Question

I got a brain teaser for you – it’s not as simple as it sounds so please read and try to solve the issue. Before you ask if it’s homework – it’s not! I just wish to see if there’s an elegant way of solving this. Here’s the issue:

X-number of friends want’s to go to the cinema and wish to be seated
in the best available groups. Best case is that everyone sits together
and worst case is that everyone sits alone. Fewer groups are preferred over more groups. Ballanced groups are preferred (3+3 is more preferred than 4+2).
Sitting alone is least preferred.

Input is the number of people going to the cinema and output should be an array of integer arrays that contains:

• Ordered combinations (most preferred are first)
• Number of people in each group

Below are some examples of number of people going to the cinema and a list of preferred combinations these people can be seated:

• 1 person: 1
• 2 persons: 2, 1+1
• 3 persons: 3, 2+1, 1+1+1
• 4 persons: 4, 2+2, 3+1, 2+1+1, 1+1+1+1
• 5 persons: 5, 3+2, 4+1, 2+2+1, 3+1+1, 2+1+1+1, 1+1+1+1+1
• 6 persons: 6, 3+3, 4+2, 2+2+2, 5+1, 3+2+1, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1

Example with more than 7 persons explodes in combinations but I think you get the point by now.

Question is: What does an algorithm look like that solves this problem?
My language by choice is C# so if you could give an answer in C# it would be fantastic!

Answer 1

I think you need to do it recursively but you need to make sure that you don’t keep partitioning the same group over and over again. This will give you exponential execution time. In my solution it looks I have O(n*n) (you can verify it for me 😉 , see the results below. Another thing is the desirablity function you mention. I don’t know how such a function could look like, but you can instead compare 2 partitions. e.g. partition 1 + 1 + 2 + 4 is less desirable then 1 + 2 + 2 + 3 because it has two ‘ones’. A general rule could be ‘a partition is less desirable if it has more of the same number of people grouped than another partition’. Makes sense, the more people sit together, the better. My solution takes this approach for comparing 2 possible groupings and I get the result that you wanted to achieve. Let me show you some results first, then the code.

        var sut = new BrainTeaser();

        for (int n = 1; n <= 6; n++) {
            StringBuilder sb = new StringBuilder();
            sb.AppendFormat("{0} person{1}: ", n, n > 1 ? "s" : "");

            var array = sut.Solve(n).Select(x => x.ToString()).ToArray();
            sb.AppendLine(string.Join(", ", array));

            Console.WriteLine(sb.ToString());
        }

1 person: 1

2 persons: 2, 1+1

3 persons: 3, 1+2, 1+1+1

4 persons: 4, 2+2, 1+3, 1+1+2, 1+1+1+1

5 persons: 5, 2+3, 1+4, 1+2+2, 1+1+3, 1+1+1+2, 1+1+1+1+1

6 persons: 6, 3+3, 2+4, 2+2+2, 1+5, 1+2+3, 1+1+4, 1+1+2+2, 1+1+1+3, 1+1+1+1+2, 1+1+1+1+1+1

performance looks to be O(n*n):

var sut = new BrainTeaser();

 for (int n = 1; n <= 40; n++) {
   Stopwatch watch = new Stopwatch();
   watch.Start();
   var count = sut.Solve(n).Count();
   watch.Stop();
   Console.WriteLine("Problem solved for {0} friends in {1} ms. Number of solutions {2}", n, watch.ElapsedMilliseconds, count);
}

