summary:We describe a class of boxes such that every sequence of boxes from this class of total volume smaller than or equal to 1 can be on-line packed in the unit cube

Januszewski, Janusz

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Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumJanusz JanuszewskiOn-line packing regular boxes in the unit cubeArchivum Mathematicum, Vol. 35 (1999), No. 2, 97--101Persistent URL: http://dml.cz/dmlcz/107686Terms of use:© Masaryk University, 1999Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 35 (1999), 97 – 101ON–LINE PACKING REGULAR BOXES IN THE UNIT CUBEJanusz JanuszewskiWe describe a class of boxes such that every sequence of boxes fromthis class of total volume smaller than or equal to 1 can be on-line packed in theunit cube.Let C be a subset of Euclidean d-space Ed and let (Cn) be a finite or infinitesequence of d-dimensional convex bodies. We say that (Cn) can be packed in C ifthere exist rigid motions σi such that sets σiCi, where i = 1, 2, . . . , have pairwisedisjoint interiors and are subsets of C. By an on-line packing we mean a packingin which we are given every Ci, where i > 1, only after the motion σi−1 has beenprovided. We are given C1 at the beginning. In other words, in the on-line packingeach set must be irreversibly put before the next set appears. A survey of resultsabout packing (respectively: on-line packing) sequences of convex bodies is givenin [1] (respectively: in [5]).By a box we understand any set of the form{(x1, . . . , xd); tj ≤ xj ≤ uj for j = 1, . . . , d} ,where tj < uj for j = 1, . . . , d. The number wj = uj − tj is called the j-thwidth of this box. By the unit cube Id we mean the set{(x1, . . . , xd); 0 ≤ xj ≤ 1 for j = 1, . . . , d} .The aim of this paper is to present a class of boxes such that each sequenceof boxes from this class of total volume smaller than or equal to 1 can be on-linepacked in the unit cube.Let q ≥ 2 be a positive integer. By a q-regular box we mean a box of the j-thwidths of the form wj = q−m−1 for j ≤ k and wj = q−m for j = k+ 1, . . . , d,where k ∈ {0, . . . , d− 1}, and where m ∈ {0, 1, . . .}. If k = 0 in this formula,then such a q-regular box is called a q-regular cube.In the paper [4] it is shown that every sequence of q-regular cubes of totalvolume not greater than 1 can be on-line packed in the unit cube. In Theorems 1and 2 we generalize this result.1991 Mathematics Subject Classification : 52C17.Key words and phrases : packing, on-line packing, box.Received May 19,1997.98 J. JANUSZEWSKITheorem 1. Let q ≥ 2 be a fixed integer. Every sequence of q-regular boxes oftotal volume smaller than or equal to 1 can be on-line packed in the unit cube Id.Proof. Let (Rn) be a sequence of q-regular boxes of total volume not greater than1. Let m be a non-negative integer and let k ∈ {0, . . . , d− 1}. By a subbox oftype (m, k) (or by a subbox, for short) we mean the set{(x1, . . . , xd); ajq−m−1 ≤ xj ≤ (aj + 1)q−m−1 for j ≤ kand ajq−m ≤ xj ≤ (aj + 1)q−m for j = k + 1, . . .d} ,where aj ∈ {0, . . . , qm+1 − 1} for j ≤ k and aj ∈ {0, . . . , qm − 1} for j =k+ 1, . . . , d (see Fig. 1, where d = 3, q = 3). Obviously, each subbox is a q-regularbox.all subboxes three subboxes three subboxesof type (0,1) of type (0,2) of type (1,0)Fig. 1We enumerate all the subboxes of type (m, k) by integers 1, . . . , qmd+k in sucha way that:(i) for k ∈ {1, . . . , d − 1} the integers (λ − 1)q + 1, . . . , (λ − 1)q + q aregiven to the subboxes of type (m, k) being subsets of the subbox of type(m, k − 1) whose number is λ,(ii) for m ≥ 1 the integers (µ − 1)q + 1, . . . , (µ− 1)q + q are given to thesubboxes of type (m, 0) being subsets of the subbox of type (m− 1, d− 1)whose number is µ.Now, we describe the packing method. We pack R1 in the first subbox congruentto it. Let k > 1. By a k-free subbox we mean a subbox whose interior has anempty intersection with σ1R1 ∪ · · · ∪ σk−1Rk−1. We pack each box Rk