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}{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \author[1]{Raghad Rowaida} \author[2]{Afsana Ahmed Munia} \affil[1]{ } \renewcommand\Authands{ and } \date{\small \em Received: 12 December 2016 Accepted: 1 January 2017 Published: 15 January 2017} \maketitle \begin{abstract} The problem of ranking players in a round- robin tournament, in which outcome of any match is a win or a loss, is to rank players according to their performances in the tournament. In this paper, we have improved previously developed MST (Majority Spanning Tree) algorithm for solving this problem, where the number of violations has been chosen as the criterion of optimality. We have compared the performance of our algorithm with the MST algorithm and GIK algorithm. \end{abstract} \keywords{ranking, round-robin tournament, upset, digraph, MST, GIK.} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \let\tabcellsep& \section[{I. Introduction}]{I. Introduction}\par he problem of ranking players in a tournament has been the subject of various research investigations. This tournament structure also arises in other environments like the problems of soliciting customer preferences of a set of products, establishing funding priorities of a set of projects \hyperref[b4]{[5]}, establishing searching priorities for a set of search engines in the internet. It is known that the results of a tournament can be represented in adigraph, G=(V, A) known as tournament graph, where vertices correspond to players and arcs correspond to match results. A tournament result is said to be upset (or violation) if a lowly-ranked player has defeated a highly-ranked player. Ali \hyperref[b0]{[1]}, Cook \hyperref[b5]{[6]}, Goddard \hyperref[b4]{[5]}, Poljak \hyperref[b2]{[3]} and many others have concentrated on the problem of determining ranks based on the results of the tournament. A constructive lower bound on the tournament ranking function was obtained in \hyperref[b3]{[4]}. In \hyperref[b1]{[2]}, a heuristic solution to optimize the number of violations has been developed. This paper presents a new version of MST algorithm which reduces the number of violations compared to MST algorithm. The problem of minimizing the number of upsets is equivalent to finding the minimum number of arcs in adigraph deletion of which results in an acyclic digraph. \section[{II. Preliminaries}]{II. Preliminaries}\par Before describing the new algorithm, we present here a brief discussion on MST algorithm \hyperref[b1]{[2]} and GIK algorithm \hyperref[b0]{[1]}. MST: For ease of discussion we recapitulate some of the definitions used in MST algorithm.\par 1. cutset(i, k, j) -is the difference between the numbers of outgoing arcs from set (i, k) to set (k + 1, j) and outgoing arcs from set (k + 1, j) to set (i, j), where set (i, j) is the set of vertices corresponding to players ranked from i to j. 2. maxwin(i, j) -is the maximum number of wins of a player in set (i, j). 3. pair(i, j) -corresponds to an upset if the player ranked j defeats the player ranked i. MST () Repeat until swap = false swap ?false for i= 1 to size-1 do for j = i + 1 to size do for k=i to j-1 do if cutset(i,k,j)< 0 swap ?true elseif cutset(i,k,j)= 0 if pair(i,j) or ( i -l , k + l) or (k,j+ l) is upset then swap ? true swap respective pair else if maxwin(i, k) < maxwm(k + 1, j) swap respective pair endif endif if swap = true then swap set (\{i, k), \{k+ 1,j\}) © 2017 Global Journals Inc. (US) ( ) \section[{G}]{G}\par This problem is knownas Minimum Feedback Arc set Problem, and is NP-hard for general digraphs \hyperref[b0]{[1]}.