Lower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners

Transcription

1 Lower Bouns for Local Monotonicity Reconstruction from Transitive-Closure Spanners Arnab Bhattacharyya Elena Grigorescu Mahav Jha Kyomin Jung Sofya Raskhonikova Davi P. Wooruff Abstract Given a irecte graph G = V, E an an integer k 1, a k-transitive-closure-spanner k-tcspanner of G is a irecte graph H = V, E H that has 1 the same transitive-closure as G an 2 iameter at most k. Transitive-closure spanners are a common abstraction for applications in access control, property testing an ata structures. We show a connection between 2-TC-spanners an local monotonicity reconstructors. A local monotonicity reconstructor, introuce by Saks an Seshahri SIAM Journal on Computing, 2010, is a ranomize algorithm that, given access to an oracle for an almost monotone function f : [m] R, can quickly evaluate a relate function g : [m] R which is guarantee to be monotone. Furthermore, the reconstructor can be implemente in a istribute manner. We show that an efficient local monotonicity reconstructor implies a sparse 2-TC-spanner of the irecte hypergri hypercube, proviing a new technique for proving lower bouns for local monotonicity reconstructors. Our connection is, in fact, more general: an efficient local monotonicity reconstructor for functions on any partially orere set poset implies a sparse 2-TC-spanner of the irecte acyclic graph corresponing to the poset. We present tight upper an lower bouns on the size of the sparsest 2-TC-spanners of the irecte hypercube an hypergri. These bouns imply tighter lower bouns for local monotonicity reconstructors that nearly match the known upper bouns. Massachusetts Institute of Technology, USA. {abhatt,elena g}@mit.eu. Pennsylvania State University, USA. {mxj201,sofya}@cse.psu.eu. Supporte by NSF CAREER awar KAIST, South Korea. kyomin@kaist.eu IBM Almaen Research Center, USA. pwooru@us.ibm.com.

2 1 Introuction Graph spanners were introuce in the context of istribute computing [?], an since then have foun numerous applications, such as efficient routing [?,?,?,?,?], simulating synchronize protocols in unsynchronize networks [?], parallel an istribute algorithms for approximating shortest paths [?,?,?], an algorithms for istance oracles [?,?]. Several variants on graph spanners have been efine. In this work, we focus on transitive-closure spanners that were introuce in [?] as a common abstraction for applications in access control, property testing an ata structures. Definition 1.1 TC-spanner. Given a irecte graph G = V, E an an integer k 1, a k-transitiveclosure-spanner k-tc-spanner of G is a irecte graph H = V, E H with the following properties: 1. E H is a subset of the eges in the transitive closure of G. 2. For all vertices u, v V, if G u, v <, then H u, v k. Thus, a k-transitive-closure-spanner or k-tc-spanner is a graph with small iameter that preserves the connectivity of the original graph. In the applications above, the goal is to fin the sparsest k-tc-spanner for a given k an G. The number of eges in the sparsest k-tc-spanner of G is enote by S k G. 1.1 Our Contributions The contributions of this work fall into two categories: 1 We show that an efficient local monotonicity reconstructor implies a sparse 2-TC-spanner of the irecte hypergri hypercube, proviing a new technique for proving lower bouns for local monotonicity reconstructors. 2 We present tight upper an lower bouns on the size of the sparsest 2-TC-spanners of the irecte hypercube an hypergri. These bouns imply tighter lower bouns for local monotonicity reconstructors for these graphs that nearly match the upper bouns given in [?]. 1.2 Lower Bouns for Local Monotonicity Reconstruction Property-preserving ata reconstruction was introuce in [?]. In this moel, a reconstruction algorithm, calle a filter, sits between a client an a ataset. A ataset is viewe as a function f : D R. The client accesses the ataset using queries of the form x D to the filter. The filter looks up a small number of values in the ataset an outputs gx, where g must satisfy some fixe structural property P. Extening this notion, Saks an Seshahri [?] efine local reconstruction. A filter is local if it allows for a local or istribute implementation: namely, if the output function g oes not epen on the orer of the queries. Definition 1.2 Local filter. A local filter for reconstructing property P is an algorithm A that has oracle access to a function f : D R, an to an auxiliary ranom string ρ the ranom see, an takes as input x D. For fixe f an ρ, A runs eterministically on input x to prouce an output A f,ρ x R. Note that a local filter has no internal state to store previously mae queries. The function gx = A f,ρ x output by the filter must satisfy the following conitions: For each f an ρ, the function g must satisfy P. If f satisfies P, then g must be ientical to f with probability at least 1 δ, for some error probability δ 1/3. The probability is taken over ρ. In answering query x D, the filter A may ask for values of f at omain points of its choice possibly aaptively using its oracle access to f. Each such access mae to the oracle is calle a lookup to istinguish 1

