text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
cube router
CM-2: 2048B/PE, plus 2,048 32-bit floating-point units
● Maspar MP-1 (1989)
16K 4-bit processors, 16-64KB/PE, 2D + Xnet router
MP-2: 16K 32-bit processors, 64KB/PE
Prof. Saman Amarasinghe, MIT.
25
6.189 IAP 2007 MIT
Shared Instruction: SIMD Architecture
● Central controller broadcasts instruc... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
V4
V5
V6
V7
S0
S1
S2
S3
S4
S5
S6
S7
A0
A1
A2
A3
A4
A5
A6
A7
Vi
Vj
Vk
Sj
Sk
Si
Aj
Ak
Ai
V. Mask
V. Length
FP Add
FP Mul
FP Recip
Int Add
Int Logic
Int Shift
Pop Cnt
Addr Add
Addr Mul
64-bitx16
4 Instruction Buffers
NIP
LIP
CIP
memory bank cycle 50 ns processor cycle 12.5 ns (8... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
A[20] B[20] A[21] B[21] A[22] B[22] A[23] B[23]
A[16] B[16] A[17] B[17] A[18] B[18] A[19] B[19]
A[12] B[12] A[13] B[13] A[14] B[14] A[15] B[15]
C[2]
C[1]
C[0]
C[8]
C[4]
C[9]
C[5]
C[10]
C[6]
C[11]
C[7]
C[0]
C[1]
C[2]
C[3]
Prof. Saman Amarasinghe, MIT.
30
6.189 IAP 2007 MIT
Vector Unit Structure
Func... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
Compiler schedules parallel execution
● Multiple parallel operations packed into one long
instruction word
● Compiler must avoid data hazards (no interlocks)
Prof. Saman Amarasinghe, MIT.
33
6.189 IAP 2007 MIT
VLIW Instruction Execution
1
IF
2
ID
IF
Successive
instructions
Cycles
3
4
5
6
EX WB
EX
EX
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
: Superscalar Processors
● Explicit Parallelism
● Shared Instruction Processors
● Shared Sequencer Processors
● Shared Network Processors
● Shared Memory Processors
● Multicore Processors
Prof. Saman Amarasinghe, MIT.
37
6.189 IAP 2007 MIT
Shared Network: Message Passing MPPs
(Massively Parallel Processors)
... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
messages is expensive, need to poll or interrupt
Prof. Saman Amarasinghe, MIT.
39
6.189 IAP 2007 MIT
Outline
● Implicit Parallelism: Superscalar Processors
● Explicit Parallelism
● Shared Instruction Processors
● Shared Sequencer Processors
● Shared Network Processors
● Shared Memory Processors
● Multicore Pr... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
Each node has 256MB-2GB local DRAM memory
● Load and stores access global memory over network
● Only local memory cached by on-chip caches
● Alpha microprocessor surrounded by custom “shell” circuitry to make it into
Y
X
Z
Image by MIT OpenCourseWare.
effective MPP node. Shell provides:
multiple stream buffers i... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
● Bus provides broadcast and serialization point for simple
snooping cache coherence protocol
● Modern microprocessors integrate support for this protocol
Prof. Saman Amarasinghe, MIT.
45
6.189 IAP 2007 MIT
Sun Starfire (UE10000)
• Up to 64-way SMP using bus-based snooping protocol
μP
μP
μP
μP
μP
μP
μP
μP ... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
008
Picochip Ambric
PC102
AM2045
Cisco
CSR-1
Intel
Tflops
Raw
Raza Cavium
XLR
Octeon
Niagara
Cell
Boardcom 1480
Xbox360
PA-8800 Opteron
Power4
Opteron 4P
Xeon MP
Tanglewood
PExtreme Power6
Yonah
P2 P3
Itanium
P4
Athlon
Itanium 2
1970
1975
1980
1985
1990
1995
2000
2005
20??
Prof. Saman Ama... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
latencies of chip-to-chip communication are drastically
reduced
ARM multi-chip core
Private IRQ
Configurable #
of hardware intr
Per-CPU
aliased
peripherals
Configurable
between 1 & 4
symmetric
CPUs
Private
peripheral
bus
Interrupt Distributor
CPU
Interface
CPU
Interface
CPU
Interface
CPU
Inter... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
Processor Elements
Or “Streaming Processor Elements”
Co-processors with dedicated 256kB of
memory (not cache)
●
IO
Dual Rambus XDR memory controllers
(on chip)
– 25.6 GB/sec of memory bandwidth
76.8 GB/s chip-to-chip bandwidth (to
off-chip GPU)
PP
P
P
U
U
M
I
C
M
I
C
Image removed due to copyright rest... | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
or migrate
back to the era of assembly programming
Prof. Saman Amarasinghe, MIT.
58
6.189 IAP 2007 MIT | https://ocw.mit.edu/courses/6-189-multicore-programming-primer-january-iap-2007/0e56d30fb5cce40a429b63435d2a039a_lec3architctre.pdf |
18.409 An Algorithmist’s Toolkit
September 24, 2009
Lecturer: Jonathan Kelner
Scribe: Shaunak Kishore
Lecture 5
1 Administrivia
Two additional resources on approximating the permanent
• Jerrum and Sinclair’s original paper on the algorithm
• An excerpt from Motwani and Raghavan’s Randomized Algorithms
2 Review of Monte... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
subset of V . In this case, it would take exponentially
many samples from V to get O( log 1/δ
) successes. Second, it could be difficult to sample uniformly from a
large and complicated set V . We will see ways to solve both these problems in two examples today.
