On the Monotonicity of Work Function in k-Server Conjecture

On the Monotonicit y of W ork F unction in k-Serv er Conjecture Ming-Zhe Chen DT ec h Inc. 5751 Sells Mill Dr Dublin, OH 43017, USA thedtec h@y aho o.com Octob er 29, 2018 Abstract This pap er presen ts a mistake in w ork fuction algorithm of k-server conjecture. That is, th e mono- tonicit y of the w ork fuction is not alwa ys true. 1 In tro duction The k-server conjecture has not b een prov ed. A lot of literature deal with the k-ser v er conjecture [3] [2] [4] [1] and references therein. In [3], the work function algorithm (WF A) is so far the b est determined algor ithm for this pro blem. In [3], there are the facts as follows (ex cerpts fr o m [3]): F act 3 F or a work function w a nd tw o configurations X, Y w ( X ) ≤ w ( Y ) + D ( X, Y ) (1) Consider a work function w and the resulting work function w ′ after request r . By F act 3 we get w ′ ( X ) = min x ∈ X { w ( X − x + r ) + d ( r , x ) } ≥ w ( X ) (2) which tra nslats to: F act 4 Let w b e a work function a nd let w ′ be the resulting work function after req uest r . Then for a ll configura tio ns X : w ′ ( X ) ≥ w ( X ) (3) But F act 4 is not true. That is, the mo notonicit y of w ork function is not true. 2 The monotonicit y of w ork function F act 4 is not true be cause (2 ) is inco rrect. F rom w ′ ( X ) = min x ∈ X { w ′ ( X − x + r ) + d ( r , x ) } (4) we ca n get (it is F act 3, here it is true): w ′ ( X ) ≤ w ′ ( X − x + r ) + d ( r , x ) (5) That is (b ecause w ′ ( X − x + r ) = w ( X − x + r )): w ′ ( X ) ≤ w ( X − x + r ) + d ( r, x ) (6) 1 Assume that Y is a co nfiguration which makes the minimum o f w ′ ( X ), so w ′ ( X ) = w ( Y ) + D ( X, Y ) (7) But we cannot ge t w ′ ( X ) ≥ w ( X ) (F act 4) from (7) based on F a ct 3 because F a ct 3 is incor r ect for this case. If F act 3 is derived from w ( X ) = min x ∈ X { w ( X − x + r ) + d ( r , x ) } , it is true. But it cannot b e used universally in all other cases b ecause F act 3 is derived under some c o nditions It is known w ( X ) = min x ∈ X { w ( X − x + r ′ ) + d ( r ′ , x ) } (8) where r ′ is the request b efore the request r . Assume that Z is a c onfiguration which makes the minim um of w ( X ), s o w ( X ) = w ( Z ) + D ( X , Z ) (9) In order for F act 4 to b e true, we have to prove the following: w ′ ( X ) = w ( Y ) + D ( X, Y ) ≥ w ( Z ) + D ( X, Z ) (10) Unfortunately , the ab ov e is not alwa ys true. W e give a concrete coun terexample a s follows. A 5-no de w eighted undirected g raph. The no de set is a, b, c , d, e . The distances (edge weigh ts) are as follows. d ( a, b ) = 1 , d ( a, c ) = 7 , d ( a, d ) = 5 , d ( a, e ) = 8 , d ( b, c ) = 4 , d ( b, d ) = 2 , d ( b, e ) = 10 , d ( c, d ) = 3 , d ( c, e ) = 9 , d ( d, e ) = 6 Consider 3-servers on this gr a ph. The initial configuratio n is abc a nd the request sequence are e, d, a, c, b, d In the folloing table we giv e v alues of work functions corresp onding to a ll 3-no de co nfig urations a nd all request sequence. T able: V a lues of W ork F unctions for 3-servers Configuratio n Request i abc abd abe acd ace ade bcd bce bde cde φ 0 0 3 9 2 10 11 5 8 11 10 e 1 16 15 9 16 10 11 14 8 11 10 d 2 18 15 13 16 14 11 14 12 11 10 a 3 18 15 1 3 16 14 11 17 15 12 18 c 4 18 20 18 1 6 14 17 17 15 18 18 b 5 18 20 18 18 16 19 17 15 1 8 17 d 6 20 20 21 18 22 19 17 19 18 17 F rom the above table we can see w edacb ( cde ) < w edac ( cde ) so the F act 4 is not always true. That is , the work function do es not have the monotonicity . In pap er [3], all theorems which are prov ed based on the mono tonicit y hav e to b e re-exa mined. In pap er [3], the e x tended cost may overestimate the o nline cost. W e still think WF A would b e k -compe titiv e. References [1] Allan Boro dim and Ran El-Y aniv. On line Computation and Comp etitive Analysis . Cambridge Universit y Press, 1 998. [2] Mark S. Mana sse et al. Competitive algorithms for ser v e r problem. Journal of Al gorithms , 11 :208–230 , 1990. 2 [3] Elias Koutso upias a nd Chr istos Papadimitriou. On the k-s erv er conjecture. Journal ACM , 42:971– 983, 1995. [4] Lawrence L. Lar more and Lames A. Orav ec. T-theor y applications to online alg orithms for the server problem. arXiv:cs/0 611088 v1 [cs. DS] 18 Nov , 200 6. 3