3 it from the client query x. A local filter is non-aaptive if the set of omain points that the filter looks up to answer an input query x oes not epen on answers given by the oracle. In [?], the authors also require that g must be sufficiently close to f: With high probability over the choice of ρ, Distg, f Bn Distf, P, where Bn is calle the error blow-up. Distg, f is the number of points in the omain on which f an g iffer. Distf, P is min g P Distg, f. If a local filter along with Definition 1.2 satisfies this conition, we call it istance-respecting Local Monotonicity Reconstructors The most stuie property in the local reconstruction moel is monotonicity of functions [?,?]. To efine monotonicity of functions, consier an n-element poset V n an let G n = V n, E be the relation graph, i.e., the Hasse iagram, for V n. A function f : V n R is calle monotone if fx fy for all x, y E. We particularly focus on posets which have the irecte hypergri graph as its relation graph. The irecte hypergri, enote H m,, has vertex set {1, 2,..., m} an ege set {x, y : unique i {1,..., } such that y i x i = 1 an for j i, y j = x j }. For the special case m = 2, H 2, is calle a hypercube an is also enote by H. A monotonicity filter nees to ensure that the output function g is monotone. For instance, if G n is a irecte line, H n,1, the filter nees to ensure that the output sequence specifie by g is sorte. To motivate monotonicity reconstructors for hypergris, consier the scenario of rolling amissions: An amissions office assigns scores to each application, such as the applicant s GPA, SAT results, essay quality, etc. Base on these scores, some complicate thir-party algorithm outputs the probability that a given applicant shoul be accepte. The amissions office wants to make sure on the fly that strictly better applicants are given higher probability, that is, probabilities are monotone in scores. A hypergri monotonicity filter may be use here. A local filter can be implemente in a istribute manner with an aitional guarantee that every copy of the filter will correct to the same monotone function of the scores. This can be one by supplying the same ranom see to each copy of the filter. [?] gives a istance-respecting local monotonicity filter for the irecte hypergri, H m,, that makes log m O lookups per query. No non-trivial monotonicity filter for the hypercube H performing o2 lookups per query is known. One of the monotonicity filters in [?] is a local filter for the irecte line H m,1 with Olog m lookups per query but a worse error blow up than in [?]. As observe in [?], this upper boun is tight. A lower boun of 2 α, on the number of lookups per query for a istance-respecting local monotonicity filter on H with error blow-up 2 β, where α, β are sufficiently small constants, appeare in [?]. Notably, all known local monotonicity filters are non-aaptive. We show how to construct sparse 2-TC-spanners from local monotonicity reconstructors with low lookup complexity. These constructions, together with our lower bouns on the size of 2-TC-spanners of the hypergri an hypercube Section 1.3, imply lower bouns on lookup complexity of local monotonicity reconstructors for these graphs with arbitrary error blow-up. We state our transformations from non-aaptive an aaptive reconstructors separately. Theorem 1.1 Transformation from non-aaptive Local Monotonicity Reconstructors to 2-TC-spanners. Let G n = V n, E be a poset on n noes. Suppose there is a non-aaptive local monotonicity reconstructor A for G n that looks up at most ln values on any query an has error probability at most δ. Then there is a 2-TC-Spanner of G n with Onln log n/ log1/δ eges. Next theorem applies even to aaptive local monotonicity reconstuctors. It takes into account how many lookups on query x are points incomparable to x. In particular, if there are no such lookups, then constructe 2-TC-spanner is of the same size as in Theorem

4 Theorem 1.2 Transformation from aaptive Local Monotonicity Reconstructors to 2-TC-spanners. Let G n = V n, E be a poset on n noes. Suppose there is an aaptive local monotonicity reconstructor A for G n that, for any query x V n, looks up at most l 1 n vertices comparable to x an at most l 2 n vertices incomparable to x, an has error probability at most δ. Then there is a 2-TC-Spanner of G n with Onl 1 n 2 l 2n log n/ log1/δ eges. In Theorem 1.1 an 1.2, when δ is sufficiently small, the bouns on the 2-TC-Spanner size become Onln an Onl 1 n 2 l 2n, respectively. As mentione earlier, all known monotonicity reconstructors are non-aaptive. It is an open question whether it is possible to give a transformation from aaptive local monotonicity reconstructors to 2-TCspanners without incurring an exponential epenence on the number of lookups mae to points incomparable to the query point. We o not know whether this epenence is an artifact of the proof or an inication that lookups to incomparable points might be helpful for aaptive local monotonicity reconstructors. In Theorems 1.5 an 1.6 Section 1.3, we present nearly tight bouns on the size of the sparsest 2- TC-spanners of the hypercube an the hypergri. Theorems 1.1 an 1.2, together with the lower bouns in Theorems 1.5 an 1.6, imply the following lower bouns on the lookup complexity of local monotonicity reconstructors for these graphs with arbitrary error blow-up. Corollary 1.3. Consier a nonaaptive local monotonicity filter with constant error probability δ. If the log filter is for functions f : H m, R, it must perform Ω 1 m lookups per query. If the filter is 2 log log m 1 for functions f : H R, it must perform Ω 2 α / lookups per query, where α Corollary 1.4. Consier an aaptive local monotonicity filter with constant error probability δ, that for every query x V n, looks up at most l 2 vertices incomparable to x. If the filter is for functions f : H m, log R, it must perform Ω 1 m lookups to vertices comparable to x per query x. If the filter is 2 l 2 2 log log m 1 for functions f : H R, it must perform Ω 2 α l 2 / comparable lookups, where α Prior to this work, no lower bouns for monotonicity reconstructors on H m, with epenence on both m an were known. Unlike the boun in [?], our lower bouns hol for any error blow-up an for non-istance-respecting filters. Our bouns are tight for non-aaptive reconstructors. Specifically, for the hypergri H m, of constant imension, the number of lookups is log m Θ, an for the hypercube H, it is 2 Θ for any error blow-up Testers vs. Reconstructors [?] obtaine monotonicity testers from 2-TC-spanners. Unlike in the application to monotonicity testing, here we use lower bouns on the size of 2-TC-spanners to prove lower bouns on complexity of local monotonicity reconstuctors. Lower bouns on the size of 2-TC-spanners o not imply corresponing lower bouns on monotonicity testers. E.g., the best monotonicity tester on H runs in O 2 time [?,?], while, as shown in Theorem 1.6, every 2-TC-spanner of H must have size exponential in. 1.3 Our Results on 2-TC-Spanners of the Hypercube an Hypergri Our main theorem gives a set of explicit bouns on S 2 H m, : Theorem 1.5 Hypergri. Let S 2 H m, enote the number of eges in the sparsest 2-TC-spanner of H m,. Then 1 for m 3, m log m Ω 2 log log m 1 = S 2 H m, m log m. 1 Logarithms are always to base 2 unless otherwise inicate. 3