from oursequence in the congruent k-free subbox with the smallest possible number.We will show that (Rn) can be on-line packed in Id by this method. Assumethe opposite; let Ri be a box from (Rn) which cannot be packed in Id. Obviously,there exists no i-free subbox congruent to Ri. Consider the family R of all i-freeON–LINE PACKING REGULAR BOXES IN THE UNIT CUBE 99subboxes of maximal volume. (In other words, S ∈ R if and only if S is i-freeand if there does not exist an i-free subbox S1 such that S ⊂ S1 and S 6= S1.)Subboxes from R have the volumes of the form q−1 Vol(Ri), q−2 Vol(Ri), . . . .We show that there are at most q − 1 subboxes of a fixed type in R. Assumethe opposite: there are at least q subboxes of type (m, k) in R. Let us denote thesesubboxes by Q1, . . . , Qz. Consider the case when k ≥ 1. From the descriptionof the packing method we conclude that q subboxes from among Q1, . . . , Qz arecontained in an i-free subbox of type (m, k − 1). Consider the case when k = 0.We can assume that m ≥ 1, because the box of type (0, 0) is nothing else but Id.In this case q subboxes from among Q1, . . . , Qz are contained in an i-free subboxof type (m− 1, d− 1). From the above consideration we conclude that q subboxesfrom among Q1, . . . , Qz do not belong to R because they are not maximal, acontradiction.A finite number of boxes has been packed in Id. Consequently, there exists afinite number of subboxes in R. This means that the total volume of subboxes inR is smaller thanVol(Ri)[(q − 1)q−1 + (q − 1)q−2 + . . .].This value is equal to Vol(Ri). Hence, the total volume of boxes R1, . . . , Ri−1is greater than 1 − Vol(Ri). Consequently, the total volume of boxes in (Rn) isgreater than 1, a contradiction. Remark. Let (Sn) be a sequence of boxes. The method of packing of (Sn) iscalled q-adic (see [2-4]), if for each positive integer n, σnSn has edges parallel tothe axes of the coordinate system and if for each j ∈ {1, . . . , d} the projectionof σnSn on the j-th axis is a segment whose both endpoints are multiples of thej-th width of Sn. Observe, that the packing method from Theorem 1 is q-adic.Obviously, the estimate 1 in Theorem 1 cannot be improved. It is an openquestion how to extend our class of q-regular boxes. For example, we cannot addthe cube of the width 12 to the class of 4-regular boxes. The reason is that one cubeof the width 12 and three boxes of the widths w1 =14 and w2 = · · · = wd = 1cannot be packed in Id. We show in Propositions 1 and 2 that some extensionsare possible. Probably, the class of q-regular boxes is however the best possible inthe sense given in Conjecture.Proposition 1. Let q ≥ 2 be an integer. Moreover, let F be the family of boxessuch that each box B ∈ F is either q-regular or the widths of B are of the formw1 = nq−1, w2 = · · · = wd = 1, where n ∈ {2, . . . , q}. Then each sequence ofboxes from F of total volume smaller than or equal to 1 can be on-line packed inId.Proof. We proceed analogously as in the proof of Theorem 1. We can regard abox B with w1 = nq−1, w2 = · · · = wd = 1 as the union of n q-regular boxeswith w1 = q−1, w2 = · · · = wd = 1. We pack such a box B in Id similarlylike in the method from Theorem 1. Just in the first free place. If such a box100 J. JANUSZEWSKIB cannot be packed, then the total volume of boxes preceding B is greater than1− nq−1 = 1− Vol (B). Let q ≥ 2 and let p1, . . . , pd be positive integers. By a (q, p1, . . . , pd)-regularbox we mean a box of the widths of the form wj = p−1j q−m−1 for j ≤ kand wj = p−1j q−m for j = k + 1, . . . , d, where k ∈ {0, . . . , d− 1}, and wherem ∈ {0, 1, . . .