Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts

We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any $O(n)$-size shortcut set cannot bring the diameter below $\Omega(n^{1/6})$, and that any $O(m)$-size shortcut set cannot bring it below $\Omega(n^{1/11})$. These improve Hesse's [Hesse03] lower bound of $\Omega(n^{1/17})$. By combining these constructions with Abboud and Bodwin's [AbboudB17] edge-splitting technique, we get additive stretch lower bounds of $+\Omega(n^{1/11})$ for $O(n)$-size spanners and $+\Omega(n^{1/18})$ for $O(n)$-size emulators. These improve Abboud and Bodwin's $+\Omega(n^{1/22})$ lower bounds.


INTRODUCTION
A spanner of an undirected unweighted graph G = (V , E) is a subgraph H that approximates the distance function of G up to some stretch. An emulator for G is defined similarly, except that H need not be a subgraph, and may contain weighted edges. In this paper we consider only additive stretch functions: dist G (u, ) ≤ dist H (u, ) ≤ dist G (u, ) + β, where β may depend on n.
Graph compression schemes (like spanners and emulators) are related to the problem of shortcutting digraphs to reduce diameter, inasmuch as lower bounds for both objects are constructed using the same suite of techniques. These lower bounds begin from the construction of graphs in which numerous pairs of vertices have shortest paths that are unique, edge-disjoint, and relatively long. Such graphs were independently discovered by Alon [4], Hesse [16], and Coppersmith and Elkin [12]; see also [1,2]. Given such a "base graph, " derived graphs can be obtained through a variety of graph products such as the alternation product discovered independently by Hesse [16] and Abboud and Bodwin [1] and the substitution product used by Abboud and Bodwin [1] and developed further by Abboud, Bodwin, and Pettie [2].
In this paper we apply the techniques developed in [1,2,4,12,16] to obtain better lower bounds on shortcutting sets, additive spanners, and additive emulators.

Citation
Shortcut Set Size Diameter Computation Time  Table 1. Upper and Lower bounds on shortcu ing sets. The lower bounds are existential, and independent of computation time.
Shortcutting Sets. Let G = (V , E) be a directed graph and G * = (V , E * ) its transitive closure. The diameter of a digraph G is the maximum of dist G (u, ) over all pairs (u, ) ∈ E * . Thorup [20] conjectured that it is possible to reduce the diameter of any digraph to poly(log n) by adding a set E ′ ⊆ E * of at most m = |E | shortcuts, i.e., G ′ = (V , E ∪ E ′ ) would have diameter poly(log n). This conjecture was confirmed for a couple special graph classes [20,21], but refuted in general by Hesse [16], who exhibited a graph with m = Θ(n 19/17 ) edges and diameter Θ(n 1/17 ) such that any diameter-reducing shortcutting requires Ω(mn 1/17 ) shortcuts. More generally, there exist graphs with m = n 1+ϵ edges and diameter n δ , δ = δ (ϵ), that require Ω(n 2−ϵ ) shortcuts to make the diameter o(n δ ); see Abboud, Bodwin, and Pettie [2, §6] for an alternative proof of this result.
On the upper bound side, it is trivial to reduce the diameter toÕ( √ n) with O(n) shortcuts or diameterÕ(n/ √ m) with O(m) shortcuts. 1 Unfortunately, the trivial shortcutting schemes are not efficiently constructible in near-linear time. In some applications of shortcuttings, efficiency of the construction is just as important as reducing the diameter. For example, a longstanding problem in parallel computing is to simultaneously achieve time and work efficiency in computing reachability. 2 Very recently, Fineman [15] proved that anÕ(n)-size shortcut set can be computed in near-optimal workÕ(m) (andÕ(n 2/3 ) parallel time) that reduces the diameter toÕ(n 2/3 ).
In this paper we prove that O(n)-size shortcut sets cannot reduce the diameter below Ω(n 1/6 ), and that O(m)-size shortcut sets cannot reduce it below Ω(n 1/11 ). See Table 1.
Our emulator lower bounds are polynomially weaker than the spanner lower bounds. Although neither bound is likely sharp, this difference reflects the rule that emulators are probably more powerful than spanners. For example, at sparsity O(n 4/3 ), the best known emulators [13] are slightly better than spanners [6]. Below the 4/3 threshold the best sublinear additive emulators [17,22] have size O(n 1+ 1 2 k +1 −1 ) and stretch function d +O(d 1−1/k ). 4 Abboud, Bodwin, and Pettie [2] showed that this tradeoff is optimal for emulators, but the best known sublinear additive spanners [11,19] are polynomially worse.
There are a certain range of parameters where emulators are known to be polynomially sparser than spanners. For pairwise distance preservers, Bodwin [8] showed that whenever ω(n 1/2 ) = |P | = o(n 2−o(1) ), any pairwise distance preserver has an ω(n + |P |) lower bound, which is worse than the trivial distance preserving emulator with size |P |. A similar separation holds for sourcewise distance preservers, where the goal is to exactly preserve distances between all vertex pairs in S ⊂ V . A trivial source-wise emulator has size |S | 2 , e.g., O(n) for |S | = √ n, but source-wise spanners with size O(n) only exist for |S | = O(n 1/4 ) [8,12].
Organization. In Section 2 we present diameter lower bounds for shortcut sets of size O(n) and O(m). Section 3 modifies the construction to give lower bounds on additive spanners and additive emulators. We conclude with some remarks in Section 4.

