<T>() { Val = val, Index = j };
ok = true;
break;
}
}
if (!ok)
{
item<T> item;
do
{
if (i == 0) throw new Exception("no solution");
items[i] = new item<T>();
item = items[--i];
item.Index++;
items[i] = item;
}
while (item.Index >= array[i].Length);
hs.Clear();
for (int j = 0; j < i; j++)
hs.Add(array[j][items[j].Index]);
i--;
}
}
return hs;
}
private struct item<T>
{
public T Val;
public int Index;
public override string ToString()
{
return string.Format("Val:{0} Index:{1}", Val, Index);
}
}