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

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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?

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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

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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)

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Multiplicative Spanners [PS, ADDJS] For every k, can quickly find a (2k-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 2k+2 – Only (2k-1,0)-spanner of G is G itself

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

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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 (1,2k-1)- spanners. Only resolved for k = 1, 2, 3, 5.

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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 (1,2k-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

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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

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Known Results Focus on (1,2k-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

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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 (1,2k-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.

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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

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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

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Our Third Result Simple lower bound for general (1,2k-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 (1,2k-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.

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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 instance- dependent lower bounds as a function of d?

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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 (1,2k-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

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Overview of Techniques

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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 specially- chosen graphs. This is crucial

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Lower Bound for (1,3)-spanners Identify vertices v as points (a,b,i) in [n 1/2 ] £ [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 = 3n. # edges = 2n 3/2

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

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Lower Bound for (1,3)-spanners Recall #vertices = 3n, #edges = 2n 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.

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

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Lower Bound for (1,3)-spanners Fix the subgraph H. Choose a path v 1, v 2, v 3 in G with v i 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.

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Example: n = 4 (1,1,1) (2,1,1) (1,2,1) (2,2,1) (1,1,3) (2,1,3) (1,2,3) (2,2,3) V1V1 V2V2 V3V3 Red lines are edges in H

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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.

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Example: n = 4 (v 1, v 2 ) and (v 2, v 3 ) are missing from H (1,1,1) (2,1,1) (1,2,1) (2,2,1) (1,1,3) (2,1,3) (1,2,3) (2,2,3) V1V1 V3V3 V2V2

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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.

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Example: n = 4 (1,1,1) (2,1,1) (1,2,1) (2,2,1) (1,1,3) (2,1,3) (1,2,3) (2,2,3) V1V1 V3V3 V2V2 G (v 1, v 3 ) = 2 but H (v 1, v 3 ) = 6

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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 1st coordinate of b and v 3 are the same –The 2nd 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.

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Example: n = 4 (1,1,1) (2,1,1) (1,2,1) (2,2,1) (1,1,3) (2,1,3) (1,2,3) (2,2,3) V1V1 V3V3 V2V2 Every path in G with G (v 1, v 3 ) < 6 contains (v 1, v 2 ) or (v 2, v 3 )

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Extension to General k Lower bound for (1,2k-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, v k+1 for which G (v 1, v k+1 ) = k, but H (v 1, v k+1 ) ¸ 3k

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Extension to Emulators Recall that a (1,2k-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 2k = O(1), so H has at most 2rk edges. Thus, r = (n 1+1/k ).

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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

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

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Lower Bound for (1,5)-approximate source-wise preserver Example 2: |S| =8, our technique implies |H| ¸ 8 Red lines indicate edges on shortest paths to and from SFor n = 8, can improve bound on |H|, but not asymptotically

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Lower Bound for (1,5)-approximate source-wise preserver Intuition: Spread out source S This is a good choiceThis is a bad choiceThere is a small H

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

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

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Prescribed Minimum Degree Left and middle vertices now have degree 8

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

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Modify middle edges so there is a unique edge connecting the clustersChoose a random vertex v 1 in level 1 v1v1 v2v2 Choose a random v 2 amongst first 2 neighbors of v 1 v3v3 v 3 is determinedv 4 is a random neighbor of v 3 v4v4 Any sparse subgraph H is likely not to contain (v 1, v 2 ) and (v 3, v 4 ) G (v 1, v 4 ) = 3, but H (v 1, v 4 ) = 7, so H is not a (1,3)-spanner

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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 (1,2k-1)-emulator has size (n 1+1/k-c(1+2/(k-1)) )

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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)

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Upper Bound for (1,2)-spanners u v Shortest path from u to v in G a a is in the dominating set Path a, w, x, y, z, v is shortest from a to v in G wxy Path a, w, x, y, z, v occurs in BFS(a), so it is in H z Path u, a, w, x, y, z, v in H H (u,v) · 1+ H (a,v) = 1 + G (a,v) · 2 + G (u,v) By triangle inequality, G (a,v) · G (u,v) + 1

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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

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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

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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?

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