Using O(n) Shortcuts
Existentially, the best known upper bound on O(n)-size shortcut sets is the trivialÕ( √ n) bound. Theorem 2.1 shows that we cannot go below Ω(n 1/6 ).
There exists a directed graph G with n vertices, such that for any shortcut set E ′ with size O(n), the graph (V , E ∪ E ′ ) has diameter Ω(n 1/6 ).
The remainder of Section 2.1 constitutes a proof of Theorem 2.1. We begin by defining the vertex set and edge set of G, and its critical pairs.
Vertices. The vertex set of G is partitioned into D + 1 layers numbered 0 through D. Define B d (ρ) to be the set of all lattice points in Z d within Euclidean distance ρ of the origin. In the calculations below we treat d as a constant. For each k ∈ {0, . . . , D}, layer-k vertices are identified with lattice points in B d (R +kr ), where r , R are parameters of the construction. A vertex can be represented by a pair (a, k), where a ∈ B d (R + rk). We want the size of all layers to be the same, up to a constant factor. To that end we fix R = dr D, so the total number of vertices is is the ratio of volume between a d-dimentional ball and its enclosing d-dimentional cube.
Edges. Define V d (r ) to be the set of all lattice points at the corners of the convex hull of B d (r ). (This excludes points that happen to lie on the boundary, but in the interior of one of its faces.) We treat elements of V d (r ) as vectors. For each layer-k vertex (a, k), k ∈ {0, . . . , D − 1}, and each vector ∈ V d (r ), we include a directed edge ((a, k), (a + , k + 1)). All edges in G are of this form.
Critical Pairs. The critical pair set is defined to be Each such pair has a corresponding path of length D, namely (a, 0) → (a + , 1) → · · · → (a + D , D). Lemma 2.2 shows that this path is unique. It was first proved by Hesse [16] and independently by Coppersmith and Elkin [12]. (Both proofs are inspired by Behrend's [7] construction of arithmetic progression-free sets, which uses ℓ 2 balls rather than convex hulls.) [12,16]) The set of critical pairs P have the following properties: • For all (x, ) ∈ P, there is a unique path from x to in G.
• For any two distinct pairs (x 1 , 1 ) and (x 2 , 2 ) ∈ P, their unique paths share no edge and at most one vertex.
P . For the first claim, let x = (a, 0) and ∈ V d (r ) be the vector for which = (a + D , D). One path from x to exists by construction. Let V d (r ) = { 1 , 2 , . . . , s }. Suppose there exists another path from x to . It must have length D because all edges join consecutive layers. Every edge on this path corresponds to a vector i , which implies that D can be represented as a linear combination k 1 1 + k 2 2 + · · · + k s s , where k 1 + · · · + k s = D and k i ≥ 0. This implies that is a non-trivial convex combination of the vectors in V d (r ), which contradicts the fact that V d (r ) is a strictly convex set.
The second claim follows from the fact that any edge in the unique x 1 -to-1 path uniquely identifies both x 1 and 1 .
For the last claim, we can express the number of critical pairs as |P | = |B d (R)| · |V d (r )|. From Bárány and Larman [5], for any constant dimension d, is strictly less than D, then |E ′ | ≥ |P |.