(cid:15)2
3 DNF Counting and an Exponentially Small Target
... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
solution out of 2n assignments.
3.1 Reducing the Sample Space
Instead of picking assignments uniformly at random and testing each clause, we will instead sample from the
set of assignments that satisfy at least one clause (but not uniformly). This algorithm illustrates the general
strategy of sampling only the impo... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
n−ki .
1To see this, understand that the negation of a DNF formula is just a CNF formula by application of De Morgan’s laws.
Therefore counting solutions to a DNF formula is equivalent to counting (non-)solutions of a CNF formula, which is the canonical
example of a #P -Hard problem.
5-2
Our probability of pickin... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
Matrix and Perfect Matchings
Given an n × n 0-1 matrix M , we construct a subgraph G of Kn,n, as follows. Let the vertices on the left be
v1, v2, . . . vn and let the vertices on the right be w1, w2, . . . wn. There is an edge between vi and wj if and
only if Mij is 1.
Suppose σ is a permutation of {1, 2, . . . n}.... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
2001: Jerrum, Sinclair and Vigoda showed how to approximate the permanent of an arbitrary graph
(and therefore for any matrix with nonnegative entries).
We will show today the result of 1989 for approximating the permanent of a dense graph.
4.2.2 General Strategy
We can’t do the na¨ıve Monte Carlo here, since the p... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
2. Then every partial matching in Mk−1 has
an augmenting path of length ≤ 3.
Proof Let m ∈ Mk−1 be a partial matching. Let u be an unmatched node of m. Now suppose that there
are no augmenting paths of length 1 starting from u in this matching m (i.e. there is no unmatched node
v such that there is an edge connecti... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
f −1(m�)| ≤ (n − k + 1)2 ≤ n2 . Thus |Mk| ≤ n2|Mk−1|.
Now we show that 1/n2 ≤ rk. Fix some m ∈ Mk. By Lemma 6, every partial matching in Mk−1 has
an augmenting path of length ≤ 3. There are at most k partial matchings in Mk−1 that can by augmented
by a path of length 1 to equal m. In addition, there are at most k(k ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
canonical paths.
Lemma 8 Let G = (V, E) be a graph for which we wish to bound Φ(G). For every v, w ∈ V , we specify a
canonical path pv,w from v to w. Suppose that for some constant b and for all e ∈ E, we have
�
v,w∈V
I[e ∈ pv,w] ≤ b|V |
that is, at most b|V | of the canonical paths run through any given edge e. ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
walk on G will converge in polynomial time.
4.2.5 The Graph Ck
We will only do Cn. It should be clear later how to extend this construction for all k. Recall that our
vertices correspond to matchings in Mn ∪ Mn−1. We show how to connect our vertices with 4 different types
of directed edges:
• Reduce (Mn −→ Mn−1): I... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
,2.
4.2.6 Canonical Paths
We still need to define the canonical paths pv,w for our graph Cn. For each node s ∈ Mn ∪Mn−1, we associate
with it a “partner” s� ∈ Mn, as follows:
• If s ∈ Mn, s� = s.
• If s ∈ Mn−1 and has an augmenting path of length 1, augment to get s�.
• If s ∈ Mn−1 and has a shortest augmenting pa... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
are perfect
matchings, d consists of a collection of disjoint, even-length, alternating (from s� or from t�) cycles of length
at least 4.
Our canonical path from s� to t� will in a sense “unwind” each cycle of d individually. Now, in order
for the path to be canonical, we need to provide some ordering on the cycles... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
of transitions.
There is only partner for the first, O(n) for the second, and O(n2) for the third. Therefore there are at most
O(n2) nodes s� that can count s� as their partner.
Now we wish to count the number of canonical paths for type B.
Lemma 11 Let T be a transition (i.e. an edge of Cn). Then the number of pair... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
and after T we agree with t. But
what we can do at this point is notice that one of these two edges (which we denote by es,t) always has
the start vertex of the current cycle as one of its end-points. Therefore by removing it we end up with a
matching again. This is further illustrated in Figure 6.
Theorem 12 The c... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
5-8
Figure by MIT OpenCourseWare.
Figure 5: Unwinding a collection of cycles (type B path).
Figure by MIT OpenCourseWare.
Figure 6: The encoding σT (s, t).
5-9
MIT OpenCourseWare
http://ocw.mit.edu
18.409 Topics in Theoretical Computer Science: An Algorithmist's Toolkit
Fall 2009
For information about citing the... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/0e8570f928771e3992f59e13c91d19bc_MIT18_409F09_scribe5.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.726 Algebraic Geometry
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Cohen-Macaulay schemes and Serre duality
In this lecture, we extend Serre dual... | https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/0ea66a0bdcf4f8127c826882531f1642_MIT18_726s09_lec25_serre_dual.pdf |
X is equidimensional (each irreducible component has dimension n) and
Cohen-Macaulay.
