SlideShare a Scribd company logo
1 of 32
Download to read offline
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
• k-Spanner [Awerbuch ‘85, Peleg, Shäffer ‘89]
Subset of edges, preserving distances up to a factor k > 1
(stretch k).
• Graph G V, E with weights ������ ∶ ������ → ℝ ≥0
     H(V, ������������ ⊆ ������): ∀(������, ������) ∈ ������ ������������������������������ ������, ������ ≤ ������ ⋅ ������(������, ������)




• Problem: Find the sparsest k-spanner of a directed
  graph.
Directed Spanners and Their Friends
Applications of spanners
• First application: simulating synchronized
  protocols in unsynchronized networks [Peleg,
  Ullman ’89]
• Efficient routing [PU’89, Cowen ’01, Thorup, Zwick ’01,
  Roditty, Thorup, Zwick ’02 , Cowen, Wagner ’04]
• Parallel/Distributed/Streaming approximation
  algorithms for shortest paths [Cohen ’98, Cohen ’00,
  Elkin’01, Feigenbaum, Kannan, McGregor, Suri, Zhang ’08]
• Algorithms for approximate distance oracles
  [Thorup, Zwick ’01, Baswana, Sen ’06]
Applications of directed spanners
• Access control hierarchies
  • Previous work: [Atallah, Frikken, Blanton, CCCS
    ‘05; De Santis, Ferrara, Masucci, MFCS’07]
  • Solution: TC-spanners [Bhattacharyya, Grigorescu,
    Jung, Raskhodnikova, Woodruff, SODA’09]
  • Steiner TC-spanners for access control:
    [Berman, Bhattacharyya, Grigorescu, Raskhodnikova,
    Woodruff, Y’ ICALP’11]
• Property testing and property reconstruction
 [BGJRW’09; Raskhodnikova ’10 (survey)]
Plan
• Approximation algorithms
  – Undirected vs. Directed
  – Framework for directed case = Sampling + LP
  – Randomized rounding
    • Directed Spanner
    • Unit-length 3-spanner
    • Directed Steiner Forest
• Combinatorial bounds on TC-Spanners
  – Upper bounds for low-dimensional posets
  – Lower bounds via linear programming
Undirected vs. Directed
• Trivial lower bound: ≥ ������ − ������ edges needed
• Every undirected graph has a (2t+1)-spanner
  with ≤ ������1+1/������ edges. [Althofer, Das, Dobkin,
  Joseph, Soares ‘93]
• Kruskal-like greedy + girth argument
       1
  => ������ −approximation
       ������

• Time/space-efficient constructions of
  undirected approximate distance oracles
  [Thorup, Zwick, STOC ‘01]
Undirected vs Directed
• For some directed graphs Ω ������2 edges
  needed for a k-spanner:




• No space-efficient directed distance oracles:
  some graphs require Ω ������2 space. [TZ ‘01]
Unit-Length Directed k-Spanner
• O(n)-approximation: trivial (whole graph)
Our ������( ������)-approximation
• Paths of stretch at most k for all edges =>
• Classify edges: thick and thin
• Take union of spanners for them
   – Thick edges: Sampling
   – Thin edges: LP + randomized rounding
Local Graph
• Local graph for an edge (a,b): Induced by
  vertices on paths of stretch ≤ ������ from a to b




• Paths of stretch ≤ ������ only use edges in local
  graphs
• Thick edges: ≥ ������ vertices in their local graph.
  Otherwise thin.
Sampling [BGJRW’09, DK11]
• Pick O( ������ ������������ ������) seed vertices at random
• Take in- and out- shortest path trees for each




• Handles all thick edges (≥ ������ vertices in their
  local graph) w.h.p.
• # of edges ≤ 2 ������ − 1 ������( ������ ������������ ������) ≤ ������������������ ⋅ Õ ������ .
Key Idea: Antispanners
• Antispanner – subset of edges, whose
  removal destroys all paths from a to b of
  stretch at most k




• Graph is spanner <=> hits all antispanners
• Enough to hit all minimal antispanners for all thin
  edges
• If ������������ is not a spanner for an edge (a,b) => ������ ∖ ������������ is
  an antispanner, can be minimized greedily
Linear Program (~dual to [DK’11])
 Hitting-set LP:   ������∈������   ������������ → ������������������
                                   ������������ ≥ 1
                           ������∈������
 for all minimal antispanners A for all thin edges.

 • # of minimal antispanners may be
   exponential in ������ => Ellipsoid + Separation
   oracle
                                    ������        1
                                                  ������ ln ������
 • We will show: ≤ ������ = ������           minimal  2

   antispanners for a fixed thin edge
 • Assume that we guessed OPT = the size of
   the sparsest k-spanner (at most ������2 values)
Oracle
Hitting-set LP:   ������∈������   ������������ ≤ ������������������
                                  ������������ ≥ 1
                          ������∈������
for all minimal antispanners A for all thin edges.




• We use a randomized oracle => in both cases
  oracle fails with exponentially small probability.
Randomized Oracle = Rounding
• Rounding: Take e w.p. ������������ = min    ������ ������������ ������ ⋅ ������������ , 1




• SMALL SPANNER: We have a set of edges of
  size ≤ ������ ������������ ⋅ ������ ������ ≤ ������������������ ⋅ ������ ������ w.h.p.
• Pr[LARGE SPANNER] ≤ e− Ω(          ������)
                                           by Chernoff.
• Pr[CONSTRAINT NOT VIOLATED] ≤ e− Ω(                   ������)
  (next slide)
Pr[CONSTRAINT NOT VIOLATED]
• Set ������: ∀������ ∈ ������ we have Pr[������ ∈ ������] = min ������ ln ������ ������������ , 1
• For a fixed minimal antispanner A, such that
   ������∈������ ������������ ≥ 1:
Pr ������ ∩ A = ∅ ≤          1 − ������ ln ������ ������������ ≤ ������ −               ������ ln ������ ������∈������ ������������   ≤ ������−       ������ ������������ ������

                  ������∈A
• #minimal antispanners for a fixed edge (������, ������) ≤
#different shortest path trees with root s in a local graph
                         ������          1
                                         ������ ln ������
              ≤ ������            = ������   2              (for a thin edge)
                                                     ������
                                                          ������ ������������ ������
• #minimal antispanners ≤ |������|������                     ������                => union bound:
                                                                                      ������
                                                                                 −         ������ ������������ ������
Pr[CONSTRAINT NOT VIOLATED] ≤ |������|������                                                  ������
Unit-length 3-spanner
• ������(������1/3 )-approximation algorithm
  – Sampling ������(������1/3 ) times
  – Dual LP + Different randomized rounding
    (simplified version of [DK’11])
• Rounding scheme (vertex-based):
  – For each vertex ������ ∈ ������: sample ������������ ∈ 0,1
  – Take all edges ������, ������ if
            min ������������ , ������������ ≤ ������(������1/3 )������(������,������)
  – Feasible solution => 3-spanner w.h.p. (see paper)
Approximation wrap-up
• Sampling + LP with randomized rounding
• Improvement for Directed Steiner Forest:
  – Cheapest set of edges, connecting pairs ������������ , ������������
  – Previous: Sampling + similar LP [Feldman,
    Kortsarz, Nutov, SODA ‘09] . Deterministic
    rounding gives ������ ������4/5+������ -approximation
  – We give ������ ������2/3+������ -approximation via
    randomized rounding
Approximation wrap-up
• Õ( ������)-approximation for Directed Spanner
• Small local graphs => better approximation
• Can we do better for general graphs?
  • Hardness: only excludes polylog(n)-
    approximation
  • Integrality gap: ������(������������/������−������ ) [DK’11]
• Can we do better for specific graphs
  • Planar graphs (still NP-hard)?
Transitive-Closure Spanners
Transitive closure TC(G) has an edge from u to v iff
              G has a path from u to v


                    G                            TC(G)
  H is a k-TC-spanner of G if H is a subgraph of
  TC(G) a k-TC-spanner ofHG if H is iffk-spanner of
   H is for which distance (u,v) ≤ k a G has a path
                        TC(G)
                    from u to v
   Shortcut edge
  consistent with
                            2-TC-spanner of G
     ordering
 [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff, SODA‘09],
 generalizing [Yao 82; Chazelle 87; Alon, Schieber 87, …]
Applications of TC-Spanners
   Data structures for storing partial products [Yao,
    ’82; Chazelle ’87, Alon, Schieber, 88]
   Constructions of unbounded fan-in circuits
    [Chandra, Fortune, Lipton ICALP, STOC‘83]
   Property testers for monotonicity and
    Lipschitzness [Dodis et al. ‘99,BGJRW’09; Jha,
    Raskhodnikova, FOCS ‘11]
 Lower bounds for reconstructors for
  monotonicity and Lipshitzness [BGJJRW’10, JR’11]
 Efficient key management in access hierarchies

              Follow references in [Raskhodnikova ’10 (survey)]
Bounds for Steiner TC-Spanners
• No non-trivial upper bound for arbitrary
  graphs

                 FACT: For a random directed
                 bipartite graph of density ½,
                 any Steiner 2-TC-spanner
                 requires Ω n2 edges.
Low-Dimensional Posets
• [ABFF 09] access hierarchies are low-
  dimensional posets
• Poset ≡ DAG
• Poset G has dimension d if G can be
  embedded into a hypergrid of dimension d
  and d is minimum.           ′
                      ������: ������ → ������ is a poset embedding if for
                      all ������, ������ ∈ ������, ������ ≼������ ������ iff ������ ������ ≼������′ ������(������).

                      Hypergrid ������ ������ has ordering
                       ������1 , … , ������������ ≼ (������1 , … , ������������ ) iff ������������ ≤ ������������
                      for all i
Main Results

Stretch k            Upper Bound                    Lower Bound
                                                                  ������
2                     ������ ������ log ������ ������                    ������log ������
                                                 ٠������
                                                            ������
                                               where ������ is a constant
≥3             ������(������ log ������−1 ������ log log ������)     Ω(������ log (������−1)/������ ������)
                      for constant d               for constant d
                        [DFM 07]


d is the poset dimension                           ������ = ������. Nothing
                                                  better known for
                                                       larger k.
2-TC-Spanner for ������                            ������

• d = 1 (so, n = m)
2-TC-spanner with ≤ ������ log ������ = ������ log ������ edges

                        …       …

                              We show this is
                            tight upto ������������ for a
• d > 1 (so, n = md )           constant ������.

                                                    log n d
2-TC-spanner with ≤ (m log m)d = n         edges
                                                      d
by taking d-wise Cartesian product of 2-TC-
spanners for a line.
Lower Bound Strategy
• Write in IP for a minimal 2-TC-spanner.
• OPT ≥ ������������ = ������������������������������������
• It is crucial that the integrality gap of the primal
  is small.
• Idea: Construct some feasible solution for the
  dual => lower bound on OPT.
• OPT ≥ ������������ = ������������������������������������ ≥ ������������
IP Formulation
• {0,1}-program for Minimal 2-TC-spanner:

   minimize
                                       ������������������
   subject to:           ������,������:������≼������


                      ������������������ ≥ ������������������������         ∀������ ≼ ������ ≼ ������
                      ������������������ ≥ ������������������������         ∀������ ≼ ������ ≼ ������
                                 ������������������������ ≥ 1   ∀������ ≼ ������
                   ������:������≼������≼������
Dual LP
• Take fractional relaxation of IP and look at its dual:


   maximize                                  ������������������
                               ������,������:������≼������


   subject to:              ������������������������ +              ������������������������ ≤ 1 ∀������ ≼ ������
                 ������:������≼������                ������:������≼������


                 ������������������ ≤ ������������������������ + ������������������������                   ∀������ ≼ ������ ≼ ������
              0 ≤ ������������������ , ������������������������ , ������������������������ ≤ 1              ∀������ ≼ ������ ≼ ������
Constructing solution to dual LP
• Now we use fact that poset is a hypergrid! For ������ ≼ ������, set
               1
  ������������������ =          , where ������(������ − ������) is volume of box with
           ������(������−������)
  corners u and v.

   maximize                                     ������������������          > ������ ������������ ������       ������

                                  ������,������:������≼������
   subject to:
                              ������������������������ +                 ������������������������ ≤ 1 ∀������ ≼ ������
                   ������:������≼������                ������:������≼������                                     ������
                                                                           ≤ ������������
                   ������������������ = ������������������������ + ������������������������                         ∀������ ≼ ������ ≼ ������
                              ������(������−������)                                    ������(������−������)
 Set ������������������������ = ������������������                      ,     ������������������������ = ������������������
                         ������ ������−������ +������(������−������)                          ������ ������−������ +������(������−������)
Constructing solution to dual LP
• Now we use fact that poset is a hypergrid! For
                               1
  ������ ≼ ������, set ������������������ =                 , where ������(������ − ������) is
                        4������ ������ ������(������−������)
  volume of box with corners u and v.

