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Slides of a talk at CMU Theory lunch (http://www.cs.cmu.edu/~theorylunch/20111116.html) and Capital Area Theory seminar (http://www.cs.umd.edu/areas/Theory/CATS/#Grigory).
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Advances in Directed Spanners
Grigory Yaroslavtsev Penn State
Joint work with
Berman (PSU), Bhattacharyya (MIT), Grigorescu (GaTech), Makarychev (IBM), Raskhodnikova (PSU),
Woodruff (IBM)
Directed Spanner Problem β’ k-Spanner [Awerbuch β85, Peleg, ShΓ€ffer β89]
Subset of edges, preserving distances up to a factor k > 1 (stretch k).
β’ Graph G V, E with weights π€ βΆ πΈ β β β₯0
H(V, π¬π― β π¬): β(π’, π£) β πΈ πππ π‘π― π’, π£ β€ π β π€(π’, π£)
β’ Problem: Find the sparsest k-spanner of a directed graph.
Directed Spanners and Their Friends
Applications of spanners
β’ First application: simulating synchronized protocols in unsynchronized networks [Peleg,
Ullman β89]
β’ Efficient routing [PUβ89, Cowen β01, Thorup, Zwick β01, Roditty, Thorup, Zwick β02 , Cowen, Wagner β04]
β’ Parallel/Distributed/Streaming approximation algorithms for shortest paths [Cohen β98, Cohen β00, Elkinβ01, Feigenbaum, Kannan, McGregor, Suri, Zhang β08]
β’ Algorithms for approximate distance oracles [Thorup, Zwick β01, Baswana, Sen β06]
Applications of directed spanners
β’ Access control hierarchies
β’ Previous work: [Atallah, Frikken, Blanton, CCCS
β05; De Santis, Ferrara, Masucci, MFCSβ07]
β’ Solution: TC-spanners [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff, SODAβ09]
β’ Steiner TC-spanners for access control: [Berman, Bhattacharyya, Grigorescu, Raskhodnikova, Woodruff, Yβ ICALPβ11]
β’ Property testing and property reconstruction [BGJRWβ09; Raskhodnikova β10 (survey)]
Plan
β’ Approximation algorithms β Undirected vs. Directed
β Framework for directed case = Sampling + LP
β Randomized rounding β’ Directed Spanner
β’ Unit-length 3-spanner
β’ Directed Steiner Forest
β’ Combinatorial bounds on TC-Spanners β Upper bounds for low-dimensional posets
β Lower bounds via linear programming
Undirected vs. Directed
β’ Trivial lower bound: β₯ π β π edges needed
β’ Every undirected graph has a (2t+1)-spanner with β€ π1+1/π‘ edges. [Althofer, Das, Dobkin, Joseph, Soares β93]
β’ Kruskal-like greedy + girth argument
=> π1
π‘ βapproximation
β’ Time/space-efficient constructions of undirected approximate distance oracles [Thorup, Zwick, STOC β01]
Undirected vs Directed
β’ For some directed graphs Ξ© π2 edges needed for a k-spanner:
β’ No space-efficient directed distance oracles: some graphs require Ξ© π2 space. [TZ β01]
Unit-Length Directed k-Spanner β’ O(n)-approximation: trivial (whole graph)
Our π ( π)-approximation
β’ Paths of stretch at most k for all edges =>
β’ Classify edges: thick and thin
β’ Take union of spanners for them
βThick edges: Sampling
βThin edges: LP + randomized rounding
Local Graph
β’ Local graph for an edge (a,b): Induced by vertices on paths of stretch β€ π from a to b
β’ Paths of stretch β€ π only use edges in local graphs
β’ Thick edges: β₯ π vertices in their local graph. Otherwise thin.
Sampling [BGJRWβ09, DK11] β’ Pick O( π π₯π§π) seed vertices at random
β’ Take in- and out- shortest path trees for each
β’ Handles all thick edges (β₯ π vertices in their local graph) w.h.p.
β’ # of edges β€ 2 π β 1 πΆ( π π₯π§π) β€ πππ β Γ π .
Key Idea: Antispanners β’ Antispanner β subset of edges, whose
removal destroys all paths from a to b of stretch at most k
β’ Graph is spanner <=> hits all antispanners β’ Enough to hit all minimal antispanners for all thin
edges β’ If πΈπ» is not a spanner for an edge (a,b) => πΈ β πΈπ» is
an antispanner, can be minimized greedily
Linear Program (~dual to [DKβ11]) Hitting-set LP: π₯ππβπΈ β πππ
π₯ππβπ
β₯ 1
for all minimal antispanners A for all thin edges.
