Ranking the vertices of an r-partite paired comparison digraph
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1 Ranking the vertices of an r-partite paired comparison digraph Mikio Kano Department of Mathematics, Akashi Technological College, Uozumi, Akashi 674, Japan Abstract A paired comparison digraph D is a weighted digraph in which the sum of the weights of arcs, if any, joining two vertices exactly one. A one-to-one mapping from V (D) onto {1, 2,..., V (D) } is called a ranking of D, and a ranking α of D is optimal if the backward length of α is minimum. We say that D is r-partite if V (D) can be partitioned into V 1 V r so that every arc of D joining a vertex of V i to a vertex of V j, where i j. We show that we can easily obtain all the optimal ranking of a certain r-partite paired comparison digraph. 1 Introduction We consider a weighted digraph D with vertex set V (D) and arc set A(D). We denote the weight of an arc vw by ε(vw), where vw joins a vertex v to a vertex w. A weighted digraph D is called a paired comparison digraph(or briefly PCD) is D satisfies the following three conditions: (i) 0 < ε(vw) 1 for every vw A(D). (ii) ε(vw) + ε(wv) = 1 if vw, wv A(D). (iii) ε(vw) = 1 if vw A(D) and wv / A(D). A digraph D can be considered as a PCD if we set the weight of each arc of D as follows: (iv) ε(vw) = ε(wv) = 0.5 if vw, wv A(D), and (v) ε(vw) = 1 if vw A(D) and wv / A(D). A PCD D is called an r-partite PCD if V (D) can be partitioned into V (D) = V 1 V r so that (vi) if v, w V i, then v and w are not joined by an arc for all i, 1 i r. An r-partite PCD D with partition V (D) = V 1 V r is called an r-partite complete PCD(see Fig.1) if 1
2 (vii) any two vertices v V i and w V j (i j) are joined by at least one arc. a 1 a 2 b 1 3 b 2 6 c 1 c 2 c 3 Fig.1. A 3-partite complete PCD. The weight of each arc is given by (iv) or (v)(i.e. 1 or 5). An r-partite complete PCD can represent the outcomes of plays between r teams with all pairs of plays of different teams, in which wa allow ties (ε(vw) = ε(wv) = 0.5) and also more than one plays between the same players(ε(vw) = ε 1 > 0 and ε(wv) = 1 ε 1 > 0 mean that v beats w with rate ε 1 and w beats v with rate 1 ε 1 ). We introduced a new method of ranking the vertices of a PCD in [2], and defined optimal rankings, by which we can rank the vertices of a PCD. In this paper, we shall show that the optimal rankings of an r-partite complete PCD can be easily obtained. Moreover, if the number of uncompared pairs of an r-partite PCD is small, then we can easily obtain the optimal rankings of it. Note that it is an Np-complete problem to obtain the optimal rankings of any PCD(see [2]). We now brifly explain our method of ranking. Let D be a PCD with n vertices. A ranking α of D is a one-to-one mapping from V (D) onto {1, 2,..., n}. For a ranking α of D, the image α(v) of v is called the ranking of v defined by α. An arc wv such that α(v) < α(w) is called a backward arc of α, and we write B(α) for the set of all backward arcs of α, that is, B(α) = {wv A(D) α(v) < α(w)}. We define the backward length B(α) of α by B(α) = ε(wv)(α(w) α(v)). wv B(α) A ranking α of D is said to be optimal if the backward length of α is minimum among the backward lengths of all rankings of D. We denote by 2