5 The upper boun in Theorem 1.5 follows from a general construction of k-tc-spanners for graph proucts for arbitrary k 2, presente in Section 4.1. The lower boun is the most technically ifficult part of our work. It is prove by a reuction of the 2-TC-spanner construction for [m] to that for the 2 [m] 1 gri an then irectly analyzing the number of eges require for a 2-TC-spanner of 2 [m] 1. We show a traeoff between the number of eges in the 2-TC-spanner of the 2 [m] 1 gri that stay within the hyperplanes {1} [m] 1 an {2} [m] 1 versus the number of eges that cross from one hyperplane to the other. The proof procees in multiple stages. Assuming an upper boun on the number of eges staying within the hyperplanes, each stage is shown to contribute a substantial number of new eges crossing between the hyperplanes. The proof of this traeoff lemma is alreay non-trivial for = 2 an is presente first in Section The proof for > 2 is presente subsequently in Section While Theorem 1.5 is most useful when m is large an is small, in Section 6, we present bouns on S 2 H m, which are optimal up to a factor of 2m an, thus, supersee the bouns from Theorem 1.5 when m is small. The general form of these bouns is a somewhat complicate combinatorial expression but they can be estimate numerically. Specifically, S 2 H m, = 2 cm poly, where c , c , c an c , each significantly smaller than the exponents corresponing to the transitive closure sizes for the ifferent m. We first prove the special case of m = 2 i.e., the hypercube in Section 5 an then generalize the arguments to general m. Specifically, we obtain the following theorem for the hypercube. Theorem 1.6 Hypercube. Let S 2 H be the number of eges in the sparsest 2-TC-spanner of H. Then Ω2 c = S 2 H = O 3 2 c, where c As a comparison point for our bouns, note that the obvious bouns on S 2 H are the number of eges in the -imensional hypercube, 2 1, an the number of eges in the transitive closure of H, which is 3 2. An ege in the transitive closure of H has 3 possibilities for each coorinate: both enpoints are 0, both enpoints are 1, or the first enpoint is 0 an the secon is 1. This inclues self-loops, so we subtract the number of vertices in H to get the esire quantity. Thus, 2 1 S 2 H 3 2. Similarly, the straightforwar bouns on the number of eges in a 2-TC-spanner of H m, in terms of the number of eges in the irecte gri an in its transitive closure are m 1 m 1 an m 2 +m Previous work on bouning S k for other families of graphs m, respectively. Thorup [?] consiere a special case of TC-spanners of graphs G that have at most twice as many eges as G, an conjecture that for all irecte graphs G on n noes there are such k-tc-spanners with k polylogarithmic in n. He prove this for planar graphs [?], but Hesse [?] gave a counterexample for general graphs by constructing a family for which all n TC-spanners nee n 1+Ω1 eges. TC-spanners were stuie for irecte trees: implicitly in [?,?,?,?,?] an explicitly in [?]. For the irecte line, [?] an later, [?] expresse S k H n,1 in terms of the inverse Ackermann function. See Section 2.2 for a efinition. Lemma 1.7 [?,?,?]. Let S k H n,1 enote the number of eges in the sparsest k-tc-spanner of the irecte line H n,1. Then S 2 H n,1 = Θn log n, S 3 H n,1 = Θn log log n, S 4 H n,1 = Θn log n an, more generally, S k H n,1 = Θnλ k n where λ k n is the inverse Ackermann function. The same boun hols for irecte trees [?,?,?]. An On log n λ k n boun on S k for H-minor-free graph families e.g., boune genus an boune tree-with graphs was given in [?]. 4

6 2 Preliminaries 2.1 Notation For a positive integer m, we enote {1,..., m} by [m]. For x {0, 1}, we use x to enote the weight of x, that is, the number of non-zero coorinates in x. Level i in a hypercube contains all vertices of weight i. The partial orer on the hypergri H m, is efine as follows: x y for two vertices x, y [m] iff x i y i for all i []. Similarly, x y, if x an y are istinct vertices in [m] satisfying x y. Vertices x an y are comparable if either y is above x that is, x y or y is below x that is, y x. We enote a path from v 1 to v l, consisting of eges v 1, v 2, v 2, v 3,..., v l 1, v l by v 1,..., v l. 2.2 The Inverse Ackermann Hierarchy Our efinition of inverse Ackermann functions is erive from the iscussion in [?]. For a given function f : R 0 R 0 such that fx < x for all x > 2, efine the function f x : R 0 R 0 to be the following: f x = min{k Z 0 : f k x 2}, where f k enotes f compose with itself k times We note that the solution to the following recursion: { 0 if n 2 T n a n + n fn T fn if n > 2 is T n = a n f n. This follows from the fact that f fn = f n 1 for n > 2. We efine the inverse Ackermann hierarchy to be a sequence of functions λ k for k 0. As the base cases, we have λ 0 n = n/2 an λ 1 n = n. For j 2, we efine λ j n = λ j 2 n. Thus, λ 2n = Θlog n, λ 3 n = Θlog log n an λ 4 n = Θlog n. Note that the λ k functions efine here coincie upto constant aitive ifferences with the λk, functions in [?] although they were formulate a bit ifferently there. Finally, we efine the inverse Ackermann function α to be αn = min{k Z 0 : λ 2k n 3}. 3 Transformation from Local Monotonicity Reconstructors to 2-TC-Spanners In this section, we prove Theorems 1.1 an From non-aaptive Local Monotonicity Reconstructors to 2-TC-Spanners Proof of Theorem 1.1. Let A be a local reconstructor given by the statement of the theorem. Let F be the set of pairs x, y with x, y in V n such that x y. Then, F is of size at most n 2. Given x, y F, let cubex, y be the set {z V n : x z y}. Define function f x,y v to be 1 on all v x an all v y, an 0 everywhere else. Also, efine function f x,y v, which is ientical to f x,y v for all v / cubex, y an 0 for v cubex, y. Both, f x,y an f x,y, are monotone functions for all x, y F. Let A ρ be the eterministic algorithm which runs A with the ranom see fixe to ρ. We say a string ρ is goo for x, y F if filter A ρ on input f x,y returns g = f x,y an on input f x,y returns g = f x,y. Now we show that there exists a set S of size s 2 log n/ log1/2δ, consisting of strings use as ranom sees by A, such that for every x, y F some string ρ S is goo for x, y. We choose S by picking strings use as ranom sees uniformly an inepenently at ranom. Since A has error probability 5