(b) The maps θi are isomorphisms for all i ≥ 0 and all coherent sheaves F on X.
This is of course meaningless if I don’t tell you what a Cohen-Macaulay scheme is. For
the moment, suffice to say that a scheme is Cohen-Macaulay if and... | https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/0ea66a0bdcf4f8127c826882531f1642_MIT18_726s09_lec25_serre_dual.pdf |
sufficiently large, we have H i(X, F (−q)) =
0 for all i < n.
(c′) For q sufficiently large, we have H i(X, OX (−q)) = 0 for all i < n.
Note that condition (c) is a sort of opposite to Serre’s vanishing theorem, which gives
the vanishing of H i(X, F (q)) for i > 0 and q sufficiently large.
Proof. Given (b), for any loca... | https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/0ea66a0bdcf4f8127c826882531f1642_MIT18_726s09_lec25_serre_dual.pdf |
, Ext P
(j∗OX , ωP ) = 0.
N −i
Remember that no matter what X is, we have Ext P
proved this in the course of constructing the dualizing sheaf ω◦
X .
N −i
(j∗OX , ωP ) = 0 for i > n: we
Proof. By Serre duality on P (and choosing an isomorphism H n(P, ωP ) = k), we may identify
∼
H i(X, OX (−q)) = H i(P, j∗OX (−q)... | https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/0ea66a0bdcf4f8127c826882531f1642_MIT18_726s09_lec25_serre_dual.pdf |
is the ideal of A defining X at x, then for all
i <
n, ExtN −i
A
(A/I, A) =
0.
Proof. This translates directly from (d) once we remember that ωP is locally free of rank 1
on P .
This is almost the local condition we are seeking, except that it still refers to the position
of X within P .
2
3 The Cohen-Macaulay... | https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/0ea66a0bdcf4f8127c826882531f1642_MIT18_726s09_lec25_serre_dual.pdf |
is projective if and only if pdA(M) = 0.
For M a module over a ring A, a regular sequence is a sequence x1, . . . , xn of elements
of A such that for i = 1, . . . , n, xi is not a zerodivisor on M/(x1, . . . , xi−1)M. For A a local
ring, the depth of M is the maximal length of a regular sequence with all xi in the m... | https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/0ea66a0bdcf4f8127c826882531f1642_MIT18_726s09_lec25_serre_dual.pdf |
.
On the other hand, we always have depthB(B) ≤ dim(B) ≤ n, so it is equivalent to
require depthB(B) = dim(B) = n.
This condition depthB (B) = dim(B) is in fact the definition of a Cohen-Macaulay lo
cal ring B. Any regular local ring is Cohen-Macaulay, since we can use generators of the
cotangent space as a regular... | https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/0ea66a0bdcf4f8127c826882531f1642_MIT18_726s09_lec25_serre_dual.pdf |
MEDIA AND BOUNDARY CONDITIONS
Media: conductivity σ, permittivity ε, permeability μ
Media are the only tools we have to create or sense EM fields
Conductivity (σ):
⎯J (current density, A/m2) =
nq v
= σ
E
n = #q’s/m3, q = charge (Coulombs),
=v
average velocity (m/s)
Semiconductors:
P{escape} ∝ e-W/kT
+ -
∞ ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/0ec2c39ffccb37e83b43f6f59868f8d7_MIT6_013S09_lec02.pdf |
nucleus
+q
Atom
-
d
⎯E
polarized
atoms
Metal plates
+
+
+
+
+
-
-
-
-
-
E
P
ρv = 0
ρp
+
+
+
+
+
-
-
-
-
-
=P
“Polarization Vector”
L2-2
MAGNETIC MATERIALS
Basic Equations:
B n
• ˆda =
0
(cid:119)∫∫S
B = μoH in vacuum
B = μH = μo (H + M)
M = “Magnetization Vector”
μ = permeability
e
+
+
e... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/0ec2c39ffccb37e83b43f6f59868f8d7_MIT6_013S09_lec02.pdf |
Teflon, petroleum 2.1
Vaseline
Paper
Polystyrene
0.99983
0.99998
0.999991
0.999991
1.000000
1.0000004
1.00002
250
600
2000
5000
100,000
Supermalloy 1,000,000
2.2 Water
2-3
Vacuum
2.6
Air
2.6
Aluminum
3.8
Fused quartz
Cobalt
4.15 Nickel
Ice
Pyrex glass
5.1
Aluminum oxide 8.8
Ethyl alcohol... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/0ec2c39ffccb37e83b43f6f59868f8d7_MIT6_013S09_lec02.pdf |
TO BOUNDARIES
Using Gauss’s Law: (
(cid:119)
∫∫
S
(
)
D D A
→
2
⊥
ˆD nda
•
1
⊥
−
∫∫(cid:119)
S
ˆ
D • n da
=
∫∫∫
v
ρ dv
A
= ρ
s
ˆn
A
(Lim A → 0, δ2 <<A)
)
:
1D
δ
2D
surface charge density ρs
surface S
Therefore:
D
1
⊥
−
D
2
⊥
= ρ
s
yields:
)
nˆ D D
−
2
1
(
•
= ρ
s
∫∫(cid:119)
A
... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/0ec2c39ffccb37e83b43f6f59868f8d7_MIT6_013S09_lec02.pdf |
/ t) • da
∂
∂
=
∫∫
A
H • ds
∫(cid:118)
c
c∫(cid:118)
i
H ds
(
H
→
1//
−
H
)
2 //
Therefore:
(
×
ˆn H H J
−
s
1
=
)
2
L
=
A∫∫(cid:119)
(
J nˆ
• s
→
a
i
J da
)
∂
A∫∫(cid:119)
i
D da
t
∂
→ 0
−
L
L2-8
PERFECT CONDUCTORS
Electric Fields:
{if σ→∞ and⎯E ≠ 0} ⇒ {
(J D/ t) da
+ ∂ ∂
} ⇒ ... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/0ec2c39ffccb37e83b43f6f59868f8d7_MIT6_013S09_lec02.pdf |
0 inside
Cooper pairs of electrons disassociate and superconductivity
