Consistent High Dimensional Rounding with Side Information
Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, Ran J. Tessler
CConsistent High Dimensional Rounding with Side Information
Orr Dunkelman ∗ and Zeev Geyzel † and Chaya Keller ‡ and Nathan Keller § and Eyal Ronen ¶ and Adi Shamir (cid:107) and Ran J. Tessler ∗∗ August 11, 2020
Abstract
In standard rounding, we want to map each value X in a large continuous space (e.g., R ) to a nearby point P from a discrete subset (e.g., Z ). This process seems to be inherentlydiscontinuous in the sense that two consecutive noisy measurements X and X of the samevalue may be extremely close to each other and yet they can be rounded to different points P (cid:54) = P , which is undesirable in many applications. In this paper we show how to makethe rounding process perfectly continuous in the sense that it maps any pair of sufficientlyclose measurements to the same point. We call such a process consistent rounding , andmake it possible by allowing a small amount of information about the first measurement X to be unidirectionally communicated to and used by the rounding process of the secondmeasurement X . In many applications such as the creation of a biometric database offingerprints, we can naturally use such a scheme by storing in each entry the user’s identity,a hashed version of the initial measurement of his fingerprints, and the side informationwhich will enable any future noisy reading of the fingerprint to be consistently hashed tothe same value.The fault tolerance of a consistent rounding scheme is defined by the maximum distancebetween pairs of measurements which guarantees that they will always be rounded to thesame point, and the goal of this paper is to study the possible tradeoffs between the amountof information provided and the achievable fault tolerance for various types of spaces. Whenthe measurements X i are arbitrary vectors in R d , we show that communicating log ( d + 1)bits of information is both sufficient and necessary (in the worst case) in order to achieveconsistent rounding for some positive fault tolerance, and when d=3 we obtain a tight upperand lower asymptotic bound of (0 .
561 + o (1)) k / on the achievable fault tolerance when wereveal log ( k ) bits of information about how X was rounded. We analyze the problem byconsidering the possible colored tilings of the space with k available colors, and obtain ourupper and lower bounds with a variety of mathematical techniques including isoperimetricinequalities, the Brunn-Minkowski theorem, sphere packing bounds, and ˇCech cohomology. ∗ Computer Science Department, University of Haifa, Israel. [email protected] . † Mobileye, an Intel company – Jerusalem, Israel. [email protected] . ‡ Department of Computer Science, Ariel University, Ariel, Israel. [email protected] . § Department of Mathematics, Bar Ilan University, Ramat Gan, Israel. [email protected] . Re-search supported by the European Research Council under the ERC starting grant agreement number 757731(LightCrypt) and by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunctionwith the Israel National Cyber Bureau in the Prime Minister’s Office. ¶ School of Computer Science, Tel Aviv University. Member of the Check Point Institute for InformationSecurity. [email protected] . (cid:107) Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Israel. [email protected] . ∗∗ Incumbent of the Lilian and George Lyttle Career Development Chair, Department of Mathematics, Weiz-mann Institute of Science, Israel. [email protected] . Supported by the Israel Science Foundation(grant no. 335/19) and by a research grant from the center for new scientists of Weizmann Institute of Science. a r X i v : . [ c s . C G ] A ug Introduction
Rounding.
Whenever we want to digitally process or store an analog value, we have toperform some kind of rounding in order to represent this value (which is typically a numberin R ) by a nearby point chosen from a discrete subset (such as the set of integers Z ). In themultidimensional generalization of this problem we are given a vector X ∈ R d , and want torepresent it by a nearby point P chosen from some discrete subset of R d (which may be Z d ,some lattice of points, or any other choice). Many rounding procedures had been proposedand studied in the literature (see, e.g., [2, 8]), but all of them are inherently discontinuous inthe sense that one can always find two inputs which are extremely close to each other (suchas 0 . . tiles , in which each tile is rounded to a different representative point (e.g., its center), and thediscontinuities happen along the tiles’ boundaries. The algorithmic aspect of the problem ofactually finding the tile that contains a given X in a given partition is a classical problem incomputational geometry, and had been studied extensively under the general name of pointlocation (see [9, 18], and the references therein).In many applications, such discontinuities are undesirable. One natural solution is to try tominimize the fraction of pairs X , X with distance ( X , X ) < (cid:15) which are mapped to differenttiles by considering foam tilings that minimize the total surface area of unit volume tiles (such atiling is called ‘foam’ since it emerges automatically in physical collections of soap bubbles). Ina beautiful FOCS’2008 paper [16] (which later appeared as a research highlight at CACM [17]),Kindler, Rao, O’Donnell and Wigderson introduced a clever new construction of such tileswhich they called spherical cubes . What makes these tiles special is that they have the O ( √ d )surface area of a ball and yet they can tile the whole R d space without gaps. Surprisingly, themain motivation for this construction came from the seemingly unrelated field of computationalhardness amplification, and it solved an interesting open problem posed by Lord Kelvin in 1887.A related class of schemes that aim at minimizing the proportion of close points mappedto different places is locality sensitive hashing schemes (see, e.g., [14]). However, they typicallydeal with situations in which the input is a vector and the output is a scalar, and thus they donot actually round inputs to nearby outputs in the same space. Our approach – consistent rounding.
In this paper we consider the more ambitious goalof completely eliminating all the discontinuities in the rounding process rather than reducingtheir fraction. We call any such scheme consistent rounding . To make it possible, we have toslightly modify the model by thinking about X and X as two consecutive noisy measurementsof the same X . When the first measurement X is rounded, we allow it to produce a few bitsof side information about how it was rounded and to provide them as an auxiliary input to theprocess that decides how to round X . The one-dimensional case.
To demonstrate the basic idea, consider the one dimensional casein which X and X are real values which have to be consistently rounded to a nearby integer. X is always rounded to the nearest integer, and it produces a single bit of side informationwhich is whether it was rounded to an even or odd integer P . When X is measured, it isrounded to the nearest integer which is of the same parity as P . To demonstrate this process,consider again the problematic inputs X = 0 . X = 0 . X is rounded to 0, We note that in statistics, the term ‘consistent rounding’ is used to denote a rounding that is consistent withsome external constraints; see [20, p. 237]. Figure 1: A 3-colored hexagonal tiling of the plane, and a maximal non-intersecting inflationof the tiles colored 1and X is also rounded to 0 since it is the closest even integer. In fact, X could be anywherebetween − .
5. It is not difficult to show (see Section 4.1.1) that thisis the highest possible fault tolerance of any one dimensional consistent rounding scheme, andthat other natural schemes (such as providing one bit of side information about whether X was rounded up or down) provide lower fault tolerance.The way we think about the problem is to consider a colored tiling of the real line with twocolors: All the values in [ − . , .
5) are colored by 1, all the values in [0 . , .
5) are colored by 2,all the values in [1 . , .
5) are colored by 1, all the values in [2 . , .
5) are colored by 2, etc. Theside information provided about X is the color of the tile in which it is located, and the waywe process X is to round it to the center of the closest tile which has the same color as thatof X . The essential property of our partition is that the minimum distance between any twotiles which have the same color is 1, and thus we can “inflate” all the tiles of a particular colorto include any erroneously measured value X up to a distance of 0 . The two-dimensional case.
To make this perspective clearer, consider the two dimensionalplane. If we use the obvious checkerboard tiling by unit squares, we need at least 4 colors (andthus 2 bits of side information) to color the tiles if we want any two tiles with the same color tohave some positive distance between them. We can reduce the number of colors to 3 (and thus,provide only log (3) = 1 .
58 bits of side information) by considering the hexagonal partition ofthe plane depicted in the left part of Figure 1. Given a two dimensional point X , we alwaysround it to the center of the hexagon in which it is located, and given X we round it to thecenter of the nearest hexagon which has the same color as the hexagon that contains X . Todetermine the fault tolerance of this consistent rounding scheme, we inflate all the hexagonaltiles of a particular color by the same amount until two of them touch each other for the first3ime, as depicted in the right part of Figure 1. As it turns out, this scheme is not optimal: eventhough the hexagonal partition is very attractive since its circle-like tiles map each point to anearby center, when we inflate the hexagons of a particular color we are stopped prematurelywhen their corners touch, leaving large gaps between the inflated hexagons. A 3-colored tilingwith a higher fault tolerance will be described in Section 4.2.1, and an asymptotically optimaltiling for a large number of colors will be described in Section 4.2.2.A fine point about consistent rounding schemes is how to break ties, and here we dealdifferently with X and X . We want to be able to deal with any X , and thus we think aboutthe tiles as being closed sets which include their boundaries. Therefore, points X which areon the boundary between tiles can have more than one possible color. We allow such ties tobe broken arbitrarily in the sense that X can be rounded to the center of any one of the tilesthat it belong to. However, when we think about X we allow it to be at a distance of strictlyless than some bound, and thus the inflated tiles (that contain all the possible X ’s we areinterested in) are open sets which have no intersections. Consequently, each X can belong toat most one inflated tile, and is rounded to the center of that tile with no possible ties. The minimal amount of side information that allows high-dimensional consistentrounding.
