Let , be positive integers, , and vectors of
nonnegative integers with and .
An -supertournament is an sized matrix , whose columns correspond to the players of the tournament (they represent the rankable objects) and the rows correspond to the comparisons of the objects.
The permitted elements of belong to the set , where means, that the player is not a participants of the match corresponding to the -th line, while means, that received points in the match corresponding to the -th line, and .
The sum (dots are taken in the count as zeros) of the elements of the -th column of is denoted by and is called the score of the th player :
The sequence is called the score vector of the tournament. The increasingly ordered sequence of the scores is called the score sequence of the tournament and is denoted by .
Using the terminology of the sports a supertournament can combine the matches of different sports. For example in Hungary there are popular chess-bridge, chess-tennis and tennis-bridge tournaments.
A sport is characterized by the set of the permitted results. For example in tennis the set of permitted results is , for chess is the set , for football is the set and in the Hungarian card game last trick is . There are different possible rules for an
individual bridge tournament, e.g. .
The number of participants in a match of a given sport is denoted by , the minimal number of the distributed points in a match is denoted by , and the maximal number of points is denoted by .
If a supertournament consists of only the matches of one sport, then we use and instead of vectors , and and omit the parameter . When the number of the players is not important, then the parameter is also omitted.
If the points can be divided into arbitrary integer partitions, then the given sport is called complete, otherwise it is called incomplete. According to this definitions chess is a complete (2,2)-sport, while football is an incomplete (2,3)-sport.
Since a set containing elements has -element subsets, an -tournament consists of matches.
If all matches are played, then the tournament is finished, otherwise it is partial.
In this chapter we deal only with finished tournaments and mostly with complete tournaments (exception is only the section on football).
Figure 25.1 contains the results of a full and complete chess+last trick+bridge supertournament. In this example
, , and . In this example the score vector of the
given supertournament is , and its score sequence is .
Figure 25.1. Point matrix of a chess+last trick-bridge tournament with players.
In this chapter we investigate the problems connected with the existence and construction of different types of supertournaments having prescribed score sequences.
At first we give an introduction to -tournaments (Section 25.2), then summarize the results on (1,1)-tournaments (Section 25.3), then for -tournaments (Section 25.4) and for general -tournaments (Section 25.5).
In Section 25.6 we deal with imbalance sequences, and in Section 25.7 with supertournaments. In Section 25.8 we investigate special incomplete tournaments (football tournaments) and finally in Section 25.9 we consider examples of the reconstruction of football tournaments.
Exercises
25.1-1 Describe known and possible multitournaments.
25.1-2 Estimate the number of given types of multitournaments.
2. 25.2 Introduction to -tournaments
Let and be nonnegative integers and let be the set of such generalized tournaments, in which every pair of distinct players is connected by at least , and at most arcs. The elements
of are called -tournaments. The vector of the outdegrees of
is called the score vector of . If the elements of are in nondecreasing order, then is called the score sequence of .
An arbitrary vector of nonnegative integers is called multigraphic vector, (or degree vector) if there exists a loopless multigraph whose degree vector is , and is called dimultigraphic vector (or score vector) iff there exists a loopless directed multigraph whose outdegree vector is .
A nondecreasingly ordered multigraphic vector is called multigraphic sequence, and a nondecreasingly ordered dimultigraphic vector is called dimultigraphic sequence (or score sequence).
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In there exists a simple graph, resp. a simple digraph with degree/out-degree sequence , then is called simply graphic, resp. digraphic.
The number of arcs of going from player to player is denoted by , and the matrix is called the point matrix or tournament of the .
In the last sixty years many efforts have been devoted to the study of both types of vectors, resp. sequences. E.g.
in the papers [27], [71], [86], [96], [100], [103], [101], [109], [130], [131], [129], [139], [204], [250], [253], [260], [262], [276], [285] the multigraphic sequences, while in the papers [1], [12], [16], [27], [41], [78], [90], [91], [95], [98], [103], [111], [148], [149], [159], [171], [183], [184], [185], [189], [190], [197], [198], [199], [211], [236], [239], [243], [279], [284], [289] the dimultigraphic sequences have been discussed.
Even in the last two years many authors investigated the conditions when is multigraphical (e.g. [21], [31], [40], [54], [82], [83], [87], [84], [116], [121], [136], [147], [150], [151], [174], [178], [213], [216], [241], [274], [280], [282], [286], [287], [292]) or dimultigraphical (e.g. [23], [105], [127], [144], [152], [166], [188], [206], [228], [227], [229], [230], [245], [247], [270], [294]).
It is worth to mention another interesting direction of the study of different kinds of tournament, the score sets [217].
In this chapter we deal first of all with directed graphs and usually follow the terminology used by K. B. Reid [237], [239]. If in the given context and are fixed or non important, then we speak simply on tournaments instead of generalized or -tournaments.