fails when the external B(T) is above a critical threshold
L2-9
SUMMARY: BOUNDARY CONDITIONS
General Boundary Conditions:
)
ˆn D D
−
2
1
)
2
ˆn B B 0
−
(
•
(
•
=
1
= ρ
s
1 1E ,D
1 1H ,B
ρs
ρs
⎯Js
(
×
(
×
ˆn E E 0... | https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/0ec2c39ffccb37e83b43f6f59868f8d7_MIT6_013S09_lec02.pdf |
6.012 - Microelectronic Devices and Circuits – Fall 2009
Inverter Analysis and Design
The inverter stage is a basic building block for digital logic circuits and memory
cells. A generic inverter stage is illustrated below on the left. It consists of two devices,
a pull-up device, which is typically either a bipolar... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
To analyze this circuit we not first that with a MOSFET pull-
down, the static input current is zero and if the stage output is connected to the input of
a similar stage, the static stage load current will also be zero, and the equation above is
simply iPU = iPD. With a resistor pull-up, the pull-up current, iPU, is (... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
- vOUT/2)vOUT,
and the desired relationship is found by solving the quadratic
(VDD - vOUT)/R = K(vIN - VT - vOUT/2)vOUT
The resulting transfer characteristic is plotted in the course text in Figure 15.8. Curves
for other pull-up devices are also shown in this section of the text.
We can identify six features of a g... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
smallest values are valid; the middle solution is unstable and will
not be realized in practice. VHI and VLO are often most easily found with the aid of the
graphical technique shown in Figure 15.5 of the course text.
To determine the switching times we must first recognize that the reason an
inverter output does not ... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
the switching
time analytically. There are two cases in which we can calculate the switching time,
3
+-vIN+-PullUpPullDownStage Load+ VqN(vOUT) vOUT however, and to the extent that we can approximate a given situation as one or the
other of these to cases, we can estimate the switching time analytically.
The first s... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
a differential equation for vOUT(t). We should be able to solve this equation
numerically, if we can not do so analytically.
If the charge store can be modeled as a linear capacitor and the charging and
dischargind currents can be modeled as constant, then
τLO→HI ≈ CL(VHI - VLO)/ICH, and τHI→LO ≈ CL(VHI - VLO)/IDCH... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
to design the circuit to have equal magnitude
charging and discharging currents, if possible.
An example to the switching transient situation in the MOSFET inverter with a
resistive pull-up that we illustrated earlier is shown in the figures below. The gate
capacitance of the loading stage(s) has been modeled as a li... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
PU = (V - vOUT)/ROFFCLVLO !VHI The dynamic power dissipation is due the fact that each low-high-low cycle the
output node is charged to roughly VDD, and then discharged to roughly zero volts. If
we can approximate the node charge store with a linear capacitor, CL, the energy
2/2, and the amount of energy stored on t... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-fall-2009/0eddab43e246fc2d62368fe37516020c_MIT6_012F09_lec14_inverter.pdf |
T. Speller
January 11, 2007
Scenarios as a creative practice
Scenarios are another creative practice increasingly being used by companies and gaining
favor from the success of Shell Oil with their future scenarios process (see references
below). We intend our system architectures to not just satisfy current requir... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/0ee4b57ff758f54fbf8bd701834d412d_scenarios.pdf |
design as much as you judge is appropriate. In this viewpoint one might consider
the selected present SA to be the result of all plausibly considered futures. By modularity
and platforming along with other SA strategies, you can achieve flexibility and its close
relative, extensibility, with minimal or modest added ... | https://ocw.mit.edu/courses/esd-34-system-architecture-january-iap-2007/0ee4b57ff758f54fbf8bd701834d412d_scenarios.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.854J / 18.415J Advanced Algorithms
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
�
�
18.415/6.854 Advanced Algorithms
November 19, 2008
Approximaion Algorithms: MAXCUT
Lecturer: Michel X. Goemans
1 MAX-CUT pro... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
Consider a locally maximum cut for this neighborhood.
