Slides

Sparse Problems in Bioinformatics
Olgica Milenkovic
Joint work with: Wei Dai and Amin Emad
University of Illinois at Urbana-Champaign
Acknowledgment: NSF CCF 1117980, NSF CCF 0729216, NSF CCF 0809895,
NSF CCF 01218764, NSF CSoI
Thanks to: Yaniv Erlich and Noam Shental
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What information theorists should know about
Bioinformatics
Bioinformatics may be a source for many interesting problems in
coding theory, statistics and signal processing, but there exists no
unifying theory of bioinformatics.
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What information theorists should know about
Bioinformatics
Bioinformatics is data-driven: without testing model/theory on
data, very little credibility for results. Data is noisy and modeling is
hard.
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What information theorists should know about
Bioinformatics
It appears to be a difficult task to reconcile the two areas...
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Something for both Sides: Sparsity
Group Testing for Experimental Design.
I
I
Group testing for Genotyping.
Group testing for synonimous coding studies.
Causal Compressive sensing and Low-Rank Completion.
I
I
Gene regulatory networks.
Synthetic lethality/Protein-Protein Interaction (PPI) inference.
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And Some Hard Problems...
Reconstructing sequences from traces and other problems
regarding sequences...
I
De novo protein sequencing via Tandem Mass Spectrometry:
reconstructing sequences based on composition multisets.
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Let’s start with group testing...
m
Test
Result
n
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%
N
Origin of group testing: blood pooling of soldiers during recruitment in WWII
(Dorfman, 1941); n tests and N test subjects, described via test matrix, with
m <<N positives.
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Mathematical Formulation I
Group testing amounts to finding indices of subjects {i1 , . . . , im }
y = xi1 xi1 ... xim ,
for given y. Here, denotes binary OR, and xi ∈ {0, 1}n denotes
signature of subject i (i.e., column of test matrix).
OR - if one positive, test positive.
I
Problem of interest: what it the smallest number of non-adaptive
measurements n needed to discover m defectives among N
subjects?
const. m2 log(N )
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Mathematical Formulation II
Two coding-theoretic paradigms (Kautz and Singleton, 1964)
I
m-separable matrices:
xj1 xj2 . . . xjl 6= xi1 ... xis , l, s ≤ m
I
m-disjunct matrices:
supp (xi ) * ∪l6=i.l≤m supp(xl )
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Applications in Bioinformatics
Out of 3 × 109 bases, about 107 single point mutations (SNPs).
GAT
GAT
GAT
GAT
GAT
GAT
ATTCGTACGGAAT
ATTCGTACTGAAT
ATTCGTACGGAAT
ATTCGTACGGAAT
GTTCGTACGGAAT
GTTCGTACGGAAT
SNPs
(Single Nucleotide Polymorphisms)
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Applications in Bioinformatics
Out of 3 × 109 bases, about 107 single point mutations (SNPs).
Asthma
Systemic Sclerosis
Lung Cancer
Type II Diabetes
Lupus
END1
Fibrilin 1
MMP1
Syn1A
Pro
Healthy individual: 2 normal genes (alleles).
Carrier of disease: 1 normal and 1 faulty gene (allele).
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We are still mixing blood, but...
Illumina high-throughput sequencer: take DNA from an individual, sequence,
determine which SNPs are present
Source: giga.ulg.ac.be
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Cost of sample preparation/sequencing is high!
Illumina high-throughput sequencer: take DNA from a number of individuals,
sequence whole pool, determine which SNPs are present based on
combinatorial designs!
(Erlich et. al. 2009, Shental et.al., 2010)
Source: giga.ulg.ac.be
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On the Cover of a Magazine
Source: Erlich Lab
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Synonymous Coding
DNA-RNA-Protein-Life
4 bases, triple codon = 64 options; only 20 amino acids.
TTA
TTG
W
Leu
W
S
W
W
Synonymous coding (SC)
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Genotyping and Group Testing: Plug-and-Play?
Compressed genotyping (Erlich et.al., IT Special Issue on Molecular Biology,
2010; Erlich and Shental, 2011)
I
Diagonally spaced ones in the test matrix (robotic arm movements
highly restricted);
I
Need sparse matrices due to simplicity of mixing precision
(Thierry-Migg et.al, STDs (shifted transversal designs)).
Source: Hamilton Robotics
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Genotyping and Group Testing: Plug-and-Play?
