A Spanner for the Day After Kevin Buchin 1 Sariel Har-Peled 2 ah 1 D´ aniel Ol´ 1 Eindhoven University of Technology 2 University of Illionis at Urbana-Champaign u d e t n S t n P o r e i t s a e t n
Geometric spanners ◮ V ⊂ R d (finite set) ◮ G = ( V, E ) network weighted with Euclidean distances 1 / 13
Geometric spanners u v ◮ V ⊂ R d (finite set) ◮ G = ( V, E ) network weighted with Euclidean distances ◮ t -path from u to v : path of length at most t · d ( u, v ) ◮ G is t -spanner if ∃ t -path between any u, v ∈ V 1 / 13
Geometric spanners u v t = 2 ◮ V ⊂ R d (finite set) ◮ G = ( V, E ) network weighted with Euclidean distances ◮ t -path from u to v : path of length at most t · d ( u, v ) ◮ G is t -spanner if ∃ t -path between any u, v ∈ V 1 / 13
Geometric spanners t = 2 ◮ V ⊂ R d (finite set) ◮ G = ( V, E ) network weighted with Euclidean distances ◮ t -path from u to v : path of length at most t · d ( u, v ) ◮ G is t -spanner if ∃ t -path between any u, v ∈ V 1 / 13
Geometric spanners t = 2 ◮ V ⊂ R d (finite set) ◮ G = ( V, E ) network weighted with Euclidean distances ◮ t -path from u to v : path of length at most t · d ( u, v ) ◮ G is t -spanner if ∃ t -path between any u, v ∈ V 1 / 13
Geometric spanners t = 2 ◮ V ⊂ R d (finite set) ◮ G = ( V, E ) network weighted with Euclidean distances ◮ t -path from u to v : path of length at most t · d ( u, v ) ◮ G is t -spanner if ∃ t -path between any u, v ∈ V 1 / 13
Fault tolerant spanners ◮ Vertex failures ◮ Fixed parameter k ≥ 0 ◮ Fault tolerant spanners withstand failures of at most k vertices ◮ Ω( kn ) edges are needed (and enough) 2 / 13
Fault tolerant spanners ◮ Vertex failures ◮ Fixed parameter k ≥ 0 ◮ Fault tolerant spanners withstand failures of at most k vertices ◮ Ω( kn ) edges are needed (and enough) Disadvantages: ◮ No guarantee if more than k vertices fail ◮ Size grows linearly with k ◮ If k = c · n , then Ω( n 2 ) edges are needed 2 / 13
Fault tolerant spanners ◮ Vertex failures ◮ Fixed parameter k ≥ 0 ◮ Fault tolerant spanners withstand failures of at most k vertices ◮ Ω( kn ) edges are needed (and enough) Disadvantages: ◮ No guarantee if more than k vertices fail ◮ Size grows linearly with k ◮ If k = c · n , then Ω( n 2 ) edges are needed How to survive catastrophic failures with fewer edges? 2 / 13
Reliable spanners t = 2 ◮ Vertex failures 3 / 13
Reliable spanners B t = 2 ◮ Vertex failures, a set B of points fail 3 / 13
Reliable spanners B 2 . 33 t = 2 ◮ Vertex failures, a set B of points fail 3 / 13
Reliable spanners B 2 . 33 B + t = 2 ◮ Vertex failures, a set B of points fail ◮ Harmed vertices B + ⊇ B ◮ Maintain 2 -paths for u, v ∈ V \ B + 3 / 13
Reliable spanners B B + t = 2 ◮ Vertex failures, a set B of points fail ◮ Harmed vertices B + ⊇ B ◮ Maintain 2 -paths for u, v ∈ V \ B + 3 / 13
Reliable spanners B v B + u t = 2 ◮ Vertex failures, a set B of points fail ◮ Harmed vertices B + ⊇ B ◮ Maintain 2 -paths for u, v ∈ V \ B + 3 / 13
Reliable spanners B v B + u t = 2 ◮ Vertex failures, a set B of points fail ◮ Harmed vertices B + ⊇ B ◮ Maintain 2 -paths for u, v ∈ V \ B + ◮ θ ∈ (0 , 1) parameter, | B + | ≤ (1 + θ ) | B | 3 / 13
Reliable spanners B v B + u t = 2 Definition ◮ Vertex failures, a set B of points fail ◮ Harmed vertices B + ⊇ B The graph G = ( V, E ) is a θ -reliable t -spanner if for any set B ⊆ V there exists a set B + ⊇ B with | B + | ≤ (1 + θ ) | B | such that the subgraph ◮ Maintain 2 -paths for u, v ∈ V \ B + induced by V \ B is a t -spanner of V \ B + . ◮ θ ∈ (0 , 1) parameter, | B + | ≤ (1 + θ ) | B | 3 / 13
Related work Results of [Bose et al. 2013] ◮ Notion of robust spanners – More general than reliable spanners – | B + | ≤ f ( | B | ) for some function f : N → R + – Several bounds on the size (number of edges) ◮ Bounds for reliable spanners – Lower bound: Ω( n log n ) edges – Upper bound: O ( n 2 ) edges (trivial) 4 / 13
