I have the following problem and I need an algorithm for this. I need to write a program that split M student (in my case they are about 170 student) to N teams (12 teams) with the same number of students in each team as much as possible (in my case 14 or 15 student in a team), and there are 3 constrains.
the first constrain is proportions of female/male between the teams should be equal as much as possible.
the second constrain is proportions of outstanding/not outstanding students between the teams should be equal as much as possible.
the second constrain is proportions of student live in the city/outside the city between the teams should be equal as much as possible.
I don't need to find the optimal split, but split that good enough and I don't have definition what is good enough, maybe the maximum different in the proportion can be an input.
I have all the info I need about the students.
Thanks!!!
First, here's a general procedure for splitting a group of M students into N teams as evenly as possible:
Assign RoundDown(M/N) students to each of the N teams, in any manner.
If N is not divisible by M, then N - (M % N) < N "extra" students remain. Assign each of them to a different team.
After doing this, team sizes differ by at most 1 (some teams got no extra students, while some may have gotten exactly 1 extra student). Notice that if we have several separate groups of students then we can perform this procedure several times in succession to build up the N teams, and provided we always add any "extra" students to the smallest teams first, we will always maintain the property that team sizes differ by at most 1.
You have 3 separate criteria, so each student is in one of 2^3 = 8 groups defined by them (e.g. the group of male, non-outstanding, city students). So you can simply perform the above procedure 8 times, once for each group.
This will result in:
Team sizes will differ by at most 1.
The number of males (or of females, or of outstanding students, etc.) in any two teams will differ by at most 4, since there are 4 groups that include males (males who are/aren't oustanding, and who live/don't live in the city), and we know that for each of these groups, the number of students from that group in a team differs by at most 1.
In practice, it's unlikely that the number of people in any category will differ by as much as 4 between teams. You can mitigate this even further by being careful about which smallest teams get extra students first -- e.g. if you have 3 extra male, non-outstanding, city students left over and there are 7 small teams, you can put them in the 3 teams that have the fewest males (or non-outstanding students, or whichever criterion you want to prioritise). The same applies if there are more extra students in some group than small teams -- if there are 4 small teams and 9 extra students in some group, the first 4 students have to go to the small teams, but the remaining 5 students can go to whichever or the remaining 8 teams gives the best, say, gender balance.
Related
Here is the real life problem:
There are limited amount of rooms that are different capacities (for example, 20 rooms in total: 6 rooms of 4 person, 4 rooms of 5 person, 7 rooms of 3 person, 3 rooms of 2 person).
There are 5 different grade of student. Let's say Grade 1,2,3,4,5. Students from different grades can't be in the same room.
Each student writes down students he wants to stay with. Each students has to write down #max capacity of room - 1 (in our example, max capacity of a room is 5, so each has to write down 4 students)
Each students writes down 1 student they don't want to stay with.
What can be the best algorithm for making the rooms, as much as preferred it can possibly be?
Thanks in advance all those who will reply.
I'm organizing a game tournament in a week. I began thinking about the best way to mathematically arrange the teams (so that they are really even, thus more competitive). So here's the data:
20 players in the event
Each player is assigned a skill level (high number = skilled)
4 teams of 5 players each (although I'd prefer to build a algorithm that takes these as
variables)
I'm using a computer to solve the problem
So, I have 20 players. I'd like to generate 4 teams with 5 players each. To do this, I'd like to generate a list of all possible team combinations. To evaluate a team combination, I:
Generate a combination of teams (a match)
Sum the total skill for each team based off the players in that team
Compare each team to each other, the highest difference between any two teams in the match is the "tolerance" level for that match. If the tolerance level is higher than a certain cap, the match is discarded
My current approach is to generate a base X number that is N digits long, where X is the number of teams I want, and N is the number of players. Then increment the base X number by 1, I'll get every possible team combination, and I can generate a list of matches that have low tolerance values.
The problem with this, as you probably know, is for 4 teams with 20 players, that's (4-1)^20 in base 3, which is 1E12 matches to check through. (This takes a long time on my computer). Is there a mathematical way to simplify this calculation to be doable in a short period of time?