Problem solved for 1 friends in 17 ms. Number of solutions 1
Problem solved for 2 friends in 49 ms. Number of solutions 2
Problem solved for 3 friends in 2 ms. Number of solutions 3
Problem solved for 4 friends in 1 ms. Number of solutions 5
Problem solved for 5 friends in 0 ms. Number of solutions 7
Problem solved for 6 friends in 2 ms. Number of solutions 11
Problem solved for 7 friends in 0 ms. Number of solutions 15
Problem solved for 8 friends in 0 ms. Number of solutions 22
Problem solved for 9 friends in 1 ms. Number of solutions 30
Problem solved for 10 friends in 1 ms. Number of solutions 42
Problem solved for 11 friends in 4 ms. Number of solutions 56
Problem solved for 12 friends in 4 ms. Number of solutions 77
Problem solved for 13 friends in 7 ms. Number of solutions 101
Problem solved for 14 friends in 9 ms. Number of solutions 135
Problem solved for 15 friends in 15 ms. Number of solutions 176
Problem solved for 16 friends in 21 ms. Number of solutions 231
Problem solved for 17 friends in 30 ms. Number of solutions 297
Problem solved for 18 friends in 43 ms. Number of solutions 385
Problem solved for 19 friends in 61 ms. Number of solutions 490
Problem solved for 20 friends in 85 ms. Number of solutions 627
Problem solved for 21 friends in 117 ms. Number of solutions 792
Problem solved for 22 friends in 164 ms. Number of solutions 1002
Problem solved for 23 friends in 219 ms. Number of solutions 1255
Problem solved for 24 friends in 300 ms. Number of solutions 1575
Problem solved for 25 friends in 386 ms. Number of solutions 1958
Problem solved for 26 friends in 519 ms. Number of solutions 2436
Problem solved for 27 friends in 677 ms. Number of solutions 3010
Problem solved for 28 friends in 895 ms. Number of solutions 3718
Problem solved for 29 friends in 1168 ms. Number of solutions 4565
Problem solved for 30 friends in 1545 ms. Number of solutions 5604
Problem solved for 31 friends in 2025 ms. Number of solutions 6842
Problem solved for 32 friends in 2577 ms. Number of solutions 8349
Problem solved for 33 friends in 3227 ms. Number of solutions 10143
Problem solved for 34 friends in 4137 ms. Number of solutions 12310
Problem solved for 35 friends in 5300 ms. Number of solutions 14883
Problem solved for 36 friends in 6429 ms. Number of solutions 17977
Problem solved for 37 friends in 8190 ms. Number of solutions 21637
Problem solved for 38 friends in 10162 ms. Number of solutions 26015
Problem solved for 39 friends in 12643 ms. Number of solutions 31185

Let me post now the 3 classes involved in the solution:

public class BrainTeaser {
    /// <summary>
    /// The possible groupings are returned in order of the 'most' desirable first. Equivalent groupings are not returned (e.g. 2 + 1 vs. 1 + 2). Only one representant
    /// of each grouping is returned (ordered ascending. e.g. 1 + 1 + 2 + 4 + 5)
    /// </summary>
    /// <param name="numberOfFriends"></param>
    /// <returns></returns>
    public IEnumerable<PossibleGrouping> Solve(int numberOfFriends) {
        if (numberOfFriends == 1) {
            yield return new PossibleGrouping(1);
            yield break;
        }
        HashSet<PossibleGrouping> possibleGroupings = new HashSet<PossibleGrouping>(new PossibleGroupingComparer());
        foreach (var grouping in Solve(numberOfFriends - 1)) {
            // for each group we create 'n+1' new groups 
            // 1 + 1 + 2 + 3 + 4 
            // Becomes
            //      (1+1) + 1 + 2 + 3 + 4  we can add a friend to the first group
            //      1 + (1+1) + 2 + 3 + 4  we can add a friend to the second group
            //      1 + 1 + (2+1) + 3 + 4  we can add a friend to the third group
            //      1 + 1 + 2 + (3+1) + 4  we can add a friend to the forth group
            //      1 + 1 + 2 + 3 + (4+1) we can add a friend to the fifth group
            //      (1 + 1 + 2 + 3 + 4) + 1  friend has to sit alone

            AddAllPartitions(grouping, possibleGroupings);
        }
        foreach (var possibleGrouping in possibleGroupings.OrderByDescending(x => x)) {
            yield return possibleGrouping;
        }
    }

    private void AddAllPartitions(PossibleGrouping grouping, HashSet<PossibleGrouping> possibleGroupings) {
        for (int i = 0; i < grouping.FriendsInGroup.Length; i++) {
            int[] newFriendsInGroup = (int[]) grouping.FriendsInGroup.Clone();
            newFriendsInGroup[i] = newFriendsInGroup[i] + 1;
            possibleGroupings.Add(new PossibleGrouping(newFriendsInGroup));
        }
        var friendsInGroupWithOneAtTheEnd = grouping.FriendsInGroup.Concat(new[] {1}).ToArray();
        possibleGroupings.Add(new PossibleGrouping(friendsInGroupWithOneAtTheEnd));
    }
}

/// <summary>
/// A possible grouping of friends. E.g.
/// 1 + 1 + 2 + 2 + 4 (10 friends). The array is sorted by the least friends in an group.
/// </summary>
public class PossibleGrouping : IComparable<PossibleGrouping> {
    private readonly int[] friendsInGroup;

    public int[] FriendsInGroup {
        get { return friendsInGroup; }
    }

    private readonly int sum;

    public PossibleGrouping(params int[] friendsInGroup) {
        this.friendsInGroup = friendsInGroup.OrderBy(x => x).ToArray();
        sum = friendsInGroup.Sum();
    }