7 at most δ, we know that for every monotone f, with probability at least 1 δ with respect to the choice of ρ, the function A f,ρ is ientical to f. Then, for fixe x, y F an uniformly ranom ρ, Pr[ρ is not goo for x, y] Pr[A ρ on input f x,y fails to output f x,y ] + Pr[A ρ on input f x,y fails to output f x,y ] 2δ. Since strings in S are chosen inepenently, Pr[no ρ S is goo for x, y] 2 δ s, which, for s = 2 log n/ log1/2δ, is at most 1/n 2 < 1/ F. By a union boun over F, Pr[for some x, y F, no ρ S is goo for x, y] < 1. Thus, there exists a set S with require properties. We construct our 2-TC-spanner H = V n, E H of G n using set S escribe above. Let N ρ x be the set consisting of x an all vertices looke up by A ρ on query x. Note that the set N ρ x is well-efine since algorithm A is assume to be non-aaptive. For each string ρ S an each vertex x V n, connect x to all comparable vertices in N ρ x other than itself an orient these eges accoring to their irection in G n. We prove H is a 2-TC-Spanner as follows. Suppose not, i.e., there exists x, y F with no path of length at most 2 in H from x to y. Consier ρ S which is goo for x, y. Define function h by setting hv = f x,y v for all v / cubex, y. Then hv = f x,y v for all v / cubex, y, by efinition of f x,y. For a v cubex, y, set hv to 1 for v N ρ x an to 0 for v N ρ y. All unassigne points are set to 0. By the assumption above, N ρ x N ρ y oes not contain any points in cubex, y. Therefore, h is well-efine. Since ρ is goo for x, y an h is ientical to f x,y for all lookups mae on query x, A ρ x = hx = 1. Similarly, A ρ y = hy = 0. But x y, so A h,ρ v is not monotone. Contraiction. The number of eges in H is at most N ρ x n ln s nln 2 log n/ log1/2δ. x V n,ρ S 3.2 From aaptive Local Monotonicity Reconstructors to 2-TC-Spanners The complication in the transformation from an aaptive filter is that the set of vertices looke up by the filter epens on the oracle that the filter is invoke on. Proof of Theorem 1.2. Define F, f x,y, f x,y, A ρ an S as in the proof of Theorem 1.1. As before, for each x V n, we efine sets N ρ x, an construct the 2-TC-Spanner H by connecting each x to comparable points in N ρ x for all ρ S an orienting the eges accoring to G n. However, now N ρ x is a union of several sets Nρ b,w x, inexe by b {0, 1} an w {0, 1} l2n. In aition, N ρ x contains x. For each x V n, b {0, 1} an w {0, 1} l2n, let Nρ b,w x V n be the set of lookups performe by A ρ on query x, assuming that the oracle answers all lookups as follows. When a lookup y is comparable to x, answer 0 if y x, b if y = x, 1 if x y. Otherwise, if y is the i th lookup mae to an incomparable point for some i [l 2 ], answer w[i]. Recall that we set N ρ x to be the union of Nρ b,w for all b {0, 1} an all w {0, 1} l2n. This completes the escription of N ρ x an construction of H. The argument that H is a 2-TC-spanner procees similarly to that in the proof of Theorem 1.1. The caveat is that an aaptive local filter might choose lookups base on the answers to previous lookups. The constructe function h sets hx = 1 an hy = 0. Further all points comparable to x are set to 0 if they are below x an 1 if they are above x. However, points incomparable to x might be comparable to y an are set to 0 or 1, epening on whether they are above or below y. Since we inclue sets of points querie uner all these possibilities in N ρ x, we can now conclue that A ρ x = hx = 1. The same applies for y. So, A h,ρ outputs a non-monotone function, witnesse by the pair x, y. Contraiction. 6

8 We procee to boun the number of eges E H in H. For each ρ S, x V n, b {0, 1}, an w {0, 1} l2n, the number of vertices in N ρ b,w x comparable to x is at most l 1n. Therefore, E H l 1 n 2 2 l 2n S O n l 1 n 2 l2n log n/ log1/δ. 4 2-TC-Spanners for Low-Dimensional Hypergris In this section, we escribe the proof of Theorem 1.5 which gives explicit bouns on the size of the sparsest 2-TC-spanner for H m,. The upper boun in Theorem 1.5 is prove in Section 4.1 an lower boun in Section Upper Boun The upper boun in Theorem 1.5 follows straightforwarly from a more general statement about TCspanners of prouct graphs presente in the following section. In the same section, we erive the upper boun in Theorem Construction for Prouct Graphs This section explains how to construct a TC-spanner of the Cartesian prouct of graphs G 1 an G 2 from TCspanners of G 1 an G 2. Since the irecte hypergri is the Cartesian prouct of irecte lines, an optimal TC-spanner constructions are known for the irecte line, our construction yiels sparse TC-spanners for the gri Corollary 4.2. We start by efining two graph proucts: Cartesian an strong. Definition 4.1 Graph proucts. Given graphs G 1 = V 1, E 1 an G 2 = V 2, E 2, a prouct of G 1 an G 2 is a new graph G with vertex set V 1 V 2. For the Cartesian graph prouct, enote by G 1 G 2, graph G contains an ege from u 1, u 2 to v 1, v 2 if an only if u 1 = v 1 an u 2, v 2 E 2, or u 1, v 1 E 1 an u 2 = v 2. For the strong graph prouct, enote by G 1 G 2, graph G contains an ege from u 1, u 2 to v 1, v 2 if an only if u 1 = v 1 an u 2, v 2 E 2, or u 1, v 1 E 1 an u 2 = v 2, or u 1, v 1 E 1 an u 2, v 2 E 2. For example, H m,2 = H m,1 H m,1 an TCH m,2 = TCH m,1 T CH m,1, where TCG enotes the transitive closure of G. Lemma 4.1. Let G 1 an G 2 be irecte graphs with k-tc-spanners S 1 an S 2, respectively. Then S 1 S 2 is a k-tc-spanner of G = G 1 G 2. Proof. Suppose u, v an u, v are comparable vertices in G 1 G 2. Then, by efinition of the Cartesian prouct, u u in G 1 an v v in G 2. Let u 1, u 2,..., u l be the shortest path in S 1 from u = u 1 to u = u l, an v 1, v 2,..., v t the shortest path in S 2 from v = v 1 to v = v t. Assume w.t.o.g. that l t. Then u 1, v 1, u 2, v 2,..., u l, v l..., u l, v t is a path in S 1 S 2 of length t k, from u, v to u, v. Therefore, S 1 S 2 is a k-tc-spanner of G = G 1 G 2. Lemma 4.1 together with previous results on the size of k-tc-spanners for the line H m,1, summarize in Lemma 1.7, imply an upper boun on the size of a k-tc-spanner of the irecte hypergri H m, : Corollary 4.2. Let S k H m, enote the number of eges in the sparsest k-tc-spanner of the irecte -imensional hypergri H m,. Then S k H m, = Om λ k m c for appropriate constant c. More precisely, S 2 H m, m log m for m 3. 7

