Randomized Algorithms CS 648 Lecture 18 Approximate Distance

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Randomized Algorithms CS 648 Lecture 18 • Approximate Distance Oracles • Algorithm for Min-cut

Randomized Algorithms CS 648 Lecture 18 • Approximate Distance Oracles • Algorithm for Min-cut : part 1 1

APPROXIMATE DISTANCE ORACLES 2

APPROXIMATE DISTANCE ORACLES 2

All-Pairs Shortest Paths • 3

All-Pairs Shortest Paths • 3

All-Pairs Shortest Paths • 4

All-Pairs Shortest Paths • 4

All-Pairs Approximate Shortest Paths • 5

All-Pairs Approximate Shortest Paths • 5

A truly magical result Approximate Distance Oracles Space Query time Preprocessing time Mikkel Thorup

A truly magical result Approximate Distance Oracles Space Query time Preprocessing time Mikkel Thorup and Uri Zwick: Approximate Distance Oracles for graphs, Journal of ACM (4), 2005 6

INSPIRATION FROM OUR DAILY LIFE 7

INSPIRATION FROM OUR DAILY LIFE 7

Air/Road Network 8

Air/Road Network 8

The Idea • 9

The Idea • 9

Formal notion of locality 10

Formal notion of locality 10

Formal notion of locality • 11

Formal notion of locality • 11

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What is the stretch ? ? ? 13

What is the stretch ? ? ? 13

3 -approximate distance oracle • Global distance info. Local distance info. 14

3 -approximate distance oracle • Global distance info. Local distance info. 14

The real challenge left • 15

The real challenge left • 15

The real challenge left 16

The real challenge left 16

Conquering the challenge • 17

Conquering the challenge • 17

Expected space of 3 -approximate distance oracle • 19

Expected space of 3 -approximate distance oracle • 19

5 -APPROXIMATE DISTANCE ORACLE Meant for only those (hopefully nonzero no. of) students whose

5 -APPROXIMATE DISTANCE ORACLE Meant for only those (hopefully nonzero no. of) students whose aim is more than just a good grade in this course. 21

3 -approximate distance oracle 22

3 -approximate distance oracle 22

5 -approximate distance oracle 23

5 -approximate distance oracle 23

5 -approximate distance oracle 24

5 -approximate distance oracle 24

PROBLEM 2 MIN-CUT 25

PROBLEM 2 MIN-CUT 25

Min-Cut • 26

Min-Cut • 26

Min-Cut • 27

Min-Cut • 27

SOME BASIC FACTS 28

SOME BASIC FACTS 28

Min-Cut • 29

Min-Cut • 29

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Algorithm for min-cut • 34

Algorithm for min-cut • 34

Algorithm for min-cut • 35

Algorithm for min-cut • 35

Algorithm for min-cut • 36

Algorithm for min-cut • 36

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