By current method also allows for the possibility of uneven players spread across the number of teams, which is preferable. If this can't be present with a highly performant algorithm, then it's okay not to use it.
Try following approach:
For teams 1 to 4: Take the strongest player from the remaining
The the same in another direction: from 4 to 1
Again 1 to 4
Again 4 to 1
In the last round use random
This works well when player skills are distributed more or less evenly. If not, then the probability of bigger differences between teams is higher.
I'm working in Ruby, but I think this question is best asked agnostic of language. It may be assumed that we have access to basic list/array functions, as well as a "random" number generator. Here's what I'd like to be able to do:
Given a collection of n teams, with n even,
Randomly pair each team with an opponent, such that every team is part of exactly one pair. Call this ROUND 1.
Randomly generate n-2 subsequent rounds (ROUND 2 through ROUND n-1) such that:
Each round has the same property as the first (every team is a
member of one pair), and
After all the rounds, every team has faced every other team exactly once.
I imagine that algorithms for doing exactly this must be well known, but as a self-taught coder I'm having trouble figuring out how to find them.
I belive You are describing a round robin tournament. The wikipedia page gives an algorithm.
If You need a way to randomize the schedule, randomize team order, round order, etc.
Well not sure if this is the most efficient algorithm but:
Randomly assign N teams into two lists of same length n/2 (List1, List2)
Starting with i = 0:
Create pairs: List1[i],List2[i] = a team pair
Repeat for i = 1-> (n/2-1)
For rounds 2-> n/2-1:
Rotate List2, so that the first team in List2 is now at the end.
Repeat steps 2 through 5, until List2 has been cycled once.
This link was very helpful to me the last time I wrote a round robin scheduling algorithm. It includes a C implementation of a first fit algorithm for round robin pairings.
http://www.devenezia.com/downloads/round-robin/
In addition to the algorithm, he has some helpful links to other aspects of tournament scheduling (balancing home and away games, as well as rotating teams across fields/courts).
Note that you don't necessarily want a "random" order to the pairings in all cases. If, for example, you were scheduling a round robin soccer league for 8 games that only had 6 teams, then each team is going to have to play two other teams twice. If you want to make a more enjoyable season for everyone, you have to start worrying about seeding so that you don't have your top 2 teams clobbering the two weakest teams in their last two games. You'd be better off arranging for the extra games to be paired against teams of similar strength/seeding.
Based on info I found through Maniek's link, I went with the following:
A simple round robin algorithm that
a. Starts with pairings achieved by zipping [0,...,(n-1)/2] and [(n-1)/2 + 1,..., n-1]. (So, if n==10, we have 0 paired with 5, 1 with 6, etc.)
b. Rotates all but one team n-2 times clockwise until all teams have played each other. (So in round 2 we pair 1 with 6, 5 with 7, etc.)
Randomly assigns one of [0,..., n-1] to each of the teams.
I have a data set of players' skill ranking, age and sex and would like to create evenly matched teams.
Teams will have the same number of players (currently 8 teams of 12 players).
Teams should have the same or similar male to female ratio.
Teams should have similar age curve/distribution.
I would like to try this in Haskell but the choice of coding language is the least important aspect of this problem.
This is a bin packing problem, or a multi-dimensional knapsack problem. Björn B. Brandenburg has made a bin packing heuristics library in Haskell that you may find useful.
You need something like...
data Player = P { skill :: Int, gender :: Bool, age :: Int }
Decide on a number of teams n (I'm guessing this is a function of the total number of players).
Find the desired total skill per team:
teamSkill n ps = sum (map skill ps) / n
Find the ideal gender ratio:
genderRatio ps = sum (map (\x -> if gender x then 1 else 0)) / length ps
Find the ideal age variance (you'll want the Math.Statistics package):
ageDist ps = pvar (map age ps)
And you must assign the three constraints some weights to come up with a scoring for a given team:
score skillW genderW ageW team = skillW * sk + genderW * g + ageW * a
where (sk, (g, a)) = (teamSkill 1 &&& genderRatio &&& ageDist) team
The problem reduces to the minimization of the difference in scores between teams. A brute force approach will take time proportional to Θ(nk−1). Given the size of your problem (8 teams of 12 players each), this translates to about 6 to 24 hours on a typical modern PC.
