Reconstruction of complete interval tournaments

📝 Abstract

Let $a, b$ and $n$ be nonnegative integers $(b \geq a, \ b > 0, \ n \geq 1) $, $\mathcal{G}_n(a,b)$ be a multigraph on $n$ vertices in which any pair of vertices is connected with at least $a$ and at most $b$ edges and \textbf{v =} $(v_1, v_2, ..., v_n)$ be a vector containing $n$ nonnegative integers. We give a necessary and sufficient condition for the existence of such orientation of the edges of $\mathcal{G}_n(a,b) $, that the resulted out-degree vector equals to \textbf{v}. We describe a reconstruction algorithm. In worst case checking of \textbf{v} requires $\Theta(n)$ time and the reconstruction algorithm works in $O(bn^3)$ time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the score sequences of tournaments are special cases $b = a = 1$ resp. $b = a \geq 1$ of our result.

💡 Analysis

Let $a, b$ and $n$ be nonnegative integers $(b \geq a, \ b > 0, \ n \geq 1) $, $\mathcal{G}_n(a,b)$ be a multigraph on $n$ vertices in which any pair of vertices is connected with at least $a$ and at most $b$ edges and \textbf{v =} $(v_1, v_2, ..., v_n)$ be a vector containing $n$ nonnegative integers. We give a necessary and sufficient condition for the existence of such orientation of the edges of $\mathcal{G}_n(a,b) $, that the resulted out-degree vector equals to \textbf{v}. We describe a reconstruction algorithm. In worst case checking of \textbf{v} requires $\Theta(n)$ time and the reconstruction algorithm works in $O(bn^3)$ time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the score sequences of tournaments are special cases $b = a = 1$ resp. $b = a \geq 1$ of our result.

📄 Content

Acta Univ. Sapientiae, Informatica, 1, 1 (2009) 71–88 Reconstruction of complete interval tournaments Antal Iv´anyi E¨otv¨os Lor´and University, Department of Computer Algebra 1117 Budapest, P´azm´any P´eter s´et´any 1/C. email: tony@compalg.inf.elte.hu Abstract. Let a, b and n be nonnegative integers (b ≥a, b > 0, n ≥ 1), Gn(a, b) be a multigraph on n vertices in which any pair of vertices is connected with at least a and at most b edges and v = (v1, v2, . . . , vn) be a vector containing n nonnegative integers. We give a necessary and sufficient condition for the existence of such orientation of the edges of Gn(a, b), that the resulted out-degree vector equals to v. We describe a reconstruction algorithm. In worst case checking of v requires Θ(n) time and the reconstruction algorithm works in O(bn3) time. Theorems of H. G. Landau (1953) and J. W. Moon (1963) on the score sequences of tournaments are special cases b = a = 1 resp. b = a ≥1 of our result. 1 Introduction Ranking of objects is a typical practical problem. One of the popular ranking methods is the pairwise comparison of the objects. If the result of a compar- ison is expressed by dividing points between the corresponding objects, then directed graphs serve as natural tools to represent the results: vertices cor- respond to the objects, arcs to the points and out-degrees serve as basis for ranking. Another natural tool to represent the results is a point table. In this paper the terminology of D. E. Knuth [9] and the pseudocode of T. H. Cormen and his coauthors [2] are used. AMS 2000 subject classifications: 05C20, 68C25 CR Categories and Descriptors: F.2 [Theory of Computation]: Subtopic – Analysis of algorithms and problem complexity Key words and phrases: score sequences, tournaments, efficiency of algorithms 71 arXiv:1003.4016v1 [cs.DM] 21 Mar 2010 72 A. Iv´anyi Let a, b and n be nonnegative integers (b ≥a, n ≥1), Tn(a, b) be a di- rected multigraph on n vertices in which any pair of vertices is connected with at least a and at most b arcs. Then Tn(a, b) is called interval or (a, b)-tournament, its vertices are called players, the out-degree sequence v = (v1, v2, . . . , vn) is called score vector and the comparisons are called matches. For the simplicity we suppose that v1 ≤v2 ≤· · · ≤vn. The increas- ingly ordered score vector is called score sequence and is denoted by s = (s1, s2, . . . , sn). If any integer partition of the points is permitted, then the tournament is complete, otherwise incomplete [7]. If a = b ≥1, then we get multitournaments Tn(a) and if a = b = 1, then we get the well-known concept of tournaments Tn. In 1953 H. G. Landau [10] proved the following popular theorem. About ten proofs are summarised by K. B. Reid [14] and two recent ones are due to J. Griggs and K. B. Reid [4], resp. to K. B. Reid and C. Q. Zhang [15]. Pirzada, Shah and Naikoo investigated similar problems [13]. Several exercises on tournaments can be found in the recent book of D. E. Knuth [8]. Theorem 1 A sequence (s1, s2, . . . , sn) satisfying 0 ≤s1 ≤s2 ≤. . . ≤sn is the score sequence of some tournament Tn(1) if and only if k X i=1 si ≥Bk, 1 ≤k ≤n, (1) with equality when k = n. In 1963 J. W. Moon in [11] proved the following generalisation of the Lan- dau’s theorem. Theorem 2 A sequence (s1, s2, . . . , sn) satisfying 0 ≤s1 ≤s2 ≤· · · ≤sn is the score sequence of some a-tournament Tn(a) if and only if k X i=1 si ≥aBk, 1 ≤k ≤n, (2) with equality when k = n. Figure 1 shows the point table of a tournament T6(2, 10). The score sequence of this tournament is s = (9,9,19,20,32,34). Reconstruction of complete interval tournaments 73 Player/Player P1 P2 P3 P4 P5 P6 Score P1 — 1 5 1 1 1 9 P2 1 — 4 2 0 2 9 P3 3 3 — 5 4 4 19 P4 8 2 5 — 2 3 20 P5 9 9 5 7 — 2 32 P6 8 7 5 6 8 — 34 Figure 1: The results of the matches of six players. We wish to decide whether there exist tournaments with a given score se- quence and if yes, then we wish to reconstruct one of them. Our problems can be formulated also as follows [3]. Let Gn be a multi- graph in which the number of connecting edges lies between a and b for any pair of vertices. Design effective algorithms to decide whether there exist an orientation of the edges guaranteeing a prescribed out-degree sequence and to reconstruct a corresponding digraph. We remark that Gy´arf´as et al. [5] and Brualdi [1] published quick algorithms for 1-tournaments. Also it is worth to remark that many enumeration type results are known. In connection with classical tournaments it is known due to P. Tetali [16] that only a few score sequences permit the reconstruction in a unique way: typical is the large number of nonisomorph reconstructions. G. P´echy and L. Sz˝ucs [12] proposed a parallel algorithm for generation of all possible score sequences of the 1-tournaments of n players. The aim of this paper is to solve the decision and reconstruction problems [6] for complete (a, b)-tournaments. 2 Necessary conditions for (a, b)-tournaments It is

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