The 4/3 Additive Spanner Exponent is Tight

Transcription

1 The 4/3 Additive Spanner Exponent is Tight Amir Abboud and Greg Bodwin Stanford University June 8, 2016

2 Our Question How much can you compress a graph into low space while still being able to approximately recover its distances? (All graphs in this talk are undirected and unweighted.) compress

3 Our Question How much can you compress a graph into low space while still being able to approximately recover its distances? (All graphs in this talk are undirected and unweighted.) dist(u, v)? compress

4 Our Question How much can you compress a graph into low space while still being able to approximately recover its distances? (All graphs in this talk are undirected and unweighted.) dist(u, v)? compress dist(u, v)

5 Overview How much can you compress a graph into low space while still being able to approximately recover its distances?

6 Overview How much can you compress a graph into low space while still being able to approximately recover its distances?...but we ll mostly keep our eye on an easier question: How much can you sparsify a graph while approximately preserving its distances?

7 Overview How much can you compress a graph into low space while still being able to approximately recover its distances?...but we ll mostly keep our eye on an easier question: How much can you sparsify a graph while approximately preserving its distances? (Distances approximately match)

8 Sparsifying Graphs While Approximately Preserving Distances Two (similar) objects are used to capture this idea:

9 Sparsifying Graphs While Approximately Preserving Distances Two (similar) objects are used to capture this idea: Multiplicative Spanners: Given a graph G, a subgraph H G is a k Multiplicative Spanner if for all node pairs (s, t). dist H (s, t) k dist G (s, t)

10 Sparsifying Graphs While Approximately Preserving Distances Two (similar) objects are used to capture this idea: Multiplicative Spanners: Given a graph G, a subgraph H G is a k Multiplicative Spanner if for all node pairs (s, t). dist H (s, t) k dist G (s, t) Additive Spanners: Given a graph G, a subgraph H G is a +k Additive Spanner if for all node pairs (s, t). dist H (s, t) k + dist G (s, t)

11 Multiplicative Spanners Multiplicative Spanners have been well understood since the 80s.

12 Multiplicative Spanners Multiplicative Spanners have been well understood since the 80s. Upper Bounds: All graphs have (2k 1) multiplicative spanners on O(n 1+1/k ) edges. [Awerbuch 85]

13 Multiplicative Spanners Multiplicative Spanners have been well understood since the 80s. Upper Bounds: All graphs have (2k 1) multiplicative spanners on O(n 1+1/k ) edges. [Awerbuch 85] Lower Bounds: There are graphs which cannot be compressed into o(n 1+1/k ) bits with (2k 1) distance error. [ Girth Conjecture Erdös 65 and others]

14 Multiplicative Spanners n 2 Matching Upper/Lower Bounds edges in spanner n 3/2 n multiplicative error More Multiplicative Error Sparser Spanners

15 Additive vs. Multiplicative Error Applications of spanners: Graph Sketching Approximation Algorithms Network Synchronization Routing Distance Oracles Labeling Schemes... In almost all applications, additive error is preferable. A 3 approximation to dist(stanford, Paris) is pretty bad. A +2 approximation is way better.

16 Additive Spanner History Originally, additive error seemed too good to be true. But then...

17 Additive Spanner History Originally, additive error seemed too good to be true. But then... Theorem: All graphs have +2 spanners on O(n 3/2 ) edges. [Aingworth, Chekuri, Indyk, Motwani 96] Same sparsity bound for +2 additive and 3 multiplicative spanners!

18 Additive Spanner History Originally, additive error seemed too good to be true. But then... Theorem: All graphs have +2 spanners on O(n 3/2 ) edges. [Aingworth, Chekuri, Indyk, Motwani 96] Same sparsity bound for +2 additive and 3 multiplicative spanners! Can we obtain sparse additive spanners in general?! Main Open Question: For all ε > 0, is there a k ε such that all graphs have +k ε additive spanners on O(n 1+ε ) edges? In other words: If we allow more additive error, then can we have sparser spanners?

19 Cause for Optimism Main Open Question: If we allow more additive error, can we have sparser spanners? Theorem: All graphs have +2 spanners on O(n 3/2 ) edges. [Aingworth, Chekuri, Indyk, Motwani SODA 96] Theorem: All graphs have +4 spanners on Õ(n 7/5 ) edges. [Chechik SODA 13] Theorem: All graphs have +6 spanners on O(n 4/3 ) edges. [Baswana, Kavitha, Mehlhorn, Pettie SODA 05] Theorem: +(2k 1) spanners need Ω(n 1+1/k ) edges (as in the Girth Conjecture, but unconditional). [Woodruff FOCS 06]

20 Past Work on Additive Spanners n 2 Upper Bounds n edges in spanner n 3/ additive error More Additive Error Sparser Spanners???

