Balanced Partitions of Vector Sequences

Transcription

1 Balanced Partitions of Vector Sequences Imre Bárány Benjamin Doerr December 20, 2004 Abstract Let d,r N and be any norm on R d. Let B denote the unit ball with respect to this norm. We show that any sequence 1, 2,... of ectors in B can be partitioned into r subsequences V 1,...,V r in a balanced manner with respect to the partial sums: For all n N, i i d. A similar bound l r, we hae i, i V l i 1 r holds for partitioning sequences of ector sets. Both results extend an earlier one of Bárány and Grinberg (1981) to partitions in arbitrarily many classes. 1 Introduction Let d, N N. We use the short-hand [N] := {1,..., N}. Let be any norm on R d and B = { R d 1} its unit ball. In this paper, we gie extensions of the Bárány Grinberg theorem to partitions into more than two classes. In its most general ersion, this theorem states the following [BG81]. Rényi Institute, Budapest, POBox 127, 1364 Hungary, and Department of Mathematics, Uniersity College London, Gower Street, London WC1E 6BT, United Kingdom. Partially supported by Hungarian National Foundation Grants T and T Mathematisches Seminar, Bereich II, Christian-Albrechts-Uniersität zu Kiel, Kiel, Germany. 1

2 Theorem 1. Let V 1,..., V N B such that 0 con(v i ) for all i [N]. Then there are i V i such that for all n [N], i 2d. The most interesting special case of Theorem 1 is that all V i are of the form V i = { i, i }, cf. [BS95] as well. In this case, Theorem 1 yields that for any sequence 1,..., N of ectors in B there are signs ε i { 1, 1} such that ε i i 2d for all n [N]. In other words, there is a i d for all n [N] partition [N] = I 1 I 2 such that i I j [n] i 1 2 and j [2]. This partitioning ersion of the Bárány Grinberg theorem was extended to partitions into r > 2 classes with error bound (r 1)d in [DS03]. In the following section, we show that the factor (r 1) can be replaced by a constant. In the third section of this paper, we show that if the stronger condition V i = 0 (instead of 0 con(v i )) holds for all i [N], then for each i [N] there are r distinct ectors il V i, l [r], such that il 5d holds for all n [N] and l [r], where r max{ i [N]}. It is worth mentioning here that the results hold for all norms in R d. This is due to the fact that proofs use linear dependences among some ectors, with the norm playing ery little role. But most liely, much better bounds are alid for particular norms. For instance, it is conjectured that for r = 2 and Euclidean norm the best bound is of order d. This was proed by Spencer [Sp86] when N = O(d), but the general case when N is arbitrary is open. In the proofs of both results below we inoe the recursie method of [DS03], which states, roughly speaing, that if one can guarantee the existence of a 2-partition with good bound on its discrepancy, then one can guarantee the existence of an r-partition with a slightly weaer bound on its discrepancy. Precisely, we hae the following: Theorem 2. Let r 2 be an integer. Let 1,..., n be a sequence of ectors and E be a set of subsets of [n]. Assume that for all integers 1 r 1 < r 2

3 and all V 0 [n] there is a V 1 V 0 such that for all E E, i r 1 i K. i V 1 E i V 0 E Then there is a partition V = V 1... V r such that for all l [r] and E E we hae i 1 r i C(r)K, i V l E i V E where C(r) is an absolute constant satisfying C(r) for all r N. Note that the assumption of the theorem is equialent to saying that for all integer 1 r, r < r with r + r = and all V 0 [n] there is a partition V V = V 0 such that for all E E i V E i r i V 0 E i K and i V E i r i V 0 E i K. The proof of Theorem 2 starts with setting V 0 = V and proceeds by partitioning V and V further. Details can be found in [DS03]). Theorem 2 was wored out only in the context of hypergraph coloring (Theorem 3.6 in [DS03]), which in our language means i = 1 d for all i [n]. Howeer, the proofs easily reeal that all results hold as well for the general setting of Theorem 2. 2 Vector Partitioning Assume V is a finite or infinite sequence of ectors 1, 2,.... We introduce the (non-standard) notation V = i=1 i. Further, for a subsequence X of V we define X = i, i X i. Theorem 3. For eery sequence V B, and for eery integer r 2, there is a partition of V into r subsequences X 1,...,X r such that for all and j X j 1 V + C(r)dB. r 3

