Lower Bounds for Additive Spanners Emulators and More

Lower Bounds for Additive Spanners, Emulators, and More David P. Woodruff MIT and Tsinghua University To appear in FOCS, 2006

The Model • G = (V, E) undirected unweighted graph, n vertices, m edges • G (u, v) shortest path length from u to v in G • Distance queries: what is G(u, v)? • Exact answers for all pairs (u, v) needs Omega(m) space • What about approximate answers?

Spanners • [A, PS] An (a, b)-spanner of G is a subgraph H such that for all u, v in V, H(u, v) · a G(u, v) + b • If b = 0, H is a multiplicative spanner • If a = 1, H is an additive spanner • Challenge: find sparse H

Spanner Application • 3 -approximate distance queries G(u, v) with small space • Construct a (3, 0)-spanner H with O(n 3/2) edges. [PS, ADDJS] do this efficiently • Query answer: G(u, v) · H(u, v) · 3 G(u, v)

Multiplicative Spanners • [PS, ADDJS] For every k, can quickly find a (2 k-1, 0)-spanner with O(n 1+1/k) edges • Assuming a girth conjecture of Erdos, cannot do better than (n 1+1/k) • Girth conjecture: there exist graphs G with Omega(n 1+1/k) edges and girth 2 k+2 – Only (2 k-1, 0)-spanner of G is G itself

Surprise, Surprise • [ACIM, DHZ]: Construct a (1, 2)-spanner H with O(n 3/2) edges! • Remarkable: for all u, v: G(u, v) + 2 G(u, v) · H(u, v) · • Query answer is a 3 -approximation, but with much stronger guarantees for G(u, v) large

Additive Spanners • Upper Bounds: – (1, 2)-spanner: O(n 3/2) edges [ACIM, DHZ] – (1, 6)-spanner: O(n 4/3) edges [BKMP] – For any constant b > 6, best (1, b)-spanner known is O(n 4/3) Major open question: can one do better than O(n 4/3) edges for constant b? • Lower Bounds: – Girth conjecture: (n 1+1/k) edges for (1, 2 k-1)spanners. Only resolved for k = 1, 2, 3, 5.

Our First Result • Lower Bound for Additive Spanners for any k without using the (unproven) girth conjecture: For every constant k, there exists an infinite family of graphs G such that any (1, 2 k-1)-spanner of G requires (n 1+1/k) edges • Matches girth conjecture up to constants • Improves weaker unconditional lower bounds by an n (1) factor

Emulators • In some applications, H must be a subgraph of G, e. g. , if you want to use a small fraction of existing internet links • For distance queries, this is not the case • [DHZ] An (a, b)-emulator of a graph G = (V, E) is an arbitrary weighted graph H on V such that for all u, v G(u, v) · H(u, v) · a G(u, v) + b • An (a, b)-spanner is (a, b)-emulator but not vice versa

Known Results • Focus on (1, 2 k-1)-emulators • Previous published bounds [DHZ] – (1, 2)-emulator: O(n 3/2), (n 3/2 / polylog n) – (1, 4)-emulator: (n 4/3 / polylog n) • Lower bounds follow from bounds on graphs of large girth

Our Second Result • Lower Bound for Emulators for any k without using graphs of large girth: For every constant k, there exists an infinite family of graphs G such that any (1, 2 k-1)-emulator of G requires (n 1+1/k) edges. • All existing proofs start with a graph of large girth. Without resolving the girth conjecture, they are necessarily n (1) weaker for general k.

Distance Preservers • [CE] In some applications, only need to preserve distances between vertices u, v in a strict subset S of all vertices V • An (a, b)-approximate source-wise preserver of a graph G = (V, E) with source S ½ V, is an arbitrary weighted graph H such that for all u, v in S, G(u, v) · H(u, v) · a G(u, v) + b

Known Results • Only existing bounds are for exact preservers, i. e. , H(u, v) = G(u, v) for all u, v in S • Bounds only hold when H is a subgraph of G • In this case, lower bounds have form (|S|2 + n) for |S| in a wide range [CE] • Lower bound graphs are complex – look at lattices in high dimensional spheres

Our Third Result • Simple lower bound for general (1, 2 k-1)approximate source-wise preservers for any k and for any |S|: For every constant k, there is an infinite family of graphs G and sets S such that any (1, 2 k-1)approximate source-wise preserver of G with source S has (|S|min(|S|, n 1/k)) edges. • Lower bound for emulators when |S| = n. • No previous non-trivial lower bounds known.

Prescribed Minimum Degree • In some applications, the minimum degree d of the underlying graph is large, and so our lower bounds are not applicable • In our graphs minimum degree is (n 1/k) • What happens when we want instancedependent lower bounds as a function of d?