Nucleic Acids Rese
Compressed genotyping (Erlich et.al., IT Special Issue on Molecular Biology,
2010; Erlich and Shental,
2011=) 40, e =0.01
reads per person
reads per person = 40, er=0
r
x 10
x 10
2
2
I Diagonally
spaced ones in the test
matrix
(robotic
arm movements
(a)
(b)
1.5
highly 1.5
restricted);
−3
measurement value
measurement value
−3
1
0.5
0
#errors: 3
0
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#errors: 0
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pool
x 10
(c)
1.5
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0.5
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#errors: 0
0
10
−3
2
measurement value
−3
x 10
20
30
40
40
x 10
(d)
1
0.5
#errors: 0
0
10
20
30
40
pool
pool
reads per person = ∞, e =0.01
reads per person = ∞, e =0
−3
r
(e)
1
0.5
#errors: 0
10
20
50
reads per person = 400, er=0
1.5
0
50
1.5
0
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0
2
measurement value
measurement value
2
reads per person = 400, er=0.01
30 CS
and
40LRMC 50
2
measurement value
−3
30
pool
x 10
50
r
(f)
1.5
1
0.5
0
#errors: 0
0
10
20
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Genotyping: The Constraints
A general theory of genotyping (Emad and M., ITW’2011, ISIT’2012):
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Non-uniform DNA sampling: availability of genetic material
m
q-­‐1 y
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Thresholds in above example: {0, 1}{2}{3}{4, 5, 6}.
Copy number variation (most people have two copies of each gene,
some have up to five copies): replication numbers
c1 , . . . , cn ∈ {1, 2, 3, 4, 5}.
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Genotyping: The Constraints
A general theory of genotyping (Emad and M., ITW’2011, ISIT’2012):
I
Semi-quantitative testing: limited readout precision
95 0 0 5 90 95 10 85 15 80 75 70 30 y
50 0
1 15 20 2 4 35 3 60 55 45 25 30 65 40 55 10 75 70 35 60 5 80 25 1 0 5 85 20 65 0 90 1
50 40 45 
Q −1

, ηQ-1,  , ηQ −1
m
∑x
k,i j
0, 1  η1 −1, η1,  , η2 −1,
j=1
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Genotyping: The Constraints
A general theory of genotyping (Emad and M., ITW’2011, ISIT’2012):
I
Two-dimensional testing (same mutation may be involved in
multiple diseases)
I
Family tree structure: diseases run in families and testing strategy
should be governed by Mendel’s law.
F
Probabilistic group testing: individuals have a certain probabilities of
being carriers.
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An Information Theoretic Approach
Group testing as an information theoretic problem (Maliytov 1980’s,
Dyachkov 2000)
m - number of positive subjects; η - thresholds; PT - probability
distribution for thresholds
C = sup PT ,η α(m, PT , η)
{i}
α(m, PT, η) = max i=1,...,m
{i}
I (XD1 , XD2 )
i
m
i
m−i
10 1 1 0 1 0 0 01
X D{i1}
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1
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An Information Theoretic Approach
Capacity lower bounds evaluated numerically.
0.8
Q=2
Q=3
Lowe r Bound on C
0.7
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0.5
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0.3
0.2
0.1
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m
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Code constructions based on disjunct and array codes.
Positives identification via belief propagation (Emad and M., 2012,
Huang and M., 2009).
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An Information theoretic Approach
Optimal thresholds - examples for q=3:
I
m
4
6
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Probability Distribution
{0.18,064,0.18}
{0.6,0.07, 0.33}
{0.1,0.8,0.1}
{0.58,0.28,0.14}
Thresholds
{0,1,2,3}{4}{5,6,7}
{0,1,2,3}{4,5}{6,7,8,9,10,11,12}
{0,1,...,7}{8}{9,10,...,16}
{0,1,...,4}{5,6}{7,8,...,20}
Code constructions based on disjunct and array codes:
[C1 C2 ...Cb ], Ci = fi (η) C
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Synonymous Coding: The Constraints
What are the practical constraints in DNA signal detection?
Signal detection (Lin and et.al., Skiena et.al., 2012) - insertion of
synonimous codes into wild type DNA is costly
W
W
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S
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S S
S S
S W
S S
Need to minimize number of W-S transitions (insertions) (Skiena et.al., 2012)
Theory of cyclic disjunct codes (Colbourn et.al, 1990s, Dyachkov et.al.
etc)
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Synonymous Coding: The Folding Constraint
Example: two synonymously coded RNA fragments fold very
differently (Vienna folding code, Zucker et.al.)