Related work Results of [Bose et al. 2013] ◮ Notion of robust spanners – More general than reliable spanners – | B + | ≤ f ( | B | ) for some function f : N → R + – Several bounds on the size (number of edges) ◮ Bounds for reliable spanners – Lower bound: Ω( n log n ) edges – Upper bound: O ( n 2 ) edges (trivial) Goal In 1-D: ◮ Construct θ -reliable 1 -spanners with as few edges as possible. In higher dimensions: ◮ Construct θ -reliable (1 + ε ) -spanners with as few edges as possible. 4 / 13
Timeline Preliminary construction in one dimension ◮ O ( n 1+ δ ) , CGWeek - YRF 2018 ◮ O ( n log n ) , one day after the arXiv version by Sariel General results in higher dimensions appeared on arXiv ◮ O ( n log c n ) , where c = O ( d ) Buchin, Har-Peled, Ol´ ah 16 Nov 2018 ◮ O ( n log 4 n log log n ) Bose, Carmi, Dujmovic, Morin 24 Dec 2018 ◮ O ( n log 2 n log log n ) Bose, Carmi, Dujmovic, Morin 6 Jan 2019 ◮ O ( n log n (log log n ) 6 ) Buchin, Har-Peled, Ol´ ah 25 Jan 2019 5 / 13
Our results In 1-D: ◮ θ -reliable 1 -spanners with O ( n log n ) edges. In higher dimensions: ◮ θ -reliable (1 + ε ) -spanners with O ( n log n (log log n ) 6 ) edges. ◮ θ -reliable (1 + ε ) -spanners with O ( n log n ) edges for points with bounded spread. max p,q ∈ V d ( p, q ) Spread of V : min p,q ∈ V, p � = q d ( p, q ) 6 / 13
Optimal construction in 1-D | V | = n = 2 ℓ Structure of blocks: 1 n ◮ Binary tree T with the points of V in the leaves 7 / 13
Optimal construction in 1-D | V | = n = 2 ℓ Structure of blocks: v n 1 i j ◮ Binary tree T with the points of V in the leaves ◮ Each node v ∈ T corresponds to a block of points { i, . . . , j } 7 / 13
Optimal construction in 1-D | V | = n = 2 ℓ Structure of blocks: v x y n 1 i j ◮ Binary tree T with the points of V in the leaves ◮ Each node v ∈ T corresponds to a block of points { i, . . . , j } ◮ Two nodes of T are neighbors if – they are in the same level and – their blocks are next to each other 7 / 13
Optimal construction in 1-D | V | = n = 2 ℓ Structure of blocks: v x y n 1 i j ◮ Binary tree T with the points of V in the leaves ◮ Each node v ∈ T corresponds to a block of points { i, . . . , j } ◮ Two nodes of T are neighbors if – they are in the same level and – their blocks are next to each other ◮ Build an expander between any neighboring blocks – expander: sparse, well connected graph ◮ O ( n ) edges per level, O ( n log n ) in total 7 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j ◮ Expanders along the path of blocks 8 / 13
Proof idea i, j ∈ V \ B + n i j 1 n i j 1 ◮ Path of blocks from i to j ◮ Expanders along the path of blocks ◮ Define the harmed set B + in a suitable way ◮ Only a few failures, otherwise i ∈ B + or j ∈ B + 8 / 13
Our results In 1-D: ◮ θ -reliable 1 -spanners with O ( n log n ) edges. In higher dimensions: ◮ θ -reliable (1 + ε ) -spanners with O ( n log n (log log n ) 6 ) edges. ◮ θ -reliable (1 + ε ) -spanners with O ( n log n ) edges for points with bounded spread. 9 / 13
Construction in R d ◮ V ⊂ R d ◮ Main tool: Locality-sensitive orderings [Chan et al. 2019] – a set Π + of a ’few’ linear orderings – ∀ p, q ∈ V exists an ordering σ ∈ Π + such that: if p ≺ σ z ≺ σ q , then z is close to either p or q ξ · d ( p, q ) q d ( p, q ) p ξ · d ( p, q ) z 10 / 13
Construction in R d ◮ V ⊂ R d ◮ Main tool: Locality-sensitive orderings [Chan et al. 2019] – a set Π + of a ’few’ linear orderings – ∀ p, q ∈ V exists an ordering σ ∈ Π + such that: if p ≺ σ z ≺ σ q , then z is close to either p or q ξ · d ( p, q ) q d ( p, q ) p ξ · d ( p, q ) z p q σ Construction: ◮ Build the 1-D construction for each σ ∈ Π + and take the union ◮ O � n log n (log log n ) 6 � edges (by setting the right parameters) 10 / 13
Construction for points with bounded spread ◮ V ⊂ R d 11 / 13
Construction for points with bounded spread ◮ V ⊂ R d ◮ Quadtree structure 11 / 13
Construction for points with bounded spread ◮ V ⊂ R d ◮ Quadtree structure 11 / 13
Construction for points with bounded spread ◮ V ⊂ R d ◮ Quadtree structure 11 / 13
Construction for points with bounded spread ◮ V ⊂ R d ◮ Quadtree structure 11 / 13
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