EDIT
An approach that may work well for you (since you don't need an exact solution in practise) is simulated annealing, or continual improvement by random permutation:
Pick teams at random.
Get a score for this configuration (see above).
Randomly swap players between two or more teams.
Get a score for the new configuration. If it's better than the previous one, keep it and recurse to step 3. Otherwise discard the new configuration and try step 3 again.
When the score has not improved for some fixed number of iterations (experiment to find the knee of this curve), stop. It's likely that the configuration you have at this point will be close enough to the ideal. Run this algorithm a few times to gain confidence that you have not hit on some local optimum that is considerably worse than ideal.
Given the number of players per team and the gender ration (which you can easily compute). The remaining problem is called n-partition problem, which is unfortunately NP-complete and thus very hard to solve exactly. You will have to use approximative or heuristic allgorithms (evolutionary algorithms), if your problem size is too big for a brute force solution. A very simple approximation would be sorting by age and assign in an alternating way.
Assign point values to the skill levels, gender, and age
Assign the sum of the points for each criteria to each player
Sort players by their calculated point value
Assign the next player to the first team
Assign players to the second team until it has >= total points than the first team or the team reaches the maximum players.
Perform 5 for each team, looping back to the first team, until all players are assigned
You can tweak the skill level, gender, and age point values to change the distribution of each.
Lets say you have six players (for a simple example). We can use the same algorithm which pairs opponents in single-elimination tournaments and adapt that to generate "even" teams based on any criteria you choose.
First rank your players best-to-worst. Don't take this too literally. You want a list of players sorted by the criteria you wish to separate them.
Why?
Let's look at single elimination tournaments for a second. The idea of using an algorithm to generate optimal single-elimination matches is to avoid the problem of the "top players" meeting too soon in the tournament. If top players meet too soon, one of the top players will be eliminated early on, making the tournament less interesting. We can use this "optimal" pairing to generate teams in which the "top" players are spread out evenly across the teams. Then spread out the the second top players, etc, etc.
So list you players by the criteria you want them separated: men first, then women... sorted by age second. We get (for example):
Player 1: Male - 18
Player 2: Male - 26
Player 3: Male - 45
Player 4: Female - 18
Player 5: Female - 26
Player 6: Female - 45
Then we'll apply the single-elimination algorithm which uses their "rank" (which is just their player number) to create "good match ups".
The single-elimination tournament generator basically works like this: take their rank (player number) and reverse the bits (binary). This new number you come up with become their "slot" in the tournament.
Player 1 in binary (001), reversed becomes 100 (4 decimal) = slot 4
Player 2 in binary (010), reversed becomes 010 (2 decimal) = slot 2
Player 3 in binary (011), reversed becomes 110 (6 decimal) = slot 6
Player 4 in binary (100), reversed becomes 001 (1 decimal) = slot 1
Player 5 in binary (101), reversed becomes 101 (5 decimal) = slot 5
Player 6 in binary (110), reversed becomes 011 (3 decimal) = slot 3
In a single-elimination tournament, slot 1 plays slot 2, 3-vs-4, 5-vs-6. We're going to uses these "pair ups" to generate optimal teams.
Looking at the player number above, ordered by their "slot number", here is the list we came up with:
Slot 1: Female - 18
Slot 2: Male - 26
Slot 3: Female - 45
Slot 4: Male - 18
Slot 5: Female - 26
Slot 6: Male - 45
When you split the slots up into teams (two or more) you get the players in slot 1-3 vs players in slot 4-6. That is the best/optimal grouping you can get.
This technique scales very well with many more players, multiple criteria (just group them together correctly), and multiple teams.
Idea:
Sort players by skill
Assign best players in order (i.e.: team A: 1st player, team B: 2nd player, ...)