  maximize                                      ������������������         > ������ ������������ ������ ������ / ������������       ������

                                  ������,������:������≼������
  subject to:
                              ������������������������ +                 ������������������������ ≤ 1 ∀������ ≼ ������
                   ������:������≼������                ������:������≼������
                                                                              ≤ ������
                 ������������������ = ������������������������ + ������������������������                         ∀������ ≼ ������ ≼ ������
                              ������(������−������)                                    ������(������−������)
 Set ������������������������ = ������������������                      ,     ������������������������ = ������������������
                         ������ ������−������ +������(������−������)                          ������ ������−������ +������(������−������)
Wrap-up
• Upper bound for Steiner 2-TC-spanner
• Lower bound for 2-TC-spanner for a hypergrid.
  – Technique: find a feasible solution for the dual LP
                              (������−1)/������
• Lower bound of Ω(������ log                 ������) for ������ ≥ 3.
  – Combinatorial
  – Holds for randomly generated posets, not explicit.
• OPEN PROBLEM:
  – Can the LP technique give a better lower bound
    for ������ ≥ 3?

More Related Content

What's hot

Isi and nyquist criterion
Isi and nyquist criterionIsi and nyquist criterion
Isi and nyquist criterionsrkrishna341
 
Detection of unknown signal
Detection of unknown signalDetection of unknown signal
Detection of unknown signalsumitf1
 
Dsp 2018 foehu - lec 10 - multi-rate digital signal processing
Dsp 2018 foehu - lec 10 - multi-rate digital signal processingDsp 2018 foehu - lec 10 - multi-rate digital signal processing
Dsp 2018 foehu - lec 10 - multi-rate digital signal processingAmr E. Mohamed
 
2.6 all pairsshortestpath
2.6 all pairsshortestpath2.6 all pairsshortestpath
2.6 all pairsshortestpathKrish_ver2
 
DSP_FOEHU - Lec 08 - The Discrete Fourier Transform
DSP_FOEHU - Lec 08 - The Discrete Fourier TransformDSP_FOEHU - Lec 08 - The Discrete Fourier Transform
DSP_FOEHU - Lec 08 - The Discrete Fourier TransformAmr E. Mohamed
 
Dsp U Lec10 DFT And FFT
Dsp U   Lec10  DFT And  FFTDsp U   Lec10  DFT And  FFT
Dsp U Lec10 DFT And FFTtaha25
 
DSP_FOEHU - Lec 03 - Sampling of Continuous Time Signals
DSP_FOEHU - Lec 03 - Sampling of Continuous Time SignalsDSP_FOEHU - Lec 03 - Sampling of Continuous Time Signals
DSP_FOEHU - Lec 03 - Sampling of Continuous Time SignalsAmr E. Mohamed
 
Dsp U Lec08 Fir Filter Design
Dsp U   Lec08 Fir Filter DesignDsp U   Lec08 Fir Filter Design
Dsp U Lec08 Fir Filter Designtaha25
 
Discrete Fourier Transform
Discrete Fourier TransformDiscrete Fourier Transform
Discrete Fourier TransformAbhishek Choksi
 
Chapter5 - The Discrete-Time Fourier Transform
Chapter5 - The Discrete-Time Fourier TransformChapter5 - The Discrete-Time Fourier Transform
Chapter5 - The Discrete-Time Fourier TransformAttaporn Ninsuwan
 
Chapter 9 computation of the dft
Chapter 9 computation of the dftChapter 9 computation of the dft
Chapter 9 computation of the dftmikeproud
 
DSP_FOEHU - Lec 07 - Digital Filters
DSP_FOEHU - Lec 07 - Digital FiltersDSP_FOEHU - Lec 07 - Digital Filters
DSP_FOEHU - Lec 07 - Digital FiltersAmr E. Mohamed
 
DSP_2018_FOEHU - Lec 07 - IIR Filter Design
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignDSP_2018_FOEHU - Lec 07 - IIR Filter Design
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignAmr E. Mohamed
 
DTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampattern
DTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampatternDTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampattern
DTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampatternChibuzo Nnonyelu, PhD
 
Fft presentation
Fft presentationFft presentation
Fft presentationilker Şin
 
DSP_FOEHU - Lec 09 - Fast Fourier Transform
DSP_FOEHU - Lec 09 - Fast Fourier TransformDSP_FOEHU - Lec 09 - Fast Fourier Transform
DSP_FOEHU - Lec 09 - Fast Fourier TransformAmr E. Mohamed
 

What's hot (20)

Isi and nyquist criterion
Isi and nyquist criterionIsi and nyquist criterion
Isi and nyquist criterion
 
Properties of dft
Properties of dftProperties of dft
Properties of dft
 
Detection of unknown signal
Detection of unknown signalDetection of unknown signal
Detection of unknown signal
 
2006 devmodel
2006 devmodel2006 devmodel
2006 devmodel
 
Dsp 2018 foehu - lec 10 - multi-rate digital signal processing
Dsp 2018 foehu - lec 10 - multi-rate digital signal processingDsp 2018 foehu - lec 10 - multi-rate digital signal processing
Dsp 2018 foehu - lec 10 - multi-rate digital signal processing
 
2.6 all pairsshortestpath
2.6 all pairsshortestpath2.6 all pairsshortestpath
2.6 all pairsshortestpath
 
DSP_FOEHU - Lec 08 - The Discrete Fourier Transform
DSP_FOEHU - Lec 08 - The Discrete Fourier TransformDSP_FOEHU - Lec 08 - The Discrete Fourier Transform
DSP_FOEHU - Lec 08 - The Discrete Fourier Transform
 
8 lti psd
8 lti psd8 lti psd
8 lti psd
 
Dsp U Lec10 DFT And FFT
Dsp U   Lec10  DFT And  FFTDsp U   Lec10  DFT And  FFT
Dsp U Lec10 DFT And FFT
 
DSP_FOEHU - Lec 03 - Sampling of Continuous Time Signals
DSP_FOEHU - Lec 03 - Sampling of Continuous Time SignalsDSP_FOEHU - Lec 03 - Sampling of Continuous Time Signals
DSP_FOEHU - Lec 03 - Sampling of Continuous Time Signals
 