P
. Every path in G ′ corresponds to some path in G. However, for pairs in P, there is only one path in G, hence any shortcut in E ′ useful for a pair (x, ) ∈ P must have both endpoints on the unique x-path in G. By Lemma 2.2, two such paths for pairs in P share no common edges, hence each shortcut can only be useful for at most one pair in P. If |E ′ | < |P | then some pair (x, ) ∈ P must still be at distance D in G ′ .
, then any shortcut set that makes the diameter < D has size Ω(n). In order to have |P | = Ω(n), it suffices to let . From the construction, by fixing d as a constant, we have Therefore, the diameter is . We can maximize D = Θ(n 1/6 ) in one of two ways, by setting d = 2, r = Θ(n 1/4 ), and R = Θ(n 5/12 ), or d = 3, r = Θ(n 1/9 ), and R = Θ(n 5/18 ). In either case, the construction leads to a graph with very similar structure: the number of vertices in each layer is Θ(n 5/6 ), and the out degrees of each vertex are Θ(n 1/6 ).
Theorem 2.1 is indifferent between d = 2 and d = 3 but that is only because the size of the shortcut set is precisely O(n). When we allow it to be O(n 1+ϵ ), for ϵ > 0, there is generally one optimum dimension.
There exists a directed graph G with n vertices, such that for any shortcut set ). In particular, by setting d = 3 the diameter lower bound becomes Ω(n

Using O(m) Shortcuts
Let G (d,r, D) denote the layered graph constructed in Section 2.1 with parameters d, D, r , and R = dr D, and let P G be its critical pair set. The total number of edges m = Θ(n|V d (r )|) is always larger than |P G | = Θ( n D |V d (r )|) by a factor of D. In order to get a lower bound for O(m) shortcuts, we use a Cartesian product combining two such graphs layer by layer, forming a sparser graph. This transformation was discovered by Hesse [16] and rediscovered by Abboud and Bodwin [1].
Let r 1 , D) and G 2 = G (d 2 ,r 2 , D) be two graphs with the same number of layers, namely D + 1. The product graph G 1 ⊗ G 2 is defined below.
Vertices. The product graph has 2D + 1 vertex layers numbered 0, . . . , 2D. The vertex set of layer Edges. Let (x, , i) be a vertex in layer i. If i is even, then for every vector ∈ V d 1 (r 1 ) we include an edge ((x, , i), (x + , , i + 1)). If i is odd, then for every vector w ∈ V d 2 (r 2 ), we include an edge ((x, , i), (x, + w, i + 1)). The total number of edges in the product graph is then Critical Pairs. By combining two graphs, we are able to construct a larger set of critical pairs, as follows.

P
. Every path in G 1 ⊗ G 2 from layer 0 to layer 2D corresponds to two paths from layers 0 to D in G 1 and G 2 , respectively. It follows from Lemma 2.2 that In G 1 ⊗ G 2 it is no longer true that pairs in P have edge-disjoint paths. They may intersect at just one edge. L 2.6. Consider two pairs (x 1 , 1 ) and (x 2 , 2 ) ∈ P. Let P 1 and P 2 be the unique shortest paths in the combined graph from x 1 to 1 and from x 2 to 2 . Then, P 1 ∩ P 2 contains at most one edge.

P
. Any two non-adjacent vertices on the unique x 1 -1 path uniquely identify x 1 and 1 . Thus, two such paths can intersect in at most 2 (consecutive) vertices, and hence one edge.

