A balancing strategy

JOURNAL

OF COMBINATORIAL

THEORY,

Series A 40, 175-178 (1985)

Note A Balancing

Strategy

JOHN E. OLSON Department

of Mathematics, University Park, Communicated

Pennsylvania Pennsylvania

State 16802

by the Managing

University,

Editors

Received September 15, 1983

Suppose x,, x2 ,..., is a sequence of vectors in Rk, llxnil < 1, where /1(x,,..., xk)ll = max,l.u,l. An algorithm is given for choosing a corresponding sequence E,, Ed,...,of numbers, E,= k 1, so that ~JE,x,+ ... +&,x,11 remains small. 6 1985 Academic Press, ITIC

For real vectors x = (x1 ,..., xk) we shall write

I/XII = I
< (2 In 2k)“’ m112.

(1)

This result proved useful in another problem, considered by Spencer and the author [l], in which the vectors come from the incidence relations for points distributed into k sets. To achieve (l), Player II (having already chosen Ed,..., E,- ,) chooses for E, the value + I for which a certain weight function W(E~X, + *.. + E,x,) is minimized. The weight function is w(x)=

i

cosh(crx,),

x =

(x1

,...,

xk),

j=l

175 0097-3165/85 $3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.

176

JOHNE.OLSON

where a = [2 ln(2k)/m]

‘j2.

Since the weight function depends on the total number of rounds m, this strategy has the peculiar feature that Player II appears to be aided by knowing how soon the game is to stop. This suggests the question: can Player II do as well as (1) without knowing m? We shall give a strategy, independent of m, that achieves a bound as good as ( 1) up to constant factor. Let x1, x2 ,..., be a sequence of vectors in Rk (ka 3), with (Ix,/1 6 1. Then one can choosenumbers E, = f 1, where each E, dependsonly on x1,..., x,, such that THEOREM.

/IEIX[ + ... + E,x,II < e’j2(2 In k - 1)“’ n”’

(2)

for all n = 1, 2,....

Proof: We shall require, and leave to the reader to prove (say, by calculus), the following inequalities for real quantities: /x+ II”+

Ix- 1\“<2(x2+s-

l)Q

($22).

(3)

(3 > 1).

(4)

(~22).

(5)

If Ial < 1, then Jx+a(“+

l)“+ Ix-

Ix-aj”Q(x+

11”

If If= i lxjlS = kg”, q 2 0, then i

IXj+ I)“+ Ixj-ll”
l(“+ \q-11”)

j=l

Let s >/ 2 be a fixed number (to be specified later as a function of k). For x = (x1 )..., xk), let

IIxII,= i lXjlS. j=l

We choose the values E, in the following way: Let s1 = 1; for n > 1 (having chosen Ei ,..., E, ~, ) let E, be that value +_1 which minimizes

II&lx1 + ... +~nX,Il,. To estimate (6), we first define, for each n, qn> 0 by the equation

IIEIXI + +.. +E,x,((.=kqs,.

(6)

A BALANCING

177

STRATEGY

We shall show that (7) Let E,X, + ... + E, _, x,-, = (y, ,..., yk) and x, = (x, ,..., xk). We have IlElXl + ... +s,x,I),=min

i /JJ~+x,/“, i i j= I j=l


Ivj-xjl”

Iv,-Xjl’

j=l


i

Iv/+

II”+

Ivj-

by (41,

II”

.j= 1

Thus kq;
c~~l4.~,+ll”+lq,~,-1l’~

by (51,

< k(q; _ ] + s - 1)s’*

by (3).

+ s - 1)“” and (7) follows. Clearly q, 6 1, hence by (7), qi
It follows that

Il&IXI + . . . + E,X,Il, ,< k(s - 1)“” rf/*. Since JlxJI < JIxJJ~‘~,we have (/E,X, + ... + c,x,(I
1)‘12 nl/‘.

(8)

We now choose that value of s> 2 which minimizes the right side of (8). For 2 2), the right side of (8) is minimized 112

and increases with s for s >/ sO. Also 2Ink-2
at

178

JOHN E.OLSON

Hence, if we take any convenient s in the interval s,6sfs,+

1,

the right side of (8) is less than its value at s1 = 2 Ink e”*(2 In k- l)l’* n”*. Thus we obtain (2).

which

is

REFERENCES 1. J. E. OLSON AND J. H. SPENCER,Balancing families of sets, J. Combin. Theory Ser. A 25 (1978), 29-37. 2. J. H. SPENCER,Balancing games, J. Combin. Theory Ser. E 23 (1977), 68-74.