When we consider higher dimensional inputs X ∈ R d , we can provide one bit ofside information about each one of its d entries separately, but for large d this requires a largenumber of side information bits. In our colored tiling formulation of the problem, it suffices toreveal the color of X in order to consistently round X , and thus if we tile the space with k colors, we need only log ( k ) bits of side information to specify this color. This naturally leadsto the question of what is the minimum number of colors needed to tile R d by bounded sizedtiles so that all the tiles of the same color will have some positive distance from each other (notethat two colors always suffice if we relax the problem by allowing an infinite series of nestedboxes, or allow tiles of the same color to touch at corners). Surprisingly, we could not find inthe literature any reference to this natural question. As we show in Section 3, there can be nosuch colored tiling with d colors, and as we show in Section 5.2.1, d + 1 colors are sufficient.Consequently, log ( d + 1) bits of side information about X are necessary and sufficient (inthe worst case) in order to obtain a consistent rounding scheme with positive fault tolerance.To prove the negative result, we use techniques borrowed from algebraic topology (namely, theˇCech cohomology and other cohomology theories), and to prove the positive result we providean explicit construction of such a colored tiling. The maximal fault tolerance that can be achieved with a given amount of sideinformation.
In addition to aiming at minimizing the amount of side information, we studythe question of maximizing the fault tolerance for a given amount of side information. InSection 4 we study the special case of the two dimensional plane. In the negative direction,we obtain several upper bounds on the fault tolerance that can be achieved, using differenttechniques from geometry and analysis (including isoperimetry, the Brunn-Minkowski inequalityand results on the circle packing problem). In the positive direction, we construct a variety oftiling schemes for various numbers of colors. In particular, while for three colors the hexagonaltiling scheme described above can tolerate an additive fault of up to 0 .
31, we present a tilingwith fault tolerance of 0 . . k of colors tends to infinity, we derive the asymptotic boundof (0 .
537 + o (1)) √ k on the achievable fault tolerance, and prove its tightness by an explicitconstruction. 4cenario Lower Bound Upper Bound Techniques Source(LB) on FT (UB) on FT3 colors 0.354 0.413 Brunn-Minkowski ineq. (UB), Sec. 4.1.1 (UB),in R Brick wall const. (LB) Sec. 4.2.1 (LB)4 colors 0.5 0.564 Brunn-Minkowski ineq. (UB), Sec. 4.1.1 (UB),in R Brick wall const. (LB) Sec. 4.2.1 (LB) k colors (0 .
537 + o (1)) √ k (0 .
537 + o (1)) √ k Circle packing (UB), Sec. 4.1.2 (UB),in R HCR const. (LB) Sec. 4.2.2 (LB)4 colors 0.175 0.365 Brunn-Minkowski ineq. (UB), Sec. 5.1.1 (UB),in R Bricks and balloons const. (LB) Sec. 5.2.2 (LB) d + 1 colors Ω(1 /d ) O (log d/ √ d ) Brunn-Minkowski ineq. (UB), Sec. 5.1.1 (UB),in R d Dimension reducing const. (LB) Sec. 5.2.1 (LB) k colors (0 .
561 + o (1)) k / (0 .
561 + o (1)) k / Sphere packing (UB), Sec. 5.1.2 (UB),in R CPB const. (LB) Sec. 5.2.3 (LB) k colors (0 .
707 + o (1)) k / (0 .
707 + o (1)) k / Sphere packing (UB), Sec. 5.1.2 (UB),in R CPB const. (LB) Sec. 5.2.3 (LB) k colors (1 + o (1)) k / (1 + o (1)) k / Sphere packing (UB), Sec. 5.1.2 (UB),in R CPB const. (LB) Sec. 5.2.3 (LB)FT – fault tolerance, LB – lower bound, UB – upper bound, Const. – constructionHCR – honeycomb of rectangles, CPB – close packing of boxesTable 1: Summary of our lower and upper bounds on the fault tolerance, for different values ofthe dimension d and the number of colors k In Section 5 we study the general case of R d , d >
2. Like in the two-dimensional case,we obtain a number of lower and upper bounds on the fault tolerance. In particular, weshow that the maximal fault tolerance achieved by a ( d + 1)-coloring of R d is between Ω(1 /d )and O (log d/ √ d ). In addition, we use the recent breakthrough results on the sphere packingproblem [5, 13, 19] to obtain tight asymptotic lower and upper bounds on the fault tolerancein dimensions 3 , , and 24. Our bounds on fault tolerance are summarized in Table 1. Applications.
The problem of consistent rounding has numerous applications. For example,consider the problem of biometric identification, in which multiple measurements of the samefingerprint are similar but not identical. It would be very helpful if all these slight variantscould be represented by the same rounded point P , and we can achieve this by storing a smallamount of side information in the biometric database (as part of the initial acquisition of thebiometric data of a new employee).Another example is the problem of developing a contact tracing app for the COVID-19pandemic, where we want to record all the cases in which two smart phones were at roughlythe same place at roughly the same time. We can do this by measuring in each phone theGPS location, the local time, and perhaps other parameters such as the ambient noise levelor the existence of sudden accelerations (to determine that the two phones were in the samemoving car). When someone tests positive for COVID-19, the health authority can releasea list of his measurements, but in order to keep the patient’s privacy, it wants to apply acryptographically strong hash function to each measurement before publishing it. Since weexpect these measurements to be slightly different for the infected and exposed persons, thehealth authority can publish the small amount of side information along with the consistently5ounded and then hashed measurements.Finally, in industrial control systems we can collect analog measurements from thousandsof sensors (temperature, pressure, flow, etc.) and the use of a consistent rounding scheme couldbe ideal in order to check the repeatability of the process in spite of slight variations. Related work.
A line of study that seems related to our work is fuzzy constructions that werewidely studied in the cryptographic literature. For example, [10] introduces the notions of fuzzyextractors and secure sketches , which enable two parties to secretly reach a consensus value frommultiple noisy measurements of some high entropy source (a recent survey of such techniquescan be found in [12]). However, such schemes concentrate on the aspects of cryptographicsecurity (which we do not consider), and produce sketches whose size depends on the numberof possible inputs (which is meaningless for real valued inputs). In this sense our consistentrounding scheme can be viewed as an exceptionally efficient reconciliation process, since it canproduce for each million entry vector of arbitrarily large real numbers a 20 bit “sketch” (inthe form of its color side information), and process this information with trivial point locationalgorithms.An actual application in which side information is used during an “error reconciliation”process can be found in the post quantum key exchange scheme
New Hope by Alkim et.al. [1].Their approach is limited to lattice points, and requires side information whose size is linear inthe number of dimensions, while our approach requires only a logarithmic number of bits.Another natural idea (which is used in many fuzzy cryptographic constructions such as fuzzy commitment [15]) is to use an error correction scheme to map inputs to nearby codewords.However, most error correction schemes cannot be applied to real valued inputs. In addition,even if we could perfectly tile the continuous input space with balls surrounding each codeword,this would not solve the problem of consistent rounding near the boundary between adjacentballs. Finally, this boundary problem becomes worse as the dimension grows, since in highdimensions almost all the points are likely to be near a boundary. Consequently, these codebased cryptographic solutions manage to solve a variety of interesting related tasks, but notthe consistent rounding problem we consider in this paper.
Open problems.
While we fully solved the question of minimizing the amount of side in-formation required for fault tolerance, several questions remain open regarding the maximaltolerance rate that can be achieved for a given amount of side information. In particular, fordimensions 2 , , ,
24 we determined the exact asymptotic fault tolerance rate when log k bitsof information are allowed, using a connection to the densest sphere packing problem. Whenonly very few bits of information are allowed, the situation is much less clear. For example,we do not even know whether the brick wall constructions we present in Section 4.2.1 havethe highest fault tolerance among rounding schemes in R with log In this section we present the basic setting that will be assumed throughout the paper.
Tiling.
We study tilings of R d , where each tile is connected , closed and bounded , and the tilesintersect only in their boundaries. We further assume that the tiling is locally finite , meaning6hat the number of tiles that intersect any bounded ball B (0 , r ) is finite. In some of the resultswe make additional assumptions on the tiles; such assumptions are stated explicitly. Coloring.
In our tiling, each tile is colored in one of k colors. We note that the naturalassumption that the tiles are unicolored is made for the sake of simplicity; all our results holdalso if the tiles contain several colors, provided that the color classes inside each tile satisfy theassumptions we made on the tiles. Fault tolerance and inflation.
In order to compute the fault tolerance of a given tiling(with respect to the L distance), we consider all tiles of the same color and inflate them (i.e.,replace the tile T by the set T (cid:48) = { y : ∃ x ∈ T, | x − y | < r } for some r >
0) until they touch eachother. Clearly, the fault tolerance is the maximal r for which such a non-intersecting inflationis possible.We denote the minimal distance between two same-colored points in different tiles by t , and so,the fault tolerance is t/ Normalization.
The d -dimensional volume of a figure T ⊂ R d is denoted by λ ( T ). Wenormalize the tiling by assuming that the volume of each tile is bounded by T (cid:48) satisfies λ ( ¯ T (cid:48) ) = λ ( T (cid:48) ), where ¯ T (cid:48) is theclosure of T (cid:48) .We choose this normalization since it complies with using the Brunn-Minkowski theorem(see Section 4.1.1) and makes computations more convenient. We note that our results canbe easily translated to results with respect to other natural normalizations. For example, theassumption that the radius of each tile is at most L distance betweeneach X ∈ R d and its rounded value is at most 1) implies that the volume of each tile is at mostthe volume of the unit ball in R d , which allows translating all our fault tolerance upper boundsto this ‘bounded radius’ setting. As for the lower bounds, they come from explicit constructionswhose fault tolerance can be recomputed with respect to any other natural normalization. Alternative metrics.