21 Past Work on Additive Spanners n 2 Upper Bounds n edges in spanner n 3/ additive error More Additive Error Sparser Spanners???

22 Past Work on Additive Spanners n 2 Upper Bounds edges in spanner n 3/2 (+2, n 3/2 ) [ACIM 96] n additive error More Additive Error Sparser Spanners???

23 Past Work on Additive Spanners n 2 Upper Bounds edges in spanner n 3/2 (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] n additive error More Additive Error Sparser Spanners???

24 Past Work on Additive Spanners n 2 Upper Bounds edges in spanner n 3/2 (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06] n additive error More Additive Error Sparser Spanners???

25 Past Work on Additive Spanners n 2 Upper Bounds edges in spanner n 3/2 (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06]????? n additive error More Additive Error Sparser Spanners???

26 Past Work on Additive Spanners edges in spanner n 2 n 3/2 Upper Bounds Lower Bounds (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06]????? n additive error More Additive Error Sparser Spanners???

27 Past Work on Additive Spanners edges in spanner n 2 n 3/2 Upper Bounds Lower Bounds (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06]????? n additive error More Additive Error Sparser Spanners???

28 Past Work on Additive Spanners edges in spanner n 2 n 3/2 Upper Bounds Lower Bounds (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06]????? n additive error More Additive Error Sparser Spanners???

29 Past Work on Additive Spanners edges in spanner n 2 n 3/2 Upper Bounds Lower Bounds (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06]????? [Girth Conjecture/Woodruff 06] n additive error More Additive Error Sparser Spanners???

30 The Critics Rave Main Open Question: For all ε > 0, is there a k ε such that all graphs have +k ε additive spanners on O(n 1+ε ) edges? This question is... a major open problem [Thorup & Zwick 05] the main existential question in the field of spanners [Baswana, Kavitha, Mehlhorn, Pettie 05] a major open question [Coppersmith & Elkin 06] fascinating [Woodruff 06] the chief open question of the field [Pettie 07] a major open problem [Chechik 13] (Solving the problem for even a specific value of ε) would be a major breakthrough [Chechik 15]

31 Our Results edges in spanner n 2 n 3/2 Upper Bounds Lower Bounds (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06]????? [Girth Conjecture/Woodruff 06] n additive error

32 Our Results n 2 Upper Bounds Lower Bounds edges in spanner n 3/2 (+2, n 3/2 ) [ACIM 96] (+4, n 7/5 ) [Chechik 13] (+6, n 4/3 ) [BKMP 06] New Sparser Spanners Don t Exist! n additive error

33 Our Results Theorem. For all ε > 0, there is a δ > 0 and a family of graphs G with no +n δ additive spanner on O(n 4/3 ε ) edges. [NEW] More additive error sparser spanners! (beyond n 4/3 ).

34 Our Results Theorem. For all ε > 0, there is a δ > 0 and a family of graphs G with no +n δ additive spanner on O(n 4/3 ε ) edges. [NEW] More additive error sparser spanners! (beyond n 4/3 ). This result is stronger than the Main Open Question in two ways: The previously-known O(n 4/3 ) upper bound is best possible ( The 4/3 Additive Spanner Exponent is Tight ).

35 Our Results Theorem. For all ε > 0, there is a δ > 0 and a family of graphs G with no +n δ additive spanner on O(n 4/3 ε ) edges. [NEW] More additive error sparser spanners! (beyond n 4/3 ). This result is stronger than the Main Open Question in two ways: The previously-known O(n 4/3 ) upper bound is best possible ( The 4/3 Additive Spanner Exponent is Tight ). Need polynomial additive error (+n δ ) to improve the exponent below 4/3. Even + log n error won t help you.

36 Back to the Real Question Remember: spanners are just one type of graph compression. The real question in this talk is: How much can you compress a graph into low space while still being able to approximately recover its distances?

37 Back to the Real Question Remember: spanners are just one type of graph compression. The real question in this talk is: How much can you compress a graph into low space while still being able to approximately recover its distances? Can your favorite type of graph compression, instead of spanners, can take us below the O(n 4/3 ) threshold? (Distance Oracles, Labeling Schemes, Emulators, Sketching,...)