Our Fourth Result • A generalization of our lower bound graphs to satisfy the minimum degree d constraint: Suppose d = n 1/k+c. For any constant k, there is an infinite family of graphs G such that any (1, 2 k-1)emulator of G has (n 1+1/k-c(1+2/(k-1))) edges. • If d = (n 1/k) recover our (n 1+1/k) bound • If k = 2, can improve to (n 3/2 – c) • Tight for (1, 2)-spanners and (1, 4)-emulators

Overview of Techniques

Additive Spanners • All previous methods looked at deleting one edge in graphs of high girth • Thus, these methods were generic, and also held for multiplicative spanners • We instead look at long paths in speciallychosen graphs. This is crucial

Lower Bound for (1, 3)-spanners • Identify vertices v as points (a, b, i) in [n 1/2] £ [3] • We call the last coordinate the level • Edges connect vertices in level i to level i+1 which differ only in the ith coordinate: (a, b, 1) connected to (a’, b, 2) for all a, a’, b (a, b, 2) connected to (a, b’, 3) for all a, b, b’ • # vertices = 3 n. # edges = 2 n 3/2

Example: n = 4 (1, 1, 1) (1, 1, 3) (2, 1, 1) (2, 1, 3) (1, 2, 1) (1, 2, 3) (2, 2, 1) (2, 2, 3)

Lower Bound for (1, 3)-spanners • Recall #vertices = 3 n, #edges = 2 n 3/2 • Consider arbitrary subgraph H with < n 3/2 edges • Let e 1, 2 = # edges in H from level 1 to 2 • Let e 2, 3 = # edges in H from level 2 to 3 • Then H has e 1, 2 + e 2, 3 < n 3/2 edges.

Example: n = 4 (1, 1, 1) (1, 1, 3) (2, 1, 1) (2, 1, 3) (1, 2, 1) (1, 2, 3) (2, 2, 1) (2, 2, 3) H has < n 3/2 = 8 edges, e 1, 2 = 3, e 2, 3 = 4

Lower Bound for (1, 3)-spanners Fix the subgraph H. Choose a path v 1, v 2, v 3 in G with vi in level i as follows: 1. Choose v 1 in level 1 uniformly at random. 2. Choose v 2 to be a random neighbor of v 1 in level 2. 3. Choose v 3 to be a random neighbor of v 2 in level 3.

Example: n = 4 V 1 (1, 1, 1) (1, 1, 3) V 3 (2, 1, 1) (2, 1, 3) V 2 (1, 2, 1) (1, 2, 3) (2, 2, 1) (2, 2, 3) Red lines are edges in H

Lower Bound for (1, 3)-spanners Pr[(v 1, v 2) and (v 2, v 3) in G H] ¸ 1 - Pr[(v 1, v 2) in H] – Pr[(v 2, v 3) in H] ¸ 1 - e 1, 2/n 3/2 - e 2, 3/n 3/2 > 0. So, there exist v 1, v 2, v 3 such that (v 1, v 2) and (v 2, v 3) are missing from H.

Example: n = 4 (1, 1, 1) (1, 1, 3) (2, 1, 1) (2, 1, 3) (1, 2, 1) V 1 (1, 2, 3) (2, 2, 1) V 2 (v 1, v 2) and (v 2, v 3) are missing from H V 3 (2, 2, 3)

Lower Bound for (1, 3)-spanners • G(v 1, v 3) = 2. • Claim: H(v 1, v 3) ¸ 6. • Proof: – Construction ensures all paths from v 1 to v 3 in G have an odd # of edges in both levels. – Pigeonhole principle: if H(v 1, v 3) < 6, some level in any shortest path has only 1 edge.

Example: n = 4 (1, 1, 1) (1, 1, 3) (2, 1, 1) (2, 1, 3) (1, 2, 1) V 1 (1, 2, 3) (2, 2, 1) V 2 G(v 1, v 3) = 2 but H(v 1, v 3) = 6 V 3 (2, 2, 3)

Lower Bound for (1, 3)-spanners • Suppose w. l. o. g. , only 1 edge e = (a, b) in level 1 • Path from v 1 to v 3 in H starts with a level 1 edge e. So, e = (v 1, b). • Edges in level i can only change the ith coordinate of a vertex. So, – The 1 st coordinate of b and v 3 are the same – The 2 nd coordinate of b and v 1 are the same • So, b = v 2 and e = (v 1, v 2). But (v 1, v 2) is missing from H. Contradiction.