Instead of the L distance (which is probably the most natural distancemetric), one may consider the fault tolerance problem with respect to other distance metrics.For example, in order to measure the fault tolerance with respect to the L ∞ distance, onehas to inflate each tile T into T (cid:48)(cid:48) = { y : ∃ x ∈ T, max i | x i − y i | < r } . It turns out that theproblem is much easier with respect to this metric. Indeed, assume that the number of colors is k = m d for some 2 ≤ m ∈ N . Consider a periodic tiling in which each basic unit is a cube withside length m that consists of m d unit-cube tiles, each colored in a different color. This tilingachieves fault tolerance of ( m − /
2. On the other hand, it is easy to see that the argumentvia the Brunn-Minkowski theorem presented in Section 4.1.1 implies that any tiling of R d with k = m d colors achieves fault tolerance of at most ( m − / L ∞ metric, andthus, the cubic tiling we described is optimal. Lower and upper bounds for other values of k can be obtained by variants of this construction and the corresponding upper bound proof.The difference between the metrics comes from the fact that in ‘cubic’ inflation (which isdone with respect to the L ∞ metric), non-intersecting inflations of cubic tiles fill the entirespace, while in inflation by balls (which we have with respect to the L metric), large gaps areleft between the inflated tiles. 7 The Minimal Number of Colors Required for Fault Tolerance
In this section we prove that the minimal number of colors required for tolerating any rate t > R d is d + 1. We provide two proofs, under different additionalnatural assumptions on the tiles. The first proof assumes that the tiles are polytopes anduses an inductive argument. The second proof assumes that the tiles and their non-emptyintersections are contractible and uses an algebraic-topologic argument. The lower bound d + 1on the number of required colors is tight; a matching construction for any d is presented inSection 5.2.1. In this section we prove the following.
Proposition 3.1
For any m > , the following holds. Let T , T , . . . be a colored tiling of R d in which the tiles are polytopes and each tile is contained in a box with side length m . If thefault tolerance of the tiling is > , then the number of colors is at least d + 1 . Informal proof.
We show by induction that there is a point x that is included in at least d + 1 tiles. This implies that if the tiling uses only d colors, there must be two same-coloredtiles that intersect in x , and thus, no fault tolerance is possible.For d = 1, as each single tile is included in a segment of length m , for any M > m , theentire segment [ − M, M ] cannot be covered by a single tile. Hence, the segment must contain apoint in which two tiles intersect.For the induction step, we consider the restriction of the tiling to a large box [ − M, M ] d .Then we look at the d ’th coordinate, consider the union of all tiles that contain a point in the‘lower’ half-box, and take the ‘upper boundary’ of this union. This gives a ‘crumpled’ variantof the equator hyperplane. By the induction hypothesis, the restriction of the tiling to this‘crumpled hyperplane’ contains a point v that is included in at least d tiles. However, there isalso at least one tile in the ‘upper half-box’ that includes v , and thus, v is included in at least d + 1 tiles. This completes the proof by induction. Formal proof.
Making the argument described above formal is somewhat cumbersome, andrequires an auxiliary notion, somewhat close to the notion of a polytopal complex [21, p. 127].
Notation.
Let k, d, m, M ∈ N , be such that d ≥ k and M > m . A crumpled ( k, d ) -flat F isa union of a finite collection of k -dimensional polytopes P , P , . . . , P (cid:96) ∈ R d that satisfies thefollowing conditions:1. F is contained in the slab [ − M, M ] k × [0 , m ] d − k .2. The projection of F onto the hyperplane H k = R k × { } d − k is [ − M, M ] k × { } d − k .3. F is connected.4. Each P i is contained in a box with side length m .5. The intersection of any P i , P j ∈ F is either empty or a polytope of dimension ≤ k − -3,-3) (3,-3)(-3,0)(-3,2)(-3,3) (3,3) Figure 2: An illustration of the inductive process of Lemma 3.2. A polygonal tiling of [ − , is depicted in ordinary lines, and the ‘crumpled equator’ to which we reduce in the inductiveprocess (that has to be included in the slab [ − , × [0 ,
2] and whose projection on the x -axismust be [ − , , M = 3 , m = 2.For example, a crumpled (1 , d )-flat is a polygonal line included in the slab [ − M, M ] × [0 , m ] d − , whose projection on the x -axis is [ − M, M ]. A crumpled (1 , M = 3 , m = 2 is demonstrated in Figure 2 (in boldface). Lemma 3.2
Let k, d, m, M be as above. Any crumpled ( k, d ) -flat F contains a point thatbelongs to at least k + 1 polytopes. Proof:
The proof goes by induction on k . For k = 1, observe that F cannot consist of asingle polytope. Indeed, as the projection of F on the x -axis is [ − M, M ], there are two points v, v (cid:48) ∈ F with | v − v (cid:48) | ≥ M . These points cannot belong to the same polytope P i , since each P i is included in a segment of length m . As F is connected, it must contain an intersectionpoint of two P i ’s.For the induction step, consider a crumpled ( k, d )-flat F . Let F (cid:48) be the union of all polytopesin F whose intersection with the set { x ∈ R d : x k ≤ } is non-empty, and let F (cid:48)(cid:48) = ∂ ( F (cid:48) ) ∩ { x ∈ R d : x k ≥ } . (That is, we look at the k ’th coordinate and take to F (cid:48) all polytopes in F that contain a pointin the ‘lower’ half-space. Then, we define F (cid:48)(cid:48) to be the intersection of the boundary of F (cid:48) withthe ‘upper’ half-space. We view F (cid:48)(cid:48) as a ‘crumpled equator flat’ of F ). This process, in the case d = 2, k = 1, is demonstrated in Figure 2, where the tiling F is depicted in ordinary lines and F (cid:48)(cid:48) is depicted by a bold line.We view F (cid:48)(cid:48) as the union of the polytopes P (cid:48) i = P i ∩ F (cid:48)(cid:48) . We claim that F (cid:48)(cid:48) is a crumpled( k − , d )-flat. Indeed: 9. F (cid:48)(cid:48) is contained in the slab [ − M, M ] k − × [0 , m ] d − k +1 , since each v ∈ ∂ ( F (cid:48) ) belongs tosome P i that contains also a point v (cid:48) with v (cid:48) k ≤
0. As each P i is included in a box of sidelength m , we have v (cid:48) k ≤ m . (The condition in all other coordinates clearly holds.)2. For any fixed value ( x , . . . , x k − ) ∈ [ − M, M ] k − , the projection of the set { v ∈ F :( v , . . . , v k − ) = ( x , . . . , x k − ) } on the k ’th axis is [ − M, M ]. In particular, F (cid:48) contains apoint whose first k coordinates are ( x , . . . , x k − , k ’th axis until we reach a point on the boundary of F (cid:48) . The resulting point is an element in F (cid:48)(cid:48) whose first k − x , . . . , x k − .Hence, the projection of F (cid:48)(cid:48) on H k − is [ − M, M ] k − × { } d − k +1 .3. F (cid:48) is connected, and hence, F (cid:48)(cid:48) = ∂ ( F (cid:48) ) ∩ { x ∈ R d : x k ≥ } is connected as well.4. Each polytope in F (cid:48)(cid:48) is contained in a box of side length m , since it is part of a polytopethat was contained in F (cid:48) .5. The intersection of any P (cid:48) i , P (cid:48) j ∈ F (cid:48)(cid:48) is either empty or a polytope, since P (cid:48) i , P (cid:48) j are partsof faces of polytopes (from F ) created by intersection with other polytopes or with thehalfspace { x ∈ R d : x k ≥ } . The dimension of any such intersection is at most k − H (cid:48) to its boundary reduces all dimensions by 1.By the induction hypothesis, H (cid:48)(cid:48) contains an intersection point v of k polytopes P (cid:48) i . Thismeans that the corresponding polytopes P i contain v as well.Furthermore, as v ∈ ∂ ( H (cid:48) ) and 0 ≤ v k ≤ m , there exists P j ∈ H \ H (cid:48) that contains v .(Here, we use the assumption that for any sufficiently small (cid:15) > H contains some w with( w , . . . , w k ) = ( v , . . . , v k − , v k + (cid:15) ) and the assumption that F consists of a finite collectionof polytopes.) Therefore, v belongs to at least k + 1 polytopes of H , completing the proof byinduction. (cid:3) Now we are ready to prove Proposition 3.1.
Proof:
Let T , T , . . . be a tiling of R d that satisfies the assumptions of the proposition. Let T (cid:48) i = T i ∩ [ − M, M ] d , and let F = ∪ T (cid:48) i . It is easy to see that F is a crumpled ( d, d )-flat. Hence,by the lemma, F contains a point v that belongs to d + 1 polytopes. The point v is included inat least d + 1 tiles T i , proving the assertion. (cid:3) Remark.