9 Proof. Let S be a k-tc-spanner for the line H m,1. By Lemma 4.1, S S, where the strong graph prouct is applie times, is a k-tc-spanner for the irecte gri H m,. By efinition of the strong graph prouct, the number of eges in the resulting spanner is ES + m m. Since the number of eges in the spanner, ES, is at least m, the main statement follows. The more precise statement for k = 2 follows from Claim 4.3 below which gives a more careful analysis of the size of the sparsest 2-TC-spanner of the line: namely, S 2 H m,1 m log m m for m 3. Claim 4.3. For all m 3, the irecte line H m,1 has a 2-TC-spanner with at most m log m m eges. Proof. Construct graph S on vertex set [m] recursively. First, efine the mile noe v mi = m 2. A eges v, v mi for all noes v < v mi an eges v mi, v for all noes v > v mi. Then recurse on the two line segments resulting from removing v mi from the current line. Procee until each line segment contains exactly one noe. This construction is implicit in, e.g., [?]. S is a 2-TC-spanner for the line H m,1, since every pair of noes u, v [m] is connecte by a path of length at most 2 via a mile noe. This happens in the stage of the recursion where u an v are separate into ifferent line segments, or one of these two noes is remove. There are t = log m stages of the recursion, an in each stage i [t] each noe that is not remove by the en of the this stage connects to the mile noe in its current line segment. Since 2 i 1 noes are remove in the ith stage, exactly m 2 i 1 eges are ae in that stage. Thus, the total number of eges in S is m t 2 t+1 t 2 m log m m. The last inequality hols for m Lower Boun In this section, we show the lower boun on S 2 H m, state in Theorem 1.5. We first treat the special case of this lower boun for = 2, since it alreay contains most of the ifficulty of the larger imensional case. The extension to arbitrary imension is presente in the subsequent section Lower Boun for = 2 In this section, we prove a lower boun on the size of a 2-TC-spanner of the 2-imensional irecte gri, state in Theorem 4.4. This is a special case of the lower boun in Theorem 1.5. Theorem 4.4. Any 2-TC-spanner of the 2-imensional gri H m,2 must have Ω eges. m 2 log 2 m loglog m One way to prove the Ωm log m lower boun on the size of a 2-TC-spanner for the irecte line H m,1, state in Lemma 1.7, is to observe that at least m 2 eges are cut when the line is halve: namely, at least one per vertex pair v, m v + 1 for all v [ m 2 ]. Continuing to halve the line recursively, we obtain the esire boun. A natural extension of this approach to proving a lower for the gri is to recursively halve the gri along both imensions, hoping that each such operation on an m m gri cuts Ωm 2 log m eges. This woul imply that the size Sm of a 2-TC-spanner of the m m gri satisfies the recurrence Sm = 4Sm/2 + Ωm 2 log m; that is, Sm = Ωm 2 log 2 m, matching the upper boun in Theorem 1.5. An immeiate problem with this approach is that in some 2-TC-spanners of the gri only Om 2 eges connect vertices in ifferent quarters. One example of such a 2-TC-spanner is the graph containing the transitive closure of each quarter an only at most 3m 2 eges crossing from one quarter to another: namely, for each noe u an each quarter q with vertices comparable to u, this graph contains an ege u, v q, where v q is the smallest noe in q comparable to u. The TC-spanner in the example above is not optimal because it has too many eges insie the quarters. The first step in our proof of Theorem 4.4 is unerstaning the traeoff between the number of eges 8

10 left noes miline right noes v R high noes & internal eges L u long internal ege block low noes & internal eges Figure 1: Illustration of the first stage in the proof of Lemma 4.5. crossing the cut an the number of eges internal to the subgris, resulting from halving the gri along some imension. The simplest manifestation of this traeoff occurs when a 2 m gri is halve into two lines. In the case of one line, there is no trae off: the Ωm boun on the number of crossing eges hols even if each half-line contains all eges of its transitive closure. Lemma 4.5 formulates the traeoff for the two-line case, while taking into account only eges neee to connect comparable vertices on ifferent lines by paths of length at most 2: Lemma 4.5 Two-Lines Lemma. Let U be a graph with vertex set [2] [m] that contains a path of length at most 2 from u to v for every u {1} [m] an v {2} [m], where u v. An ege u, v in U is calle internal if u 1 = v 1, an crossing otherwise. If U contains at most m log2 m contain at least m log m 16 log log m crossing eges. 32 internal eges, it must Note that if the number of internal eges is unrestricte, a 2-TC-spanner of H m,2 may have only m crossing eges. log m Proof. The proof procees in 2 log log m stages ealing with pairwise isjoint sets of crossing eges. In each stage, we show that U contains at least m 8 crossing eges in the prescribe set. In the first stage, ivie U into log 2 m m blocks, each of length log 2 m : namely, a noe v 1, v 2 is in block i ]. Call an ege long if it starts an ens in ifferent blocks, an short otherwise. [ if v 2 i 1 m log 2 m + 1, i m log 2 m Assume, for contraiction, that U contains fewer than m 8 long crossing eges. Call a noe v 1, v 2 low if v 1 = 1 high if v 1 = 2, an left if v 2 [ ] m 2 right otherwise. Also, call an ege u, v low-internal if u 1 = v 1 = 1 an high-internal if u 1 = v 1 = 2. Let L be the set of low left noes that are not incient to long crossing eges. Similarly, let R be the set of high right noes that are not incient to long crossing eges. Since there are fewer than m 8 long crossing eges, L > m 4 an R > m 4. A noe u L can connect to a noe v R via a path of length at most 2 only by using a long internal ege. Observe that each long low-internal ege can be use by at most such pairs u, v: one low log 2 m noe u an high noes v from one block. This is illustrate in Figure 1. Analogously, every long highinternal ege can be use by at most such pairs. Since L R > m2 16 pairs in L R connect via m log 2 m paths of length at most 2, graph U contains more than m2 16 log2 m m = m log2 m 16 long internal eges, which is a contraiction. In each subsequent stage, call blocks use in the previous stage megablocks, an enote their length by B. Subivie each megablock into log 2 m blocks of equal size. Call an ege long if it starts an ens in 9 m