Example: n = 4 (1, 1, 1) (1, 1, 3) (2, 1, 1) (2, 1, 3) (1, 2, 1) V 1 (1, 2, 3) (2, 2, 1) V 2 V 3 (2, 2, 3) Every path in G with G(v 1, v 3) < 6 contains (v 1, v 2) or (v 2, v 3)

Extension to General k • Lower bound for (1, 2 k-1)-spanners same: • Vertices are points in [n 1/k]k £ [k+1] • Edges only connect adjacent levels i, i+1, and can change the ith coordinate arbitrarily • If subgraph H has less than n 1+1/k edges, there are vertices v 1, vk+1 for which G(v 1, vk+1) = k, but H(v 1, vk+1) ¸ 3 k

Extension to Emulators • Recall that a (1, 2 k-1)-emulator H is like a spanner except H can be weighted and need not be a subgraph. • Observation: if e=(u, v) is an edge in H, then the weight of e is exactly G(u, v). • Reduction: Given emulator H with less than r edges, can replace each weighted edge in H by a shortest path in G. The result is an additive spanner H’. • Our graphs have diameter 2 k = O(1), so H’ has at most 2 rk edges. Thus, r = (n 1+1/k).

Extension to Preservers • An (a, b)-approximate source-wise preserver of a graph G with source S ½ V, is an arbitrary weighted graph H such that for all u, v in S, G(u, v) · H(u, v) · a G(u, v) + b • Use same lower bound graph • Restrict to subgraph case. Can apply “diameter argument” • Choose a “hard’’ set S of vertices, based on |S|, whose distances to preserve

Lower Bound for (1, 5)-approximate source-wise preserver Example 1: |S| Graph =4, |H| formust n= 8: be at least 6 Red lines indicate edges on shortest paths to and from S

Lower Bound for (1, 5)-approximate source-wise preserver Example 2: |S| =8, our technique implies |H| ¸ 8 Rednlines For = 8, indicate can improve edges bound on shortest on |H|, paths but nottoasymptotically and from S

Lower Bound for (1, 5)-approximate source-wise preserver Intuition: “Spread out” source S This There is aisbad good a small choice H

Other Extensions • For (1, 2 k-1)-approximate source-wise preservers, we achieve (|S|min(|S|, n 1/k)) • Prescribed minimum degree d – Insert Kd, ds to ensure the minimum degree constraint is satisfied, while preserving the distortion property

Prescribed Minimum Degree n = 16, degree Suppose we insist = 4, on care minimum about (1, 3)-spanners degree 8

Prescribed Minimum Degree Left and middle vertices now have degree 8

Prescribed Minimum Degree Add a new level so everyone has degree 8. What happens to the distortion?

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Prescribed Minimum Degree • (1, 2)-spanners require (n 3/2 – c) edges if the minimum degree is n 1/2 + c • Corresponding O(n 3/2 -c log n) upper bound • General result: if min degree is n 1/k+c, any (1, 2 k-1)-emulator has size (n 1+1/k-c(1+2/(k 1)))

Upper Bound for (1, 2)-spanners • A set S is dominating if for all vertices v 2 V, there is an s 2 S such that (s, v) is an edge in G • If minimum degree n 1/2+c , then there is a dominating S of size O(n 1/2 –c log n) • For v 2 V, BFS(v) denotes the shortest-path tree in G rooted at v • H = [v in S BFS(v). Then |H| = O(n 3/2 – c log n)

Upper Bound for (1, 2)-spanners a u w x y z v Path u, a, w, x, y, z, v in H By triangle inequality, Gu(a, v) +1 Path z, vv isoccurs shortest from to Shortest from v· in a G G(u, v) a y, is the dominating set Path a, a, w, w, x, x, y, in z, path into BFS(a), sovitinis. Gin H H(u, v) · 1+ H(a, v) = 1 + G(a, v) · 2 + G(u, v)

Upper Bound Recap • If minimum degree n 1/2+c , then there is a dominating S of size O(n 1/2 –c log n) • H = [v in S BFS(v). • |H| = O(n 3/2 – c log n) • H is a (1, 2)-spanner

Summary of Results • Unconditional lower bounds for additive spanners and emulators beating previous ones by n (1), and matching a 40+ year old conjecture, without proving the conjecture • Many new lower bounds for approximate source-wise preservers and for emulators with prescribed minimum degree. In some cases the bounds are tight

Future Directions • Moral: – One can show the equivalence of the girth conjecture to lower bounds for multiplicative spanners, – However, for additive spanners are lower bounds are just as good as those provided by the girth conjecture, so the conjecture is not a bottleneck. • Still a gap, e. g. , (1, 4)-spanners: O(n 3/2) vs. (n 4/3) • Challenge: What is the size of additive spanners?