38 General Graph Compression with Additive Error bits in compressed graph n 2 n 3/2 Upper Bounds Lower Bounds (+2, n 3/2 ) [ACIM 96] (+4, n 4/3 ) [Dor, Halperin, Zwick 96]????? [Girth Conjecture] n additive error

39 General Graph Compression with Additive Error bits in compressed graph n 2 n 3/2 Upper Bounds Lower Bounds (+2, n 3/2 ) [ACIM 96] (+4, n 4/3 ) [Dor, Halperin, Zwick 96] [NEW] Sparser Compression Doesn t Exist! n additive error

40 Our Main Result Theorem. For all ε > 0, there is a δ > 0 and a family of graphs G that cannot be compressed into O(n 4/3 ε ) bits such that their distances can be recovered within +n δ error. [NEW] The answer is NO! Your favorite type of graph compression also can t improve the size exponent below 4/3.

41 Our Main Result Theorem. For all ε > 0, there is a δ > 0 and a family of graphs G that cannot be compressed into O(n 4/3 ε ) bits such that their distances can be recovered within +n δ error. [NEW] The answer is NO! Your favorite type of graph compression also can t improve the size exponent below 4/3. Rest of this talk: A sketch of the spanner lower bound proof A sketch of the generalization to arbitrary compression Some good open questions

42 Proof of the Spanner Lower Bound.

43 Starting Point: +1 Pairwise Lower Bounds We will talk about a graph G and a set of node pairs P, such that: (1) G is dense, and (2) Deleting any edge from G stretches the distance of some pair (u, v) P by +1.

44 Starting Point: +1 Pairwise Lower Bounds We will talk about a graph G and a set of node pairs P, such that: (1) G is dense, and (2) Deleting any edge from G stretches the distance of some pair (u, v) P by +1. Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(n 2 ε ) edges that is the union of shortest paths that are unique edge-disjoint length exactly n δ (endpoints of these paths = P) [Alon 01, Coppersmith and Elkin 06]

45 Starting Point: +1 Pairwise Lower Bounds Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(n 2 ε ) edges that is the union of unique edge-disjoint shortest paths of length exactly n δ. [Alon 01, Coppersmith and Elkin 06]

46 Starting Point: +1 Pairwise Lower Bounds Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(n 2 ε ) edges that is the union of unique edge-disjoint shortest paths of length exactly n δ. [Alon 01, Coppersmith and Elkin 06] When ε = δ = 0, think of a biclique (P = {a, b, c} {x, y, z}): a b c x y z

47 Starting Point: +1 Pairwise Lower Bounds Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(n 2 ε ) edges that is the union of unique edge-disjoint shortest paths of length exactly n δ. [Alon 01, Coppersmith and Elkin 06] When ε = δ = 0, think of a biclique (P = {a, b, c} {x, y, z}): a b c x y z For longer paths, this is a simple rephrasing of some basic facts from Additive Combinatorics ( there exist very dense sum-free sets of integers ).

48 Roadmap (for now) Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths

49 Roadmap (for now) Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths FAIL Spanner Lower Bounds

50 First move: subdivide the edges s t n δ length

51 First move: subdivide the edges s X t n δ length

52 First move: subdivide the edges A +1 detour s OPT X t n δ length

53 First move: subdivide the edges A +1 detour s OPT X t n δ length Subdivide each edge n δ times

54 First move: subdivide the edges +n δ detour A +1 detour s OPT X t n δ length Subdivide each edge n δ times

55 First move: subdivide the edges +n δ detour A +1 detour s OPT X t n δ length Subdivide each edge n δ times Graph is now a +n δ pairwise lower bound (but it s sparse)

56 Next move: clique replacement Replace each original node v with a clique on deg(v) nodes. Connect every edge entering v to a different clique node.

57 Next move: clique replacement Replace each original node v with a clique on deg(v) nodes. Connect every edge entering v to a different clique node. An example path in G now looks like this: n δ n δ n δ s n δ n δ t

58 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property:

59 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property: If a spanner is missing all of the clique edges used by a path, then...

60 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property: If a spanner is missing all of the clique edges used by a path, then... the path is stretched by +1 at each clique, for a total of +n δ distance error

61 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property: If a spanner is missing all of the clique edges used by a path, then... the path is stretched by +1 at each clique, for a total of +n δ distance error or maybe the path changes course and tries to go through some new cliques. But then it must travel an extra n δ -long path to get there, and so it suffers +n δ error.