Dsp U Lec08 Fir Filter Design
Dsp U   Lec08 Fir Filter DesignDsp U   Lec08 Fir Filter Design
Dsp U Lec08 Fir Filter Design
 
Discrete Fourier Transform
Discrete Fourier TransformDiscrete Fourier Transform
Discrete Fourier Transform
 
Sampling
SamplingSampling
Sampling
 
Chapter5 - The Discrete-Time Fourier Transform
Chapter5 - The Discrete-Time Fourier TransformChapter5 - The Discrete-Time Fourier Transform
Chapter5 - The Discrete-Time Fourier Transform
 
Chapter 9 computation of the dft
Chapter 9 computation of the dftChapter 9 computation of the dft
Chapter 9 computation of the dft
 
DSP_FOEHU - Lec 07 - Digital Filters
DSP_FOEHU - Lec 07 - Digital FiltersDSP_FOEHU - Lec 07 - Digital Filters
DSP_FOEHU - Lec 07 - Digital Filters
 
DSP_2018_FOEHU - Lec 07 - IIR Filter Design
DSP_2018_FOEHU - Lec 07 - IIR Filter DesignDSP_2018_FOEHU - Lec 07 - IIR Filter Design
DSP_2018_FOEHU - Lec 07 - IIR Filter Design
 
DTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampattern
DTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampatternDTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampattern
DTFT and ULA: mathematical similarities of DTFT spectrum and ULA beampattern
 
Fft presentation
Fft presentationFft presentation
Fft presentation
 
DSP_FOEHU - Lec 09 - Fast Fourier Transform
DSP_FOEHU - Lec 09 - Fast Fourier TransformDSP_FOEHU - Lec 09 - Fast Fourier Transform
DSP_FOEHU - Lec 09 - Fast Fourier Transform
 

Viewers also liked

Business Proposal
Business ProposalBusiness Proposal
Business Proposallnmathews
 
Liberal Arts Internbridge2b
Liberal Arts Internbridge2bLiberal Arts Internbridge2b
Liberal Arts Internbridge2blnmathews
 
l1-Embeddings and Algorithmic Applications
l1-Embeddings and Algorithmic Applicationsl1-Embeddings and Algorithmic Applications
l1-Embeddings and Algorithmic ApplicationsGrigory Yaroslavtsev
 
Tips for Excellent Presentation
 Tips for Excellent Presentation Tips for Excellent Presentation
Tips for Excellent Presentationgroup6csw
 
Time Management Msu Fall 2009
Time Management Msu Fall 2009Time Management Msu Fall 2009
Time Management Msu Fall 2009lnmathews
 
Karangan SPM- Rasuah
Karangan SPM- RasuahKarangan SPM- Rasuah
Karangan SPM- RasuahM D
 

Viewers also liked (8)

Business Proposal
Business ProposalBusiness Proposal
Business Proposal
 
Liberal Arts Internbridge2b
Liberal Arts Internbridge2bLiberal Arts Internbridge2b
Liberal Arts Internbridge2b
 
l1-Embeddings and Algorithmic Applications
l1-Embeddings and Algorithmic Applicationsl1-Embeddings and Algorithmic Applications
l1-Embeddings and Algorithmic Applications
 
Tips for Excellent Presentation
 Tips for Excellent Presentation Tips for Excellent Presentation
Tips for Excellent Presentation
 
Metric Embeddings and Expanders
Metric Embeddings and ExpandersMetric Embeddings and Expanders
Metric Embeddings and Expanders
 
Time Management Msu Fall 2009
Time Management Msu Fall 2009Time Management Msu Fall 2009
Time Management Msu Fall 2009
 
5.democracy
5.democracy5.democracy
5.democracy
 
Karangan SPM- Rasuah
Karangan SPM- RasuahKarangan SPM- Rasuah
Karangan SPM- Rasuah
 

Similar to Advances in Directed Spanners

Chap10 slides
Chap10 slidesChap10 slides
Chap10 slidesHJ DS
 
Randomness conductors
Randomness conductorsRandomness conductors
Randomness conductorswtyru1989
 
Average Sensitivity of Graph Algorithms
Average Sensitivity of Graph AlgorithmsAverage Sensitivity of Graph Algorithms
Average Sensitivity of Graph AlgorithmsYuichi Yoshida
 
Waveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptxWaveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptxKIRUTHIKAAR2
 
presentation
presentationpresentation
presentationjie ren
 
Divide and conquer surfing lower bounds
Divide and conquer  surfing lower boundsDivide and conquer  surfing lower bounds
Divide and conquer surfing lower boundsRajendran
 
Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...
Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...
Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...Taiji Suzuki
 
Waveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptxWaveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptxKIRUTHIKAAR2
 
7 convolutional codes
7 convolutional codes7 convolutional codes
7 convolutional codesVarun Raj
 
Digital communication
Digital communicationDigital communication
Digital communicationmeashi
 
Decomposition and Denoising for moment sequences using convex optimization
Decomposition and Denoising for moment sequences using convex optimizationDecomposition and Denoising for moment sequences using convex optimization
Decomposition and Denoising for moment sequences using convex optimizationBadri Narayan Bhaskar
 
Continuous and Discrete-Time Analysis of SGD
Continuous and Discrete-Time Analysis of SGDContinuous and Discrete-Time Analysis of SGD
Continuous and Discrete-Time Analysis of SGDValentin De Bortoli
 
Modèle de coordination du groupe de robots mobiles
Modèle de coordination du groupe de robots mobilesModèle de coordination du groupe de robots mobiles
Modèle de coordination du groupe de robots mobilesAkrem Hadji
 
Significant scales in community structure
Significant scales in community structureSignificant scales in community structure
Significant scales in community structureVincent Traag
 
Cheatsheet recurrent-neural-networks
Cheatsheet recurrent-neural-networksCheatsheet recurrent-neural-networks
Cheatsheet recurrent-neural-networksSteve Nouri
 
PowerPoint Presentation
PowerPoint PresentationPowerPoint Presentation
PowerPoint Presentationbutest
 

Similar to Advances in Directed Spanners (20)

Chap10 slides
Chap10 slidesChap10 slides
Chap10 slides
 
Randomness conductors
Randomness conductorsRandomness conductors
Randomness conductors
 
Average Sensitivity of Graph Algorithms
Average Sensitivity of Graph AlgorithmsAverage Sensitivity of Graph Algorithms
Average Sensitivity of Graph Algorithms
 
1535 graph algorithms
1535 graph algorithms1535 graph algorithms
1535 graph algorithms
 
Chap10 slides
Chap10 slidesChap10 slides
Chap10 slides
 
Optimisation random graph presentation
Optimisation random graph presentationOptimisation random graph presentation
Optimisation random graph presentation
 
Waveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptxWaveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptx
 
presentation
presentationpresentation
presentation
 
7 227 2005
7 227 20057 227 2005
7 227 2005
 
Divide and conquer surfing lower bounds
Divide and conquer  surfing lower boundsDivide and conquer  surfing lower bounds
Divide and conquer surfing lower bounds
 
Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...
Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...
Minimax optimal alternating minimization \\ for kernel nonparametric tensor l...
 
Waveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptxWaveform_codingUNIT-II_DC_-PPT.pptx
Waveform_codingUNIT-II_DC_-PPT.pptx
 
7 convolutional codes
7 convolutional codes7 convolutional codes
7 convolutional codes
 
Digital communication
Digital communicationDigital communication
Digital communication
 
Decomposition and Denoising for moment sequences using convex optimization
Decomposition and Denoising for moment sequences using convex optimizationDecomposition and Denoising for moment sequences using convex optimization
Decomposition and Denoising for moment sequences using convex optimization
 
Continuous and Discrete-Time Analysis of SGD
Continuous and Discrete-Time Analysis of SGDContinuous and Discrete-Time Analysis of SGD
Continuous and Discrete-Time Analysis of SGD
 
Modèle de coordination du groupe de robots mobiles
Modèle de coordination du groupe de robots mobilesModèle de coordination du groupe de robots mobiles
Modèle de coordination du groupe de robots mobiles
 
Significant scales in community structure
Significant scales in community structureSignificant scales in community structure
Significant scales in community structure
 
Cheatsheet recurrent-neural-networks
Cheatsheet recurrent-neural-networksCheatsheet recurrent-neural-networks
Cheatsheet recurrent-neural-networks
 
PowerPoint Presentation
PowerPoint PresentationPowerPoint Presentation
PowerPoint Presentation
 

Recently uploaded

Build your next Gen AI Breakthrough - April 2024
Build your next Gen AI Breakthrough - April 2024Build your next Gen AI Breakthrough - April 2024
Build your next Gen AI Breakthrough - April 2024Neo4j
 
Key Features Of Token Development (1).pptx
Key  Features Of Token  Development (1).pptxKey  Features Of Token  Development (1).pptx
Key Features Of Token Development (1).pptxLBM Solutions
 
Gen AI in Business - Global Trends Report 2024.pdf
Gen AI in Business - Global Trends Report 2024.pdfGen AI in Business - Global Trends Report 2024.pdf
Gen AI in Business - Global Trends Report 2024.pdfAddepto
 
Understanding the Laravel MVC Architecture
Understanding the Laravel MVC ArchitectureUnderstanding the Laravel MVC Architecture
Understanding the Laravel MVC ArchitecturePixlogix Infotech
 
My INSURER PTE LTD - Insurtech Innovation Award 2024
My INSURER PTE LTD - Insurtech Innovation Award 2024My INSURER PTE LTD - Insurtech Innovation Award 2024
My INSURER PTE LTD - Insurtech Innovation Award 2024The Digital Insurer
 
Scanning the Internet for External Cloud Exposures via SSL Certs
Scanning the Internet for External Cloud Exposures via SSL CertsScanning the Internet for External Cloud Exposures via SSL Certs
Scanning the Internet for External Cloud Exposures via SSL CertsRizwan Syed
 
Kotlin Multiplatform & Compose Multiplatform - Starter kit for pragmatics
Kotlin Multiplatform & Compose Multiplatform - Starter kit for pragmaticsKotlin Multiplatform & Compose Multiplatform - Starter kit for pragmatics
Kotlin Multiplatform & Compose Multiplatform - Starter kit for pragmaticscarlostorres15106
 
SQL Database Design For Developers at php[tek] 2024
SQL Database Design For Developers at php[tek] 2024SQL Database Design For Developers at php[tek] 2024
SQL Database Design For Developers at php[tek] 2024Scott Keck-Warren
 
Beyond Boundaries: Leveraging No-Code Solutions for Industry Innovation
Beyond Boundaries: Leveraging No-Code Solutions for Industry InnovationBeyond Boundaries: Leveraging No-Code Solutions for Industry Innovation
Beyond Boundaries: Leveraging No-Code Solutions for Industry InnovationSafe Software
 
Streamlining Python Development: A Guide to a Modern Project Setup
Streamlining Python Development: A Guide to a Modern Project SetupStreamlining Python Development: A Guide to a Modern Project Setup
Streamlining Python Development: A Guide to a Modern Project SetupFlorian Wilhelm
 
"Federated learning: out of reach no matter how close",Oleksandr Lapshyn
"Federated learning: out of reach no matter how close",Oleksandr Lapshyn"Federated learning: out of reach no matter how close",Oleksandr Lapshyn
"Federated learning: out of reach no matter how close",Oleksandr LapshynFwdays
 
APIForce Zurich 5 April Automation LPDG
APIForce Zurich 5 April  Automation LPDGAPIForce Zurich 5 April  Automation LPDG
APIForce Zurich 5 April Automation LPDGMarianaLemus7
 
Pigging Solutions in Pet Food Manufacturing
Pigging Solutions in Pet Food ManufacturingPigging Solutions in Pet Food Manufacturing
Pigging Solutions in Pet Food ManufacturingPigging Solutions
 
New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024
New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024
New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024BookNet Canada
 
Bun (KitWorks Team Study 노별마루 발표 2024.4.22)
Bun (KitWorks Team Study 노별마루 발표 2024.4.22)Bun (KitWorks Team Study 노별마루 발표 2024.4.22)
Bun (KitWorks Team Study 노별마루 발표 2024.4.22)Wonjun Hwang
 
Connect Wave/ connectwave Pitch Deck Presentation
Connect Wave/ connectwave Pitch Deck PresentationConnect Wave/ connectwave Pitch Deck Presentation
Connect Wave/ connectwave Pitch Deck PresentationSlibray Presentation
 