We note that if one assumes that the tiles are convex polytopes, then a muchsimpler argument shows that at least d + 1 colors are needed. Indeed, start with a point insome tile, and move in some direction until you reach the boundary of the tile. The pointson this boundary belong to two tiles. Move along this boundary face until you reach a pointon its boundary (i.e., now we reduce from dimension d − d − d + 1 tiles. This argument fails in nonconvexpolytopes since the corners of a box nested inside a bigger box touch only two tiles. Recall that a set in R d is called contractible if is can be continuously shrunk to a point in R d .(The formal definition is that the identity map is homotopic to some constant map.)10nformally, in this section we prove that if the tiles and their non-empty intersections arefinite unions of contractible sets, then at least d + 1 colors are required for fault tolerance. Dueto the possibility of pathologies, the formal statement is a bit more cumbersome: Proposition 3.3
Let T , T , . . . be a colored tiling of R d in which the tiles and all their non-empty intersections are disjoint unions of finitely many contractible sets. Assume that the tilingis locally finite and that all T i ’s are bounded. In addition, assume that each T i has an openneighborhood U i such that for any set of indices I, (cid:92) i ∈ I U i (cid:54) = ∅ ⇔ (cid:92) i ∈ I T i (cid:54) = ∅ , and the U i ’s and their non-empty intersections are disjoint unions of finitely many contractibles.Then the number of colors is at least d + 1 . A similar method proves an analogous statement for colored tilings of the sphere S d : Proposition 3.4
Let T , T , . . . , T N be a colored tiling of S d in which the tiles and all theirnon-empty intersections are disjoint unions of finitely many contractible sets. Assume thateach T i has an open neighborhood U i such that for any set of indices I, (cid:92) i ∈ I U i (cid:54) = ∅ ⇔ (cid:92) i ∈ I T i (cid:54) = ∅ , and the U i ’s and their non-empty intersections are disjoint unions of finitely many contractibles.Then the number of colors is at least d + 1 . Remark 3.5
We stress that for most natural tilings the additional assumption on the existenceof the neighborhoods U i follows from the existence of T i ’s with the corresponding properties.However, there are topological pathologies in which this is not the case. The proof of Propositions 3.3 and 3.4 uses the notion of ˇCech cohomology and classical resultsregarding its properties. For the ease of reading, we begin with an intuitive explanation of theproof ideas, and then present the formal proof.
Intuitive proof.
The d ’th (singular) cohomology group is a topological invariant of a manifoldwhich roughly counts “non trivial holes” of dimension d . A classical result asserts that the d ’thcohomology group of a d -dimensional compact oriented manifold like S d is R . (This correspondsto the intuitive understanding that S d has one d -dimensional hole.) The de-Rham cohomologyand the ˇCech cohomology are analytic and algebro-geometric/combinatorial invariants, thatin many cases agree with their topological cousin. In particular, the d ’th de-Rham and ˇCechcohomologies of S d are equal to R as well.The d ’th ˇCech cohomology with respect to an open cover of the manifold depends on prop-erties of intersections of d + 1 sets in that cover. In general, it depends on the sets which formthe cover, however, it is known that if these sets and their non-empty intersections are finitedisjoint unions of contractibles, then the cohomology groups remain the same, independentlyof the cover. In particular, if the d ’th ˇCech cohomology with respect to such a cover is nontrivial, then there must be d + 1 sets with a non-empty intersection.Hence, for our cover U , U , . . . , we know that its d ’th ˇCech cohomology is R . This readilycompletes the proof of the proposition for S d , as this implies that there must be a point thatbelongs to at least d + 1 of the U i ’s.The proof in R d works in essentially the same way, with cohomology groups replaced bycohomology groups with compact support. 11 ormal proof. For the proof we recall the notion of ˇCech cohomology with values in theconstant sheaf R , and describe the slightly less standard concept of ˇCech cohomology withcompact support. Definitions.
Let S be either R d or a compact manifold such as S d . Let U = { U , U , . . . } bean open cover of S. If S is compact, we assume the collection to be finite. If S is R d , we assumeit to be locally finite and assume in addition that each U i is bounded. • A q − simplex σ = { U i , . . . , U i k } of U is an ordered collection of q + 1 different sets chosenfrom U , such that q (cid:92) k =0 U i k (cid:54) = ∅ . • For a q − simplex σ = ( U i k ) k ∈{ ,...,q } , the j ’th partial boundary is the ( q − ∂ j σ := ( U i k ) k ∈{ ,...,q }\{ j } , obtained by removing the j ’th set from σ . • A q − cochain of U is a function which associates to any q − simplex a real number. The q − cochains form a vector space denoted by C q ( U , R ) , with operations( λf + µg )( σ ) = λf ( σ ) + µg ( σ ) , where λ, µ ∈ R , f, g ∈ C q ( U , R ) , σ is a q − simplex . Similarly, we define C qc ( U , R ) , as the vector space of q − cochains with compact support ,meaning those cochains which assign 0 to all q − simplices, except for finitely many. • There is a differential map δ q : C q ( U , R ) → C q +1 ( U , R ) whose application to f ∈ C q ( U , R )is the ( q + 1) − cochain δ q ( f ) whose value at a ( q + 1) − simplex σ is( δ q f )( σ ) = q +1 (cid:88) j =0 ( − j f ( ∂ j σ ) . The restriction of δ q to C qc ( U , R ) maps it to C q +1 c ( U , R ). • It can be easily seen that δ q +1 ◦ δ q = 0 . • The q ’th ˇCech cohomology group (with compact support) of S with respect to the cover U and values in R is ˇ H q ( U , R ) := Ker( δ q ) / Image( δ q − ) , ( ˇ H qc ( U , R ) := Ker( δ q | C qc ( U , R ) ) / Image( δ q − | C q − c ( U , R ) )) . • A cover (by open sets) is good if all its sets as well as their multiple intersections areeither empty or contractible. It is almost good if all non empty intersections are unionsof finitely many disjoint contractible components.12 lassical results we use.
The first result we use is the following:
Theorem 3.6 If S is a compact smooth orientable manifold (such as S d ), and U is a good oran almost good finite cover, then ˇ H i ( U , R ) (cid:39) H idR ( S ) , where the right hand side is the standard de-Rham cohomology group.Similarly, if S = R d and U is a locally finite good or almost good cover whose sets are bounded,then ˇ H ic ( U , R ) (cid:39) H idR,c ( S ) , where the right hand side is the i ’th de-Rham cohomology group with compact support. For further reading about de-Rham cohomology, with or without compact support, we referthe reader to [4, Sec. 1]. For further reading about the ˇCech cohomology, we refer to [4,Sec. 8]. In particular, Theorem 3.6, for the compact case and good covers is Theorem 8.9 there.The passage to almost good covers is straightforward: In the paragraph which precedes theproof, it is explained that the obstructions to the isomorphism between ˇCech and de-Rhamcohomologies are given by products of the i ’th de-Rham cohomology groups, for i ≥ , of thedifferent intersections (cid:84) qk =0 U i k . Since those intersections are disjoint unions of contractibles,their higher cohomology groups vanish, hence there is no obstruction to the isomorphism.Regarding the case S = R d , the proof in [4, Sec. 8] requires a few small changes: In thestatement of Proposition 8.5 there, one needs to replace the de-Rham complex of the manifoldwith the de-Rham complex with compact support, and the direct product with direct sum.The maps r, δ which appear there will still be well defined by our local finiteness assumptionon the cover, and the assumption that U i ’s are bounded. The proof requires no change. Then,the double complex in the definition of Proposition 8.8 should also be defined using direct sumrather than direct product, but again there is no change in the proof. Given these changes indefinitions, the proof of Theorem 8.9 (also for the almost good case) is unchanged.The second standard result, which is a consequence of Poincar´e duality, is the following: Theorem 3.7
For a compact smooth oriented manifold S of dimension d (such as S d ), H ddR ( S ) (cid:39) R . Similarly, for S = R d , H ddR,c ( R d ) (cid:39) R . See, for example, [4, Sec. 7] for the compact case, and [4, Sec. 4] for R d . Theorems 3.6 and 3.7 yield:
Corollary 3.8 If S is a compact smooth orientable manifold (such as S d ), and U is a good oran almost good finite cover, then ˇ H d ( U , R ) = R . Similarly, if S = R d and U is a locally finite good or almost good cover whose sets are bounded,then ˇ H dc ( U , R ) = R . roof of Propositions 3.3 and 3.4. We show that there must exist d + 1 T i ’s whoseintersection is non-empty. This clearly implies that for any fault tolerance, at least d + 1 colorsare needed.Assume on the contrary that any ( d + 1)-intersection of the T i ’s is empty. Let U i be as in thestatement of the theorem. Then by definition, they form an almost good cover. All intersectionsof at least d + 1 U i ’s are empty by our assumptions. Therefore, there are no d − simplices, and so C d ( U , R ) = 0 . Thus, in the compact case, ˇ H d ( U , R ) = 0 . But on the other hand, by Corollary3.8, ˇ H d ( U , R ) (cid:39) R , a contradiction. In the case of R d the same argument works, with ˇ H dc ( U , R ) in place of ˇ H d ( U , R ). In this section we consider tilings of the plane by tiles T , T , . . . of area at most 1. Each pointin the plane is colored in one of k ≥ t between two points of the same color that belong to different tiles. (The maximum is takenover all possible tilings that satisfy the mild regularity conditions stated in Section 2 and overall possible colorings.) Clearly, the fault tolerance of a rounding scheme based on such a coloredtiling is t/ t , using the Brunn-Minkowski inequality, the circlepacking problem, and the Minkowski-Steiner formula, respectively. In the other direction, weprove lower bounds on t by a series of explicit tilings. In particular, we obtain a tight asymptoticestimate for t , namely, t = ( β + o (1)) √ k , where β = (cid:113) / √ ≈ . The basic idea behind our upper bound proofs is as follows. Assume we have a colored tiling ofthe plane, with minimal distance t . Pick a single color – say, black – and consider all black tilesinside a large square S . We obtain a new collection of tiles T (cid:48) , T (cid:48) , . . . , T (cid:48) m that covers part of S .Now, the assumption that the minimal distance between two same-colored points in differenttiles is t implies that if we inflate each black tile T (cid:48) i into the set T (cid:48)(cid:48) i = { x : ∃ y ∈ T (cid:48) i , | x − y | < t/ } , (1)then the inflations T (cid:48)(cid:48) i are pairwise disjoint. Hence, the sum of their areas essentially cannotexceed the area of the large square, and this allows bounding t from above. The inflations T (cid:48)(cid:48) i have a convenient representation in terms of the Minkowski sum of sets in R . Definition 4.1
For two sets
A, B ⊂ R d , the Minkowski sum of A, B is A + B = { a + b : a ∈ A, b ∈ B } . In terms of this definition, we have T (cid:48)(cid:48) i = T (cid:48) i + B (0 , t/ , (2)14here B (0 , t/
2) is a disk of radius t/ T (cid:48)(cid:48) i , using the classical Brunn-Minkowski (BM) inequality (see, e.g., [3]). Recallthe inequality asserts the following. Theorem 4.2 (Brunn-Minkowski)
Let
A, B be compact sets in R d . Then λ ( A + B ) /d ≥ λ ( A ) /d + λ ( B ) /d , where λ ( X ) is the volume of X (formally, the d -dimensional Lebesgue measure of X ).The inequality is tight if and only if A, B are positive homothetic, i.e., are equal up to translationand dilation by a positive factor.
