applied algorithms and optimization
DESCRIPTION
Applied Algorithms and Optimization. Gabriel Robins Department of Computer Science University of Virginia www.cs.virginia.edu/robins. “Make everything as simple as possible, but not simpler.” - Albert Einstein (1879-1955). Algorithms. Solution. exact. approximate. Speed. fast. - PowerPoint PPT PresentationTRANSCRIPT
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Gabriel Robins
Department of Computer Science
University of Virginiawww.cs.virginia.edu/robins
Applied Algorithms and Optimization
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“Make everything as simple as possible, but not simpler.”- Albert Einstein (1879-1955)
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Solutionexact approximate
fast
slow
Spee
d Short & sweet Quick & dirty
Slowly but surely Too little, too late
Algorithms
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Complexity
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Fabrication PhysicalLayout
StructuralDesign
xyw
z
Requirements
e.g., “secure communication”
LogicDesign
Z = x + y w
VLSI DesignFunctional
Design
C(M) = Mp mod N
Design Specification
Dataencryption
PhysicalLayout
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Placement & Routing
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Trends in Interconnecttime
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2
Steiner Trees
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Steiner TreesSteiner Trees
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Iterated 1-Steiner AlgorithmQ: Given pointset S, which point p minimizes |MST(S È p)| ? Algorithmic idea: Iterate!
Theorem: Optimal for £ 4 points
Theorem: Solutions cost < 3/2 · OPT
Theorem: Solutions cost £ 4/3 · OPT for “difficult” pointsets
In practice: Solution cost is within 0.5% of OPT on average
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Group Steiner Problem
Theorem: o(log # groups) · OPT approximation is NP-hardTheorem: Efficient solution with cost = O((# groups)e) · OPT " e>0
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Graph Steiner Problem
Algorithm: “Loss-Contracting” polynomial-time approximationTheorem: 1 + (ln 3)/2 ≈ 1.55 · OPT for general graphsTheorem: 1.28 · OPT for quasi-bipartite graphsCurrently best-known; won the 2007 SIAM Outstanding Paper Prize
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Bounded Radius Trees
Algorithm:
Input:• points / graph• any e > 0
Output: tree T with• radius(T) £ (1+e) ·
OPT• cost(T) £ (1+2/e) ·
OPT
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Low-Degree Spanning Trees
MST 1: cost = 8max degree = 8
MST 2: cost = 8max degree = 4
Theorem: max degree 4 is always achievable in 2D
Theorem: max degree 14 is always achievable in 3D
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Low-Skew Trees
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A
B
Circuit Testing
Theorem: # leaves / 2 probes are necessary Theorem: # leaves / 2 probes are sufficientAlgorithm: linear time
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Improving Manufacturability
Theorem: extremal density windows all lie on Hanan gridAlgorithms: efficient fill analyses and generation for VLSIEnabled startup company: Blaze DFM Inc. - www.blaze-dfm.com
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Landmine Detection
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Moving-Target TSP
Origin
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3
2
1
4
Theorem: “waiting” can never helpAlgorithms: · efficient exact solution for 1-dimension
· efficient heuristics for other variants
Moving-Target TSP
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Robust Paths
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Minimum Surfaces
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time
Evolutionary Trees
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protein
DNA
Polymerase Chain Reaction (PCR)
BiologicalSequences
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Discovering New Proteins
flyNK
gpPAF
bovOP
ratPOT
ratCCKA
humD2humA2a
hamA1ahamB2
bovH1
ratNK1
flyNPYmusGIR
humSSR1
humC5a
ratRTA
ratG10d
chkP2y
dogCCKB
dogAd1
ratD1
ratNPYY1
ratNTR
humTHR
humMAShumEDG1
hum5HT1a
musTRH
humIL8
RBS11musdelto
ratBK2humMRG
humfMLF
musEP2
ratV1a
herpesECcrnvHH2
cmvHH3
bovLOR1ratANG
dogRDC1
humRSC
chkGPCR
musP2uratODOR
ratLH
ratCGPCR
humACTHhumMSH
musEP3humTXA2
humM1
musGnRH
bovETAmusGRP
flyNK
gpPAF
bovOP
ratPOT
ratCCKA
humD2humA2a
hamA1ahamB2
bovH1
ratNK1
flyNPYmusGIR
humSSR1
humC5a
ratRTA
ratG10d
chkP2y
dogCCKB
dogAd1
ratD1
ratNPYY1
ratNTR
humTHR
humMAShumEDG1
hum5HT1a
musTRH
humIL8
RBS11musdelto
ratBK2humMRG
humfMLF
musEP2
ratV1a
herpesECcrnvHH2
cmvHH3
bovLOR1ratANG
dogRDC1
humRSC
chkGPCR
musP2uratODOR
ratLH
ratCGPCR
humACTHhumMSH
musEP3humTXA2
humM1
musGnRH
bovETAmusGRP
????
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Primer Selection Problem
Input: set of DNA sequencesOutput: minimal set of covering primers
Theorem: NP-completeTheorem: W(log # sequences)·OPT within P-timeHeuristic: O(log # sequences)·OPT solution
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Genome Tiling Microarrays
Algorithms: efficient DNA replication timing analysesPapers in Science, Nature, Genome Research
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Radio-Frequency Identification
Generalized “Yoking Proofs”
Physically Unclonable Functions
Inter-Tag Communication
1 Tag: 75% 2 Tags: 94% 3 Tags: 98% 4 Tags: 100%
Multi Tags
Tagging BulkMaterials
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UVa Computer Science
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“Gabe aiming to solve a tough problem”for details see www.cs.virginia.edu/robins/dssg
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Lets Collaborate!
• What I offer:• Practical problems & ideas• Experience & mentoring• Infrastructure & support
• What I need:• PhD students• Dedication & hard work• Creativity & maturity
Goal: your success!
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Proof: Low-Degree MST’s
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“You want proof? I’ll give you proof!”
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• Compute MST’ over P’
Proof: Low-Degree MST’s
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2
3
4
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7 8
Idea: |MST’(P)| = |MST(P)|
Output: MST’ over P
Theorem: max MST degree £ 4
Input: pointset PFind: MST(P)
• Perturb region 5-8 points,yielding pointset P’
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“I think you should be more explicit here in step two.”
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Low-Degree MST’s in 3D
Idea: |MST’(P)| = |MST(P)|
• Perturb boundary pointsto yield pointset P’
• Compute MST’ over P’• Output: MST’ over P
Theorem: max MST degree in 3D is £ 6 + 8 = 14Theorem: lower bound on max MST degree in 3D is ³ 13
Input: 3D pointset PFind: MST(P)
Partition space:• 6 square pyramids• 8 triangular pyramids
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On the flight deck of the nuclear aircraft carrier USS Eisenhowerout in the Atlantic ocean
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38On the bridge of the nuclear aircraft carrier USS Eisenhower
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39At the helm of the SSBN nuclear missile submarine USS Nebraska
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40Refueling a B-1 bomber in mid-air from a KC-135 tanker
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41Aboard an M-1 tank at the National Training Center, Fort Erwin
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42At U.S. Strategic Command Headquarters, Colorado Springs
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43Pentagon meeting with U.S. Secretary of Defense Bill Perry
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44Patch of the Defense Science Study Group (DSSG)
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UVa Computer Science
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UVa Computer Science
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Density Analysis
Input:• n´n layout• k rectangles• w´w window Algorithms:
O(n2) time O(k2)
++
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nwkk logk
wn O
Theorem: extremal density windows all lie on Hanan grid
Output: allextremal density w´w windows