P
. Assume the diameter of (V , E∪E ′ ) is strictly less than 2D. Every useful shortcut connects vertices that are at distance at least 2. By Lemma 2.6, such a shortcut can only be useful for one pair in P. Thus, if the diameter of (V , E ∪ E ′ ) is less than 2D, |E ′ | ≥ |P |.
By construction, the size of |P | is There exists a directed graph G with n vertices and m edges such that for any shortcut set E ′ with size O(m), the graph (V , E ∪ E ′ ) has diameter Ω(n 1/11 ).

LOWER BOUNDS ON ADDITIVE SPANNERS AND EMULATORS
We now establish better bounds on O(n)-size additive spanners and emulators. In Section 3.1, we give an +Ω(n 1/13 ) stretch lower bound on spanners. Using a different construction, we improve this in Section 3.2 to +Ω(n 1/11 ). In Section 3.3 we show how to adapt the +Ω(n 1/13 )-spanner lower bound from Section 3.1 to prove that O(n)-size emulators have stretch +Ω(n 1/18 ).
Recall the definition of additive spanners and emulators.
Definition 3.1. Let G = (V , E) be an unweighted undirected graph. A subgraph H = (V , E ′ ), E ′ ⊆ E, is said to be a spanner for G with additive stretch β if for any two vertices u, ∈ V , Observe that we can assume w.l.o.g. that if (u, ) ∈ E ′ then w(u, ) = dist G (u, ).

3.2.
There exists an undirected graph G with n vertices, such that any spanner for G with O(n) edges has +Ω(n 1/13 ) additive stretch.
In this section we regard G (d,r, D) to be an undirected graph. We begin with the undirected graph G 0 = G (d 1 ,r 1 , D) ⊗ G (d 2 ,r 2 , D) , then modify it in the edge subdivision step and the clique replacement step to obtain G.

The Edge Subdivision
Step. Every edge in G 0 is subdivided into D edges, yielding G E . This step makes the graph very sparse since most of the vertices in G E now have degree 2.

The Clique Replacement
Step. Consider a vertex u in G E that comes from one of the interior layers of G 0 , i.e., layers 1, . . . , 2D − 1, not 0 or 2D. Note that u has degree δ 1 + δ 2 , with δ 1 = Θ r 2 edges leading to the following layer (or vice versa). We replace each such u with a complete bipartite clique K δ 1 ,δ 2 , where each clique vertex becomes attached to one non-clique edge formerly attached to u. The final graph is denoted G.
Critical Pairs. The set P of critical pairs for G is identical to the set of critical pairs for G 0 . For each (x, ) ∈ P, the unique x-path in G is called a critical path. From the construction, the number of vertices, edges, and critical pairs in G is Lemma 3.3 is key to relating the size of the spanner with the pair set P.

P
. Every clique has δ 1 vertices on one side and δ 2 vertices on the other side. Each vertex on the δ 1 side corresponds to a vector ∈ V d 1 (r 1 ) and each vertex on the δ 2 side corresponds to a vector w ∈ V d 2 (r 2 ). Each clique edge uniquely determines a pair of vectors ( , w), and hence exactly one critical pair in P. 3.4. Every spanner of G with additive stretch +(2D − 1) must contain at least D|P | clique edges.

P
. For the sake of contradiction suppose there exists a spanner H containing at most D|P | − 1 clique edges. By the pigeonhole principle there exists a pair (x, ) ∈ P such that at least D clique edges are missing in H .
Let P (x, ) be the unique shortest path from x to in G, and let P ′ (x, ) be a shortest path from x to in H . Since G 0 is formed from G by contracting all bipartite cliques and replacing subdivided edges with single edges, we can apply the same operations on P ′ (x, ) to get a path P ′′ (x, ) in G 0 . We now consider two cases: is the unique shortest path from x to in G 0 , then P ′ (x, ) suffers at least a +2 stretch on each of the D missing clique edges, so |P ′ (x, ) | ≥ |P (x, ) | + 2D.
is not the unique shortest path from x to in G 0 , then it must traverse at least two more edges than the shortest x-path in G 0 (because G 0 is bipartite), each of which is subdivided D times in the formation of G. Thus |P ′ (x, ) | ≥ |P (x, ) | + 2D. In either case, P ′ (x, ) has at least +2D additive stretch and H cannot be a +(2D − 1) spanner.