62 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property: If a spanner is missing all of the clique edges used by a path, then... the path is stretched by +1 at each clique, for a total of +n δ distance error or maybe the path changes course and tries to go through some new cliques. But then it must travel an extra n δ -long path to get there, and so it suffers +n δ error. Lemma: If a subgraph is missing all of the clique edges used by a path, then it stretches that path distance by at least +n δ.

63 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property: n δ n δ n δ s n δ n δ t

64 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property: n δ n δ X X n δ n δ X s +1 X n δ X t X n δ

65 After the clique replacement step Error Analysis These graphs have a nice hard-to-span property: n δ n δ X X n δ n δ X s or +n δ +1 X n δ X t X n δ and then +0

66 After the clique replacement step Density Analysis If you want a good spanner, you must keep at least one clique edge per path, which is P = n 2 δ ε edges.

67 After the clique replacement step Density Analysis If you want a good spanner, you must keep at least one clique edge per path, which is P = n 2 δ ε edges. We subdivided each edge n δ times, so: nodes in spanner Edges in original graph n δ = n 2 ε+δ nodes.

68 After the clique replacement step Density Analysis If you want a good spanner, you must keep at least one clique edge per path, which is P = n 2 δ ε edges. We subdivided each edge n δ times, so: nodes in spanner Edges in original graph n δ = n 2 ε+δ nodes. Uninteresting Theorem: Any +N δ spanner of the graph we just built has Ω(N) edges.

69 New Roadmap Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths

70 New Roadmap Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths FAIL Spanner Lower Bounds

71 New Roadmap Additive Combinatorics Sparser graphs made of unique 2-path-disjoint shortest paths Dense graphs made of unique edge-disjoint shortest paths FAIL Spanner Lower Bounds

72 New Roadmap Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths FAIL Sparser graphs made of unique Spanner Lower Bounds 2-path-disjoint shortest paths SUCCESS

73 Back to the start Our original graphs weren t quite what we wanted.

74 Back to the start Our original graphs weren t quite what we wanted. Remember the properties of the starting graph: Each pair in P has a unique shortest path These shortest paths are edge disjoint These shortest paths all have length exactly n δ P = n 2 ε δ (And therefore, E(G) = n 2 ε )

75 Back to the start Each pair in P has a unique shortest path These shortest paths are edge disjoint These shortest paths all have length exactly n δ P = n 2 ε δ This determines the number of edges in our spanner lower bound better keep this (And therefore, E(G) = n 2 ε ) This determines the number of nodes in our spanner lower bound I wish this were smaller

76 Back to the start Each pair in P has a unique shortest path These shortest paths are edge disjoint These shortest paths all have length exactly n δ P = n 2 ε δ This determines the number of edges in our spanner lower bound better keep this (And therefore, E(G) = n 2 ε ) This determines the number of nodes in our spanner lower bound I wish this were smaller We will need to change one of the other properties if we want to reduce E(G) while keeping P...

77 Back to the start Each pair in P has a unique shortest path These shortest paths are edge disjoint This one! These shortest paths all have length exactly n δ P = n 2 ε δ This determines the number of edges in our spanner lower bound better keep this (And therefore, E(G) = n 2 ε ) This determines the number of nodes in our spanner lower bound I wish this were smaller What other property can we change in order to reduce E(G) while keeping P?

78 Do we really need our paths to be edge disjoint?

79 Do we really need our paths to be edge disjoint? In our edge-extended, clique-replaced graphs, we need our paths to be clique edge disjoint. That is important, to make sure that our spanner has to keep a total of P different clique edges.

80 Do we really need our paths to be edge disjoint? In our edge-extended, clique-replaced graphs, we need our paths to be clique edge disjoint. That is important, to make sure that our spanner has to keep a total of P different clique edges. One clique edge corresponds to a pair of edges entering/leaving a node in the original graph.

81 Do we really need our paths to be edge disjoint? In our edge-extended, clique-replaced graphs, we need our paths to be clique edge disjoint. That is important, to make sure that our spanner has to keep a total of P different clique edges. One clique edge corresponds to a pair of edges entering/leaving a node in the original graph. So we only need our original paths to be 2-path disjoint, not edge disjoint. Definition: A pair of paths p 1, p 2 in a graph G is 2-path disjoint if there is no length-2 path in G that is a subpath of both p 1 and p 2.