Proposition 4.3
Let T , T , . . . be a k -colored tiling of the plane, with tiles of area ≤ andminimal distance t . Then t ≤ (cid:18) √ π + o (1) (cid:19) · ( √ k − ≈ . √ k − . Proof:
Consider a square S such that λ ( S ) = n (for some ‘large’ n ). By the pigeonholeprinciple, there exists a color (say, black) that covers at least n /k of the area of S . Look atthe black tiles whose intersection with S is non-empty, and denote their intersections with S by T (cid:48) , T (cid:48) , . . . , T (cid:48) m . Hence, we have m ‘black’ subsets of S , each of area at most 1, whose totalarea is at least n /k .For each T (cid:48) i , define T (cid:48)(cid:48) i = T (cid:48) i + B (0 , t/ T (cid:48)(cid:48) i are disjoint.Furthermore, they are included in S + B (0 , t/
2) whose area is less than ( n + t ) . By taking asufficiently large n , we may assume (cid:88) i λ ( T (cid:48)(cid:48) i ) ≤ (1 + o (1)) n . (3)By the Brunn-Minkowski inequality, we have ∀ i : (cid:113) λ ( T (cid:48)(cid:48) i ) ≥ (cid:113) λ ( T (cid:48) i ) + √ π · t , and thus, ∀ i : λ ( T (cid:48)(cid:48) i ) ≥ λ ( T (cid:48) i ) + √ π · t · (cid:113) λ ( T (cid:48) i ) + πt / . Summing over i and using (3), we get(1 + o (1)) n ≥ m (cid:88) i =1 λ ( T (cid:48)(cid:48) i ) ≥ m (cid:88) i =1 λ ( T (cid:48) i ) + √ π · t · m (cid:88) i =1 (cid:113) λ ( T (cid:48) i ) + mπt / . (4)Note that we have (cid:80) i λ ( T (cid:48) i ) ≥ n /k (by assumption), (cid:80) i (cid:112) λ ( T (cid:48) i ) ≥ n /k (since ∀ ≤ x ≤ √ x ≥ x ), and m ≥ n /k (since ∀ i : λ ( T (cid:48) i ) ≤ o (1)) n ≥ n k · (1 + √ π · t + πt /
4) = n k · (1 + √ πt/ . Dividing by n and rearranging, we get (1 + o (1))( √ k − ≥ √ πt/
2, and hence, t ≥ (cid:18) √ π + o (1) (cid:19) · ( √ k − ≈ . √ k − , as asserted. (cid:3) iscussion. Substituting specific values of k , the upper bounds that follow from Proposi-tion 4.3 are t ≤ .
826 for k = 3, t ≤ .
128 for k = 4 and t ≤ . √ k for a large k .These upper bounds are lossy in two ways. One source of loss is the application of theBrunn-Minkowski inequality. Here, the inequality is tight if the tiles are disks , and the fartherthey are from disks, the larger is the loss. Another source of loss is the space left between theinflations, that is not taken into account in the proof.Interestingly, there is a dichotomy between these two sources of loss. As follows from thecircle packing problem, when the tiles are disks (and so, there is no loss in the BM inequality),the space between the inflations (and so, the loss of the second type) is relatively large. Thespace between the inflations can be made smaller if the tiles are taken to be polygons with afew vertices. However, this comes at the expense of increased loss in the BM inequality, as isdemonstrated in Section 4.1.3. Optimality of our 1-dimensional rounding scheme.
The argument described above givesan easy proof of the optimality of the 1-dimensional rounding scheme presented in the intro-duction. Indeed, consider a 2-colored tiling of the line and look at the segment I = [ − n, n ]for some large n . By the pigeonhole principle, we may assume that black tiles cover at leasthalf of I . By the 1-dimensional Brunn-Minkowski inequality, for each black tile T (cid:48) i ⊂ I and thecorresponding inflation T (cid:48)(cid:48) i = T (cid:48) i + ( − t/ , t/ λ ( T (cid:48)(cid:48) i ) ≥ λ ( T (cid:48) i ) + t . As ∀ i : λ ( T (cid:48) i ) ≤ n tiles. Since the T (cid:48)(cid:48) i ’s are pairwise disjoint and included in [ − n − , n + 1],we obtain 2 n + 2 ≥ (cid:88) i λ ( T (cid:48)(cid:48) i ) ≥ (cid:88) i λ ( T (cid:48) i ) + (cid:88) i t ≥ n + nt, and thus, t ≤ ( n + 2) /n . By letting n tend to infinity, we obtain t ≤
1, implying that the faulttolerance of any two-colored tiling of the line is at most 1 / This upper bound is interesting mainly for a large number k of colors, where the size and formof the original tiles is negligible with respect to the size of the inflations. In this setting, wecan represent each tile by a single point from it, and consider our problem as an instance of thewell-known circle packing problem (see, e.g., [7]), as we know that circles of radius t/ Theorem 4.4 (Fejes-T´oth)
The maximal number of points that can be placed in a square ofarea a , such that the minimal distance between two points is , is at most a/ √ . By rescaling, the theorem implies:
Corollary 4.5
The maximal number of points that can be placed in a square of area a , suchthat the minimal distance between two points is t , is at most a √ t . Proposition 4.6
Let T , T , . . . be a tiling of the plane in k colors, with tiles of area ≤ andminimal distance t . Then t ≤ (cid:32)(cid:115) √ o (1) (cid:33) · √ k ≈ . √ k. roof: Like in the proof of Proposition 4.3, we begin with a tiling T , T , . . . with minimaldistance t , consider the intersection with a square S of area n , pick a single color (say, black)whose intersection with S has area of at least n /k , and obtain disjoint black tiles T (cid:48) , . . . , T (cid:48) m inside a square S of area (1 + o (1)) n , whose total area is at least n /k . We pick a single point x i from each tile T (cid:48) i . By assumption, the minimal distance between the x i ’s is at least t . ByCorollary 4.5, this implies m ≤ (1 + o (1)) · n √ t . On the other hand, m ≥ n /k , since the total area of the black tiles is at least n /k and thearea of each tile is at most 1. Therefore, n k ≤ (1 + o (1)) · n √ t , and consequently, t ≤ (cid:16)(cid:113) √ + o (1) (cid:17) · √ k , as asserted. (cid:3) Discussion.
For small values of k , this upper bound is rather weak – e.g., for k = 3 we obtain t ≤ .
861 (compared to t ≤ .
826 via the BM inequality). This happens, since we completelyneglect the tiles and consider only the area added in the inflation process. For a large k , whenthe area is dominated by the inflation step, the upper bound t ≤ (1 .