An Improved O(n)-Size Spanner Lower Bound
The construction from Section 3.1 is versatile, inasmuch as it extends to polynomial densities (Corollary 3.5) and emulator lower bounds (Section 3.3). However, it can be improved, slightly, for the specific case of O(n)-size additive spanners. It turns out that the the Cartesion product step (generating G 0 from G (d 1 ,r 1 , D) ⊗ G (d 2 ,r 2 , D) ) is inefficient, and that we can do better with a simple replacement step. By its nature, the proof of Theorem 3.6 needs to explictly keep track of the leading absolute constant in the size of the spanner, i.e., it has at most c 0 n = O(n) edges. (In contrast, the proof of Theorem 3.2 can easily accommodate any O(n)-size bound by tweaking r , R, D by constant factors.) T 3.6. For any parameter c 0 > 1 and sufficiently large n there exists an undirected n-vertex graph G such that any spanner for G with at most c 0 n edges has +Ω(n 1/11 c −18/11 0 ) additive stretch.
In Lemmas 3.7 and 3.8 we construct the inner and outer graphs, then discuss how to combine them using a substitution product.
. Fix a parameter c > 1. There exists sufficiently large q, L such that q = Θ(L 2 c 6 ) and a graph G I = (V I , E I ) with a set of critical pairs P I ⊆ V I ×V I satisfying the following.
(3) ∀(u, ) ∈ P I , the shortest path between u and is unique and has length Lc.
(4) ∀(u 1 , 1 ), (u 2 , 2 ) ∈ P, the unique shortest paths between u 1 and 1 and between u 2 and 2 are edge-disjoint. Moreover, dist G I (u 1 , 2 ) ≥ Lc and dist G I (u 2 , 1 ) ≥ Lc. (As a consequence, We use almost the same construction as in Theorem 2.1, except that the graph will be undirected and we will pay closer attention to the density. In this proof d = O(1) represents the density of the graph, not the geometric dimension. We will ultimately choose d = 4c. The graph G I we construct consists of Lc + 1 layers, numbered by 0, 1, . . . , Lc.
By choosing q/d = (Lc)ξ d d 3/2 , we have q = L 2 c 2 ξ 2 d d 4 = Θ(L 2 c 6 ) and the total number of vertices in G can be upper bounded by the number of layers times the size of the last layer: (Lc)(2 q/d) 2 = 4qLc/d. Thus, condition 1 is satisfied whenever d ≥ 4c.
Again, we use a similar construction to Theorem 2.1 to obtain our outer graph.

P
. Consider the following (L + 1)-layer graph. Vertices in the k-th layer are identified with points in the 3-dimensional integer lattice inside the ball of radius Lr + kr around the origin. Here r is the minimum value such that |V 3 (r )| ≥ q. From Bárány and Larman [5] we have r = Θ(q 2/3 ).
We label each vertex with its coordinate and its layer number: (a, k) ∈ B 3 (Lr + kr ) × [L + 1]. Fix an arbitrary subset V ′ 3 (r ) ⊆ V 3 (r ) of any q vectors. For each vertex (a, k) in the k-th layer (0 ≤ k < L), and for every vector x ∈ V ′ 3 (r ), the edge ((a, k), (a + , k + 1)) is added to the graph. It is straightforward to check that |V 0 | ≈ L k=0 η 3 (2(Lr + kr )) 3 = Θ(L 4 q 2 ). For each vector ∈ V ′ 3 (r ) and each layer-0 vertex (a, 0), the vertex pair ((a, 0), (a + L , L)) is added to the critical pair set P, hence |P | = Θ((Lr ) 3 q) = Θ(L 3 q 3 ). By the same argument as in the proof of Lemma 2.2, there is exactly one shortest path of length L connecting (a, 0) and (a +L , L). Moreover, no edge belongs to more than one critical path.
Recall that we are aiming for lower bounds against spanners with size c 0 n. Once c 0 is fixed, we choose a c = Θ(c 0 ) and invoke Lemma 3.7 to construct an inner graph G I with parameters q, L. Once q, L are fixed we invoke Lemma 3.8 to build the outer graph G 0 . Our final graph G is formed from G 0 , G I through the inner graph replacement step and the edge subdivision step, as follows.