3 OR(D) the set of all optimal rankings of D, and our method of ranking the vertices of D is one making use of π(v) = 1 OR(D) α OR(D) α(v) for all v V (D). Of course, v is stronger than w if π(v) < π(w). In particular, the champion is the player whose value of π ia minimum. We denote a ranking α of D by α = [v 1, v 2,..., v n ] if V (D) = {v 1,..., v n } and α(v i ) = i for all i, 1 i n. For a ranking α of D and a subset X of V (D), we define the restriction α X : X {1, 2,..., X } by α X (x) = {v X α(v) α(x)} for all x X. The score(positive score)σ + (v) of v V (D) is the sum of the weights of all arcs vw, w V (D)/{v}. Then σ + (v) = σ + D (v) = ε(vw). vw A(D) The negative score σ (v) can be defined analogously. For a ranking α of an r-partite PCD D with partition V (D) = V 1 V r, we define a function Ψ(α, v) on V (D) by Ψ(α, v) = Ψ D (α, v) = σ + (v) + V t α Vt (v), where v V t for all v V (D). Our main theorem is the following: Theorem 1 Let D be an r-partite complete PCD with partite sets V 1,..., V r, and put V (D) = n. Then a ranking α = [v 1,..., v n ] of D is optimal if and only if the following two conditions hold. (1) For every U {V 1,..., V n }, put α U = [u 1,..., u k ]. Then σ + (u 1 ) σ + (u k ). (2) Ψ(α, v 1 ) Ψ(α, v n ). For example, let D be a 3-partite complete PCD given in Fig.1, and α be an optimal ranking of D. Set A = {a 1, a 2 }, B = {b 1, b 2 } and C = {c 1, c 2, C 3 }. Since σ + (a 1 ) = 2C σ + (a 2 ) = 2.5, σ + (b 1 ) = σ + (b 2 ) = 2.5, σ + (c 1 ) = 3.5C σ + (c 2 ) = 2, and σ + (c 3 ) = 2, we have α A = [a 2, a 1 ], α B = [b 1, b 2 ] of [b 2, b 1 ], and α C = [c 1, c 3, c 2 ]. Thus Ψ(α, a 2 ) = 3.5, Ψ(α, a 1 ) = 2, Ψ(α, b 1 ) = 3.5, Ψ(α, b 2 ) = 2.5(or Ψ(α, b 2 ) = 3.5, and Ψ(α, b 1 ) = 2.5), Ψ(α, c 1 ) = 5.5, Ψ(α, c 3 ) = 3 and Ψ(α, c 2 ) = 1. Therefore α = [c 1, u, w, c 3, b 2, a 1, c 2 ], where 3
4 {u, w} = {a 2, b 1 }, or α = [c 1, u, w, c 3, b 1, a 1, c 2 ], where {u, w} = {a 2, b 2 }. In particular, OR(D) = 4. We conclude this section by giving a conjecture. A PCD D is said to be ranking equal if π(v) = ( V (D) + 1)/2 for all v V (D). A PCD D is said to be balanced(regular) if σ + (v) = σ (v) for all v V (D). Conjecture. Let D be a PCD with the weight of every arc 1(i.e. D is an oriented digraph.) Then D is ranking equal if and only if D is balanced. We can prove that every balanced PCD is ranking equal, and show that the condition that the weight of every arc is 1 is necessary(see [2]). 2 Proof of Theorem 1 Let D be a PCD. We define a function µ : V (D) V (D) {0, 1} by { 1 if v and w are joined by an arc, µ(vw) = µ(wv) = 0 otherwise. For a ranking α of D such that α(v) = k and α(w) = k + m > k, let αm k denote the ranking defined by k + m if x = v, αm(x) k = k if x = w, α(x) otherwise. Lemma 1[2]. Let α be a ranking of D such that α(v) = k and α(w) = k + m > k. Then B(αm) k B(α) = m(σ + (v) σ + (w)) + m( (µ(wx) µ(vx)) + k+m<α(x) k<α(y)<k+m (α(y) k)(µ(wy) µ(vy))) Lemma 2. Let D be an r-partite complete PCD with partite sets V 1,..., V r, and let X, Y {V 1,..., V r }, X Y. Then (1) If v, w X, α(w) = k + m > k, then B(α k m) B(α) = m(σ + (v) σ + (w)). (2) If v X, w Y, α(v) = k and α(w) = k + m > k and there is no vertex u X Y such that α(v) < α(u)α(w), then B(α k m) B(α) = m(ψ(α, v) Ψ(α, w)). 4
5 Proof. (1) It is obvious that µ(vz) = µ(wz) for all z V (D). Hence (1) is an easy consequence of Lemma 1. (2) Since µ(vz) = µ(wz) for all z V (D)/(X Y ), µ(vx) = 0 and µ(wx) = 1 for all x X, and µ(vy) = 1 and µ(wy) = 0 for all y Y, we have by Lemma 1 that B(α k m) B(α) = m(σ + (v) σ + (w)) + m( {x X k + m < α(x)} {y Y k + m < α(y)}) = m(σ + (v) + X α X (v) (σ + (w) + Y α Y (w))) = m(ψ(α, v) Ψ(α, w)). Proof of Theorem 1. We first prove the necessity. Assume α is an optimal ranking of D. Then (1) of the theorem fpllpws immediately from Lemma 2. We next prove (2). Suppose that there exist v, w V (D) such that α(v) < α(w) and Ψ(α, v) < Ψ(α, w). By (1), we may assume v V s, w V t and s t. Choose vertices v 1 V s and w 1 V t so that α(v) α(w 1 ) < α(w 1 ) α(w) and there are no vertices x V s V t such that α(v 1 ) < α(x) < α(w 1 ). Then we have σ + (v) σ + (v 1 ) and σ + (w 1 ) σ + (w) by (1), and so ΨΨ(α, v) Ψ(α, v 1 ) and Ψ(α, w 1 ) Ψ(α, w). If α(v 1 ) = k and α(w 1 ) = k + m, then we obtain by Lemma 2 that 0 B(α k m) B(α) = m(ψ(α, v 1 ) Ψ(α, w 1 )). Hence Ψ(α, v) Ψ(α, w), a contradiction. Consequently (2) is proved. We next prove the sufficiency. Let α be a ranking which satisfies the conditions (1) and (2), and β be an optimal ranking. Note that β also satisfies the conditions (1) and (2) since we proved the necessity. Suppose α U = [u 1,..., u t, y,...] and x y for some U {V 1,..., V r }. Then σ + (x) = σ + (y), and we difine a ranking α of D by α(y) if u = x α (u) = α(x) if u = y. α(u) otherwise It is clear that α U = [u 1,..., u t, y,...], and it follows from Lemma 2 that B(α ) = B(α). By repeating this procedure, we can get a ranking γ such that B(γ) = B(α), γ U = β U for all U {V 1,..., V r }, and γ satisfies the conditions (1) and (2). It is easy to see that for any vertices v, w V i, 1 i r, we have Ψ(γ, v) = Ψ(β, v), and Ψ(γ, v) Ψ(γ, w) if v w. Therefore, we obtain B(γ) = B(β) by (2) of Lemma 2, and conclude that α is an optimal ranking. 5
6 An r-partite complete PCD D with partite sets V 1,..., V r is called a complete PCD if V i = 1 for all i, 1 i r. Then any two vertices of a complete PCD are joined by at least one arc. Corollary[2]. Let D be a complete PCD and α = [v 1,..., v r ] be a ranking of D. Then α is an optimal ranking if and only if σ + (v 1 ) σ + (v n ). Proof. Since Ψ(α, v) = σ + (v) for all v V (D), the corollary follows immediately from Theorem 1. When we want ro rank the teams V 1,..., V r instead of player v 1,..., v n, we can rank the teams as follows by using the corollary mentioned above. Let D be an r-partite complete PCD with partite sets V 1,..., V r. We first construct a complete PCD D with vertex set V (D ) = {V 1,..., V r } in which the weight of each arc V i V j is given by ε(v i V j ) = 1 V i V j ε(uw) where the summation is over all uw A(D) such that u V i and w V j. Applying the corollary to D, we can obtain all the optimal rankings of D. It is easy to see that a semicomplete PCD(see[2]) is an r-partite complete PCD each of whose partite sets consists of one vertex or two vertices. Hence, the theorem in [2], by which we can get all the optimal ranking of a semicomplete PCD, is also a corollary of Theorem 1. 3 An r-partite PCD Let D be an r-partite PCD with partite sets V 1,..., V r. An unordered pair {v, w} of vertices of D is called an uncompared pair of D if v and w are not joined by arcs an if v and w are contained in distinct partite sets. Let U (D) denote the set of all uncompared pairs of D. An r-partite complete PCD obtained from D by adding exactly one of arcs vw and wv for every uncompared pair {v, w} of D is called an r-partite completion of D(which is called an r-partite completeness of D in [2]). It is clear that if U (D) = t, then there exist 2 t r-partite completion of D, and we denote by C (D) the set of all r-partite completion of D. For example, let D be a 3-partite PCD given in Fig. 2. Then U (D) = {{a 2, b 1 }, {a 1, c 1 }} and C (D) = {D + a 2 b 1 + a 1 c 1, D + a 2 b 1 + c 1 a 1, D + b 1 a 2 + a 1 c 1, D + b 1 a 2 + c 1 a 1 }. In this section we shall U (D) small. In order to do so, we need the next theorem. 6
