A Spanner for the Day After Kevin Buchin 1 Sariel Har-Peled 2 ah 1 D - - PowerPoint PPT Presentation

A Spanner for the Day After

Kevin Buchin1 Sariel Har-Peled2 D´ aniel Ol´ ah1

1Eindhoven University of Technology 2University of Illionis at Urbana-Champaign

S t u d e n t P r e s e n t a t i

• n

Geometric spanners

◮ V ⊂ Rd (finite set) ◮ G = (V, E) network weighted with Euclidean distances

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Geometric spanners

u v

◮ V ⊂ Rd (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

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Geometric spanners

u v

◮ V ⊂ Rd (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

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t = 2

Geometric spanners

◮ V ⊂ Rd (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

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t = 2

Geometric spanners

◮ V ⊂ Rd (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

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t = 2

Geometric spanners

◮ V ⊂ Rd (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

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t = 2

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)

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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(n2) edges are needed

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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(n2) edges are needed

How to survive catastrophic failures with fewer edges?

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Reliable spanners

◮ Vertex failures

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t = 2

Reliable spanners

B

◮ Vertex failures, a set B of points fail

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t = 2

Reliable spanners

B

◮ Vertex failures, a set B of points fail

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t = 2 2.33

Reliable spanners

B+ B

◮ Vertex failures, a set B of points fail ◮ Harmed vertices B+ ⊇ B ◮ Maintain 2-paths for u, v ∈ V \B+

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t = 2 2.33

Reliable spanners

B+ B

◮ Vertex failures, a set B of points fail ◮ Harmed vertices B+ ⊇ B ◮ Maintain 2-paths for u, v ∈ V \B+

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t = 2

Reliable spanners

u v B+ B

◮ Vertex failures, a set B of points fail ◮ Harmed vertices B+ ⊇ B ◮ Maintain 2-paths for u, v ∈ V \B+

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t = 2

Reliable spanners

u v B+ B

◮ 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|

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t = 2

Reliable spanners

u v B+ B

◮ 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|

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t = 2

Definition

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 induced by V \B is a t-spanner of V \B+.

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(n2) edges (trivial)

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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(n2) 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.

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Timeline

Preliminary construction in one dimension

◮ O(n1+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 logc n), where c = O(d)

Buchin, Har-Peled, Ol´ ah 16 Nov 2018

◮ O(n log4 n log log n)

Bose, Carmi, Dujmovic, Morin 24 Dec 2018

◮ O(n log2 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

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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. Spread of V : maxp,q∈V d(p, q) minp,q∈V, p=q d(p, q)

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Optimal construction in 1-D

Structure of blocks: |V | = n = 2ℓ

1

n

◮ Binary tree T with the points of V in the leaves

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Optimal construction in 1-D

Structure of blocks: |V | = n = 2ℓ

1

n i j v

◮ Binary tree T with the points of V in the leaves ◮ Each node v ∈ T corresponds to a block of points {i, . . . , j}

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Optimal construction in 1-D

Structure of blocks: |V | = n = 2ℓ

1

n i j v x y

◮ 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

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Optimal construction in 1-D

Structure of blocks: |V | = n = 2ℓ

1

n i j v x y

◮ 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

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j

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Proof idea

i, j ∈ V \ B+

1

n j i

1

n i j

◮ Path of blocks from i to j ◮ Expanders along the path of blocks

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Proof idea

i, j ∈ V \ B+ j i

1

n

1

n i j

◮ 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+

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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.