Inner Graph Replacement
Step. For every vertex (a, k) ∈ G 0 (0 < k < L), we replace (a, k) with a copy of inner graph G I as follows.
Recall that the critical pair set for G I has size q. We regard the sources of these q pairs to be input ports and the sinks to be output ports. Let G I, (a,k) be the copy of G I substituted for (a, k) in the outer graph. For each i ∈ V ′ 3 (r ) = { 1 , 2 , . . . , q } and each critical path of G 0 passing through (a − i , k − 1), (a, k), (a + i , k + 1), we reattach one endpoint of (a − i , k − 1) to the ith input port of G I, (a,k) and reattach one endpoint of (a + i , k + 1) to the ith output port of G I, (a,k) . Let G * be the result of this process.

The Edge Subdivision
Step. Every edge in G * that was inherited from G 0 (i.e., not inside any copy of G I ) is subdivided into a path of L/2 edges. The outcome of this process is G.
Observe that for every critical pair (x, ) from G 0 , there is a unique shortest path between x and in G of length 1 2 L 2 +(L−1)Lc, where 1 2 L 2 edges come from the subdivision step and the remaining ones come from copies of G I . Moreover, any two unique shortest paths are edge-disjoint.

P
. Suppose there exists a spanner H of G with additive stretch +(L − 2) but has strictly less than 1 2 L 2 + cL L−1 2 |P | edges. By the pigeonhole principle there must exist a critical pair (x, ) ∈ P with unique shortest path P (x, ) that is in one of the following two cases: (1) H is missing an edge in P (x, ) introduced in the edge subdivision step, or (2) H is missing at least one critical edge from P (x, ) in at least half ((L − 1)/2) of the copies of G I along P (x, ) .
Let P ′ (x, ) be a shortest path connecting x and in H . If (1) holds, then P ′ (x, ) traverses at least two more subdivided edges than P (x, ) and at least the same number of copies of G I , hence |P ′ (x, ) | ≥ |P (x, ) | + L, a contradiction. If (2) holds, then for every inner graph that has a missing edge on P (x, ) , P ′ (x, ) traverses at least two more edges. Since there are at least (L − 1)/2 such inner graphs, |P ′ (x, ) | ≥ |P (x, ) | + (L − 1). In either case, P ′ (x, ) has at least +(L − 1) additive stretch so H cannot be a +(L − 2) spanner. P T 3.6. Given the density parameter c 0 , we will choose a larger parameter c = Θ(c 0 ) (defined precisely below) and construct the inner graph G I (Lemma 3.7) with at most qL vertices, at least cqL edges, and q critical pairs, with q = Θ(L 2 c 6 ). Once q, L are fixed, we construct the outer graph G 0 using Lemma 3.8. After the replacement and subdivision steps, G has |V | = Θ(L 5 q 3 ) vertices and |P | = Θ(L 3 q 3 ) critical pairs. This implies that there is some absolute constant λ > 1 such that λL L−1 2 |P | ≥ |V |. Now, by Lemma 3.9, any spanner of G with at most cL L−1 2 |P | edges has additive stretch at least +(L − 1). We choose c = λc 0 , so cL L−1 2 |P | ≥ c 0 |V |. Therefore, any spanner of G with at most c 0 |V | edges has additive stretch at least +(L − 1). Since q = Θ(L 2 c 6 ) = Θ(L 2 c 6 0 ), it follows that |V | = Θ(L 11 c 18 0 ). Thus, we conclude that any spanner of G with c 0 n edges has additive stretch +Ω(|V | 1/11 c −18/11 0 ).

R
. The construction from Theorem 3.6 cannot be easily traslated into an emulator lower bound. The reason is that the number of critical pairs is always sublinear in the number of vertices. A distance preserving emulator of linear size always exists in this type of construction.