82 Do we really need our paths to be edge disjoint? Edge disjoint paths imply clique edge disjoint paths (black clique edges unused).

83 Do we really need our paths to be edge disjoint? Edge disjoint paths imply clique edge disjoint paths (black clique edges unused). 2-path disjoint paths also imply clique edge disjoint paths (all edges used).

84 New Graphs Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(N 3/2 ε ) edges that is the union of unique 2-path disjoint shortest paths of length exactly N δ. [NEW]

85 New Graphs Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(N 3/2 ε ) edges that is the union of unique 2-path disjoint shortest paths of length exactly N δ. [NEW] After we do the edge extension and clique replacement steps from before... A spanner must keep at least one clique edge per pair in P (just like before), or else it stretches one of the distances by +n δ.

86 New Graphs Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(N 3/2 ε ) edges that is the union of unique 2-path disjoint shortest paths of length exactly N δ. [NEW] After we do the edge extension and clique replacement steps from before... A spanner must keep at least one clique edge per pair in P (just like before), or else it stretches one of the distances by +n δ. Clique edges that must be kept in spanner = P = n 2 ε δ (just like before)

87 New Graphs Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(N 3/2 ε ) edges that is the union of unique 2-path disjoint shortest paths of length exactly N δ. [NEW] After we do the edge extension and clique replacement steps from before... A spanner must keep at least one clique edge per pair in P (just like before), or else it stretches one of the distances by +n δ. Clique edges that must be kept in spanner = P = n 2 ε δ (just like before) Nodes in spanner = Edges in original graph n δ n 3/2 ε+δ (much better than before!)

88 New Graphs Lemma: For all ε > 0, there is a δ > 0 and a graph G on Ω(N 3/2 ε ) edges that is the union of unique 2-path disjoint shortest paths of length exactly N δ. [NEW] After we do the edge extension and clique replacement steps from before... A spanner must keep at least one clique edge per pair in P (just like before), or else it stretches one of the distances by +n δ. Clique edges that must be kept in spanner = P = n 2 ε δ (just like before) Nodes in spanner = Edges in original graph n δ n 3/2 ε+δ (much better than before!) Lower bound now follows from the calculation (n 3/2 ) 4/3 = n 2.

89 Summary of Spanner Lower Bound Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths

90 Summary of Spanner Lower Bound Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths FAIL Spanner Lower Bounds

91 Summary of Spanner Lower Bound Additive Combinatorics Sparser graphs made of unique 2-path-disjoint shortest paths Dense graphs made of unique edge-disjoint shortest paths FAIL Spanner Lower Bounds

92 Summary of Spanner Lower Bound Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths FAIL Sparser graphs made of unique Spanner Lower Bounds 2-path-disjoint shortest paths SUCCESS

93 Summary of Spanner Lower Bound Additive Combinatorics Dense graphs made of unique edge-disjoint shortest paths FAIL Sparser graphs made of unique Spanner Lower Bounds 2-path-disjoint shortest paths SUCCESS NEXT Incompressibility

94 Incompressibility We will next sketch how to prove the generalization: Theorem: There is no function that compresses graphs into n 4/3 ε bits in such a way that distances can be recovered within +n δ error.

95 Incompressibility We will next sketch how to prove the generalization: Theorem: There is no function that compresses graphs into n 4/3 ε bits in such a way that distances can be recovered within +n δ error. The main idea is information theoretic: we show that there are n 4/3 ε bits worth of graphs, such that any pair of graphs disagree on a distance by +n δ.

96 Incompressibility We will next sketch how to prove the generalization: Theorem: There is no function that compresses graphs into n 4/3 ε bits in such a way that distances can be recovered within +n δ error. The main idea is information theoretic: we show that there are n 4/3 ε bits worth of graphs, such that any pair of graphs disagree on a distance by +n δ. Thus, no two of these graphs can collide on a representation, and the lower bound follows from the Pigeonhole Principle.

97 Incompressibility Imagine we have a switch for each pair in P (so n 4/3 switches in total), which we can turn on or off in any combination. Each combination defines a graph. p 1 p 2 p 3... p P Switch j on = keep all clique edges used by p j Switch j off = remove all clique edges used by p j

98 Incompressibility Imagine we have a switch for each pair in P (so n 4/3 switches in total), which we can turn on or off in any combination. Each combination defines a graph. p 1 p 2 p 3... p P Switch j on = keep all clique edges used by p j Switch j off = remove all clique edges used by p j Since our shortest paths are clique-edge-disjoint, we may turn the switches on or off in any combination they don t interfere with each other. So n 4/3 bits worth of graphs.