Coding theoretic approach to RNA folding (M. and Kashyap, 2007)
A
A
U
C
G
GC
CGC
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G
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C
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C
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CCG
AU
A UC
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C C AA
U U
G
G U GA A
UCCAC
CG
CU CU
U
A GU CG
GC
U
G
C
U
A
AC CU
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Compressive Sensing in Bioinformatics
x ∈ Rn : when is reconstruction possible? Gelfand, Kashin, 1977; Bresler et.
al., 1996; Donoho et. al., 2004; Candés, Romberg, Tao, 2005
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CS Signal Reconstruction
l0 -minimization:
Find a signal x̂ such that |supp (x̂)| = kx̂k0 ≤ K and y = Φx̂.
l1 -minimization:
min kx̂k1 subject to y = Φx̂.
Greedy algorithms:
OMP [Tropp, 2004], SP [Dai & Milenkovic, 2008], CoSaMP [Needell & Tropp,
2008],
IHT [Blumensath and Davis, 2009]· · ·
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Applications of CS to Bioinformatics
Compressive sensing DNA microarrays (Dai, Sheikh, Barniuk and
M., 2009)
Network analysis (example: vertex distance matrix, gene
expressions, protein interaction affinities etc): gene regulatory
networks and protein-protein interaction networks.
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Gene Regulatory Networks
Transcription factors are proteins, coded by genes.
Transcription factors supress or promote the activity of
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A Simple Dynamical System (DS) Model
Gene expressions: concentration of mRNA corresponding to
genes - X(t) = (X1 (t), . . . , XN (t))
Linear/nonlinear dynamical system model:
X(t + 1) = F (A(t)X(t)) + n(t)
A(t) ∈ RN is sparse, with some column/row weight distribution.
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Compressive Sensing for DS Analysis
Granger causality.
Compressive sensing and Granger causality.
Application in inference of causal gene interactions.
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Causal Compressive Sensing: Granger Causality I
Granger [1960] (Nobel Prize in Economics): how can one deduce
is a random process X causally influences another random
process Y?
Autoregressive model for X :
en = a1 Xn−1 + a2 Xn−2 + ... + am Xn−m
X
en ||
M SEX = ||Xn − X
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Causal Compressive Sensing: Granger Causality I
Granger [1960] (Nobel Prize in Economics): how can one deduce
is a random process X causally influences another random
process Y?
Does including Y in the autoregressive model reduce the
estimation error?
eY,n = a1 Xn−1 + ... + am Xn−m + b0 Yn + b1 Yn−1 + ... + bs Yn−s
X
eY,n ||
M SEX,Y = ||Xn − X
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Causal Compressive Sensing: Granger Causality II
Compressive sensing corresponds to a sparse linear model:
g = Φx = x1 φ1 + x2 φ2 + ... + xN φN
Compressive sensing corresponds to a combination of linear
models:
x
g = [ΦX ΦY ]
= x1 φX,1 + ... + xN φX,N + y1 φY,1 + ... + yN φY,N
y
Sensing vectors may be delayed versions of the same random
process:
φt = φ(t0 − t).
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Causal Compressive Sensing: Granger Causality II
Compressive sensing corresponds to a sparse linear model:
g = Φx = x1 φ1 + x2 φ2 + ... + xN φN
In principle, can make the model non-linear:
x
g = [F (ΦX )H(ΦY )]
=
y
xf (φX,1 ) + ... + xN f (φX,N ) + y1 f (φY,1 ) + ... + yN f (φY,N )
g = x1 f 1 (φX,1 , ..., φX,N ) + ... + xN f N (φX,1 , ..., φX,N ).
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Causal Gene Interactions I
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Causal Gene Interactions II
Gene expression data: time series of gene activity levels
φ(t1 ), φ(t2 ), ..., φ(tm ).
Target gene T: yT (t1 ), yT (t2 ), ..., yT (tm ). Causal gene: C.
Compressive sensing linear model: activity of T an unknown
sparse combination of activities of genes other than C:
yT = Φ/C x + r/C
Compressive sensing linear model: activity of T an unknown
sparse combination of activities of genes that include C:
yT = Φ+C x + r+C
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Causal Gene Interactions II
Gene expression data: time series of gene activity levels
φ(t1 ), φ(t2 ), ..., φ(tm ).
Target gene T: yT (t1 ), yT (t2 ), ..., yT (tm ). Causal gene: C.
Compressive sensing linear model: activity of T an unknown
sparse combination of activities of genes other than C:
yT = Φ/C x + r/C
Compressive sensing linear model: activity of T an unknown
sparse combination of activities of genes that include C:
yT = Φ+C x + r+C
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Causal Gene Interactions III
Determining if C causally influences T:
Residuals: r/C > r+C ;
Gene C is in the list of “strongest” contributors (large xC vs. other
entries of x).
Remark: Self-loops and indirect causality cannot be captured.
