Everything you always wanted to know about spanners * * But were afraid to ask

1. Everything you always wanted to know about spanners**But were afraid to ask Seth Pettie University of Michigan, Ann Arbor

2. What is a spanner? Spanner (English) : wrench Spanner (German) : voyeur, peeping tom. Spanner (CS) : sparse subgraph that preserves distances up to some stretch.

3. Graph SpannersPeleg-Schäffer’89 Spanner : sparse subgraph that preserves distances up to some stretch. Given possibly weighted input graph G = (V, E, w) Find a sparse subgraphH = (V, E(H)) such that distH(u,v) ≤ t∙dist(u,v) H is called a t-spanner of G

4. The Greedy AlgorithmAlthöfer, Das, Dobkin, Joseph, Soares’93 Greedy(G, k): (stretch 2k–1) H ←  Examine each (u,v) in increasing order by w(u,v). If distH(u,v) > (2k–1)∙w(u,v) H ← H  {(u,v)} Return H Execution for stretch=3

5. heaviest edge on the cycle Claim: the greedy spanner H has girth at least 2k+1. (girth = length of shortest cycle) Proof by contradiction. Let (u,v) ∈ H be the last edge added to form a length-2k cycle.

6. Behavior of the Greedy Algorithm • mg(n) = max # edges in graph w/n vertices, and girth g • Some observations: • |Greedy(G,k)|m2k+1(n) • |Greedy(G,k)|m2k+1(n) for some G • m2k+1(n) ≤ 2∙m2k+2(n)

7. Upper Bounds: m2k+1(n), m2k+2(n) n1+1/k (1) Repeatedly discard any vertex of degree n1/k (2) Examine k-neighborhood of any vertex:

8. The Girth ConjectureErdos’63, Bondy-Simonovits’74, Bollobas’78 • Conjecture: m2k+2(n) = (n1+1/k) • Confirmed for k = 1, 2, 3, 5 • In general, m2k+2(n) = (n1+1/(3k/2–1)) (Lazebnik, Ustimenko, Woldar’96)

9. The Girth Conjecture • Conjecture: m2k+2(n) = (n1+1/k) • Confirmed for k = 1, 2, 3, 5 ((n2) edges, girth 4)

10. The Girth ConjectureReiman’58, Brown’66, Erdoős-Rényi-Sós’66, Wenger’91 • Conjecture: m2k+2(n) = (n1+1/k) • Confirmed for k = 1,2,3,5 ((n3/2) edges, girth 6) Incidence matrix of a projective geometry:

11. (1) We can’t beat the girth bound (2) We can achieve the girth bound (Althöfer et al.’93) Is there anything else to say about spanners? • Computation time: • (Althöfer et al.’93): O(mn1+1/k) is slow • (Baswana-Sen’03): An O(kn1+1/k)-size (2k–1)spanner in O(km) time.

12. The girth bound, restated If G is an unweighted graph with girth g, the only (g–2)-spanner of G is G. • If (u,v) ∉ H, distH(u,v) ≥ (g-1)∙dist(u,v) Why measure stretch multiplicatively? Defn. H is an f-spanner of unweighted G if distH(u,v) ≤ f(dist(u,v)) f(d) = d + bAdditiveb-spanner f(d) = (1+e)d + b(1+e,b)-spanner f(d) = d + O(d1-e) Sublinear additive spanner

13. The girth bound, restated If G is an unweighted graph with girth g, the only (g–2)-spanner of G is G. • If (u,v) ∉ H, distH(u,v) ≥ (g-1)∙dist(u,v) What if we only care about certain vertex-pairs? • Defn. H is a pairwisef-spanner for vertex pairs P distH(u,v) ≤ f(dist(u,v)) holds for every (u,v) ∈ P. (Note: it makes sense to consider no stretch: f(d)=d.)

14. A big open problem: are there +Õ(1) spanners with size n4/3–e?

16. Assuming the girth conjecture: Any additive 2k–2 spanner has size W(n1+1/k) (Woodruff’06) Any additive 2k–2 spanner has size W(n1+1/k)

17. Additive Spanners: Lower Bounds

18. Lower Bounds on Additive SpannersWoodruff’06 Vertices in k+1 columns named: {1,…,N1/k}k((k+1)N total)

19. Lower Bounds on Additive SpannersWoodruff’06 Edges in layer i connect vertices that may only differ in their ith coordinate. (kN1+1/k edges in total)

20. Lower Bounds on Additive SpannersWoodruff’06 Spanner size  N1+1/k some shortest path excluded

21. Lower Bounds on Additive SpannersWoodruff’06 Spanner path < 3k some layer crossed just once

22. Lower Bounds on Additive SpannersWoodruff’06 e = e (contradiction)

23. Additive Spanners: Upper Bounds

24. Additive 6-SpannersBaswana, Kavitha, Mehlhorn, Pettie’09 (edges not shown)

25. Additive 6-SpannersBaswana, Kavitha, Mehlhorn, Pettie’09 Sample n2/3cluster centers uniformly at random.

26. Additive 6-SpannersBaswana, Kavitha, Mehlhorn, Pettie’09 Sample n2/3cluster centers uniformly at random. Every vertex includes 1 edge to an adjacent center.