O(n)-Size Emulators
The difference between emulators and spanners is that emulators can use weighted edges not present in the original graph. In this section, our lower bound graph, G, is constructed exactly as in Section 3.1, but with different numerical parameters. 3.10. There exists an undirected graph G with n vertices such that any emulator with O(n) edges has +Ω(n 1/18 ) additive stretch.
Before proving Theorem 3.10 we first argue that the size of low-stretch emulators is tied to the number of critical pairs |P | for G. 3.11. Every emulator for G with additive stretch +(2D − 1) requires at least |P |/2 edges.

P
. Let H be an emulator with additive stretch +(2D − 1). Without loss of generality, we may assume that any (u, ) ∈ E(H ) has weight precisely dist G (u, ). (It is not allowed to be smaller, and it is unwise to make it larger.) We proceed to convert H into a spanner H ′ that has the same stretch +(2D − 1) on all pairs in P, then apply Lemma 3.4.
Initially H ′ is empty. Consider each (x, ) ∈ P one at a time. Let P (x, ) be the shortest path in H and P ′ (x, ) be the corresponding path in G. Include the entire path P ′ (x, ) in H ′ . After this process is complete, for any (x, ) ∈ P, dist H ′ (x, ) = dist H (x, ), and H ′ is a spanner with at most n + 2D|H | edges. In particular, it has at most 2D|H | clique edges since each weighted edge in some P (x, ) contributes at most 2D clique edges to H ′ . By Lemma 3.4, the number of clique edges in H ′ is at least D|P |, hence |H | ≥ |P |/2. P T 3.10. In order to get |P | = Ω(n), it suffices to set δ 1 ≥ δ 2 ≥ D 2 . Now, we have (by definition of δ 1 and δ 2 ) The exponent is minimized when d 1 = d 2 = 3. This implies n = Ω(D 18 ) and hence D = O(n 1/18 ). These parameters can be achieved asymptotically by setting D = Θ(n 1/18 ), δ 1 = δ 2 = D 2 , r = Θ(n 2/27 ), and R = Θ(n 7/54 ). Using the same proof technique as in [1,2], it is possible to extend our emulator lower bound to any compressed representation of graphs usingÕ(n) bits. 3.13. Consider any mapping from n-vertex graphs toÕ(n)-length bitstrings. Any algorithm for reconstructing an approximation of dist G , given the bitstring encoding of G, must have additive error +Ω(n 1/18 ).

P
. For each subset T ⊆ P construct the graph G T by removing all clique edges from G that are on the critical paths of pairs in T . Because all clique edges are missing, for all (x, ) ∈ T we have d G T (x, ) ≥ d G (x, ) + 2D. On the other hand, for all (x, ) T , d G T (x, ) = d G (x, ).
There are 2 |P | such graphs. If we represent all such graphs with bitstrings of length |P | − 1 then by the pigeonhole principle two such graphs G T and G T ′ are mapped to the same bitstring. Let (x, ) be any pair in T \T ′ . Since dist G T (x, ) ≥ dist G T ′ (x, ) + 2D, the additive stretch of any such scheme must be at least 2D. Alternatively, any scheme with stretch 2D − 1 must use bitstrings of length at least length |P |. Now, by setting d = 3 with D =Θ(n 1/18 ), r 1 = r 2 =Θ(n 2/27 ) and R 1 = R 2 =Θ(n 7/54 ), we have |P | =Θ(n). Thus anyÕ(n)-length encoding must recover approximate distances with stretch +Ω(n 1/18 ).

CONCLUSION
Our constructions, like [1,2,12,16], are based on looking at the convex hulls of integer lattice points in Z d lying in a ball of some radius. Whereas Theorems 3.10 and 3.13 hold for d = 3, Theorems 2.1, 2.8, and 3.2 are indifferent between dimensions d = 2 and d = 3, but that is only because d must be an integer.
Suppose we engage in a little magical thinking, and imagine that there are integer lattices in any fractional dimension, and moreover, that some analogue of Bárány and Larman's [5] bound holds in these lattices. If such objects existed then we could obtain slightly better lower bounds. For example, setting d = 1 + √ 2 in the proof of Theorem 2.1, we would conclude that any O(n)size shortcut set cannot reduce the diameter below Ω(n 1/(3+2 √ 2) ), which is an improvement over Ω(n 1/6 ) as 3 + 2 √ 2 < 5.83.