Are Multi-Cloud and Serverless Good or Bad?
Are Multi-Cloud and Serverless Good or Bad?Are Multi-Cloud and Serverless Good or Bad?
Are Multi-Cloud and Serverless Good or Bad?Mattias Andersson
 
"Debugging python applications inside k8s environment", Andrii Soldatenko
"Debugging python applications inside k8s environment", Andrii Soldatenko"Debugging python applications inside k8s environment", Andrii Soldatenko
"Debugging python applications inside k8s environment", Andrii SoldatenkoFwdays
 
Advanced Test Driven-Development @ php[tek] 2024
Advanced Test Driven-Development @ php[tek] 2024Advanced Test Driven-Development @ php[tek] 2024
Advanced Test Driven-Development @ php[tek] 2024Scott Keck-Warren
 
Unblocking The Main Thread Solving ANRs and Frozen Frames
Unblocking The Main Thread Solving ANRs and Frozen FramesUnblocking The Main Thread Solving ANRs and Frozen Frames
Unblocking The Main Thread Solving ANRs and Frozen FramesSinan KOZAK
 

Recently uploaded (20)

Build your next Gen AI Breakthrough - April 2024
Build your next Gen AI Breakthrough - April 2024Build your next Gen AI Breakthrough - April 2024
Build your next Gen AI Breakthrough - April 2024
 
Key Features Of Token Development (1).pptx
Key  Features Of Token  Development (1).pptxKey  Features Of Token  Development (1).pptx
Key Features Of Token Development (1).pptx
 
Gen AI in Business - Global Trends Report 2024.pdf
Gen AI in Business - Global Trends Report 2024.pdfGen AI in Business - Global Trends Report 2024.pdf
Gen AI in Business - Global Trends Report 2024.pdf
 
Understanding the Laravel MVC Architecture
Understanding the Laravel MVC ArchitectureUnderstanding the Laravel MVC Architecture
Understanding the Laravel MVC Architecture
 
My INSURER PTE LTD - Insurtech Innovation Award 2024
My INSURER PTE LTD - Insurtech Innovation Award 2024My INSURER PTE LTD - Insurtech Innovation Award 2024
My INSURER PTE LTD - Insurtech Innovation Award 2024
 
Scanning the Internet for External Cloud Exposures via SSL Certs
Scanning the Internet for External Cloud Exposures via SSL CertsScanning the Internet for External Cloud Exposures via SSL Certs
Scanning the Internet for External Cloud Exposures via SSL Certs
 
Kotlin Multiplatform & Compose Multiplatform - Starter kit for pragmatics
Kotlin Multiplatform & Compose Multiplatform - Starter kit for pragmaticsKotlin Multiplatform & Compose Multiplatform - Starter kit for pragmatics
Kotlin Multiplatform & Compose Multiplatform - Starter kit for pragmatics
 
SQL Database Design For Developers at php[tek] 2024
SQL Database Design For Developers at php[tek] 2024SQL Database Design For Developers at php[tek] 2024
SQL Database Design For Developers at php[tek] 2024
 
Beyond Boundaries: Leveraging No-Code Solutions for Industry Innovation
Beyond Boundaries: Leveraging No-Code Solutions for Industry InnovationBeyond Boundaries: Leveraging No-Code Solutions for Industry Innovation
Beyond Boundaries: Leveraging No-Code Solutions for Industry Innovation
 
Streamlining Python Development: A Guide to a Modern Project Setup
Streamlining Python Development: A Guide to a Modern Project SetupStreamlining Python Development: A Guide to a Modern Project Setup
Streamlining Python Development: A Guide to a Modern Project Setup
 
"Federated learning: out of reach no matter how close",Oleksandr Lapshyn
"Federated learning: out of reach no matter how close",Oleksandr Lapshyn"Federated learning: out of reach no matter how close",Oleksandr Lapshyn
"Federated learning: out of reach no matter how close",Oleksandr Lapshyn
 
APIForce Zurich 5 April Automation LPDG
APIForce Zurich 5 April  Automation LPDGAPIForce Zurich 5 April  Automation LPDG
APIForce Zurich 5 April Automation LPDG
 
Pigging Solutions in Pet Food Manufacturing
Pigging Solutions in Pet Food ManufacturingPigging Solutions in Pet Food Manufacturing
Pigging Solutions in Pet Food Manufacturing
 
New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024
New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024
New from BookNet Canada for 2024: BNC BiblioShare - Tech Forum 2024
 
Bun (KitWorks Team Study 노별마루 발표 2024.4.22)
Bun (KitWorks Team Study 노별마루 발표 2024.4.22)Bun (KitWorks Team Study 노별마루 발표 2024.4.22)
Bun (KitWorks Team Study 노별마루 발표 2024.4.22)
 
Connect Wave/ connectwave Pitch Deck Presentation
Connect Wave/ connectwave Pitch Deck PresentationConnect Wave/ connectwave Pitch Deck Presentation
Connect Wave/ connectwave Pitch Deck Presentation
 
Are Multi-Cloud and Serverless Good or Bad?
Are Multi-Cloud and Serverless Good or Bad?Are Multi-Cloud and Serverless Good or Bad?
Are Multi-Cloud and Serverless Good or Bad?
 
"Debugging python applications inside k8s environment", Andrii Soldatenko
"Debugging python applications inside k8s environment", Andrii Soldatenko"Debugging python applications inside k8s environment", Andrii Soldatenko
"Debugging python applications inside k8s environment", Andrii Soldatenko
 
Advanced Test Driven-Development @ php[tek] 2024
Advanced Test Driven-Development @ php[tek] 2024Advanced Test Driven-Development @ php[tek] 2024
Advanced Test Driven-Development @ php[tek] 2024
 
Unblocking The Main Thread Solving ANRs and Frozen Frames
Unblocking The Main Thread Solving ANRs and Frozen FramesUnblocking The Main Thread Solving ANRs and Frozen Frames
Unblocking The Main Thread Solving ANRs and Frozen Frames
 

