statistics - Select x random elements from a weighted list in C# (without replacement)

Question

Update: my problem has been solved, I updated the code source in my question to match with Jason's answer. Note that rikitikitik answer is solving the issue of picking cards from a sample with replacement.

I want to select x random elements from a weighted list. The sampling is without replacement. I found this answer: https://stackoverflow.com/a/2149533/57369 with an implementation in Python. I implemented it in C# and tested it. But the results (as described below) were not matching what I expected. I've no knowledge of Python so I'm quite sure I made a mistake while porting the code to C# but I can't see where as the code in Pythong was really well documented.

I picked one card 10000 times and this is the results I obtained (the result is consistent accross executions):

Card 1: 18.25 % (10.00 % expected)
Card 2: 26.85 % (30.00 % expected)
Card 3: 46.22 % (50.00 % expected)
Card 4: 8.68 % (10.00 % expected)

As you can see Card 1 and Card 4 have both a weigth of 1 but Card 1 is awlays picked way more often than card 4 (even if I pick 2 or 3 cards).

Test data:

var cards = new List<Card>
{
    new Card { Id = 1, AttributionRate = 1 }, // 10 %
    new Card { Id = 2, AttributionRate = 3 }, // 30 %
    new Card { Id = 3, AttributionRate = 5 }, // 50 %
    new Card { Id = 4, AttributionRate = 1 }, // 10 %
};

Here is my implementation in C#

public class CardAttributor : ICardsAttributor
{
    private static Random random = new Random();

    private List<Node> GenerateHeap(List<Card> cards)
    {
        List<Node> nodes = new List<Node>();
        nodes.Add(null);

        foreach (Card card in cards)
        {
            nodes.Add(new Node(card.AttributionRate, card, card.AttributionRate));
        }

        for (int i = nodes.Count - 1; i > 1; i--)
        {
            nodes[i>>1].TotalWeight += nodes[i].TotalWeight;
        }

        return nodes;
    }

    private Card PopFromHeap(List<Node> heap)
    {
        Card card = null;

        int gas = random.Next(heap[1].TotalWeight);
        int i = 1;

        while (gas >= heap[i].Weight)
        {
            gas -= heap[i].Weight;
            i <<= 1;

            if (gas >= heap[i].TotalWeight)
            {
                gas -= heap[i].TotalWeight;
                i += 1;
            }
        }

        int weight = heap[i].Weight;
        card = heap[i].Value;

        heap[i].Weight = 0;

        while (i > 0)
        {
            heap[i].TotalWeight -= weight;
            i >>= 1;
        }

        return card;
    }

    public List<Card> PickMultipleCards(List<Card> cards, int cardsToPickCount)
    {
        List<Card> pickedCards = new List<Card>();

        List<Node> heap = GenerateHeap(cards);

        for (int i = 0; i < cardsToPickCount; i++)
        {
            pickedCards.Add(PopFromHeap(heap));
        }

        return pickedCards;
    }
}

class Node
{
    public int Weight { get; set; }
    public Card Value { get; set; }
    public int TotalWeight { get; set; }

    public Node(int weight, Card value, int totalWeight)
    {
        Weight = weight;
        Value = value;
        TotalWeight = totalWeight;
    }
}

public class Card
{
    public int Id { get; set; }
    public int AttributionRate { get; set; }
}

Answer 1

As some people have mentioned in the comments, create a list of the cards in the exact proportion that you want:

var deck = new List<Card>();

cards.ForEach(c => 
{
    for(int i = 0; i < c.AttributionRate; i++)
    {
         deck.Add(c);
    }
}

Shuffle:

deck = deck.OrderBy(c => Guid.NewGuid()).ToList();

And pick x cards:

var hand = deck.Take(x)

Of course this only works if AttributionRate is an int. Otherwise, you would have to tinker with the deck generation a bit.

I get the following results for 10,000 runs taking 5 at a time:

Card 1: 9.932% 
Card 2: 30.15% 
Card 3: 49.854% 
Card 4: 10.064% 

Another result:

Card 1: 10.024%
Card 2: 30.034%
Card 3: 50.034% 
Card 4: 9.908% 

EDIT:

I've braved the bitwise operations and I have taken a look at your code. After adding a generous amount of barbecue sauce on my fried brain, I noticed a few things:

First, Random.Next(min,max) will include min in the random pool, but not max. This is the reason for the higher than expected probability for Card 1.

After doing that change, I implemented your code and it appears to be working when you draw 1 card.

Card 1: 10.4%  
Card 2: 32.2% 
Card 3: 48.4% 
Card 4: 9.0% 

Card 1: 7.5%
Card 2: 28.1%
Card 3: 50.0% 
Card 4: 14.4% 

HOWEVER, your code will not work when you draw more than 1 card because of this statement:

heap[i].Weight = 0;

That line, and the recalculation loop after that, essentially removes all instances of the drawn card from the heap. If you happen to draw four cards, then the percentage becomes 25% for all cards since you're basically drawing all 4 cards. The algorithm, as it is, is not completely applicable to your case.

I suspect you would have to recreate the heap every time you draw a card, but I doubt it would still perform as well. If I were to work on this though, I would just generate 4 distinct random numbers from 1 to heap[1].TotalWeight and get the 4 corresponding cards from there, although the random number generation in this case might become unpredictable (rerolling) and thus inefficient.