The first question is: how one can characterize the set of the score sequences of the -tournaments? Or, with other words, for which sequences of nonnegative integers does exist an -tournament whose outdegree sequence is . The answer is given in Section 25.5.
If is an -tournament with point matrix , then let and be defined as
investigated the construction problem of a minimal size graph having a prescribed degree set [233], [291]. In a similar way we follow a mini-max approach formulating the following questions: given a sequence of nonnegative integers,
• How to compute and how to construct a tournament characterized by ? In Subsection 25.5.3 a formula to compute , and in 25.5.4 an algorithm to construct a corresponding tournament is presented.
• How to compute and ? In Subsection 25.5.4 we characterize and , and in Subsection 25.5.5 an algorithm to compute and is described, while in Subsection 25.5.8 we compute and in linear time.
• How to construct a tournament characterized by and ? In Subsection 25.5.10 an algorithm to construct a corresponding tournament is presented and analyzed.
We describe the proposed algorithms in words, by examples and by the pseudocode used in [57].
3. 25.3 Existence of -tournaments with prescribed score sequence
The simplest supertournament is the classical tournament, in our notation the -tournament.
Now, we give the characterization of score sequences of tournaments which is due to Landau [159]. This result has attracted quite a bit of attention as nearly a dozen different proofs appear in the literature. Early proofs tested the readers patience with special choices of subscripts, but eventually such gymnastics were replaced by more elegant arguments. Many of the existing proofs are discussed in a survey written by K. Brooks Reid [236]. The
proof we give here is due to Thomassen [270]. Further, two new proofs can be found in the paper due to Griggs and Reid [95].
Theorem 25.1 (Landau [159]) A sequence of nonnegative integers is the score vector of a -tournament if and only if for each subset
with equality, when .
This theorem, called Landau theorem is a nice necessary and sufficient condition, but its direct application can require the test of exponential number of subsets.
If instead of the nonordered vector we consider a nondecreasingly ordered sequence , then due to the monotonity the inequalities (25.2), called Landau inequalities, we get the following consequence.
Corollary 25.2 (Landau [159]) A nondecreasing sequence of nonnegative integers is the score sequence of some -tournament, if and only if
for , with equality for .
Proof. Necessity If a nondecreasing sequence of nonnegative integers is the score sequence of an -tournament , then the sum of the first scores in the sequence counts exactly once each arc in the subtournament induced by plus each arc from to . Therefore the sum is at least , the number of arcs in . Also, since the sum of the scores of the vertices counts each arc of the tournament exactly once, the sum of the scores is the total number of arcs, that is, .
Sufficiency (Thomassen [270]) Let be the smallest integer for which there is a nondecreasing sequence of nonnegative integers satisfying Landau's conditions (25.3), but for which there is no -tournament with score sequence . Among all such , pick one for which is as lexicografically small as possible.
First consider the case where for some ,
By the minimality of , the sequence is the score sequence of some tournament . Further,
for each , with the equality when . Therefore, by the minimality of , the sequence is the score sequence of some tournament . By forming the disjoint union of and and adding all arcs from to , we obtain a tournament with score sequence . Now, consider the case where each inequality in (25.3) is strict when (in particular ). Then the sequence satisfies (25.3) and by the minimality of , is the score sequence of some tournament . Let and be the vertices with scores and respectively. Since the score of is larger than that of , then according to Lemma 25.5 [47] has a path from to of length . By reversing the arcs of , we obtain a tournament with score sequence , a contradiction.
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Landau's theorem is the tournament analog of the Erdős-Gallai theorem for graphical sequences [71]. A tournament analog of the Havel-Hakimi theorem [102], [109] for graphical sequences is the following result.
Theorem 25.3 (Reid, Beineke [238]) A nondecreasing sequence of nonnegative integers, , is the score sequence of an -tournament if and only if the new sequence
arranged in nondecreasing order, is the score sequence of some -tournament.
Proof. See [238].
4. 25.4 Existence of an -tournament with prescribed score sequence
For the -tournament Moon [184] proved the following extension of Landau's theorem.
Theorem 25.4 (Moon [184], Kemnitz, Dulff [144]) A nondecreasing sequence of nonnegative integers is the score sequence of an -tournament if and only if
for , with equality for .
Proof. See [144], [184].
Later Kemnitz and Dulff [144] reproved this theorem.
The proof of Kemnitz and Dulff is based on the following lemma, which is an extension of a lemma due to Thomassen [270].
Lemma 25.5 (Thomassen [270]) Let be a vertex of maximum score in an -tournament . If is a vertex of different from , then there is a directed path from to of length at most 2.
Proof. ([144]) Let be all vertices of such that . If then
for the length of path . Otherwise if there exists a vertex , such that
then . If for all then there are arcs which implies
, a contradiction to the assumption that has maximum score.