Lemma 1 If (S : S¯) is a local maximum for the MOVE neighborhood, then w(S : S¯) ≥ 2 w(E) ≥
1 OP T .
2
1
Proof of lemma 1: Look at a vertex i ∈ V . Let Ci be the set of all edges (i, j) ∈ E that are
part of the cut (S : S¯) (that is if i ∈ S then j ∈ S ¯ and vice... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
solution with only k/2 edges in the cut.
(b) The local search algorithm based on the MOVE neighborhood for MAX-CUT takes expo
nentially many steps in the worst-case. This is true even for graphs that are 4-regular (each
vertex has exactly 4 neighbors) (Haken and Luby [1]). For 3-regular graphs the algorithm is
poly... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
In particular,
MAX-CUT with the MOVE neighborhood is PLS-complete [5]. Furthermore, it follows from
Johnson et al. [3] that the obvious local search algorithm is not an efficient way of finding
a local optimum for a PLS-complete problem; indeed, for any PLS-complete problem, there
exist instances for which the local s... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
= E[f |A]P r(A) + E[f |A¯](1 − P r(A))
|
|
{E[f |A], E[f |A¯]
}.
≤ max
In our setting, we consider the vertices in a specific order, say v1, v2,
already decided/conditioned on the position (i.e. whether or not they are in S) of v1,
Now, condition on whether vi ∈ S. Letting f = w(S : S¯), we get:
, and suppose we ha... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
, we notice that we will place vi on the side
, vi−1}.
of the cut that maximizes the total weight between vi and the previous vertices {v1, v2,
This is therefore a simple greedy algorithm.
· · ·
Remarks:
(a) The performance guarantee of the randomized algorithm is no better than 0.5; just consider
the complete gr... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
(e)xe
s.t.
e∈E
0 ≤ xe ≤ 1 ∀e ∈ E
� �
xe −
xe ≤ |F | − 1 ∀cycle C ⊆ E ∀F ⊆ C, |F | odd.
e∈F
e∈C\F
This isa relaxation of the maximum cut problem, and thus provides an upper bound on the value
of the optimum cut. We could try to solve this linear program and devise a scheme to “round” the
possibly fractional so... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
The idea is to use semidefinite programming to get a more useful relaxation of the maximum cut
problem. This is due to Goemans and Williamson [2].
Instead of defining variables on the edges as we did in the previous section, let’s use variables on
the vertices to denote which side of the cut a given vertex is. This le... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
T Ly =
n
� n
�
yiyj lij =
n
� �
y 2
i
w(i, k) −
�
yiyj w(i, j)
i=1 j=1
i=1
k�=i
(i,j)∈E
= 2w(E) −
�
yiyj w(i, j) = 4 ⎝ �
(i,j)∈E
(i,j)∈E
⎛
w(i, j)
1 − yiyj
2
⎞
⎠ ,
and thus
�
(i,j)∈E
w(i, j)
1 − yiyj
2
=
1
4
y T Ly.
Lec18-4
�
�
Thus the maximum cut value is thus equal to
max{
y T LY... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
equal to
1. It is easy to see that the coverse is also true: if Y � 0, rank(Y ) = 1 and Yii = 1 for all i then
Y = yyT where y ∈ {−1, 1}n . Thus we can reformulate the problem as:
max
s.t.
•
L Y
1
4
rank(Y ) = 1,
∀i ∈ V : Yii = 1,
Y � 0.
This is almost a semidefinite program except that the rank condition is ... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
strong duality). Our semidefinite programming relaxation of
MAXCUT is particularly simple and indeed satisfies both the primal and dual regularity conditions:
(a) Primal regularity conditions ∃Y � 0 s.t. Yii = 1 ∀i. This condition is obviously satisfied
(consider Y = I).
Lec18-5
(b) Dual regularity condition: First ... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
, we have that OP T ≥ 0.87856.
SDP
In order to prove this theorem, we will propose an algorithm which derives a cut from the solution
to the semidefinite program. To describe this algorithm, we first need some preliminaries. From the
Cholesky’s decomposition, we know that:
Y � 0 ⇔ ∃V ∈ Rk×n , k = rank(Y ) ≤ n, s.t. Y... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
pp. 822-839.
[5] A.A. Sch¨affer and M. Yannakakis, “Simple local search problems that are hard to solve”, SIAM
Journal on Computing, 20, 56–87, 1991.
Lec18-7 | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/0eef690adf10b98bffd65d5415516412_lec18.pdf |
6.858 Lecture 6
Capabilities and
other Protection Mechanisms
What's the problem the authors of "confused deputy" encountered?
• Their system had a Fortran compiler, /sysx/fort (in Unix filename syntax)
• They wanted the Fortran compiler to record usage statistics,
but where?
o Created
a special statistics ... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
Not so convenient:
can't just open
the
file
like
any
other.
• What if we make gcc setuid to some non-‐root user
(owner
of stats
file)?
o Hard
to
access
user's
original files.
• What
if gcc is setuid-‐root? (Bad
idea, but let's
figure
out why..)
o Lots
of potential... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
be
also
the
privileges
for accessing
it.
2. Complex
permission checks: hard for privileged app to replicate. With simpler
checks, privileged apps might be able to correctly check if another
user should
have access to some object.
What are examples of ambient authority?