11 ifferent blocks, but stays within one megablock. Assume, for contraiction, that U contains fewer than m 8 long crossing eges. Call a noe v 1, v 2 left if it is in the left half of its megablock, that is, if v 2 l+r 2 whenever v 1, v 2 is in a megablock [2] {l,..., r}. Call it right otherwise. Consier megablocks containing fewer than B 4 long crossing eges each. By an averaging argument, at least m 2B megablocks are of this type. Recall that there are m B megablocks in total. Within each such megablock more than B 4 low left noes an more than B 4 high right noes have no incient long crossing eges. By the argument from the first stage, each such megablock contributes more than B2 16b long internal eges, where b = B is the size of the blocks. Hence log 2 m there must be more than B2 16b m 2B = m log2 m 32 long internal eges, which is a contraiction to the fact that U contains at most m log2 m 32 internal eges. We procee to the next stage until each block is of length 1. Therefore, the number of stages, t, satisfies m log 2t m = 1. That is, t = log m 2 log log m, an each stage contributes m 8 new crossing eges, as esire. Next we generalize Lemma 4.5 to unerstan the traeoff between the number of internal eges an crossing eges resulting from halving a 2-TC-spanner of a 2l m gri with the usual partial orer. Lemma 4.6. Let S be a 2-TC-spanner of the irecte [2l] [m] gri. An ege u, v in S is calle internal if u 1, v 1 [l] or u 1, v 1 {l + 1,..., 2l}, an crossing otherwise. If S contains at most lm log2 m 64 internal eges, it must contain at least crossing eges. lm log m 32 log log m Proof. For each i [l], we match the lines {i} [m] an {2l i + 1} [m]. Observe that a path of length at most 2 between the matche lines cannot use any eges with both enpoints in {i + 1,..., 2l i} [m]. We moify S to ensure that there are no eges with only one enpoint in {i + 1,..., 2l i} [m] for all i [l], an then apply Lemma 4.5 to the matche pairs of lines. Call the [l] [m] subgri an all vertices an eges it contains low, an the remaining {l + 1,..., 2l} [m] subgri an its vertices an eges high. Transform S into S as follows: change each low internal ege u, v to u, u 1, v 2, change each high internal ege u, v to v 1, u 2, v, an finally change each crossing ege i 1, j 1, 2l i 2 + 1, j 2 to i, j 1, 2l i + 1, j 2, where i = mini 1, i 2. Intuitively, we are projecting the eges in S to be fully containe in one of the matche pairs of lines, while preserving whether the ege is internal or crossing. Crossing eges are projecte onto the outer matche pair of lines chosen from the two pairs that contain the enpoints of a given ege. Clearly, S contains at most the number of internal crossing eges as S. Observe that S contains a path of length at most 2 from u to v for every comparable pair u, v where u is low, v is high, an u an v belong to the same pair of matche lines. Inee, since S is a 2-TC-spanner, it contains either the ege u, v or a path u, w, v. In the first case, S also contains u, v. In the secon case, if u, w is a crossing ege S contains u, v 1, w 2, v, an if u, w is an internal ege S contains u, u 1, w 2, v. As claime, each ege in S belongs to one of the matche pairs of lines. Finally, we apply Lemma 4.5. If S contains at most lm log2 m 64 internal eges, then so oes S, an so at least half i.e., l 2 of the matche line pairs each contain at most m log 2 m 32 internal eges. By Lemma 4.5, m log m each of these pairs contributes at least 16 log log m crossing eges. Thus S lm log m must contain at least 32 log log m crossing eges. Since S contains as many crossing eges as S, the lemma follows. Now we prove Theorem 4.4 by recursively halving H m,2 along the horizontal imension. Some resulting l m subgris may violate Lemma 4.6, but we can guarantee that the lemma hols for a constant fraction of the recursive steps for which l m. This is sufficient for obtaining the lower boun in the theorem. Proof of Theorem 4.4. Assume m is a power of 2 for simplicity. For each step i {1,..., 1 2 log m}, partition H m,2 into the following 2 i 1 equal-size subgris: {1,..., l i } 10

12 [m], {l i + 1,..., 2l i } [m],..., {m l i + 1,..., m} [m] where l i = m/2 i 1. For each of these subgris, efine internal an crossing eges as in Lemma 4.6. Now, suppose that there exists a step i such that at least half of the 2 i 1 subgris have > l im log 2 m 64 internal eges. Since at a fixe i, the subgris are isjoint, there are 2 i 1 Ωl i m log 2 m = Ωm 2 log 2 m eges in S, proving the theorem. On the other han, suppose that for every i {1,..., 1 2 log m}, at least half of the 2i 1 subgris have l im log 2 m 64 internal eges. Then, applying Lemma 4.6, the number of crossing eges in those subgris is l im log m 32 log log m. Counting over all steps i an for all appropriate subgris from those steps, the number of eges in S is boune by Ω m 2 log m log m log log m = Ω m 2 log2 m log log m Lower Boun for general In this section, we exten the above proof to establish lower bouns on S 2 H m, for arbitrary 2. The main technical result is a traeoff lemma between internal an crossing eges with respect to two 1-imensional hyperplanes. An important part of the generalization is the appropriate efinition of the notions of blocks an megablocks, so that the iterative argument in the proof of Lemma 4.5 applies in the high-imensional setting. The following theorem implies the lower boun expression in Theorem 1.5: Theorem 4.7. Any 2-TC-spanner of H m, has at least m 32 log m 2 log log m 1 eges. The main ingreient in the proof is the Two-Hyperplanes Lemma, an analogue of the Two-Lines Lemma Lemma 4.5 for imensions. The main ifficulty in extening the proof of the Two-Lines lemma to work for two hyperplanes is in generalizing the efinitions of blocks an megablocks, so that, on one han, each stage in the proof contributes a substantial number of crossing eges an, on the other han, the crossing eges contribute in separate stages are pairwise isjoint. Lemma 4.8 Two-Hyperplanes Lemma. Let U be a graph with vertex set [2] [m] 1 that contains a path of length at most 2 from u to v for every u {1} [m] 1 an v {2} [m] 1, where u v. As in Lemma 4.5, an ege u, v in U is calle internal if u 1 = v 1, an crossing otherwise. Then, if U contains less than m 1 log m internal eges, it must contain m 1 8 log m 2 log log m 1 crossing eges. Proof. As for Lemma 4.5, the proof procees in several stages. The stages are inexe by 1-tuples i log m in {0, 1,..., log log m 1} 1. Then, the number of stages is log m 1. log log m We show below that each stage contributes at least m 1 separate eges to the set of crossing eges, thus proving our lemma As in the proof of Lemma 4.5, at each stage vertices are partitione into megablocks an blocks. In stage i = i 1,..., i 1, we partition U into log m i 1+ +i 1 equal-size megablocks inexe by b = b 1,..., b 1, where b j [log i j m] for all j [ [ 1]. ] m A vertex v is in a megablock b if v j+1 b j 1 + 1, b m log ij j for each j [ 1]. So, m log ij m initially when i = 0, there is only one megablock, an each time i increases by 1 in one coorinate, the volume of the megablocks shrinks by a factor of log m. Each megablock b is further partitione into log m 1 equal-size [ blocks inexe by c [log ] m] 1. l A vertex v in a megablock b lies in block c if v b min j+1 c j 1 j + 1, c l j log m j for each log m j [ 1], where b min enotes the smallest vertex containe in megablock b an l j enotes the length of b in the the j th imension. Note that vertices 1, v 2,..., v an 2, v 2,..., v belong to the same megablock. At the last stage, each block contains only two vertices iffering by the first coorinate. Next, we specify the set of crossing eges contribute at each stage. A crossing ege u, v in U is sai to be long in stage i if: 11