074 + o (1)) √ k is tight, aswill be shown in Section 4.2.2. This upper bound is applicable under the additional condition that the tiles T i are convex. Ituses another well-known result, called the Minkowski-Steiner formula . Theorem 4.7 (Minkowski-Steiner)
Let A be a convex compact set in R . Then for anycircle B (0 , r ) , λ ( A + B (0 , r )) = λ ( A ) + (cid:96) ( ∂A ) r + πr , (5) where (cid:96) ( ∂ ( A )) is the (1-dimensional) length of the boundary of A . By substituting (5) into the proof of Proposition 4.3, we obtain(1 + o (1)) n ≥ m (cid:88) i =1 λ ( T (cid:48)(cid:48) i ) = m (cid:88) i =1 λ ( T (cid:48) i ) + t · m (cid:88) i =1 (cid:96) ( ∂ ( T (cid:48) i )) + mπt / . (6)To proceed, we may bound the length of the boundary of each T (cid:48) i in terms of the area λ ( T (cid:48) i ). Ifwe do not make further assumptions on the T (cid:48) i ’s, then the best possible bound of this type isthe isoperimetric inequality , which asserts (cid:96) ( ∂ ( A )) ≥ √ π (cid:112) λ ( A ) , for any region A in the plane bounded by a closed curve. Plugging this into (6) yields exactlythe same upper bound as Proposition 4.3 (which comes by no surprize, as the tightness case ofthe isoperimetric inequality is circles, just like the tightness case in the proof of Proposition 4.3).However, if we further assume that each tile is a convex polygon, then we can obtain animproved bound, as function of the number of vertices in each such polygon. We use anotherclassical result, going back to an Ancient Greec mathematician:17 heorem 4.8 (Zenodorus) Among all polygons on n vertices with the same area, the perime-ter is minimized for the regular n -gon. Recall that the perimeter of a regular l -gon P l of area λ ( P l ) is (cid:96) ( ∂ ( P l )) = 2 √ l (cid:112) cot( π/l ) · (cid:112) λ ( P l ) . Plugging this into (6), we obtain the following.
Proposition 4.9
Let T , T , . . . be a tiling of the plane in k colors, with tiles of area ≤ andminimal distance t . Assume than all tiles are convex polygons with at most l vertices. Then t ≤ (1 + o (1)) · − α l + (cid:113) α l − − k ) ππ , where α l = 2 √ l (cid:112) cot( π/l ) . Proof:
First, we follow the proof sketch presented above until Equation (6), which asserts(1 + o (1)) n ≥ m (cid:88) i =1 λ ( T (cid:48)(cid:48) i ) = m (cid:88) i =1 λ ( T (cid:48) i ) + t · m (cid:88) i =1 (cid:96) ( ∂ ( T (cid:48) i )) + mπt / . Using the theorem of Zenodorus, we obtain(1 + o (1)) n ≥ m (cid:88) i =1 λ ( T (cid:48) i ) + t · α l · m (cid:88) i =1 (cid:113) λ ( T (cid:48) i ) + mπt / , where α l = √ l √ cot( π/l ) . As m ≥ n /k and (cid:80) i (cid:112) λ ( T (cid:48) i ) ≥ (cid:80) i λ ( T (cid:48) i ) ≥ n /k , this implies(1 + o (1)) n ≥ n k · (cid:18) t · α l + πt / (cid:19) , and consequently, (1 + o (1)) k ≥ α l t + π t . Solving the quadratic inequality, we obtain t ≤ (1 + o (1)) · − α l + (cid:113) α l − − k ) ππ , as asserted. (cid:3) Discussion.
For small values of l , the upper bound obtained in Proposition 4.9 is ratherstrong. For example, for l = 3 we obtain t ≤ .
707 for 3 colors and t ≤ .
985 for four colors.As the constructions we obtain below, based on rectangular tiles, satisfy t = 1 / √ t = 1 for four colors, Proposition 4.9 shows that these constructions cannot be improvedusing triangular tiles.As l increases, the bound of Proposition 4.9 becomes weaker and approaches the bound ofProposition 4.3, since circles (for which Proposition 4.3 is tight) can be approximated to anyprecision by convex polygons with a sufficiently large number of vertices.18 k = 3 1 2 3 43 4 1 21 2 3 43 4 1 2 k = 4 1 2 3 4 5 14 5 1 2 31 2 3 4 5 1 k = 5 Figure 3: The brick wall construction for k = 3 , , k − In this subsection we present two series of explicit tilings. The first series provides the bestlower bounds on t for a small number of colors we aware of, and the second series shows thetightness of the asymptotic upper bound t ≤ (1 .
074 + o (1)) √ k . In the brick wall construction with k colors, demonstrated in Figure 3, each tile is a rectanglewith side lengths (cid:112) ( k − / (cid:112) / ( k −
2) (and so, the area of each tile is 1). The construc-tion is periodic, where the basic unit is two rows of adjacent rectangles, colored in a round robinfashion. For an even k , the second row is placed exactly below the first row, and the sequenceof colors is shifted by k/
2. For an odd k , the second row is indented by half a brick (makingthe construction look like a brick wall), and the sequence of colors is shifted by ( k + 1) / (cid:112) ( k − /
2. (This distance is attained both in the vertical and in the horizontal directions.Having the same minimal distance in both directions is the optimization that dictates the sidelengths of the bricks.)In particular, we obtain the lower bounds t ≥ / √ t ≥ t ≥ (cid:112) / In the honeycomb of rectangles construction with k = m colors, each tile is a rectangle withside lengths a and 1 /a , where a = (cid:32) m − m + 1 m − m (cid:33) / . (7)The construction is periodic, where the basic unit is composed as follows. First, we constructa basic ‘large rectangle’, which is an m -by- m square block of tiles, using all the k = m colors(in arbitrary order). Then, the basic unit of the construction is two ‘fat rows’ of adjacent largerectangles, where the second row is indented by half a large rectangle. The coloring of eachlarge rectangle is the same. The construction, for k = 16, is demonstrated in Figure 4.19 Figure 4: The honeycomb of rectangles construction for k = 16 colors. The boundaries of thebasic ‘large rectangles’ are depicted in bold. The placement of the tiles in each single colorcorresponds to the honeycomb lattice.Note that the set of tiles colored in some single color forms the shape of a honeycomb lattice(including the centers of the hexagons). This is why we call the construction ‘honeycomb ofrectangles’.It is easy to see that the minimal horizontal and diagonal distances between two same-colored tiles are ( m − a and (cid:115)(cid:18) m − a (cid:19) + (cid:16)(cid:16) m − (cid:17) a (cid:17) , respectively. By choosing a such that the two distances are equal, we obtain (7), and theasymptotic lower bound t ≥ (1 + o (1)) (cid:18) (cid:19) / · m ≈ . √ k, that matches the upper bound obtained above. In this section we consider tilings of R d , for d ≥
3, by tiles T , T , . . . of volume at most 1. Aswas shown in Section 3, the minimal number of colors that allows for any fault tolerance is d + 1.First, we obtain two upper bounds on the fault tolerance, by generalizing the bounds viathe Brunn-Minkowski inequality and via the circle packing problem presented in Section 4.1.In the other direction, we present an explicit tiling with d + 1 colors that attains a minimaldistance of t = d − √ (and hence, fault tolerance of t/ R thatattains a larger minimal distance, and a k -color tiling of R that attains the optimal asymptoticfault tolerance. 20 .1 Upper bounds on the fault tolerance in R d We present two upper bounds. The first – via the Brunn-Minkowski inequality – is most effectivefor a small number of colors. The second – via sphere packing – is most effective in dimensions3 , , and 24, for which the sphere packing problem was solved completely. Our bound is a straightforward generalization of Proposition 4.3. We start with a large cube S of volume n d , find a color whose intersection with the cube has volume at least n d /k , considerthe tiles T (cid:48) , . . . , T (cid:48) m in that color, and use the fact that by assumption, inflations of these tilesby Minkowski sum with the ball B (0 , t/
2) are pairwise disjoint.
Proposition 5.1
Let T , T , . . . be a tiling of R d in k colors, with tiles of volume ≤ andminimal distance t . Then t ≤ (cid:32) d/ /d √ π + o (1) (cid:33) · ( k /d − , where Γ( · ) is the Gamma function. Proof:
The proof is almost identical to the proof of Proposition 4.3. Instead of (4), we obtain(1 + o (1)) n d ≥ m (cid:88) i =1 d (cid:88) j =0 (cid:18) dj (cid:19) λ ( T (cid:48) i ) j/d ( b t/ ) − j/d , where b t/ is the volume of the d -dimensional ball B (0 , t/ ≤ λ ( T (cid:48) i ) ≤
1, we have (cid:80) i λ ( T (cid:48) i ) j/d ≥ (cid:80) i λ ( T (cid:48) i ) ≥ n d /k , and hence we obtain(1 + o (1)) n d ≥ n d k · (cid:16) b /dt/ (cid:17) d . This implies (1 + o (1))( k /d − ≥ b /dt/ = π / Γ( d/ /d · t , and consequently, t ≤ (cid:32) d/ /d √ π + o (1) (cid:33) · ( k /d − , as asserted. (cid:3) Discussion.