Advances in Directed Spanners

  • 1. Advances in Directed Spanners Grigory Yaroslavtsev Penn State Joint work with Berman (PSU), Bhattacharyya (MIT), Grigorescu (GaTech), Makarychev (IBM), Raskhodnikova (PSU), Woodruff (IBM)
  • 2. Directed Spanner Problem • k-Spanner [Awerbuch ‘85, Peleg, Shäffer ‘89] Subset of edges, preserving distances up to a factor k > 1 (stretch k). • Graph G V, E with weights ������ ∶ ������ → ℝ ≥0 H(V, ������������ ⊆ ������): ∀(������, ������) ∈ ������ ������������������������������ ������, ������ ≤ ������ ⋅ ������(������, ������) • Problem: Find the sparsest k-spanner of a directed graph.
  • 3. Directed Spanners and Their Friends
  • 4. Applications of spanners • First application: simulating synchronized protocols in unsynchronized networks [Peleg, Ullman ’89] • Efficient routing [PU’89, Cowen ’01, Thorup, Zwick ’01, Roditty, Thorup, Zwick ’02 , Cowen, Wagner ’04] • Parallel/Distributed/Streaming approximation algorithms for shortest paths [Cohen ’98, Cohen ’00, Elkin’01, Feigenbaum, Kannan, McGregor, Suri, Zhang ’08] • Algorithms for approximate distance oracles [Thorup, Zwick ’01, Baswana, Sen ’06]
  • 5. Applications of directed spanners • Access control hierarchies • Previous work: [Atallah, Frikken, Blanton, CCCS ‘05; De Santis, Ferrara, Masucci, MFCS’07] • Solution: TC-spanners [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff, SODA’09] • Steiner TC-spanners for access control: [Berman, Bhattacharyya, Grigorescu, Raskhodnikova, Woodruff, Y’ ICALP’11] • Property testing and property reconstruction [BGJRW’09; Raskhodnikova ’10 (survey)]
  • 6. Plan • Approximation algorithms – Undirected vs. Directed – Framework for directed case = Sampling + LP – Randomized rounding • Directed Spanner • Unit-length 3-spanner • Directed Steiner Forest • Combinatorial bounds on TC-Spanners – Upper bounds for low-dimensional posets – Lower bounds via linear programming
  • 7. Undirected vs. Directed • Trivial lower bound: ≥ ������ − ������ edges needed • Every undirected graph has a (2t+1)-spanner with ≤ ������1+1/������ edges. [Althofer, Das, Dobkin, Joseph, Soares ‘93] • Kruskal-like greedy + girth argument 1 => ������ −approximation ������ • Time/space-efficient constructions of undirected approximate distance oracles [Thorup, Zwick, STOC ‘01]
  • 8. Undirected vs Directed • For some directed graphs Ω ������2 edges needed for a k-spanner: • No space-efficient directed distance oracles: some graphs require Ω ������2 space. [TZ ‘01]
  • 9. Unit-Length Directed k-Spanner • O(n)-approximation: trivial (whole graph)
  • 10. Our ������( ������)-approximation • Paths of stretch at most k for all edges => • Classify edges: thick and thin • Take union of spanners for them – Thick edges: Sampling – Thin edges: LP + randomized rounding
  • 11. Local Graph • Local graph for an edge (a,b): Induced by vertices on paths of stretch ≤ ������ from a to b • Paths of stretch ≤ ������ only use edges in local graphs • Thick edges: ≥ ������ vertices in their local graph. Otherwise thin.
  • 12. Sampling [BGJRW’09, DK11] • Pick O( ������ ������������ ������) seed vertices at random • Take in- and out- shortest path trees for each • Handles all thick edges (≥ ������ vertices in their local graph) w.h.p. • # of edges ≤ 2 ������ − 1 ������( ������ ������������ ������) ≤ ������������������ ⋅ Õ ������ .
  • 13. Key Idea: Antispanners • Antispanner – subset of edges, whose removal destroys all paths from a to b of stretch at most k • Graph is spanner <=> hits all antispanners • Enough to hit all minimal antispanners for all thin edges • If ������������ is not a spanner for an edge (a,b) => ������ ∖ ������������ is an antispanner, can be minimized greedily
  • 14. Linear Program (~dual to [DK’11]) Hitting-set LP: ������∈������ ������������ → ������������������ ������������ ≥ 1 ������∈������ for all minimal antispanners A for all thin edges. • # of minimal antispanners may be exponential in ������ => Ellipsoid + Separation oracle ������ 1 ������ ln ������ • We will show: ≤ ������ = ������ minimal 2 antispanners for a fixed thin edge • Assume that we guessed OPT = the size of the sparsest k-spanner (at most ������2 values)
  • 15. Oracle Hitting-set LP: ������∈������ ������������ ≤ ������������������ ������������ ≥ 1 ������∈������ for all minimal antispanners A for all thin edges. • We use a randomized oracle => in both cases oracle fails with exponentially small probability.
  • 16. Randomized Oracle = Rounding • Rounding: Take e w.p. ������������ = min ������ ������������ ������ ⋅ ������������ , 1 • SMALL SPANNER: We have a set of edges of size ≤ ������ ������������ ⋅ ������ ������ ≤ ������������������ ⋅ ������ ������ w.h.p. • Pr[LARGE SPANNER] ≤ e− Ω( ������) by Chernoff. • Pr[CONSTRAINT NOT VIOLATED] ≤ e− Ω( ������) (next slide)
  • 17. Pr[CONSTRAINT NOT VIOLATED] • Set ������: ∀������ ∈ ������ we have Pr[������ ∈ ������] = min ������ ln ������ ������������ , 1 • For a fixed minimal antispanner A, such that ������∈������ ������������ ≥ 1: Pr ������ ∩ A = ∅ ≤ 1 − ������ ln ������ ������������ ≤ ������ − ������ ln ������ ������∈������ ������������ ≤ ������− ������ ������������ ������ ������∈A • #minimal antispanners for a fixed edge (������, ������) ≤ #different shortest path trees with root s in a local graph ������ 1 ������ ln ������ ≤ ������ = ������ 2 (for a thin edge) ������ ������ ������������ ������ • #minimal antispanners ≤ |������|������ ������ => union bound: ������ − ������ ������������ ������ Pr[CONSTRAINT NOT VIOLATED] ≤ |������|������ ������
  • 18. Unit-length 3-spanner • ������(������1/3 )-approximation algorithm – Sampling ������(������1/3 ) times – Dual LP + Different randomized rounding (simplified version of [DK’11]) • Rounding scheme (vertex-based): – For each vertex ������ ∈ ������: sample ������������ ∈ 0,1 – Take all edges ������, ������ if min ������������ , ������������ ≤ ������(������1/3 )������(������,������) – Feasible solution => 3-spanner w.h.p. (see paper)