4 Proof. Assume = r 1 + r 2 (with positie integers r 1, r 2 ). We are going to construct a partition of V into subsequences Y 1 and Y 2 such that for each and for j = 1, 2, Y j r j V + db. This implies the theorem ia Theorem 2. For the construction of Y 1, Y 2 we use a modified ersion of the method of floating ariables as gien in [BG81]. Define V = { 1, 2,..., +d }, = 0, 1, 2,.... We are going to construct mappings β : V [ r 1, r 2 ] and subsets W V with the following properties (for all ): (i) V β () = 0, (ii) β () { r 1, r 2 } wheneer W, (iii) W = and W W +1. The construction is by induction on. For = 0, W 0 = and β 0 = 0 clearly suffice. Now assume that β and W hae been constructed and satisfy (i) to (iii). The d + 1 ectors in V +1 \ W are linearly dependent, so there are α() R not all zero such that V +1 \W α() = 0. Putting β ( +d+1 ) = 0, we hae β () + (β () + tα()) = 0 W V +1 \W for all t R. For t = 0 all coefficients lie in [ r 1, r 2 ]. Hence for a suitable t = t, all coefficients still belong to [ r 1, r 2 ], and β () + tα() { r 1, r 2 } for some = V +1 \W. Set now W +1 = W { } and β +1 () = β (), if W, and β +1 () = β () + t α(), if V +1 \ W. Now W +1 and β +1 satisfy the requirements. Moreoer, β +1 () = β () for all W. We now define the subsequences Y 1 and Y 2. Put i into Y 1 if i W and β ( i ) = r 2 for some, and put i into Y 2 if i W and β ( i ) = r 1 for 4

5 some. As β () = β +1 () once W, this definition is correct for all ectors that appear in some W. The remaining (at most d) ectors can be put into Y 1 or Y 2 in any way. Set γ() = r 2, if Y 1, and γ() = r 1, if Y 2. Clearly, r 2 Y 1 r 1 Y 2 db for all d. For > d we hae, with = h + d, r 2 Y 1 r 1 Y 2 = γ() = γ() β h () V h V h V h = V h (γ() β h ()) = V h \W h (γ() β h ()). The last sum contains at most d non-zero terms, each haing norm at most. Thus r 2 Y 1 r 1 Y 2 db for eery. Adding this to the triial equation r 1 Y 1+r 1 Y 2 = r 1 V (expressing that Y 1, Y 2 form a partition of V ), we obtain Y 1 r 1 V + db for eery. 3 Vector Selection Let now V 1,..., V N be a sequence of finite subsets of B such that r for all i [N]. An r selection of (V i ) is a mapping χ : [N] [r] R d such that χ(i, [r]) is an r element subset of V i for all i [N]. For such a χ, we define its discrepancy with respect to (V i ) by disc(χ, (V i ) i [N] ) = max max n [N] l [r] ( χ(i, l) 1 ). V i Theorem 4. There is an r selection with discrepancy at most 5d. 5

6 We mention that this theorem also holds for infinite sequences of finite subsets of B. To proe the theorem, we apply the following lemma twice. Lemma 5. Let r N, r 2. Let V 1,..., V N B such that r for all i [N]. Then for all [r] there are U i V i such that U i = for all i [N] and max n [N] ( U i V i ) 2d. Proof. We gie an algorithm for the construction of the sets U i. For each i [N], V i put x i =. We iteratiely change these numbers to zeros and ones in such a way that U i := { V i x i = 1} gies the desired solution. For the start let n = 1. What we do is the following: View those x i such that x i / {0, 1} and i n as ariables. If there is exactly one solution to the linear system (x i ) = 0 (1) i [N] V i x i =, i [N] (2) [ ] x i [0, 1], i [N], V i, then increase n by one and try again. Otherwise our existing solution may be changed in such a way that at least one more ariable x i becomes 0 or 1. If n reaches N and no solution can be found, then stop and change the remaining non-integral alues of x i to 0 or 1 in such a way that (2) is still fulfilled. Assume that in some step of this iteration no solution can be found. Then there are at least as many constraints containing ariables as there are ariables. Let q be the number of constraints of type (2) that contain a ariable. Then the total number of constraints containing ariables is at most d + q, and the number of ariables is at least 2q. Hence q d holds if no non-triial solution can be found, and at most q + d 2d of the x i, i n, are not in {0, 1}. Denote the set of these pairs (i, ) by I. Since the remaining x i, 6