7 Theorem 2 Let D ba an r-partite complete PCD with n vertices, and let α be a ranking of D. Then B(α) = Ψ(α, v)α(v) 1 6 n(n2 1). v V (D) Proof. We begin with a new notation. A function ε : V (D) V (D) [0, 1] is defined by { ε(vw) if vw A(D), ε = 0 otherwise. It is trivial that σ + D (v) = x V (D) ε(vx). a 2 b 1 a 2 b a 1 b 2 a 1 b 1 7 c 1 c 2 c 3 c 1 c 2 c 3 Fig.2. A 3-partite complete PCD D andd + a 2 b 1 + a 1 c 1. The weight of each arc is 1 or 0.5. We prove the theorem by induction on n. If n = 1 or 2, then the theorem holds at once. Suppose that the equation holds for n = k, and let n = k+1 3. Let w be the vertex such that α(w) = n, and assume that w V t. Put U = V t /{w}, V (D) = X U {w}(disjoint union) and H = D w, which is a PCD obtained from D by deleting w together with its incident arcs. Note that V (H) = X U. By the inductive hypothesis, we have B D (α) = B H (α) + ε(wv)(n α(v)) = v V (D) v V (H) Ψ H (α, v)α(v) 1 6 k(k2 1) + x i nx ε)(wx)(n α(x)). Since ε(wx) + ε(xw) = 1 for all x X, we obtain ε(wx)(n α(x)) = n ε(wx) + x X( ε(xw) 1)α(x) = nσ + D (w) + ε(xw)α(x) x i nx α(x). 7
8 Hence B D (α) = Ψ D (α, x)α(x) ε(xw)α(x) + Ψ D (α, u)α(u) u U u U = v V (D) α(u) 1 6 k(k2 1) + nσ + D (w) + Ψ D (α, v)α(v) 1 6 k(k2 1) 1 k(k + 1) 2 (by Ψ D (α, w) = nσ + D (w) and u U ε(xw)α(x) α(x) α(u) + α(x) = 1 k(k + 1)) 2 = v Ψ D (α, v)α(v) 1 k(k + 1) 2 = v Ψ D (α, v)α(v) 1 6 n(n2 1). For an r-partite PCD D, we denote by l(d) the backward length of an optimal ranking of D. Namely, l(d) = B(α) for α OR(D). Lemma 3[2]. Let D be an r-partite PCD, and let C (D) = {D 1,..., D t }. Then l(d) = min{l(d 1 ),..., l(d t )}. Moreover, if {D i C (D) l(d i ) = l(d)} = {D a,..., D c }, then OR(D) = OR(D a ) OR(D c ) (disjoint union). By Lemma 3 and Theorems 1 and 2, we can easily obtain all the optimal rankings of an r-partite PCD D if D has a small number of uncompared pairs. For example, let D be a 3-partite PCD given in Fig. 2. Then C (D) = {D 1 = D + a 2 b 1 + a 1 c 1, D 2 = D + a 2 b 1 + c 1 a 1. It follows from Theorems 1 and 2 that D 3 = D + b 1 a 2 + a 1 c 1, D 4 = D + b 1 a 2 + c 1 a 1 }. α 1 = [c 1, a 1, b 2, a 2, c 3, b 1, c 2 ] OR(D 1 ), l(d 1 ) = B(α 1 ) = (7 2 1)/6 = 13, α 2 = [c 1, a 2, b 2, c 3, a 1, b 1, c 2 ] OR(D 2 ), l(d 2 ) = 9, α 3 = [c 1, a 1, b 1, c 3, b 2, a 2, c 2 ] OR(D 3 ), l(d 3 ) = 12, α 4 = [c 1, b 1, a 1, c 3, b 2, a 2, c 2 ] OR(D 4 ) and l(d 4 ) = 10. 8
9 Hence, by Lemma 3, we have OR(D) = OR(D 2 ) = {α = [c 1, a 2, b 2, c 3, u, w, c 2 ] {u, w} = {a 1, b 1 }}. We conclude this section with a remark on forward optimal rankings. Let α be a ranking of a PCD D. An arc vw of D is called a forward arc of α if α(v) < α(w). We write F (α) for the set of all forward arcs of α,and define the forward length F (α) od α by F (α) = ε(vw)(α(w) α(v)). vw F (α) We say that α is a forward optimal ranking of D if F (α) is maximum(not minimum). Some results on forward optimal rankings can be found in [1]. For a ranking α of an r-partite PCD D with partite sets V 1,..., V r, we define a function Φ(α, v) on V (D) by Φ(α, v) = σ (v) + V t α Vt (v), where v V t for all v V (D). Then the following lemma holds, which can be proved as Theorem 2. Lemma 4. Let D be an r-partite complete PCD with n vertices, and let αbe a ranking of D. Then F (α) = v V (D) Ψ(α, v)α(v) 1 6 n(n2 1). Note that it seems to be difficult to characterize forward optimal rankings of an r-partite complete PCD. References [1] M. Kano and A. Sakamto, Ranking the vertices of a weighted digraph using th elength of forward arcs, Networks,13 (1983), [2] M. Kano and A. Sakamto, Ranking the vertices of a paired comparison digraph, SIAM Algebraic Discrete Methods,6 (1985),
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