}. Denote by Bp the box{(x1, . . . , xd); 0 ≤ xj ≤ p−1j for j = 1, . . . , d}.Obviously, there is an affine image T (Id) equal to Bp. Observe that if a box B is q-regular, then the affine image T (B) is (q, p1, . . . , pd)-regular. Thus, from Theorem1 we conclude that each sequence of (q, p1, . . . , pd)-regular boxes of total volumesmaller than or equal to∏di=1 p−1i can be on-line packed in Bp. Consequently,we obtain the following result.Proposition 2. Every sequence of (q, p1, . . . , pd)-regular boxes of total volumesmaller than or equal to 1 can be on-line packed in the unit cube.Denote by w(B) the greatest width of a box B.Conjecture. Let F be a family of boxes such that: (i) F contains a cube; (ii)for each ε > 0 and for each box B ∈ F there exists a homotetic copy k1B of Bsuch that k1B ∈ F and w(k1B) < ε; (iii) for each box B ∈ F there exists ahomotetic copy k2B such that k2B ∈ F and w(k2B) = 1, (iv) each sequenceof boxes from F of total volume not greater than 1 can be on-line packed in Id.Then there exists an integer q ≥ 2 such that all the boxes from F are q-regular.Another interesting question is about the conection between usual packing andon-line packing in the unit cube sequences of boxes of total volume not greaterthan 1.Problem 1. Let F be a family of boxes such that the conditions (i)− (iii) fromConjecture are satisfied and such that each sequence of boxes from F of totalvolume not greater than 1 can be packed in Id. Does there exist an integer q ≥ 2such that all the boxes from F are q-regular?Problem 2. Let F be a family of boxes such that each sequence of boxes fromF of total volume not greater than 1 can be packed in Id. Let (Sn) be a sequenceof boxes from F of total volume smaller than or equal to 1. Can (Sn) be on-linepacked in Id?Finally, we present a theorem about packing another class of boxes. Let p1,. . . , pd and q1, . . . , qd be positive integers. Let m ∈ {0, 1, . . .}. By (p1, q1,. . . , pd, qd)-regular box we mean a box of the j-th widths of the form wj = p−1j q−mj ,for j = 1, . . . , d.ON–LINE PACKING REGULAR BOXES IN THE UNIT CUBE 101Theorem 2. Every sequence of (p1, q1, . . . , pd, qd)-regular boxes of total volumenot greater than 1 can be on-line packed in the unit cube.Proof. The proof is similar to the proof of Theorem 1. We can divide the unitcube into regular subboxes. Let m ∈ {0, 1, . . .}. By a regular subbox of size m wemean the set{(x1, . . . , xd); ajp−1j q−mj ≤ xj ≤ (aj + 1)p−1j q−mj for j = 1, . . . , d},where aj ∈ {0, . . . , pjqmj − 1}. We enumerate all the subboxes. The subboxes ofsize 0 are enumerated from 1 to∏di=1 pj . We enumerate other subboxes in such away that the integers (λ− 1)∏di=1 qi+ 1, . . . , λ∏di=1 qi are given to the subboxesof size m ≥ 1 being subsets of the subbox of size m − 1 whose number is λ.Let (Sn) be a sequence of (p1, q1, . . . , pd, qd)-regular boxes of total volumesmaller than or equal to 1. We pack S1 in the first subbox congruent to it. Letk > 1. By a k-free subbox we mean a subbox with such a property that no interiorpoint of it is covered by σ1S1 ∪ · · · ∪ σk−1Sk−1. We pack each box Sk from oursequence in the congruent k-free subbox with the smallest possible number. Wecan now proceed analogously to the proof of Theorem 1. Consequently, (Sn) canbe on-line packed in Id. References[1] Groemer, H., Covering and packing by sequences of convex sets, Discrete Geometry andConvexity, Annals of the New York Academy of Science 440 (1985), 262-278.[2] Januszewski, J., Lassak, M., Rote, G., Woeginger, G., On-line q-adic covering by the methodof the n-th segment and its application to on-line covering by cubes, Beitr. Alg. Geom. 37(1996) No. 1, 51-56.[3] Kuperberg, W., Problem 74: Ein Intervallüberdeckungsspiel, Math. Semesterber. 41 (1994),207-210.[4] Lassak, M., On-line packing sequences of segments, cubes and boxes, Beitr. Alg. Geom. 38(1997), 377–384.[5] Lassak, M., A survey of algorithms for on-line packing and covering by sequences of convexbodies, Bolyai Society Mathematical Studies 6 (1997), 129–157.E-mail

On-line packing regular boxes in the unit cube

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On-line packing regular boxes in the unit cube

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Institute of Mathematics AS CR, v. v. i.