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Construction in Rd

◮ V ⊂ Rd ◮ 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

p q d(p, q) ξ · d(p, q) ξ · d(p, q) z

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Construction in Rd

◮ V ⊂ Rd ◮ 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

p q d(p, q) ξ · d(p, q) 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)

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Construction for points with bounded spread

◮ V ⊂ Rd

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure ◮ Compute WSPD

(well-separated pair decomposition)

• ver quadtree

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure ◮ Compute WSPD

(well-separated pair decomposition)

• ver quadtree

◮ Expander between

cells u and v if:

– {u, v} ∈ WSPD – par(u) = par(v)

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure ◮ Compute WSPD

(well-separated pair decomposition)

• ver quadtree

◮ Expander between

cells u and v if:

– {u, v} ∈ WSPD – par(u) = par(v)

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure ◮ Compute WSPD

(well-separated pair decomposition)

• ver quadtree

◮ Expander between

cells u and v if:

– {u, v} ∈ WSPD – par(u) = par(v)

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Construction for points with bounded spread

◮ V ⊂ Rd ◮ Quadtree structure ◮ Compute WSPD

(well-separated pair decomposition)

• ver quadtree

◮ Expander between

cells u and v if:

– {u, v} ∈ WSPD – par(u) = par(v)

◮ Size: O(n log n)

(only if V has bounded spread)

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q ◮ Climbing up from leaf cells to u and v using the expanders

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q ◮ Climbing up from leaf cells to u and v using the expanders

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q ◮ Climbing up from leaf cells to u and v using the expanders

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q ◮ Climbing up from leaf cells to u and v using the expanders

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q ◮ Climbing up from leaf cells to u and v using the expanders

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ q p u v

◮ (u, v) ∈ WSPD separates p and q ◮ Climbing up from leaf cells to u and v using the expanders

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Proof idea

Harmed set B+

◮ Set α = 1 − θ 2 ◮ B+ := {p ∈ V : ∃cell such that p ∈ cell and |B ∩ cell| ≥ α|Vcell|} ◮ We have |B+| ≤ 1 α · |B| ≤ (1 + θ)|B|

(1 + ε)-path between p, q ∈ V \ B+ p′ q′ q p u v

◮ (u, v) ∈ WSPD separates p and q ◮ Climbing up from leaf cells to u and v using the expanders

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Summary

Constructions of reliable spanners One dimension

◮ O(n log n) edges (optimal)

Higher dimensions

◮ O

• n log n (log log n)6

edges

◮ O(n log n) edges for points with bounded spread (optimal)

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Summary

Constructions of reliable spanners One dimension

◮ O(n log n) edges (optimal)

Higher dimensions

◮ O

• n log n (log log n)6

edges

◮ O(n log n) edges for points with bounded spread (optimal)

Thank you!

S t u d e n t P r e s e n t a t i

• n

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Harmed points in 1-D

How do we define B+? B+ := Sα(B)

1

n

j i

◮ [i : j] := {i, i + 1, . . . , j} ◮ Set α ∈ (0, 1) based on θ ◮ Sα(B): α-shadow of B ◮ i ∈ Sα(B) ⇐

⇒ ∃j such that |[i : j] ∩ B| ≥ α|[i : j]|

• r |[j : i] ∩ B| ≥ α|[j : i]|

1

n S 1

2 (B)

Bipartite expanders

X Γ(X) L R

◮ γ ∈ (0, 1) parameter ◮ Two sets with |L| = |R| = m ◮ X ⊆ L with |X| ≥ γ · m ◮ |Γ(X)| > (1 − γ) · m

Expander of size O(m/γ2) with properties (i) ∀X ⊆ L, with |X| ≥ γ · m, we have that |Γ(X)| > (1 − γ) · m (ii) ∀Y ⊆ R, with |Y | ≥ γ · m, we have that |Γ(Y )| > (1 − γ) · m

Spanners in 1-D

V = {1, 2, . . . , n} is one dimensional

1 2 3 n n - 1 1 2 3 n n - 1 1 2 3 n n - 1

1-spanner with size O(n)... BUT not a reliable spanner!

• P. Bose, V. Dujmovi´

c, P. Morin, and M. Smid. Robust geometric

• spanners. SIAM Journal on Computing, 42(4):1720–1736, 2013. doi:

10.1137/120874473. URL https://doi.org/10.1137/120874473.

• T. M. Chan, S. Har-Peled, and M. Jones. On locality-sensitive orderings

and their applications. In 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, pages 21:1–21:17, 2019. URL https://doi.org/10.4230/LIPIcs.ITCS.2019.21.

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