27. Additive 6-SpannersBaswana, Kavitha, Mehlhorn, Pettie’09 Sample n2/3cluster centers uniformly at random. Put all edges adjacent to unclustered vertices in the spanner.

28. The Path-Buying AlgorithmBaswana, Kavitha, Mehlhorn, Pettie’09 Overview (1) There are n4/3 cluster pairs (C,C’). (2) Each pair “wants” distH(C, C’) = dist(C, C’). (3) Each pair can “buy” O(1) edges to achieve (2). To compute an additive 6-Spanner: H ← edges selected by clustering procedure Evaluate every shortest path P Ifcost(P) < value(P) then H ← H∪P

29. The Path-Buying AlgorithmBaswana, Kavitha, Mehlhorn, Pettie’09 (Cu,C’) contributes 1 to value(P) cost(P) = number of missing edges on P (roughly number of clusters incident to P.) value(P) = number of pairs (C,C’) such that distP(C,C’) < distH(C,C’)

30. The Path-Buying AlgorithmBaswana, Kavitha, Mehlhorn, Pettie’09 • If P is bought…great! Then distH(u,v) = dist(u,v) • If P is not bought… there exist cluster C’ on P: dist(Cu,C’) and dist(C’,Cv) well-approximated by H.

31. Pairwise Spanners: Upper Bounds

32. Pairwise Distance PreserversCoppersmith, Elkin’06 Given vertex pairs P, want to find spanner H such that distH(u,v) = dist(u,v) for all (u,v) ∈ P (Coppersmith-Elkin’06): If H is chosen in a natural way,

33. Pairwise Distance PreserversCoppersmith, Elkin’06 branch point for (green,blue) and (green,red) branch points for (red,blue)

34. Pairwise Distance PreserversCoppersmith, Elkin’06

35. Pairwise Distance PreserversCoppersmith, Elkin’06

36. Sublinear Additive Spanners

37. Sublinear Additive Error

38. Sublinear Additive Error

39. A -spannerThorup-Zwick’06 C1 = set of n2/3 level-1centers. Include BFS tree from v to radius dist(v,C1)–1

40. A -spannerThorup-Zwick’06 C2 = set of n1/3 level-1centers. For each v∈ C1, include BFS tree from v with radius dist(v,C2)–1. Include BFS tree from each v∈C2 too all other vertices.

41. Why it is a -spannerThorup-Zwick’06 The analysis:d = dist(u,v) Start walking along a shortest u–v path If you can’t walk further, you’re adjacent to a w∈ C1 (a) Walk steps toward v in BFS(w), if possible (b) Walk steps to an x∈ C2 then walk from x to v in BFS(x).

42. Spanners vs. Compact Routing Store Õ(ne)-size routing tables at each node Route message from A to B using only information discovered at routing tables. (Thorup-Zwick’01): Õ(n1/k)-size tables & 4k–5 stretch. (Gavoille-Sommer’11): O(1)-additive routing is impossible: O(ne)-size tables implies W(n(1–e)/2) additive stretch

43. Spanners vs. Distance Oracles Build O(n1+e)-size data structure in order to answer approximate distance queries in O(1) time. (Thorup-Zwick’01): O(n1+1/k) size with (2k–1)d stretch. (Patrascu-Roditty’10):O(n5/3)-size with 2d+1 stretch. Conditional lower bound that < 2 multiplicative stretch is impossible in O(1) query time. Lot’s of followup work, alternate constructions, etc.

44. Graph Spanners vs. Geometric Spanners

45. Some open problems • Existential: • Do f(k)-additive spanners exist with size O(n1+1/k)? f(k)=2k–2 would be optimal. • Do no(1)-additive spanners exist with size O(n)? • Are there -spanners with size O(n1+e)? • Computational • What spanners can be constructed in O(m) time? (Baswana et al.’09):(kd + k-1)-spanners with size O(n1+1/k). • New applications of spanners? • (Kapralov-Panigrahy’12): Build Õ(ne-2)-size spectral sparsifiers using spanners as a “black box.”

46. The End