• Unix UIDs, GIDs.
• Fire... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
in various
applications.
• Overall
plan:
o Break up an application into smaller components.
o Reduce privileges of components that are most vulnerable to attack.
o Carefully
design interfaces so one component can't compromise another.
• Why is this difficult?
o Hard
to reduce privileges of code ("sand... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
project directory.
o Risk: Bob could replace the file with a symlink to Alice's private file.
o Alice's process will implicitly use Alice's ambient privileges to open.
o Can think of this
as
sandboxing an individual file
operation.
What sandboxing plans (mechanisms) are out there (advantages, limitations)?
• O... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
isolation.
§ Finer-‐grained
isolation is often hard
to get right
(Javascript,
NaCl).
§ E.g., Native
Client
uses both a fine-‐grained
sandbox + OS-‐level
sandbox.
o Will look at these in more detail in later lectures.
Plan 0: Virtualize everything
(e.g., VMs).
• Run untrustworthy code in... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
V
Object -> Permissions -> Allow?
How
would you sandbox a program on a DAC system (e.g., Unix)?
• Must
allocate a new
principal
(user ID):
o Otherwise,
existing principal's privileges will be used implicitly!
• Prevent process from reading/writing other files:
o Change
permissions on every file syst... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
"deputy" for multiple principals.
•
• One solution: check if group permissions allow access (manual, error-‐prone).
o Alternative solution: explicitly specify
privileges
for each operation.
§ Capabilities
can help: capability (e.g., fd) combines object +
privileges.
Unix
§ Some
by
name)
features
incompat.
w... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
o When
process reads low-‐integrity
data, it becomes low integrity too.
o Transitive, prevents adversary from indirectly tampering with files.
• Not immediately useful for sandboxing: only a fixed number of levels.
SElinux.
Idea: system administrator specifies a system-‐wide security policy.
•
• Policy
file ... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
some server?
• Note:
quite
is
seccomp_filter
about
talks
Capsicum
paper
different
the
from
regular/old
regular/old
seccomp.
seccomp,
and
the
]
Is it a good idea to separate policy from application code?
• Depends
on overall goal.
• Potentially good if user/admin wants to look at or change policy.
• Problematic i... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
sandbox's
access
to
objects
in
global
namespaces
• Hard
to
control
Kernel
changes.
•
Just to
double-‐check:
why
do we
need kernel changes?
.
o Can
we implement everything in a library (and LD_PRELOAD it)?
• Represent more things as file descriptors: processes (pdfork).
o Good idea in general... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
up the first "..", which goes to "/foo"
Look up the second "..", which goes to "/"
## should return a cap
• Do Unix permissions still apply?
can't
o Yes
o But intent is that sandbox shouldn't rely on Unix permissions.
because
you
files
in
have
acce
just
cap
dir
all
a
s
for
dir.
• For file
descriptors, add
... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
pipe, lease
file.
(Even a sandbox can
create
a sandbox.)
Advantages: any process can create a new sandbox.
•
Advantages: fine-‐grained control of access to resources (if they map to FDs).
• Files, network
sockets,
processes.
Disadvantage:
weak story
for keeping track of access
to
persi... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
open lots of files all over the place may be cumbersome.
•
tcpdump.
o 2-‐line version: just
cap_enter() after opening
all FDs.
o Used procstat to
look at resulting
capabilities.
o 8-‐line version: also restrict
stdin/stdout/stderr.
o Why?
E.g., avoid reading
stderr log,
changing termina... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
, lazy initialization seems to be a problem.
§ No general-‐purpose
solution
-‐-‐ either
change
code or initialize
early.
o Suggested plan: sandbox and see what breaks.
§ Might
be subtle: gzip
compression level bug.
• What
are the security guarantees it
provides?
o Guarantees
prov... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
icum_and_Casper.pdf
§
§
o There's a port of Capsicum
to Linux (but not in upstream kernel repo).
What applications wouldn't be a good fit for Capsicum?
• Apps that need to control access to non-‐kernel-‐managed
objects.
o E.g.: X server state,
DBus,
HTTP
origins in a web browser,
etc.
o... | https://ocw.mit.edu/courses/6-858-computer-systems-security-fall-2014/0f397e7b504aa9b146763705619d49ba_MIT6_858F14_lec6.pdf |
15.083J/6.859J Integer Optimization
Lecture 9: Duality II
1 Outline
• Solution of Lagrangean dual
• Geometry and strength of the Lagrangean dual
2 The TSP
�
xe = 2,
i ∈ V,
e∈δ({i})
�
xe ≤ |S| − 1,
e∈E(S)
S ⊂ V, S =(cid:5)
∅, V,
min
s.t.
xe ∈ {0, 1}.
�
cexe
e∈E
�
xe = 2,
i ∈ V \ {1},
e∈δ({i})
�
xe =... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/0f3bef63990e2ad5c559713b2dc7fa6e_MIT15_083JF09_lec09.pdf |
gradients
• Prop: f : (cid:8)n (cid:9)
→ (cid:8) is concave if and only if for any x
∗ ∈ (cid:8)n
s ∈ (cid:8)n such that
f (x) ≤ f (x
∗ ) + s
(cid:3)
(x − x
∗ ).