β’ # of minimal antispanners may be exponential in π => Ellipsoid + Separation oracle
β’ We will show: β€ ππ= π
1
2π ln πminimal
antispanners for a fixed thin edge β’ Assume that we guessed OPT = the size of
the sparsest k-spanner (at most π2 values)
Oracle Hitting-set LP: π₯ππβπΈ β€ πππ
π₯ππβπ
β₯ 1
for all minimal antispanners A for all thin edges.
β’ We use a randomized oracle => in both cases oracle fails with exponentially small probability.
β’ Rounding: Take e w.p. ππ = min π π₯π§π β π₯π , 1
β’ SMALL SPANNER: We have a set of edges of size β€ π₯ππ β π π β€ πππ β π π w.h.p.
β’ Pr[LARGE SPANNER] β€ eβ Ξ©( π) by Chernoff.
β’ Pr[CONSTRAINT NOT VIOLATED] β€ eβ Ξ©( π) (next slide)
Randomized Oracle = Rounding
Pr[CONSTRAINT NOT VIOLATED]
β’ Set π: βπ β πΈ we have Pr [π β π] = min π ln π π₯π , 1
β’ For a fixed minimal antispanner A, such that π₯π β₯ 1πβπ΄ :
Pr π β©A = β β€ 1 β π ln π π₯ππβA
β€ πβ π ln π π₯ππβπ΄ β€ πβ π ππ π
β’ #minimal antispanners for a fixed edge (π, π) β€
#different shortest path trees with root s in a local graph
β€ ππ= π
1
2π ln π (for a thin edge)
β’ #minimal antispanners β€ |πΈ|ππ
ππ π₯π§ π => union bound:
Pr[CONSTRAINT NOT VIOLATED] β€ |πΈ|πβπ
ππ π₯π§ π
Unit-length 3-spanner
β’ π (π1/3)-approximation algorithm β Sampling π (π1/3) times
β Dual LP + Different randomized rounding (simplified version of [DKβ11])
β’ Rounding scheme (vertex-based): β For each vertex π’ β π: sample ππ’ β 0,1
β Take all edges π’, π£ if
min ππ’, ππ£ β€ π (π1/3)π₯(π’,π£)
β Feasible solution => 3-spanner w.h.p. (see paper)
Approximation wrap-up β’ Sampling + LP with randomized rounding
β’ Improvement for Directed Steiner Forest:
βCheapest set of edges, connecting pairs π π , π‘π
β Previous: Sampling + similar LP [Feldman,
Kortsarz, Nutov, SODA β09] . Deterministic
rounding gives π π4/5+π -approximation
βWe give π π2/3+π -approximation via randomized rounding
Approximation wrap-up
β’ Γ( π)-approximation for Directed Spanner
β’ Small local graphs => better approximation
β’ Can we do better for general graphs?
β’ Hardness: only excludes polylog(n)-approximation
β’ Integrality gap: π΄(ππ/πβπ) [DKβ11]
β’ Can we do better for specific graphs
β’ Planar graphs (still NP-hard)?
H is a k-TC-spanner of G if H is a k-spanner of TC(G)
Transitive-Closure Spanners
Transitive closure TC(G) has an edge from u to v iff
G has a path from u to v
H is a k-TC-spanner of G if H is a subgraph of TC(G) for which distanceH(u,v) β€ k iff G has a path
from u to v
G TC(G)
2-TC-spanner of G
Shortcut edge consistent with
ordering [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff, SODAβ09], generalizing [Yao 82; Chazelle 87; Alon, Schieber 87, β¦]
Applications of TC-Spanners
Data structures for storing partial products [Yao,
β82; Chazelle β87, Alon, Schieber, 88]
Constructions of unbounded fan-in circuits [Chandra, Fortune, Lipton ICALP, STOCβ83]
Property testers for monotonicity and Lipschitzness [Dodis et al. β99,BGJRWβ09; Jha, Raskhodnikova, FOCS β11]
Lower bounds for reconstructors for monotonicity and Lipshitzness [BGJJRWβ10, JRβ11]
Efficient key management in access hierarchies
Follow references in [Raskhodnikova β10 (survey)]
Bounds for Steiner TC-Spanners
β’ No non-trivial upper bound for arbitrary graphs
FACT: For a random directed bipartite graph of density Β½, any Steiner 2-TC-spanner requires Ξ© n2 edges.
Low-Dimensional Posets
β’ [ABFF 09] access hierarchies are low-dimensional posets
β’ Poset β‘ DAG
β’ Poset G has dimension d if G can be embedded into a hypergrid of dimension d and d is minimum.