Proof of Theorem 25.4 [47]. The necessity of condition (25.7) is obvious since there are arcs among any vertices and there are arcs among all vertices.
To prove the sufficiency of (25.7) we assume that the sequence is a counterexample to the theorem with minimum and smallest with that choice of . Suppose first that there exists an integer
, such that
Because the minimality of , the sequence is the score sequence of some -tournament .
Consider the sequence defined by , . Because of by assumption it follows that
which implies . Since is nondecreasing also is a nondecreasing sequence of nonnegative integers.
For each with it holds that
with equality for since by assumption
Therefore the sequence fulfils condition (25.8), by the minimality of , is the score sequence of some -tournament . By forming the disjoint union of and we obtain a -tournament
with score sequence in contradiction to the assumption that is counterexample.
Now we consider the case when the inequality in condition (25.8) is strict for each , . This implies in particular .
The sequence is a nondecreasing sequence of nonnegative integers which fulfils condition (25.8). Because of the minimality of , is the score sequence of some -tournament . Let denote a vertex of with score and a vertex of with score . Since has maximum score in there is a directed path from to of length at most 2 according to Lemma 25.5 [47]. By reversing the arcs of the path we obtain an -tournament with score sequence . This contradiction completes the proof.
5. 25.5 Existence of an -tournament with prescribed score sequence
In this section we show that for arbitrary prescribed sequence of nondecreasingly ordered nonnegative integers there exists an -tournament
5.1. 25.5.1 Existence of a tournament with arbitrary degree sequence
Since the numbers of points are not limited, it is easy to construct a -tournament for any .
Lemma 25.6 If , then for any vector of nonnegative integers there exists a loopless directed multigraph with outdegree vector so, that .
Proof. Let and for , and let the remaining values be equal to zero.
Using weighted graphs it would be easy to extend the definition of the -tournaments to allow arbitrary real values of , , and . The following algorithm, Naive-Construct works without changes also for input consisting of real numbers.
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We remark that Ore in 1956 [197], [198], [199] gave the necessary and sufficient conditions of the existence of a tournament with prescribed indegree and outdegree vectors. Further Ford and Fulkerson [78][Theorem11.1]
published in 1962 necessary and sufficient conditions of the existence of a tournament having prescribed lower and upper bounds for the indegree and outdegree of the vertices. Their results also can serve as basis of the existence of a tournament having arbitrary outdegree sequence.
5.2. 25.5.2 Description of a naive reconstructing algorithm
Sorting of the elements of is not necessary.
Input. : the number of players ;
: arbitrary sequence of nonnegative integer numbers.
Output. : the point matrix of the reconstructed tournament.
Working variables. : cycle variables.
Naive-Construct
1 FOR TO 2 FOR TO 3 4 5 FOR TO 6 7 RETURN
The running time of this algorithm is in worst case (in best case too). Since the point matrix has elements, this algorithm is asymptotically optimal.
5.3. 25.5.3 Computation of
This is also an easy question. From now on we suppose that is a nondecreasing sequence of nonnegative
integers, that is . Let .
Since is a finite set for any finite score vector , exists.
Lemma 25.7 (Iványi [127]) If , then for any sequence there exists a -tournament such that
and is the smallest upper bound for , and is the smallest possible upper bound for .
Proof. If all players gather their points in a uniform as possible manner, that is
then we get , that is the bound is valid. Since player has to gather points, the pigeonhole principle [24], [25], [64] implies , that is the bound is not improvable. implies
. The score sequence
shows, that the upper bound is not improvable.
Corollary 25.8 (Iványi [128]) If , then for any sequence holds . Proof. According to Lemma 25.7 [49] is the smallest upper bound for .
5.4. 25.5.4 Description of a construction algorithm
The following algorithm constructs a -tournament having for any . Input. : the number of players ;
: arbitrary sequence of nonnegative integer numbers.
Output. : the point matrix of the tournament.
Working variables. : cycle variables;
: the number of the ―larger part‖ in the uniform distribution of the points.
Pigeonhole-Construct
1 FOR TO 2 3 4 FOR TO
5 6 7 FOR TO 8
9 10 RETURN
The running time of Pigeonhole-Construct is in worst case (in best case too). Since the point matrix has elements, this algorithm is asymptotically optimal.
5.5. 25.5.5 Computation of and
Let be the sum of the first elements of . be the binomial
coefficient . Then the players together can have points only if . Since the score of player is , the pigeonhole principle implies .
These observations result the following lower bound for :
If every player gathers his points in a uniform as possible manner then
These observations imply a useful characterization of .
Lemma 25.9 (Iványi [127]) If , then for arbitrary sequence there exists a -tournament having as its outdegree sequence and the following bounds for and :
Proof. (25.15) follows from (25.13) and (25.14), (25.16) follows from the definition of .
