Optimal Groomings with Grooming Ratios Six and Seven. Issue 9 (14th May 2015)

• Record Type: Journal Article
• Authors: Ge, Gennian; Kolotoğlu, Emre; Wei, Hengjia
• Is Part Of: Journal of combinatorial designs. Volume 23:Issue 9(2015:Sep.)
• Page Start: 400
• Page End: 415
• Publication Date: 2015-05-14
• Subjects: Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21428
• Languages: English
• ISSNs: 1063-8539

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Grooming uniform all‐to‐all traffic in optical (SONET) rings with grooming ratio <italic>C</italic> requires the determination of a decomposition of the complete graph into subgraphs each having at most <italic>C</italic> edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal <italic>C</italic>‐groomings has been considered for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1m9c40g3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21428:jcd21428-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>3</mml:mn><mml:mo>≤</mml:mo><mml:mi>C</mml:mi><mml:mo>≤</mml:mo><mml:mn>9</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>, and completely solved for <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1m9c40fj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21428:jcd21428-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>3</mml:mn><mml:mo>≤</mml:mo><mml:mi>C</mml:mi><mml:mo>≤</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. For <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1m9c40d0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21428:jcd21428-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>, it has been shown that the lower bound for the drop cost of an optimal <italic>C</italic>‐grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1m9c40cf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21428:jcd21428-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>, there are infinitely many unsettled orders; especially the case <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1m9c40wq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10638539:media:jcd21428:jcd21428-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mspace width="4.44443pt" /><mml:mo>(</mml:mo><mml:mo form="prefix">mod</mml:mo><mml:mspace width="0.28em" /><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> is far from complete. In this paper, we show that the lower bound for the drop cost of a 6‐grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7‐grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.</p> </abstract>