For a large number k (cid:29) d of colors, Proposition 5.1 gives the upper bound t ≤ ( (cid:112) /πe + o (1)) k /d (see (8) below). This bound is not far from being tight. Indeed, itsdependence on k is correct, as it can be easily matched by a periodic cubic tiling, in whicheach tile is a cube with side length 1 and the basic unit is a large cube with side length k /d that contains each color in exactly one tile (in the same order). Moreover, even regarding thecoefficient of k /d , the optimal asymptotic upper bounds for d = 3 , ,
24 which we obtain belowvia the sphere packing problem, improve over this bound by only a small factor.21o estimate the upper bound on t for a large d and k = d + 1 colors, note that( d + 1) /d − o (1)) ln( d ) d , and Γ( d/ /d = (cid:18) √ e + o (1) (cid:19) √ d. (8)Therefore, the bound we obtain in this case is t ≤ (cid:32)(cid:114) πe + o (1) (cid:33) ln d √ d , which implies that the fault tolerance decreases to zero as d tends to infinity. For comparison,the lower bound we obtain in Section 5.2.1 is t ≥ Ω(1 /d ).For the ‘smallest’ case d = 3 , k = 4, in which the bound of Proposition 5.1 is much strongerthan the bound we obtain via sphere packing, the bound is t ≤ . / √ π · (4 / − ≈ . . For comparison, the best construction we have in this setting satisfies t = 0 . The possibility of obtaining an upper bound via the Minkowski-Steiner formula.
In Section 4.1.3 we obtained an upper bound on t , under the additional assumption that the tilesare convex polygons with a bounded number of vertices, using the Minkowski-Steiner formula.This formula has a higher-dimensional analogue: λ ( A + B (0 , r )) = λ ( A ) + λ d − ( ∂A ) r + d − (cid:88) j =2 λ j ( A ) r j + 2 π d/ d Γ( d/ r d , (9)where λ d − ( ∂ ( A )) is the ( d − A , and λ j ( A ) arecontinuous functions of A called mixed volumes .In order to obtain an upper bound on the fault tolerance using (9), one has to bound themixed volumes λ j ( A ) from below in terms of λ ( A ), which seems to be a challenging task. In order to generalize the argument of Proposition 4.6 to R d , we need a generalization to d > Question 5.2
What is the maximal number of points that can be placed in a d -dimensionalcube of volume v , such that the minimal distance between two points is 1? However, as far as we know, this question is open for all d > sphere packing problem,which asks for the maximal possible density of a set of non-intersecting congruent spheres in R d . Definition 5.3
The density of a sphere packing (i.e., collection of pairwise disjoint congruentspheres) P = ∪ P i in R d is lim sup r →∞ λ ( B (0 , r ) ∩ (cid:83) P ) λ ( B (0 , r )) . Notation 5.4
Denote the maximal density of a sphere packing in R d by δ d , and the volume ofthe unit ball B (0 , ⊂ R d by v d = π d/ / Γ( d/ . Proposition 5.5
Let T , T , . . . be a tiling of R d in k colors, with tiles of volume ≤ andminimal distance t . Then t ≤ (cid:16) δ d /v d ) /d + o (1) (cid:17) · k /d = (cid:32) d/ /d · δ /dd √ π + o (1) (cid:33) · k /d , where Γ( · ) is the Gamma function. Note that the asymptotic upper bound of Proposition 5.5 is stronger than the asymptotic upperbound that follows from Proposition 5.1 by the constant factor ( δ d ) /d . Proof:
Let T be a k -colored tiling of R d that satisfies the assumptions of the proposition, andconsider the sequence of balls { B (0 , n ) } n =1 , , ,... . By the pigeonhole principle, there exists acolor (say, black) such that for each n (cid:96) in an infinite subsequence { n (cid:96) } (cid:96) =1 , ,... , the intersectionof the black tiles with the ball B (0 , n (cid:96) ) has volume of at least λ ( B (0 , n (cid:96) )) /k = n d(cid:96) · v d k . As the volume of each tile is at most 1, we know that for each n (cid:96) , the number of black tilesthat intersect B (0 , n (cid:96) ) is at least n d(cid:96) v d /k .Pick some value n (cid:96) , denote the intersections of black tiles with B (0 , n (cid:96) ) by T (cid:48) , T (cid:48) , . . . , and takeone point x i from each tile T (cid:48) i . As the minimal distance between two black points in differenttiles is t , balls of radius t/ x i are pairwise disjoint. Hence, their total volumeis at least n d(cid:96) · v d k · ( t/ d · v d . On the other hand, each such ball is contained in the ball B (0 , (1 + o (1)) n (cid:96) ) (since its volumeis at most 1, and it contains a point in B (0 , n (cid:96) )). This implies that for a sufficiently large (cid:96) ,the total volume of these balls must be smaller than δ d · λ ( B (0 , (1 + o (1)) n (cid:96) )), as otherwise, theinfinite collection of the balls B ( x i , t/
2) (where for each ball B (0 , n (cid:96) ) we select x i ’s in the waydescribed above, respecting the x i ’s selected for smaller values of n (cid:96) ) would be a sphere packingof R d whose density is larger than δ d . Therefore, for a sufficiently large n (cid:96) , we have n d(cid:96) · v d k · ( t/ d · v d ≤ (1 + o (1)) δ d · n d(cid:96) v d , or equivalently, t ≤ (2 + o (1))( δ d /v d ) /d · k /d , as asserted. (cid:3) iscussion. In the two last decades, there has been a tremendous progress in the research ofthe sphere packing problem. In 2005, Hales [13] solved the problem for d = 3, proving a 17’thcentury conjecture of Kepler. Three years ago, in a beautiful short paper, Viazovska [19] solvedthe problem for d = 8, and shortly after, Cohn, Kumar, Miller, Radchenko, and Viazovska [5]used Viazovska’s method along with other tools to solve the problem for d = 24. For otherdimensions, the problem is still open. We can use the results of [5, 13, 19] to obtain tight upperbounds on t in dimensions 3 , , and 24. • For d = 3, Hales [13] showed that δ = π √ ≈ . t ≤ (2 / + o (1)) k / ≈ (1 .
122 + o (1)) k / . • For d = 8, Viazovska [19] showed that δ = π ≈ . t ≤ ( √ o (1)) k / ≈ (1 .
414 + o (1)) k / . • For d = 24, Cohn et al. [5] showed that δ = π ≈ . t ≤ (2 + o (1)) k / .As we show in Section 5.2, these bounds are asymptotically tight.Using the same method, we can transform any upper bound for the sphere packing problemto an upper bound on the fault tolerance of a rounding scheme in the corresponding dimension.A list of such conjectured bounds for d ≤
10 can be found in [6]. R d In this section we present three explicit constructions that yield lower bounds on t . The firstconstruction considers d + 1 colors in R d . The second construction is specific to 4-colored tilingof R but yields a larger lower bound. The third construction shows that the asymptotic upperbound t ≤ (2 / + o (1)) k / in R is tight. We exemplify the construction in R , but it will be apparent how to generalize it to higherdimensions. Informal description of the construction.
Informally, the construction works as follows.
Step 1:
We begin with dividing R into cubes with side length a (to be determined below),depicted in Figure 5(a). Then, we make the ‘interior’ of each cube into a tile and give all thesetiles the color 1. In order to keep a minimal distance of t between two points colored 1 indifferent tiles, we must leave a neighborhood of width t/ t . The part of such a wall includedin a basic cube is shown in Figure 5(b). Step 2:
We make the ‘interior’ of each wall (i.e., fattened facet) into a tile and give all thesetiles the color 2, as is demonstrated in Figure 5(b). (Note that each tile contains points from twoadjacent basic cubes.) In order to keep a minimal distance of t between two points colored 2in different tiles, we must leave a neighborhood of t/ √ Step 3:
We make the ‘interior’ of each edge of the skeleton into a tile and give all these tilesthe color 3, as is shown in Figure 5(d). (Note that each tile contains points from four adjacent24 a) t / t / (b) t / t √ t √ t ◦ (c) t / t √ t √ t ◦ (d) Figure 5: The dimension reducing construction. Part (a) shows the basic cubes we start with.Part (b) shows a fattened wall of width t/
2, and its interior that gets the color 2. Part (c)shows where the minimal distance between two tiles colored 2 is attained. Part (d) presents (infull lines) the interiors of the fattened edges that get the color 3 and shows where the minimaldistance between two such tiles is attained.basic cubes.) In order to keep a minimal distance of t between two points colored 3 in differenttiles, we must leave an additional neighborhood of t/ √ Step 4:
We make the neighborhoods of the corners into tiles and give them the color 4. (Notethat each tile contains points from 8 adjacent basic cubes.) We have to make sure that thedistance between each two such ‘fattened corners’ is at least t , and this requirement dictatesthe choice of t , as is explained below.It is clear that the resulting 4-colored tiling can be generalized into a ( d + 1)-colored tilingof R d . 25 ormal definition of the construction. For the sake of formality, we give the exact defi-nitions of the tiles below.For ( x, y, z ) ∈ R , let f ( x, y, z ) ∈ [0 , a/ be a motonone non-decreasing ordering of { ¯ x, ¯ y, ¯ z } ,where ¯ b = min {| b − an | : n ∈ Z } . That is, we measure the minimal distance to a wall in eachcoordinate and arrange these minimal distances in a non-decreasing order. For example, forany a ≥
1, the point (2 . a, . a, . a ) is mapped by f to (0 . a, . a, . a ). • The tiles colored 1 consist of all points ( x, y, z ) in a basic cube such that f ( x, y, z ) > t/ t/ • The tiles colored 2 consist of points ( x, y, z ) in a basic cube such that f ( x, y, z ) ≤ t/ f ( x, y, z ) > t/ t/ √
2. (These are the points that are close to a wall only in a singlecoordinate – the ‘interiors’ of the fattened facets.) Note that each basic cube contains sixsuch tiles, and each such tile contains points of two adjacent basic cubes. • The tiles colored 3 consist of points ( x, y, z ) in a basic cube such that f ( x, y, z ) ≤ t/ f ( x, y, z ) ≤ t/ t/ √
2, and f ( x, y, z ) > t/ t/ √
2. (These are the points that areclose to a wall in two coordinates – the ‘interiors’ of the fattened edges.) Note that eachbasic cube contains 12 such tiles (one for each edge), and each such tile contains pointsof four adjacent basic cubes. • The tiles colored 4 consist of the rest of the points (i.e., the fattened neighborhoods ofthe vertices). Note that each basic cube contains 8 such tiles (one for each vertex), andeach such tile contains points of eight adjacent basic cubes.