99 Incompressibility n δ n δ n δ s n δ n δ t If switch (s, t) is on, then dist(s, t) is [something].

100 Incompressibility n δ X n X δ n δ n δ X s or +n δ +1 X n δ X t X n δ and then +0 If switch (s, t) is off, then dist(s, t) is at least [something] +n δ.

101 Incompressibility Theorem: There is no function that compresses graphs into n 4/3 ε bits in such a way that distances can be recovered within +n δ error.

102 Incompressibility Theorem: There is no function that compresses graphs into n 4/3 ε bits in such a way that distances can be recovered within +n δ error. Can turn all P n 4/3 switches on or off in any combination; each one defines a different graph.

103 Incompressibility Theorem: There is no function that compresses graphs into n 4/3 ε bits in such a way that distances can be recovered within +n δ error. Can turn all P n 4/3 switches on or off in any combination; each one defines a different graph. If two graphs disagree on a switch, then they disagree on a distance by +n δ.

104 Incompressibility Theorem: There is no function that compresses graphs into n 4/3 ε bits in such a way that distances can be recovered within +n δ error. Can turn all P n 4/3 switches on or off in any combination; each one defines a different graph. If two graphs disagree on a switch, then they disagree on a distance by +n δ. Incompressibility bound now follows from information theory.

105 Remaining Open Questions

106 Are Additive Spanners Optimal Compression Schemes?

107 Are Additive Spanners Optimal Compression Schemes? +2 error: there are spanners on O(n 3/2 ) edges, and there is no compression scheme that uses O(n 3/2 ε ) bits. YES!

108 Are Additive Spanners Optimal Compression Schemes? +2 error: there are spanners on O(n 3/2 ) edges, and there is no compression scheme that uses O(n 3/2 ε ) bits. YES! +6 (or more) error: there are spanners on O(n 4/3 ) edges, and there is no compression scheme that uses O(n 4/3 ε ) bits. YES!

109 Are Additive Spanners Optimal Compression Schemes? +2 error: there are spanners on O(n 3/2 ) edges, and there is no compression scheme that uses O(n 3/2 ε ) bits. YES! +6 (or more) error: there are spanners on O(n 4/3 ) edges, and there is no compression scheme that uses O(n 4/3 ε ) bits. YES! +4 error: Best spanners: Õ(n 7/5 ) edges [Chechik 13] Best compression: Õ(n 4/3 ) bits [Dor, Halperin, Zwick 96] No compression scheme that uses O(n 4/3 ε ) bits. MAYBE?

110 How powerful are linear-sized spanners? In particular: what is the smallest constant δ such that all graphs have +O(n δ ) spanners on O(n) edges?

111 How powerful are linear-sized spanners? In particular: what is the smallest constant δ such that all graphs have +O(n δ ) spanners on O(n) edges? Current upper bound: δ 3/7 [B. - Vassilevska Williams 16] Lower bound from a very naive application of our framework: δ 1/21 (but this is definitely improvable).

112 How powerful are linear-sized spanners? In particular: what is the smallest constant δ such that all graphs have +O(n δ ) spanners on O(n) edges? Current upper bound: δ 3/7 [B. - Vassilevska Williams 16] Lower bound from a very naive application of our framework: δ 1/21 (but this is definitely improvable). Intuitively, the goal here is to optimize the tradeoff between δ and ε in our main theorems.

113 Open Questions Are Additive Spanners optimal compression schemes? In particular: what is the right sparsity for +4 spanners?

114 Open Questions Are Additive Spanners optimal compression schemes? In particular: what is the right sparsity for +4 spanners? How powerful are linear-sized spanners? Tricks to optimize the δ vs ε tradeoff in our main theorems?

115 Open Questions Are Additive Spanners optimal compression schemes? In particular: what is the right sparsity for +4 spanners? How powerful are linear-sized spanners? Tricks to optimize the δ vs ε tradeoff in our main theorems? For what other objects can we get lower bounds? Can measure spanner error in many different ways, not just additive and multiplicative.

116 Open Questions Are Additive Spanners optimal compression schemes? In particular: what is the right sparsity for +4 spanners? How powerful are linear-sized spanners? Tricks to optimize the δ vs ε tradeoff in our main theorems? For what other objects can we get lower bounds? Can measure spanner error in many different ways, not just additive and multiplicative. Thanks for your attention!