\par Assuming the number of players in the tournament to be n, complexity of the MST algorithm can be derived as follows: In the k-loop, calculation of cutset value requires O(n) operations. Each of the i, j and k-loop will be done at most n times for a single swap, which will reduce the number of violations by 1. The amount of computation for this is at most O(n 4 ). Since there can be at most O(n 2 ) violations initially, the algorithm requires at most O(n 6 ) calculations. \section[{GIK: This algorithm is based on the IK algorithm [ ].}]{GIK: This algorithm is based on the IK algorithm [ ].}\par When applying the IK algorithm to rank a tournament, two basic steps are executed in case of a tie. The first attempts to break the tie by restoring the players, while the second (which is applied when the first step fails) randomly ranks the players involved in the tie. The GIK algorithm differs from the IK procedure in these two steps. The restoring method is different, and if this restoring method does not resolve the ties, an attempt is made to rank the players in a manner that will reduce the overall number of violations.\par The GIK algorithm appears below. > . . . >P k ) and R, = (Q 1 > Q 2 > . ?> Q J , then R1||R2 , = (P 1 > P 2 > . . . >P k > Q 1 > Q 2 > . . . > Q J ). \section[{The GIK Algorithm}]{The GIK Algorithm}\par Let R = á´?", A = \{P \section[{III. THE NEW ALGORITHM}]{III. THE NEW ALGORITHM}\par In this Section we propose A new version of MST algorithm that results in minimum number of upset compared to the MST algorithm and GIK algorithm for ranking players in a round-robin tournament [].\par We consider only simple connected digraphs G=(V,A). Spanning trees of any digraph are denoted by T. A directed cutset(V i ,V j ) is defined as(V i ,V j )=\{(k,l)|k ? V i, l ? V j \}\par For improvement of the algorithm we introduce the following symbols and functions: Sa -start of setA Ea -end of setA Sb -start of setB Eb -end of setB Sc -start of setC Ec -end of setC Cutset(A,B)-is the difference between the numbers of outgoing arcs from set A to set B and outgoing arcs from set B to set A.\par Cutset(A,C)-is the difference between the numbers of outgoing arcs from set A to set C and outgoing arcs from set B to set A.\par Cutset(B,C)-is the difference between the numbers of outgoing arcs from set B to set C and outgoing arcs from set C to set B.\par Procedure: Improved MST 3. \section[{4.}]{4.}\par 5. \section[{6.}]{6.}\par 7. 8. \section[{9.}]{9.}\par 10. \section[{11.}]{11.}\par 12. 13.\par 14. 15. 16. \section[{IV. Experimental Results}]{IV. Experimental Results}\par The new\textunderscore MST Algorithm has been compared with MST Algorithm and the GIK algorithm on the basis of a set of randomly generated tournaments of sizes ranging from 5 to 50 players. All algorithms have been programmed in C and runs were made on a core i3 machines. We have been measured both in terms of violations and computational time. Here new\textunderscore MST gives better result compared to MST and GIK with respect to number of violations. \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{} \end{longtable} \par \caption{\label{tab_1}}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.85\textwidth}} and D = D\textbackslash \{Pi\}. If |D| = á´?", then go to\\ (2).Otherwise go to (4).\\ Let i=i + 1, and go to (12).\\ Execute procedure Arrange on the ranking R.\\ End.\\ If A = á´?", then to go (15); otherwise determine\\ the current scores of players in A.