Assign worst players in order
Loop on 2
Evaluate possible corrections and perform them (i.e.: if team A has a total skill of 19 with a player with skill 5 and team B has a total skill of 21 with a player with skill 4, interchange them)
Evaluate possible corrections on gender distribution and perform them
Evaluate possible corrections on age distribution and perform them
Almost trivial approach for two teams:
Sort all player by your skill/rank assessment.
Assign team A the best player.
Assign team B the next two best players
Assign team A the next two best players
goto 3
End when you're out of players.
Not very flexible, and only works on one column ranking, so it won't try to get similar gender or age profiles. But it does make fair well matched teams if the input distribution is reasonably smooth. Plus it doesn't always end with team A have the spare player when there are an odd number.
Well,
My answer is not about scoring strategies of teams/players because all the posted are good, but I would try a brute force or a random search approach.
I don't think it's worth create a genetic algorithm.
Regards.
I have a project for school where I have to come up with an algorithm for scheduling 4 teams to play volleyball on one court, such that each team gets as close to the same amount of time as possible to play.
If you always have the winners stay in and rotate out the loser, then the 4th ranked team will never play and the #1 team always will.
The goal is to have everybody play the same amount of time.
The simplest answer is team 1 play team 2, then team 3 play team 4 and keep switching, but then team 1 never gets to play team 3 or 4 and so on.
So I'm trying to figure out an algorithm that will allow everybody to play everybody else at some point without having one team sit out a lot more than any other team.
Suggestions?
How about this: Make a hashtable H of size NC2, in this case, 6. It looks like:
H[12] = 0
H[13] = 0
H[14] = 0
H[23] = 0
H[24] = 0
H[34] = 0
I am assuming it would be trivial to generate the keys.
Now to schedule a game, scan through the hash and pick the key with the lowest value (one pass). The teams denoted by the key play the game and you increment the value by one.
EDIT:
To add another constraint that no team should wait too long, make another hash W:
W[1] = 0
W[2] = 0
W[3] = 0
W[4] = 0
After every game increment the W value for the team that did not play, by one.
Now when picking up the least played team if there are more than one team combo with low play score, take help from this hash to determine which team must play next.
well you should play 1-2 3-4, 1-3 2-4, 1-4 2-3 and then start all over again.
If there are N teams and you want all pairs of them to play once, then there are "N choose 2" = N*(N-1)/2 games you need to run.
To enumerate them, just put the teams in an ordered list and have the first team play every other team, then have the second team play all the teams below it in the list, and so on. If you want to spread the games out so teams have similar rest intervals between games, then see Knuth.
Check out the wikipedia entry on round robin scheduling.
pretend it's a small sports league, and repeat the "seasons"...
(in most sports leagues in Europe, all teams play against all other teams a couple of times during a season)
The REQUIREMENTS for the BALANCED ROUND ROBIN algorithm, for the Team championship scheduling may be found here:
Constellation Algorithm - Balanced Round Robin
The requirements of the algorithm can be defined by these four constraints:
1) All versus all
Each team must meet exactly once, and once only, the other teams in the division/ league.
If the division is composed of n teams, the championship takes place in the n-1 rounds.
2) Alternations HOME / AWAY rule
The sequence of alternations HOME / AWAY matches for every teams in the division league, should be retained if possible. For any team in the division league at most once in the sequence of consecutive matches HAHA, occurs the BREAK of the rhythm, i.e. HH or AA match in the two consecutive rounds.
3) The rule of the last slot number
The team with the highest slot number must always be positioned in the last row of the grid. For each subsequent iteration the highest slot number of grid alternates left and right position; left column (home) and right (away).
The system used to compose the league schedule is "counter-clockwise circuit." In the construction of matches in one round of the championship, a division with an even number of teams. If in a division is present odd number of teams, it will be inserted a BYE/Dummy team in the highest slot number of grid/ring.
4) HH and AA non-terminal and not initial
Cadence HH or AA must never happen at the beginning or at the end of the of matches for any team in the division.