ADVANCED ALGORITHMS Lecture 18: optimization, linear programming 1 - - PowerPoint PPT Presentation

SLIDE 1

ADVANCED ALGORITHMS

Lecture 18: optimization, linear programming

1

SLIDE 2

ANNOUNCEMENTS

➤ HW 4 is due on Monday, November 5 ➤ Project meetings… 2

SLIDE 3

RANDOMIZED ALGORITHMS — SUMMARY

3 ➤ Typical use of randomness: ➤ “many” good solutions — think of checking if p(x) != q(x) ➤ Probability of failure — accuracy / running time ➤ Expected run time — quicksort, median selection ➤ Random variables (key to analysis), expectation, linearity

_=

'

(

random recurrence

)

1- I

SLIDE 4

KEY EXAMPLES

4 ➤ Balls and bins (random variables, expectation, etc.) ➤ Sampling — “concentration” of random variables ➤ Key result about sampling: if we want to estimate 0/1 outcome to an error +/- s, we need ~ 1/s2 samples

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SLIDE 5

CONCENTRATION OF RANDOM VARIABLES

5 ➤ Meta result: random variables do not deviate much from their expectation ➤ Markov’s inequality ➤ “Standard deviation” (also variance): amount we expect the variable to deviate from expectation ➤ Chebychev’s inequality ➤ Stronger bounds for sums of independent variables = . X >

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SLIDE 6

PARTING THOUGHTS

6 ➤ Is there “true randomness”? (semi-positive answers) ➤ Can randomness help solve “truly hard” problems? IT . . . . T #

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SLIDE 7

OPTIMIZATION

7

• Code
• ptimization

descent .

• Gradient

inane .

• Genetic

algorithms .

• bjective

, constraints

• ptimize

.

SLIDE 8

OPTIMIZATION WITH CONSTRAINTS

8 ➤ {x1, x2, …, xn} are variables — values in some domain D ➤ find maximum value of f(x) subject to
 g1(x) ≥ 0 g2(x) ≥ 0 …. Meta problem — can phrase many problems as optimization (why?) {

• ,

B . ↳

• bjective function

.

}

→ constraints .

• Decades
• f

research

• n

how to solve problems efficiently .

SLIDE 9

EXAMPLE: SCHEDULING JOBS ON MACHINES

9 ➤ What are variables? ➤ What are constraints? ➤ What is the objective?
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SLIDE 10

EXAMPLE: SCHEDULING JOBS ON MACHINES

10

variables

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if

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SLIDE 11

Xij Elo ,

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.

SLIDE 12

EXAMPLE: MATCHING VERTICES

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SLIDE 13

EXAMPLE: MATCHING VERTICES

12

SLIDE 14

EXAMPLE: SHORTEST PATHS

13

• Variables

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• Constraint

?

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SLIDE 15

EXAMPLE: SHORTEST PATHS

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SLIDE 16

EXAMPLE: SPANNING TREE

15

SLIDE 17

EXAMPLE: SPANNING TREE

16

SLIDE 18

EXAMPLE: MAXIMUM FLOW

17

SLIDE 19

ZOOMING OUT

18 ➤ What are variables? ➤ What are constraints? ➤ What is the objective? ➤ Why?


SLIDE 20

WHEN CAN WE SOLVE OPTIMIZATION?

19 ➤ Linear constraints, objective ➤ Convexity