7 i n, are not changed anymore, our final solution x satisfies ( x i ) = (x i ) + ( x i x i ) V i V i (i,) I = ( x i x i ). (i,) I Since I 2d, we conclude V i ( x i ) 2d for all n [N]. Since x i {0, 1}, putting U i := { V i x i = 1} gies the desired solution. Proof of the theorem. Let us assume first that = r for all i [N]. Then, by the aboe lemma, for all integers r 1, r 2 such that r = r 1 + r 2 there are U (1) i U (2) i = V i such that U (j) i = r j and ( r U (j) j r V i i ) 2d. Hence from Theorem 2, we obtain an r selection (actually an r partition) of (V i ) such that ( ) χ(i, l) 1 r 2 C(r)d V i for all n [N], l [r]. If > r for some i, apply the Lemma 5 (with = r) to obtain Ṽi V i such that Ṽi = r and ( r Ṽi V i ) 2d. By the aboe, there is an r selection for (Ṽi) such that ( χ(i, l) 1 r ) 2 C(r)d Ṽi for all n [N], l [r]. Note that, triially, χ is also an r selection for (V i ). It satisfies ( ) χ(i, l) 1 V i ( χ(i, l) 1 r 2 C(r)d + 1 r 2d Ṽ i ) + ( ) 1 1 r Ṽ i V i for all n [N] and l [r]. By noting that C(2) = 1 and C(r) for all r N, we obtain the constant of 5. 7

8 We may remar that a closer inspection of C(r) for small r yields better constants. For example, easy calculations by hand or Lemma 3.5 in [DS03] show that C(r) for r 10 (for r = 7 obsere that C(7) r max{1 + 3 C(3), 1 +C(4)}). Hence the bound C(r) implies C(r) for 4 r all r N, leading to a constant of 4.2 instead of 5. The following is an immediate consequence of Theorem 4. Corollary 6. Let r, N N. For i [N] let V i B such that V i = 0 and. Then there is a selection of (V i ) such that χ(i, l) 5d for all n [N] and l [r]. This answers a question of Emo Welzl concerning multi-class extensions of Theorem 1 posed at the Oberwolfach Seminar on Discrepancy Theory and its Applications in March It is clear that the stronger assumption V i = 0 is necessary. Already for d = 1 and r = 2, the sequence V i = { 1 2, 1} shows that 0 con(v i) does not suffice. Acnowledgment We than the organizers of the Oberwolfach Seminar on Discrepancy Theory and its Applications (March 2004) as well as the Oberwolfach crew for proiding us with surroundings that resulted in this paper. The first named author is grateful to Microsoft Research (Redmond, WA) as part of the research on this paper was carried out on a ery pleasant and fruitful isit there. For the same nice reason, the second author would lie to than Joel Spencer and the Courant Institute of Mathematical Sciences (New Yor City). References [BG81] I. Bárány and V. S. Grinberg. On some combinatorial questions in finite-dimensional spaces. Linear Algebra Appl., 41:1 9,

9 [BS95] J. Bec and V. T. Sós. Discrepancy theory. In R. Graham, M. Grötschel, and L. Loász, editors, Handboo of Combinatorics, pages Elseier, [DS03] B. Doerr and A. Sriasta. Multicolour discrepancies. Combinatorics, Probability and Computing, 12: , [Sp86] J. Spencer. Balancing ectors in the max norm. Combinatorica, 6:55 65,