13 i u an v lie in the same megablock, an ii If u lies in block c 1,..., c 1 an v lies in block c 1,..., c 1, then c j < c j for all j [ 1]. We claim that if i i, the sets of long crossing eges in stages i an i are isjoint. To see this, let j be an inex such that i j i j ; suppose without loss of generality that i j < i j. Then, the length of the megablocks in the j th imension for stage i is at most the length of the blocks in the j th imension for stage i. Hence, conition ii above implies that long crossing eges in stage i must have enpoints in ifferent megablocks of stage i, an so violate conition i for being a long crossing ege in stage i. It remains to show that every stage contributes at least m 1 long crossing eges. For the sake of 2 +2 contraiction, suppose that the number of long crossing eges at some stage i is < m 1. Let B = 2 +2 m 1 /log m i 1+ +i 1 be the volume of the megablocks restricte to one of the two hyperplanes. By an averaging argument, at least m 1 B 2B megablocks contain < long crossing eges otherwise, there woul 2 +1 be at least m 1 long crossing eges. But we show next that if a megablock contains < B long crossing eges, then there are B log m imply that the total number of internal eges is m 1 internal eges with both enpoints insie the megablock. This woul 2B B log m = m 1 log m , a contraiction. Suppose then that a megablock contains < B long crossing eges. Let Low be the set of vertices 2 +1 in the megablock with each coorinate at most the average value of that coorinate in the megablock, an High the set of vertices with each coorinate greater than the average value of that coorinate. Then Low B, High B, an each vertex in Low is comparable to each vertex in High. By the boun 2 2 on the number of long crossing eges, there must exist a set L of at least vertices in Low not incient 2 +1 B to any long crossing ege, an a set R of at least vertices in High not incient to any long crossing 2 +1 eges. L lies in the lower hyperplane, R in the upper hyperplane, an each vertex in L is comparable to each vertex in R. Call a crossing ege short if it satisfies conition i, but violates conition ii above. A path in U of length at most 2 from a vertex in L to a vertex in R must consist of one internal ege an one short B crossing ege. The number of short crossing eges incient to a given vertex v is at most 1, by log m counting, for each of the 1 block inices, the number of vertices in the megablock that share the value of B that block inex with v. So, each internal ege helps connect at most 1 pairs of vertices. Since log m B 2 B pairs of vertices nee to be connecte by a path, there must exist at least 2 log m B = internal eges. B B log m The analogue of Lemma 4.6 in imensions Lemma 4.9 an the rest of the proof of Theorem 4.7 are straightforwar generalizations of the 2-imensional case. Lemma 4.9. Let S be a 2-TC-spanner of the irecte [2l] [m] 1 gri. An ege u, v in S is calle internal if u 1, v 1 [l] or u 1, v 1 {l + 1,..., 2l}, an crossing otherwise. If S contains less than lm 1 log m internal eges, it must contain at least l m 1 log m 8 2 log log m crossing eges. Proof sketch. We can generalize the proof of Lemma 4.6 in a straightforwar way. For each i [l], instea of matching the lines, we match the hyperplanes {i} [m] 1 an {2l i + 1} [m] 1. Proof of Theorem 4.7. Assume m is a power of 2 for simplicity. For each step i {1,..., 1 2 log m}, partition H m, into the following 2 i 1 equal-size subgris: {1,..., l i } [m] 1, {l i +1,..., 2l i } [m] 1,..., {m l i +1,..., m} [m] where l i = m/2 i 1. For each of these subgris, efine internal an crossing eges as in Lemma 4.9. Now, suppose that there exists a step i such that at least half of the 2 i 1 subgris have l im 1 log m internal eges. Since at a fixe i, the subgris are

14 isjoint, there are at least 2 i 2 l im 1 log m = m log m eges in S, which is enough to prove the theorem On the other han, suppose that for every i {1,..., 1 2 log m}, at least half of the 2i 1 subgris have < l im 1 log m log m is l im 1 8 internal eges. Then, applying Lemma 4.9, the number of crossing eges in those subgris 1. 2 log log m Counting over all steps i an for all appropriate subgris from those steps, the number of eges in S is lower-boune by log m 2 2 i 2 lim 1 1 log m 8 2 log log m = m TC-Spanners of the Hypercube log m 2 log log m 1. In this section we prove Theorem 1.6, namely, we analyze the size of the sparsest 2-TC-spanner of the -imensional hypercube H. Lemma 5.1 presents the upper boun on S 2 H. Lemma 5.3 presents the lower boun. The upper an lower bouns iffer only by a factor of O 3, an are ominate by the same combinatorial expression. A numerical approximation to this expression is given in Lemma 5.5. Remark 5.1 at the en of the section explains why our ranomize construction in Lemma 5.1 yiels a 2-TC-spanner of H of size within O 2 of the optimal. 5.1 Upper Boun Lemma 5.1. There is a 2-TC-spanner of H with O 3 max min :i<j k:i k j { k k k i max j i i, k j } eges. Proof. Consier the following probabilistic construction that connects all comparable vertices at levels i an j of H by paths of length at most 2: Given levels i, j {0, 1,..., }, i < j, 1. Initialize the set E to. 2. Let k = argmin k:i k j k j i k i max { k i, k j }. 3. Let S be a set of 3 k k i vertices chosen uniformly at ranom from the set of k vertices that are j i in weight level k = k. 4. For each vertex v S, set E to E {x, v : x = i x v} {v, y : y = j v y}. That is, connect v to all comparable vertices in levels i an j. 5. Output E. Claim 5.2. For all 0 i < j, with probability at least 1 2, E contains a path of length at most 2 between any pair of vertices x, y such that x y, x = i, an y = j. Proof. Consier any particular pair of vertices x, y such that x y, x = i, an y = j. The number of vertices in level k that are greater than x an less than y is exactly j i k i. So, the probability that S oes not contain such a vertex is: 1 j i k i / 3 k k i j i k e 3. The number of comparable pairs x, y is i i j. So, by the union boun, the probability that there exists an x, y such that no vertex v S satisfies x v y is at most i i j e e 3 <