1
, there exists a vector
Slide 4
• Def: f concave. A vector s such that for all x ∈ (cid:8)n:
f (x) ≤ f (x
∗ ) + s
(cid:3)
(x − x
∗ ),
∗
is ca... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/0f3bef63990e2ad5c559713b2dc7fa6e_MIT15_083JF09_lec09.pdf |
s is a subgradient of the
function Z(·) at λ ∗ if and only if Z(λ ∗ ) is a convex combination of the vectors
f k , k ∈ E(λ ∗ ).
�
�
3.2 The subgradient algorithm
Input: A nondifferentiable concave function Z(λ).
Output: A maximizer of Z(λ) subject to λ ≥ 0.
Algorithm:
1. Choose a starting point λ1 ≥ 0; let t = 1. ... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/0f3bef63990e2ad5c559713b2dc7fa6e_MIT15_083JF09_lec09.pdf |
�Z(λt) is rarely met. Typically, the algorithm is
stopped after a fixed number of iterations.
3.3 Example
�
�
• Z(λ) = min 3 − 2λ, 6 − 3λ, 2 − λ, 5 − 2λ, − 2 + λ, 1, 4 − λ, λ, 3 ,
• θt = 0.8t .
Slide 8
2
•
λt
s t Z(λt)
1.5.00 −3 −9.00
2.2.60 −2 −2.20
3.1.32 −1 −0.68
2 −0.66
4.1.83
1 −0.99
5.1.01
1 −0.6... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/0f3bef63990e2ad5c559713b2dc7fa6e_MIT15_083JF09_lec09.pdf |
y + λ(cid:3) z,
• Geometrically, the hyperplane Z(λ) = y + λ(cid:3) z lies below the set Y .
• Theorem:
∀(y, z) ∈ Y.
ZD = min y
s.t.
(y, 0) ∈ conv(Y ).
4.1 Figure
4.2 Example again
X = {(1, 0)(cid:3) , (2, 0)(cid:3) , (1, 1)(cid:3) , (2, 1)(cid:3) , (0, 2)(cid:3) , (1, 2)(cid:3) , (2, 2)(cid:3) , (1, 3)(cid:3) ,... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/0f3bef63990e2ad5c559713b2dc7fa6e_MIT15_083JF09_lec09.pdf |
gradient s t; λt = max{λj +
j
θtsj
t , 0}, where
θt = g
Zˆ − ZLP(λ)
||st||2
with Zˆ an upper bound on ZD, and 0 < g < 2. With λ = λt solve minx∈X f (x)+
λ(cid:3) g(x) to obtain the optimal value Zt and an optimal solution x t .
�
�
4. (Solution update) Update
x ← αx
t + (1 − α)x
where 0 < α <1.
5. (Improving st... | https://ocw.mit.edu/courses/15-083j-integer-programming-and-combinatorial-optimization-fall-2009/0f3bef63990e2ad5c559713b2dc7fa6e_MIT15_083JF09_lec09.pdf |
Lecture Notes for LG’s Diff. Analysis
trans. Paul Gallagher
Feb. 23, 2015
1 The Sobolev Inequality
Suppose that u 2 C 1
∇u is \small", does this imply that u is small?
c (Rn). Clearly, if ∇u = 0, then u = 0. So, we can ask if
Question 1. If u 2 C 1
c (Rn) and
∫
j∇uj = 1, is there a bound for sup juj?
Answer 1. If n = 1,... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/0f7233291fe1c944d7170aeb6f822e42_MIT18_156S16_lec8.pdf |
∥∇u∥L1. As in the scaling example, pick (cid:17) a
smooth bump function, and de(cid:12)ne (cid:17)(cid:21) as before. Then
n(cid:0)1
1
̸
∫
j(cid:17)(cid:21)jp = (cid:21)n
∫
(
(cid:17)pdx (cid:20) (cid:21)n
∫
(∫
)
p
j∇(cid:17)jdx
)
p
= (cid:21)n
(cid:21)(cid:0)n
(
= (cid:21)n
(cid:21)(cid:0)n+1
j(∇(cid:17))(x=(cid:21))... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/0f7233291fe1c944d7170aeb6f822e42_MIT18_156S16_lec8.pdf |
(cid:1) (cid:1) (cid:1) dxn(cid:0)1
j
@nu(x1;
(cid:1) (cid:1) (cid:1)
; xn)
jdxndx1 (cid:1) (cid:1) (cid:1) dxn(cid:0)1
(cid:20)
(cid:20)
Rn(cid:0)1
∫
R
j∇uj
Rn
Then we can use the Loomis-Whitney theorem which we proved in the
homework:
Theorem 1.2 (Loomis-Whitney). If U (cid:26) Rn is open, and j(cid:25)j(U )
all j, t... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/0f7233291fe1c944d7170aeb6f822e42_MIT18_156S16_lec8.pdf |
∫
Rn u(cid:0)1 n (cid:20)
∫
Rn
j∇uj.
We will proceed by induction.