The choice of t . The construction makes sure that for i = 1 , ,
3, the distance between twopoints colored i in different tiles is at least t . In order to guarantee the same condition forthe color 4 as well, we have to choose t such that the distance between two ‘neighborhoods ofcorners’ will be at least t . By the construction, this amounts to the inequality a − (cid:18) t t √ (cid:19) ≥ t, or equivalently, t ≤ a √ . Similarly, in R d we obtain t ≤ a d − √ . The choice of a . An easy computation shows that the largest tiles are those colored 1. Hence,in order to make the volume of all tiles ≤
1, we have to choose a such that the volume of eachsuch tile is 1. By construction, this amounts to the equality1 = ( a − t ) = a · (cid:18) −
12 + 2 √ (cid:19) , or equivalently, a = 1 + √ ≈ . R d we obtain a = 1 + d − √ . The lower bound on t . Summarizing the above, the value of t we obtain in R is t = a √ √ ≈ . .
212 3434 2121 4343 1212 3434 2121 4343
Figure 6: The bricks and balloons constructionIn R d , we obtain t = d − √ . This gives lower bounds of 0 .
131 and d − √ on the faulttolerance in R and R d , respectively. The lower bound in R is superseded by the ‘bricks andballoons’ construction described below. The lower bound in R d , which is Ω(1 /d ), is not veryfar from the upper bound we obtained in Section 5.1.1 – namely, O (log d/ √ d ). The bricks and balloons (B&B) construction, demonstrated in Figure 6, is a periodic tiling of R , colored in 4 colors. In order to present the tiling, we need an auxiliary notation. Notation.
Consider the slab D = R × R × [ z , z ] ⊂ R . We say that a tiling T , T , . . . of D is a fattened plane tiling if there exists a tiling T (cid:48) , T (cid:48) , . . . of the plane such that ∀ i : T i = T (cid:48) i × [ z , z ]. The layers of B&B.
The B&B construction is based on two fattened plane tilings: • The brick wall layer – a fattening of a brick wall tiling of the plane. The underlyingplane tiling is a periodic tiling, in which the basic unit consists of four rows of adjacentrectangles, where the even rows are indented by half a brick, making the constructionlook like a brick wall. In the odd rows, the colors 1,2 are used alternately, and in the evenrows, the colors 3,4 are used alternately. Furthermore, the colors in the third and fourthrows are shifted by one, so that a rectangle colored i is not placed right under anotherrectangle colored i , see Figure 6. The side lengths of the rectangles and the width of theslab will be specified below. • The balloon layer – a fattening of another periodic tiling of the plane. This tiling is similarin its high-level structure to the brick wall layer, but instead of rectangles, the tiles are‘balloons’ consisting of a regular octagon and a square built on one of its edges. The basicunit of the tiling is four columns of adjacent balloons, where the even columns are shiftedin such a way that the balloons tile the plane. In the odd columns, the colors 1,2 are usedalternately, and in the even columns, the colors 3,4 are used alternately. Furthermore, the27olors in the third and fourth columns are shifted by one, so that a balloon colored i isnot placed near another balloon colored i (see Figure 6). The side lengths of the balloonsand the width of the slab will be specified below. The structure of B&B.
The B&B tiling is periodic, where the basic unit consists of a brickwall layer and a balloon layer, placed one on top of the other in the way presented in Figure 6.(Note that once the two layers are placed, the coloring of one layer fully determines the coloringof the other.)
The side lengths and the slab heights.
Denote the side length of the balloon by 2 a . It iseasy to see that the side lengths of the rectangles are (2 + √ a and (4 + 2 √ a . Furthermore,the area of both a balloon and a rectangle is equal to (12 + 8 √ a . Hence, in order to makeall tiles equal-volume, we choose the height of all slabs (both in the brick wall layers and in theballoon layers) to be the same value z .To find the minimal distance between two same-colored tiles, we consider several cases: • Two same-colored balloons in the same layer: The minimal distance between two suchballoons is (cid:112) √ a ≈ . a . • Two same-colored bricks in the same layer: Here, the minimal distance is (2 + √ a . • A balloon and a same-colored brick at an adjacent layer: Here, the minimal distance is a . • A balloon/brick and a same colored balloon/brick two layers apart: Here, the minimaldistance is z .Hence, the minimal distance between two same-colored tiles is min { a, z } , and in order to opti-mize it we choose z = a . The lower bound on t . Since z = a , the volume of each tile is (12+8 √ a · a = (12+8 √ a .Thus, in order to make the volume of all tiles equal to 1, we fix a = (cid:18)
112 + 8 √ (cid:19) / ≈ . . Therefore, this construction satisfies t = a ≈ .
35, and hence, provides a lower bound of t/ ≈ .
175 on the fault tolerance. This improves significantly over the fault tolerance of thedimension reducing construction (i.e., 0 . . This construction, a k -colored tiling of R where k (cid:29)
3, is a natural general-ization to R of the ‘honeycomb of rectangles’ construction presented in Section 4.2.2. Theidea behind the construction is to choose an optimal sphere packing in R , and construct afattened plane tiling in which the tiles in each color are placed at the centers of the spheres ofthe packing. (Recall that as the number of colors is large, the size of each tile is negligible withrespect to the size of its inflation, and hence, we can treat the tiles as single points.)We use the classical HCP lattice (one of the most common closed packings , see [7]), whichcorresponds to a periodic sphere packing in which the basic unit is two hexagonal layers of28pheres, where in the top layer, each sphere is placed on top in the hollow between threespheres in the bottom layer. The coordinates of the centers of these spheres are:( r, r, r ) , (3 r, r, r ) , (5 r, r, r ) , . . . , (2 r, r + √ r, r ) , (4 r, r + √ r, r ) , (6 r, r + √ r, r ) , . . . for the bottom layer, and(2 r, r + √ r, r + 2 √ r ) , (4 r, r + √ r, r + 2 √ r ) , . . . , ( r, r + 4 √ r, r + 2 √ r ) , (3 r, r + 4 √ r, r + 2 √ r ) , . . . for the top layer. The structure of the tiling.
Assume that the number of colors is k = m . Each tile isa box with side lengths ( a, b, c ) to be determined below, and the basic unit is a ‘large box’,which is an m × m × m cubic block of tiles, using all the k = m colors (in arbitrary order).Then, the basic unit of the construction is a two-layer fattened plane tiling, in which each layeris a fattened copy of the ‘honeycomb of rectangles’ tiling. The upper layer is shifted by m a in the x -coordinate and by √ m a in the y -coordinate, so that the corners of the large boxeslie in the coordinates of the sphere centers described above (for r = m a ). A quick calculationshows that in order to make this possible, the proportion ( a : b : c ) should be approximately (2 : √ √ /
3) (where we neglect the size of each tile with respect to the size of the inflation,which can be absorbed in an 1 + o (1) multiplicative factor in the final value of t ). The volumeof each tile is clearly abc . In order to make the volumes of all tiles equal to 1, we need a · √ a · √ a = 1 , and thus, a = (6 / √ / = 2 / ≈ . / , / / / , / / / ) ≈ (1 . , . , . , and the minimal distance between two same-colored points in different tiles is( m − a = (2 / + o (1)) k /d , which matches the upper bound proved in Section 5.1.1. Generalization to higher dimensions.
A similar tiling can be constructed to match any lattice sphere packing , assuming the number of colors k is sufficiently large with respect to d .Hence, any dense lattice sphere packing can be translated into a lower bound on the asymptoticfault tolerance of rounding schemes in the corresponding dimension. In particular, as the E8lattice and the
Leech lattice which attain the maximal possible density of sphere packings indimension 8 and 24 (respectively) are lattice packings, they can be translated to box tilingsshowing that the asymptotic upper bounds on the fault tolerance in R and R proved inSection 5.1.1 are tight. In this paper we demonstrated that the natural problem of how to round points in R d withoptimal fault tolerance using a tiny amount of side information is tied to deep problems on space29artitions and sphere packings. We proposed a new way to look at the problem as colored tilingsof the space, and related it to a generalized type of error correcting code in which we inflate tilesinto Minkowski sums rather than points into balls. We obtained a number of tight asymptoticbounds for a large number of colors, as well as lower and upper bounds for small numbers ofcolors, but finding the optimal solutions for particular numbers of colors and dimensions is stillwide open. Acknowledgements
The authors are grateful to Stephen D. Miller for valuable suggestions regarding the spherepacking problem.
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