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Results - Synthetic Data I
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Results I
Reduction in MSE at least 80% (coefficient of determination);
Identified 85% of causal relationships.
Almost all errors caused by problems in CS reconstruction algorithms
(no RIP, no incoherence).
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Results II
[Gardner, 2004]: SOS network of E.coli
R/C
dinI
lexA
recA
recF
rpoD
rpoH
rpoS
ssb
uCD
dinI
0
0 (1)
0 (1)
0
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0 (1)
0 (?)
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lexA
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Results II
[Galkin et.al., 2011]: SOS network of E.coli - the role of dinI
F9.large.jpg 1280×980 pixels
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Formal definition of the LRMC Problem
Let X ∈ Rm×n be a low-rank matrix: r min (m, n).
One does not have full information of X, but only knows a subset
of entries.
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Observation subset Ω ⊂ [m] × [n].
Matrix Completion: Given low-rank assumption and Ω & XΩ , X =?
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The Framework
“l0 -approach”:
rank X 0
subject to PΩ X 0 = XΩ .
(P 0) minimize
“l1 -approach”:
(P 1) minimize
0
X ∗
subject to PΩ X 0 = XΩ ,
where kX 0 k∗ =
P
σi (X 0 ).
(P 1) ≡ (P 0) if X singular vectors “sufficiently spread” :
uncorrelated with the standard basis.
m = |Ω| = c(n + m) r log (max(n, m)) for unique recovery.
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Protein-Protein Interaction (PPI) Prediction
Sequence based methods: assumption that interacting proteins
belong to spacially confined conserved regions.
I
I
Sequence alignment.
Phylogenetic tree analysis.
Probabilistic network inference: assumption that interacting
proteins form special network motifs.
I
I
Graphelet models.
Bayesian models.
References: many, just to mention a few...
I
I
I
I
Grama et.al., 2010
Kim et.al. 2006.
Valencia and Pazos, 2010.
Przulj et.al., 2006-2010.
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Protein-Protein Interactions
Let X ∈ Rm×n be a matrix of interaction probabilities between
proteins, say
Protein \ Protein
PR1
PR2
PR3
PR4
PR1
*
0.765
0.99
?
PR2
0.765
*
0.112
0.5
PR3
0.99
0.112
*
?
PR4
?
0.5
?
*
Can you predict the missing entries?
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Why LRMC for PPI inference?
Low-rank assumption can be justified in many ways:
I
Not all pairwise interactions are independent.
I
There is a relatively small number of degrees of freedom - ”factors”
that influences protein binding (compared to number of proteins).
Otherwise, binding would be physically impossible.
For small sampling rates, the solution may not be unique. Pick
“dense subsets” of proteins.
I
I
The dependencies may not be linear (easy to handle in proposed
framework).
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PPI Prediction via Completion
STRING database: Yeast Proteins (roughly 1200 out of 6000 proteins
used in experiment)
PPpair
YBL032W:YDL220C
YDL220C:YDR510W
YDL220C:YLL039C
YDR381W:YLR418C
YBL032W:YLL002W
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Affinity
1.00
1.00
1.00
1.00
1.00
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Biologically Plausible?
Yes
?
?
Yes
?
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Traces of Sequences
Genomic and Proteomic Sequences
Channel N
Substitution, Deletion/Insertion channel: Levenshtein, 1990s;
Mitzenmacher et.al., 2008.
Substring extraction: Genome assembly problem.
Multiset of prefixes/suffixes: Protein mass spectrometry problem.
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Traces of Sequences I
Problem formulation: given S = s1 s2 s3 ...sn , si ∈ {0, 1}.
Traces:
{{s1 }, {s1 , s2 }, {s2 }, {s2 , s3 }, . . . , {sn }, {sn , sn−1 }, . . . , {s1 , s2 , . . . , sn }}.
Example: 0001 gives {0, 0, 0, 1, 02 , 02 , 01, 03 , 02 1, 03 1}
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Traces of Sequences II
When can the sequence be reconstructed? (Acharya et.al., ISIT
2010)
A string is reconstructable iff its length n satisfies:
n ≤ 7.
n ≥ 8 and n + 1 is a prime or twice a prime.
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Two Announcements
IMA Workshop on Group Testing in Biology, 2012.
ISIT Tutorial on Bioinformatics (with Sharon Aviran, UC Berkeley),
2012.
Milenkovic et al. (UIUC)
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Thank you!
Milenkovic et al. (UIUC)
CS and LRMC
May 2012
54 / 54
Milenkovic et al. (UIUC)
CS and LRMC
May 2012
54 / 54