  • 19. Approximation wrap-up • Sampling + LP with randomized rounding • Improvement for Directed Steiner Forest: – Cheapest set of edges, connecting pairs ������������ , ������������ – Previous: Sampling + similar LP [Feldman, Kortsarz, Nutov, SODA ‘09] . Deterministic rounding gives ������ ������4/5+������ -approximation – We give ������ ������2/3+������ -approximation via randomized rounding
  • 20. Approximation wrap-up • Õ( ������)-approximation for Directed Spanner • Small local graphs => better approximation • Can we do better for general graphs? • Hardness: only excludes polylog(n)- approximation • Integrality gap: ������(������������/������−������ ) [DK’11] • Can we do better for specific graphs • Planar graphs (still NP-hard)?
  • 21. Transitive-Closure Spanners Transitive closure TC(G) has an edge from u to v iff G has a path from u to v G TC(G) H is a k-TC-spanner of G if H is a subgraph of TC(G) a k-TC-spanner ofHG if H is iffk-spanner of H is for which distance (u,v) ≤ k a G has a path TC(G) from u to v Shortcut edge consistent with 2-TC-spanner of G ordering [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff, SODA‘09], generalizing [Yao 82; Chazelle 87; Alon, Schieber 87, …]
  • 22. Applications of TC-Spanners  Data structures for storing partial products [Yao, ’82; Chazelle ’87, Alon, Schieber, 88]  Constructions of unbounded fan-in circuits [Chandra, Fortune, Lipton ICALP, STOC‘83]  Property testers for monotonicity and Lipschitzness [Dodis et al. ‘99,BGJRW’09; Jha, Raskhodnikova, FOCS ‘11]  Lower bounds for reconstructors for monotonicity and Lipshitzness [BGJJRW’10, JR’11]  Efficient key management in access hierarchies Follow references in [Raskhodnikova ’10 (survey)]
  • 23. Bounds for Steiner TC-Spanners • No non-trivial upper bound for arbitrary graphs FACT: For a random directed bipartite graph of density ½, any Steiner 2-TC-spanner requires Ω n2 edges.
  • 24. Low-Dimensional Posets • [ABFF 09] access hierarchies are low- dimensional posets • Poset ≡ DAG • Poset G has dimension d if G can be embedded into a hypergrid of dimension d and d is minimum. ′ ������: ������ → ������ is a poset embedding if for all ������, ������ ∈ ������, ������ ≼������ ������ iff ������ ������ ≼������′ ������(������). Hypergrid ������ ������ has ordering ������1 , … , ������������ ≼ (������1 , … , ������������ ) iff ������������ ≤ ������������ for all i
  • 25. Main Results Stretch k Upper Bound Lower Bound ������ 2 ������ ������ log ������ ������ ������log ������ Ω ������ ������ where ������ is a constant ≥3 ������(������ log ������−1 ������ log log ������) Ω(������ log (������−1)/������ ������) for constant d for constant d [DFM 07] d is the poset dimension ������ = ������. Nothing better known for larger k.
  • 26. 2-TC-Spanner for ������ ������ • d = 1 (so, n = m) 2-TC-spanner with ≤ ������ log ������ = ������ log ������ edges … … We show this is tight upto ������������ for a • d > 1 (so, n = md ) constant ������. log n d 2-TC-spanner with ≤ (m log m)d = n edges d by taking d-wise Cartesian product of 2-TC- spanners for a line.
  • 27. Lower Bound Strategy • Write in IP for a minimal 2-TC-spanner. • OPT ≥ ������������ = ������������������������������������ • It is crucial that the integrality gap of the primal is small. • Idea: Construct some feasible solution for the dual => lower bound on OPT. • OPT ≥ ������������ = ������������������������������������ ≥ ������������
  • 28. IP Formulation • {0,1}-program for Minimal 2-TC-spanner: minimize ������������������ subject to: ������,������:������≼������ ������������������ ≥ ������������������������ ∀������ ≼ ������ ≼ ������ ������������������ ≥ ������������������������ ∀������ ≼ ������ ≼ ������ ������������������������ ≥ 1 ∀������ ≼ ������ ������:������≼������≼������
  • 29. Dual LP • Take fractional relaxation of IP and look at its dual: maximize ������������������ ������,������:������≼������ subject to: ������������������������ + ������������������������ ≤ 1 ∀������ ≼ ������ ������:������≼������ ������:������≼������ ������������������ ≤ ������������������������ + ������������������������ ∀������ ≼ ������ ≼ ������ 0 ≤ ������������������ , ������������������������ , ������������������������ ≤ 1 ∀������ ≼ ������ ≼ ������
  • 30. Constructing solution to dual LP • Now we use fact that poset is a hypergrid! For ������ ≼ ������, set 1 ������������������ = , where ������(������ − ������) is volume of box with ������(������−������) corners u and v. maximize ������������������ > ������ ������������ ������ ������ ������,������:������≼������ subject to: ������������������������ + ������������������������ ≤ 1 ∀������ ≼ ������ ������:������≼������ ������:������≼������ ������ ≤ ������������ ������������������ = ������������������������ + ������������������������ ∀������ ≼ ������ ≼ ������ ������(������−������) ������(������−������) Set ������������������������ = ������������������ , ������������������������ = ������������������ ������ ������−������ +������(������−������) ������ ������−������ +������(������−������)
  • 31. Constructing solution to dual LP • Now we use fact that poset is a hypergrid! For 1 ������ ≼ ������, set ������������������ = , where ������(������ − ������) is 4������ ������ ������(������−������) volume of box with corners u and v. maximize ������������������ > ������ ������������ ������ ������ / ������������ ������ ������,������:������≼������ subject to: ������������������������ + ������������������������ ≤ 1 ∀������ ≼ ������ ������:������≼������ ������:������≼������ ≤ ������ ������������������ = ������������������������ + ������������������������ ∀������ ≼ ������ ≼ ������ ������(������−������) ������(������−������) Set ������������������������ = ������������������ , ������������������������ = ������������������ ������ ������−������ +������(������−������) ������ ������−������ +������(������−������)
  • 32. Wrap-up • Upper bound for Steiner 2-TC-spanner • Lower bound for 2-TC-spanner for a hypergrid. – Technique: find a feasible solution for the dual LP (������−1)/������ • Lower bound of Ω(������ log ������) for ������ ≥ 3. – Combinatorial – Holds for randomly generated posets, not explicit. • OPEN PROBLEM: – Can the LP technique give a better lower bound for ������ ≥ 3?