15 So, for every i an j, there exists a choice of S such that comparable pairs from the two weight levels are connecte by a path of length at most 2. Let E be the set of eges returne by the algorithm when this S is chosen. We set E = 0 i<j E. By Claim 5.2, {0, 1}, E is a 2-TC-spanner of H. Now, we show that the size of E is as claime in the lemma statement. The main observation is that in step 4, for any specific v S, {x, v : x = i x v} {v, y : y = j v y} is exactly k i + k j. Therefore, for all 0 i < j, E 3 min k:i k j k j i k i k + i k j 6 min k:i k j k j i max k i { k, i Since E = 0 i<j E, where the sum has O2 terms, the claime boun follows. 5.2 Lower Boun Lemma 5.3. Any 2-TC-spanner of H has Ω max min :i<j k:i k j } k. j { k k k i max j i i, k j } eges. Proof. Let S be a 2-TC-spanner for H. We will count the eges in S that occur on paths connecting two particular weight levels of H. Let P be the pairs {v 1, v 2 : v 1 = i, v 2 = j, v 1 v 2 }. We will lower boun e, the number of eges in the paths of length at most 2 in S that connect the pairs P. Let e k,l enote the number of eges in S that connect vertices in level k to vertices in level l. Then e = e + j 1 k=i+1 e i,k + e k,j. We say that a vertex v covers a pair of vertices v 1, v 2 if S contains the eges v 1, v an v, v 2 or, for the special case v = v 1, if S contains v 1, v 2. Let V k be the set of vertices of weight k that cover pairs in P. Let α k be the fraction of pairs in P that are covere by a vertex in V k. Since each pair in P must be covere by a vertex in levels i to j 1, j 1 k=i α k 1. For any vertex v V k, let in v be the number of incoming eges from vertices of weight i incient to v an let out v be the number of outgoing eges to vertices of weight j incient to v. For each k {i + 1,..., j 1}, since each vertex v V k v V k covers in v out v pairs, i in v out v α k P = α k. 1 i j We upper boun v V k in v out v as a function of e i,k + e k,j, an then use Equation 1 to lower boun e i,k + e k,j. For all k {i + 1,..., j 1}, variables in v an out v satisfy the following constraints: in v e i,k + e k,j, out v e i,k + e k,j. v V k in v k v V k i, v V k out v k v V k. j The last two constraints hol because in v an out v count the number of eges to a vertex of weight k from from vertices of weight i an from a vertex of weight k to vertices of weight j, respectively. Using these bouns we obtain v V k in v out v v V k k out v = i 14 k i v V k out v k e i,k + e k,j. i

16 Similarly, v V k in v out v k Let s i,k,j = Therefore, e = e + i i v V k { j min k j 1 k=i+1 j ei,k + e k,j. Therefore, for all k {i + 1,..., j 1}: { } k k in v out v e i,k + e k,j min,. i j i, k j }. From Equation 1, e i,k + e k,j α k s i,k,j for all k {i + 1,..., j 1}. e i,k + e k,j α i i i j + j 1 k=i+1 j 1 α k s i,k,j α k s i,k,j Since this hols for arbitrary i an j, the number of eges in the 2-TC-spanner S max min s i,k,j. :i<j k:i k j Finally, a simple algebraic manipulation finishes the proof. { Claim 5.4. s i,k,j = k k k i max j i i, k j }. Proof. Take the ratio of the two sies: s i,k,j { k k k i max j i i, k } = j i i k j i j k i k i k j = i i k k=i j i j i k i k i k j k = 1. min k:i k j s i,k,j The first equality follows from the fact that maxx, y minx, y = x y. The last equality can be prove either by expaning the binomial coefficients into factorials, or by realizing that both i j i i j i k i an k k k i j k count the number of ways i re balls, j k blue balls, an k i green balls can be place into slots, each of which can hol one ball at most. This completes the proof of the claim. This completes the proof of the lemma. The following lemma gives a hanle on the expression capturing the size of a 2-TC-Spanner. Lemma 5.5. Let s = max min, k j }. Then s = 2 c, where c :i<j k:i k j k j i k i max { k i Proof. We use the fact that n cn = 2 H b c o n1n, where o n 1 is a function of n that tens to zero as n tens to infinity, an H b p = p log p 1 p log1 p is the binary entropy function. Substituting i = α, j = β an k = γ in the resulting expression for s, an taking the logarithm of both sies, we get log 2 s = max min 0 α<β 1 α γ β [ H b γ H b γ α β α β α + max H b α γ γ, H b 1 β 1 γ In other wors, log 2 s = c where c is a constant. We can check numerically that c γ ] Remark 5.1. We note that if the first maximum in the expression for s is replace with the sum then Lemma 5.1 hols for O s instea of O 3 s while Lemma 5.3 hols for Ω/s instea of Ωs. The proofs of these moifie statements are similar. We o not have an analogue of Lemma 5.5 for the moifie expression for s. Observe that the moifie bouns iffer by a factor of O 2 instea of O 3. This emonstrates that our ranomize construction yiels a 2-TC-spanner of H of size within O 2 of the optimal. 15