applied algorithms and optimization

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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

38On the bridge of the nuclear aircraft carrier USS Eisenhower

39At the helm of the SSBN nuclear missile submarine USS Nebraska

40Refueling a B-1 bomber in mid-air from a KC-135 tanker

41Aboard an M-1 tank at the National Training Center, Fort Erwin

42At U.S. Strategic Command Headquarters, Colorado Springs

43Pentagon meeting with U.S. Secretary of Defense Bill Perry

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