π: πΊ β πΊβ² is a poset embedding if for all π₯, π¦ β πΊ, π₯ βΌπΊ π¦ iff π π₯ βΌπΊβ² π(π¦). Hypergrid π π has ordering π₯1, β¦ , π₯π βΌ (π¦1, β¦ , π¦π) iff π₯π β€ π¦π
for all i
Main Results
Stretch k Upper Bound Lower Bound
2 π π logπ π Ξ© π
πΌlogπ
π
π
where πΌ is a constant
β₯ 3 π(π logπβ1 π log log π) for constant d
[DFM 07]
Ξ©(π log (πβ1)/π π) for constant d
π = π. Nothing better known for
larger k.
d is the poset dimension
2-TC-Spanner for π π β’ d = 1 (so, n = m)
2-TC-spanner with β€ π logπ =π log π edges
β¦ β¦
β’ d > 1 (so, n = md)
2-TC-spanner with β€ (m logm)d= nlog n
d
d edges
by taking d-wise Cartesian product of 2-TC-spanners for a line.
We show this is
tight upto πΆπ for a
constant πΆ.
Lower Bound Strategy
β’ Write in IP for a minimal 2-TC-spanner.
β’ OPT β₯ πΏπ = πΏπππ’ππ
β’ It is crucial that the integrality gap of the primal is small.
β’ Idea: Construct some feasible solution for the dual => lower bound on OPT.
β’ OPT β₯ πΏπ = πΏπππ’ππ β₯ πΏπ΅
IP Formulation
β’ {0,1}-program for Minimal 2-TC-spanner:
minimize
subject to: π₯π’π€ β₯ ππ’π€π£ βπ’ βΌ π€ βΌ π£ π₯π€π£ β₯ ππ’π€π£ βπ’ βΌ π€ βΌ π£ βπ’ βΌ π£
π₯π’π£π’,π£:π’βΌπ£
ππ’π€π£ β₯ 1
π€:π’βΌπ€βΌπ£
Dual LP
β’ Take fractional relaxation of IP and look at its dual:
maximize
subject to: βπ’ βΌ π£ π¦π’π£ β€ ππ’π€π£ + ππ’π€π£ βπ’ βΌ π€ βΌ π£ 0 β€ π¦π’π£ , ππ’π€π£ , ππ’π€π£ β€ 1 βπ’ βΌ π€ βΌ π£
π¦π’π£π’,π£:π’βΌπ£
ππ’π£π€ + ππ€π’π£ β€ 1
π€:π€βΌπ’π€:π£βΌπ€
Constructing solution to dual LP
β’ Now we use fact that poset is a hypergrid! For π’ βΌ π£, set
π¦π’π£ = 1
π(π£βπ’), where π(π£ β π’) is volume of box with
corners u and v.
maximize subject to: βπ’ βΌ π£ π¦π’π£ = ππ’π€π£ + ππ’π€π£ βπ’ βΌ π€ βΌ π£
π¦π’π£π’,π£:π’βΌπ£
ππ’π£π€ + ππ€π’π£ β€ 1
π€:π€βΌπ’π€:π£βΌπ€
> ππ₯π§π π
Set ππππ = ππππ½(πβπ)
π½ πβπ +π½(πβπ), ππππ = πππ
π½(πβπ)
π½ πβπ +π½(πβπ)
β€ ππ π
Constructing solution to dual LP
β’ Now we use fact that poset is a hypergrid! For
π’ βΌ π£, set π¦π’π£ = 1
4π ππ(π£βπ’), where π(π£ β π’) is
volume of box with corners u and v.
maximize subject to: βπ’ βΌ π£ π¦π’π£ = ππ’π€π£ + ππ’π€π£ βπ’ βΌ π€ βΌ π£
π¦π’π£π’,π£:π’βΌπ£
ππ’π£π€ + ππ€π’π£ β€ 1
π€:π€βΌπ’π€:π£βΌπ€
> ππ₯π§π π / ππ π
Set ππππ = ππππ½(πβπ)
π½ πβπ +π½(πβπ), ππππ = πππ
π½(πβπ)
π½ πβπ +π½(πβπ)
β€ π
Wrap-up β’ Upper bound for Steiner 2-TC-spanner
β’ Lower bound for 2-TC-spanner for a hypergrid.
β Technique: find a feasible solution for the dual LP
β’ Lower bound of Ξ©(π log (πβ1)/π π) for π β₯ 3. β Combinatorial
β Holds for randomly generated posets, not explicit.
β’ OPEN PROBLEM:
β Can the LP technique give a better lower bound for π β₯ 3?