\\ If A = á´?", then go to (15); otherwise determine D,\\ the dominant set.\\ If |D|>1, then to go (6).\\ Letting P denote the only player in D, form the\\ ranking R = R || P , let A = A\textbackslash \{P\} and go to\\ (3).\\ If from the last time of updating the current\\ scores of A [step (2)], set A has changed, then\\ go to(2).\end{longtable} \par {\small\itshape [Note: 1 , P 2 , . ., P n \}. If |D| >2, then go to (9). Let P 1 and P 2 , denote the players in D with P 1 > P 2 . Let R= R|| P 1 ||P 2 , and A=A\textbackslash \{P 1 , P 2 \}. Go to (2). If R = á´?", then go to (11).]} \caption{\label{tab_2}}\end{figure} \begin{figure}[htbp] \noindent\textbf{1} \par \begin{longtable}{P{0.10625\textwidth}P{0.22037037037037036\textwidth}P{0.17708333333333334\textwidth}P{0.16527777777777777\textwidth}P{0.18101851851851852\textwidth}} No of player\tabcellsep Initial upset\tabcellsep GIK\tabcellsep MST\tabcellsep New MST\\ 5\tabcellsep 3.66\tabcellsep 2.66\tabcellsep 1.66\tabcellsep 1.66\\ 10\tabcellsep 24.00\tabcellsep 13.33\tabcellsep 9.00\tabcellsep 8.66\\ 15\tabcellsep 47.33\tabcellsep 39.33\tabcellsep 25.66\tabcellsep 24.66\\ 20\tabcellsep 89.00\tabcellsep 38.33\tabcellsep 25.33\tabcellsep 22.00\\ 25\tabcellsep 194.33\tabcellsep 109.66\tabcellsep 76.33\tabcellsep 72.33\\ 30\tabcellsep 106.66\tabcellsep 94.66\tabcellsep 67.33\tabcellsep 61.00\\ 40\tabcellsep 482.00\tabcellsep 138.66\tabcellsep 88.66\tabcellsep 79.00\\ 50\tabcellsep 585.66\tabcellsep 515.00\tabcellsep 439.00\tabcellsep 418.33\end{longtable} \par \caption{\label{tab_3}Table 1 :}\end{figure} \begin{figure}[htbp] \noindent\textbf{2} \par \begin{longtable}{P{0.10917431192660551\textwidth}P{0.3002293577981651\textwidth}P{0.15206422018348625\textwidth}P{0.25344036697247707\textwidth}P{0.03509174311926606\textwidth}} No of player\tabcellsep GIK\tabcellsep MST\tabcellsep New MST\\ 5\tabcellsep 0.0013\tabcellsep 0.0030\tabcellsep 0.0010\\ 10\tabcellsep 0.0103\tabcellsep 0.0090\tabcellsep 0.0110\\ 15\tabcellsep 0.0173\tabcellsep 0.0093\tabcellsep 0.0680\\ 20\tabcellsep 0.0226\tabcellsep 0.0236\tabcellsep 0.5756\\ 25\tabcellsep 0.0266\tabcellsep 0.0563\tabcellsep 88.5506\\ 30\tabcellsep 0.0320\tabcellsep 0.0216\tabcellsep 1268.37\\ 40 50\tabcellsep \multicolumn{2}{l}{0.043 0.054 V. Conclusion 0.1913 4.147}\tabcellsep 24877.110 63415.8188\tabcellsep Year 2017\end{longtable} \par {\small\itshape [Note: GThis page is intentionally left blank]} \caption{\label{tab_4}Table 2 :}\end{figure} \footnote{© 20 7 Global Journa ls Inc. (US) 1} \backmatter \begin{bibitemlist}{1} \bibitem[Kaykobad et al. ()]{b1}\label{b1} ‘A New Algorithm for Ranking Players of a Round Robin Tournament’. Mohammad Kaykobad , Q N U Ahmed , A T M Khalid , Rezwan-Al Bakhtiar . \textit{Computer Ops Res} 1995. 22 (2) p. . \bibitem[Poljak and Czygrinowy]{b3}\label{b3} ‘Constructive quasi-ramsey numbers and tournament ranking’. S Poljak , A Czygrinowy , V . \textit{SIAM J. Discrete Math} 12 (1) p. . \bibitem[Cook et al. ()]{b5}\label{b5} ‘Heuristics for ranking players in a round-robin tournament’. W D Cook , I Golan , M Kress . \textit{Computers Ops Res} 1988. 15 p. . \bibitem[Ali et al. ()]{b0}\label{b0} ‘On the minimum violations ranking of a tournament’. W D Ali , M Cook , Kress . \textit{M qmt Sci} 1986. 32 p. . \bibitem[Goddard ()]{b4}\label{b4} ‘Ranking tournaments and group decision making’. S T Goddard . \textit{Mgmt Sci} 1983. 29 p. . \bibitem[Poljak et al. ()]{b2}\label{b2} ‘Tournament ranking with expected profit in polynomial time’. S Poljak , V Rodl , J Spencer . \textit{SIAM JI Disc. Math.I} 1988. (3) . \end{bibitemlist} \end{document}