It is worth to remark, that if is integer and the scores are identical, then the lower and upper bounds in (25.15) coincide and so Lemma 25.9 [50] gives the exact value of .
In connection with this lemma we consider three examples. If
, then and , that is is
twice larger than . In the other extremal case, when and
, then , , so is times larger, than .
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If , then Lemma 25.9 [50] gives the bounds . Elementary calculations show that Figure 25.2 contains the solution with minimal , where .
Figure 25.2. Point matrix of a -tournament with for .
In 2009 we proved the following assertion.
Theorem 25.10 (Iványi [127]) For a nondecreasing sequence of nonnegative integers is the score sequence of some -tournament if and only if
where
The theorem was proved by Moon [184], and later by Kemnitz and Dolff [144] for -tournaments is the special case of Theorem 25.10 [51]. Theorem 3.1.4 of [133] is the special case . The theorem of Landau [159] is the special case of Theorem 25.10 [51].
5.6. 25.5.6 Description of a testing algorithm
The following algorithm Interval-Test decides whether a given is a score sequence of an -tournament or not. This algorithm is based on Theorem 25.10 [51] and returns if is a score sequence, and returns otherwise.
Input. : minimal number of points divided after each match;
: maximal number of points divided after each match.
Output. : logical variable ( shows that is an -tournament).
Local working variable. : cycle variable;
: the sequence of the values of the loss function.
Global working variables. : the number of players ; : a nondecreasing sequence of nonnegative integers;
: the sequence of the binomial coefficients;
: the sequence of the sums of the smallest scores.
Interval-Test
1 FOR TO 2 3 IF 4 5 RETURN 6 IF 7
8 RETURN 9 RETURN
In worst case Interval-Test runs in time even in the general case (in the best case the running time of Interval-Test is ). It is worth to mention, that the often referenced Havel–Hakimi algorithm [100], [109] even in the special case decides in time whether a sequence is digraphical or not.
5.7. 25.5.7 Description of an algorithm computing and
The following algorithm is based on the bounds of and given by Lemma 25.9 [50] and the logarithmic search algorithm described by D. E. Knuth [152][page 410].
Input. No special input (global working variables serve as input).
Output. : (the minimal );
: (the maximal ).
Local working variables. : cycle variable;
: lower bound of the interval of the possible values of ; : upper bound of the interval of the possible values of . Global working variables. : the number of players ;
: a nondecreasing sequence of nonnegative integers;
: the sequence of the binomial coefficients;
: the sequence of the sums of the smallest scores;
: logical variable (its value is , when the investigated is a score sequence).
MinF-MaxG
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5.8. 25.5.8 Computing of and in linear time
Analyzing Theorem 25.10 [51] and the work of algorithm MinF-MaxG one can observe that the maximal value of and the minimal value of can be computed independently by the following Linear-MinF-MaxG.
Input. No special input (global working variables serve as input).
Output. : (the minimal ).
: (the maximal ).
Local working variables. : cycle variable.
Global working variables. : the number of players ; : a nondecreasing sequence of nonnegative integers;
: the sequence of the binomial coefficients;
: the sequence of the sums of the smallest scores.
Linear-MinF-MaxG
Proof. Lines 01, 05, 06, and 16 require only constant time, lines 02–06, 07–10, and 11–15 require time, so the total running time is .
5.9. 25.5.9 Tournament with and
The following reconstruction algorithm Score-Slicing2 is based on balancing between additional points (they are similar to ―excess‖, introduced by Brauer et al. [36]) and missing points introduced in [127]. The greediness of the algorithm Havel–Hakimi [100], [109] also characterizes this algorithm.
This algorithm is an extended version of the algorithm Score-Slicing proposed in [127].
The work of the slicing program is managed by the following program Mini-Max. Input. : the number of players ;
: a nondecreasing sequence of integers satisfying (25.17).
Output. : the point matrix of the reconstructed tournament.
Local working variables. : cycle variables.
Global working variables. : provisional score sequence;
: the partial sums of the provisional scores;
: matrix of the provisional points.
Mini-Max
1 MinF-MaxG Initialization 2 3 4 FOR TO 5 FOR TO 6 7 FOR TO 8 9
10 IF Score slicing for players 11 FOR DOWNTO 12 Score-Slicing2 13 IF Score slicing for 2 players 14 15
16 RETURN
5.10. 25.5.10 Description of the score slicing algorithm
The key part of the reconstruction is the following algorithm Score-Slicing2 [127].
During the reconstruction process we have to take into account the following bounds:
Input. : the number of the investigated players ;
: prefix of the provisional score sequence ; : matrix of provisional points;
Output. : number of missing points : prefix of the provisional score sequence.
Output. : number of missing points : prefix of the provisional score sequence.