    public int Sum {
        get { return sum; }
    }

    /// <summary>
    /// determine which group is more desirable. Example:
    /// Consider g1: 1 + 2 + 3 + 4 vs g2: 1 + 1 + 2 + 2 + 4  
    /// Group each sequence by the number of occurrences:
    /// 
    /// group   | g1   | g2
    /// --------|-------------
    /// 1       | 1    | 2
    /// ----------------------
    /// 2       | 1    | 2
    /// ----------------------
    /// 3       | 1    | 0
    /// ----------------------
    /// 4       | 1    | 1
    /// ----------------------
    /// 
    /// Sequence 'g1' should score 'higher' because it has 'less' 'ones' (least desirable) elements. 
    /// 
    /// If both sequence would have same number of 'ones', we'd compare the 'twos'.
    /// 
    /// </summary>
    /// <param name="other"></param>
    /// <returns></returns>
    public int CompareTo(PossibleGrouping other) {
        var thisGroup = (from n in friendsInGroup group n by n).ToDictionary(x => x.Key,
                                                                             x => x.Count());

        var otherGroup = (from n in other.friendsInGroup group n by n).ToDictionary(x => x.Key,
                                                                                    x => x.Count());

        return WhichGroupIsBetter(thisGroup, otherGroup);
    }

    private int WhichGroupIsBetter(IDictionary<int, int> thisGroup, IDictionary<int, int> otherGroup) {
        int maxNumberOfFriendsInAGroups = Math.Max(thisGroup.Keys.Max(), otherGroup.Keys.Max());

        for (int numberOfFriendsInGroup = 1;
             numberOfFriendsInGroup <= maxNumberOfFriendsInAGroups;
             numberOfFriendsInGroup++) {
            // zero means that the current grouping does not contain a such group with 'numberOfFriendsInGroup'
            // in the example above, e.g. group '3'
            int thisNumberOfGroups = thisGroup.ContainsKey(numberOfFriendsInGroup)
                                         ? thisGroup[numberOfFriendsInGroup]
                                         : 0;
            int otherNumberOfGroups = otherGroup.ContainsKey(numberOfFriendsInGroup)
                                          ? otherGroup[numberOfFriendsInGroup]
                                          : 0;

            int compare = thisNumberOfGroups.CompareTo(otherNumberOfGroups);

            if (compare != 0) {
                // positive score means that the other group has more occurrences. e.g. 'this' group might have 2 groups with each 2 friends,
                // but the other solution might have 3 groups with each 2 friends. It's obvious that (because both solutions must sum up to the same value)
                // this 'solution' must contain a grouping with more than 3 friends which is more desirable.
                return -compare;
            }
        }
        // they must be 'equal' in this case.
        return 0;
    }

    public override string ToString() {
        return string.Join("+", friendsInGroup.Select(x => x.ToString()).ToArray());
    }
}

public class PossibleGroupingComparer : EqualityComparer<PossibleGrouping> {
    public override bool Equals(PossibleGrouping x, PossibleGrouping y) {
        return x.FriendsInGroup.SequenceEqual(y.FriendsInGroup);
    }

    /// <summary>
    /// may not be the best hashcode function. for alternatives look here: http://burtleburtle.net/bob/hash/doobs.html
    /// I got this code from here: http://stackoverflow.com/questions/3404715/c-sharp-hashcode-for-array-of-ints
    /// </summary>
    /// <param name="obj"></param>
    /// <returns></returns>
    public override int GetHashCode(PossibleGrouping obj) {
        var array = obj.FriendsInGroup;

        int hc = obj.FriendsInGroup.Length;
        for (int i = 0; i < array.Length; ++i) {
            hc = unchecked(hc*314159 + array[i]);
        }
        return hc;
    }
}

Now to the solution:

The brainteaser class does the recursion. One trick in this class is to use a custom comparer (PossibleGroupingComparer) in the hashset. This will make sure that when we calculate new groupings (e.g. 1+1+2 vs 2+1+1), those will be treated as the same (our set will contain only one representant for each equivalent grouping). This should reduce exponential runtime to O(n^2).

The next trick is that ordering the result is possible because our PossibleGroupings class implements IComparable. The implementation of the Compare() method uses the idea mentioned above. This method essentially contains the salt in this solution and if you want to have it grouped differently you should only modify this method.

I hope you can understand the code otherwise let me know. I tried to make it readable and didn’t care much about performance. You could for instance order the groupings only before you return them to the caller, the ordering within the recursions doesn’t bring much.

One comment though: A typical scenario might be that a cinema has ‘already’ booked many seats and won’t allow for ‘any’ partition. Here you need to get all partitions and then check one-by-one if it’s possible to use for the current cinema. That works, but costs unnecessary CPU. Instead, we could use the input to reduce the number of recursions and improve the overall execution time. Maybe someone wants to post a solution for this 😉