∫
Rn
jujn=(n(cid:0)1) (cid:20)
(cid:20)
(cid:20)
(cid:20)
∫ (∫
R
∫ [∫
Rn(cid:0)
1
R
∫
∫
Rn(cid:0)1
)
dxn
1=(n 1)
(cid:0)
juju1=(n(cid:0)1)
n
]
dx1
[∫
n
n
(cid:0)2
(cid:0)1
(cid:0)
n
1
j
j
u n(cid:0)2
(cid:1) (cid:1) (cid:1) dx
n
(cid:0)1
]
j
(∫
j
un
Rn(... | https://ocw.mit.edu/courses/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016/0f7233291fe1c944d7170aeb6f822e42_MIT18_156S16_lec8.pdf |
Chapter 9 Notes, Part 1 - Inference for Proportion and Count Data
We want to estimate the proportion p of a population that have a specific attribute, like
“what percent of houses in Cambridge have a mouse in the house?”
We are given X1, . . . , Xp where Xi’s are Bernoulli, and P (Xi = 1) = p.
Xi is 1 if house i ... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/0fa15fa0462a921af5d493f45a3f0809_MIT15_075JF11_chpt09a.pdf |
width 2E:
pˆ − E ≤ p ≤ pˆ + E
so
(cid:114)
E = zα/2
pˆqˆ
n
which means n =
zα/2
(cid:16)
E
2
(cid:17)
pˆq. ˆ
Question: We’ll take ˆpqˆ to be its largest possible value, (1/2) × (1/2). Why do we do this?
Why don’t we just use the ˆp and ˆq that we measure from the data?
So, we need:
n =
(cid:16)
zα/2
E
(cid... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/0fa15fa0462a921af5d493f45a3f0809_MIT15_075JF11_chpt09a.pdf |
(cid:17)
p2
1−p2
(cid:16)
p1
1−p1
→ we’ll use this one
“relative risk”
“odds ratio”
2
For large samples, we’ll use the CLT:
pˆ1 − pˆ2 − (p1 − p2)
Z = -
pˆ1qˆ1
n1
pˆ2qˆ2+
n2
≈ N (0, 1)
where ˆp1 = X/n1 and ˆp2 = Y /n2.
To test
H0 : p1 − p2 = δ0,
H1 : p1 − p2 = δ0,
we can just compute zscore... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/0fa15fa0462a921af5d493f45a3f0809_MIT15_075JF11_chpt09a.pdf |
Data Networks
Lecture 1
Introduction
Eytan Modiano
Eytan Modiano
Slide 1
6.263: Data Networks
• Fundamental aspects of network Design and Analysis:
– Architecture
Layering
Topology design
– Protocols
Pt.-to-Pt.
Multiple access
End-to-end
– Algorithms
Error recovery
Routing
Flow Control
– Analysis tool... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
istributed routing algorithms, optimal routing
Flow Control - Window/Credit Schemes
Flow Control - Rate Based Schemes
Transport layer and TCP/IP
ATM Networks
Special topic: Optical Networks, Wireless networks
Final Exam during final exam week. Date and time to be announced.
13
14
15
16
17
18
19
20
21
22... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
bursty
– E.g., Interactive sessions, file transfers, email
• Connection oriented services
– Long sustained session
– Orderly and timely delivery of packets
– E.g., Telnet, FTP
• Connectionless services
– One time transaction (e.g., email)
• QoS
Eytan Modiano
Slide 8
Switching Techniques
• Circuit Switching ... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
lengths
λ = arrival rate of messages
R = channel rate in bits per second
X = message transmission delay = L/R
– R must be large enough to keep X small
– Bursty traffic => λx << 1 => low utilization
• Example
λ = 1 message per second
– L = 1000 bytes (8000 bits)
–
– X < 0.1 seconds (delay requirement)
– => R ... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
to set-up virtual circuit
– Once established, local virtual circuit numbers can then be used to
represent the virtual circuits on a given link: VC number changes from
link to link
• Merits of virtual circuits
– Save on route computation
Need only be done once
at start of session
– Save on header size
– Facilitate ... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
Virtual link for
reliable packets
Virtual bit pipe
DLC
DLC
DLC
DLC
phys. int. phys. int.
phys. int. phys. int.
Data link
Control
physical
interface
Physical link
subnet
node
subnet
node
External
site
Data link
Control
physical
interface
External
Site
Eytan Modiano
Slide 18
Layers
• Presentation layer
– Provid... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
error-free transmission of packets across a
single link
– Framing
Determine the start and end of packets
– Error detection
Determine which packets contain transmission errors
– Error correction
Retransmission schemes (Automatic Repeat Request (ARQ))
Eytan Modiano
Slide 22
Physical Layer
• Responsible for transmiss... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
driver
token ring
Protocol
token ring
driver
Ethernet
token ring
Eytan Modiano
Slide 25
Encapsulation
user data
Appl
header
user data
TCP
header
application data
TCP segment
IP
header
TCP
header
IP datagram
application data
Ethernet
header
14
IP
header
20
TCP
header
20
application data
Ethernet
trailer
4
Ethernet fr... | https://ocw.mit.edu/courses/6-263j-data-communication-networks-fall-2002/0feb60d4d98f77a9c08991b4960139f3_Lecture1.pdf |
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