how to win coding competitions: secrets of champions week ... filehow to win coding competitions:...

How to Win Coding Competitions: Secrets of Champions

Week 3: Sorting and Search AlgorithmsLecture 1: Introduction to Sorting

Maxim BuzdalovSaint Petersburg 2016

Introductory example

Find Cormen on this bookshelf

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Too slow to be practical :/

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2 / 8


Find Cormen when books are in alphabetical order

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Much faster if sorted! :)

3 / 8



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a


3 / 8



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3 / 8

Sorting – Formally

I Given a set S with total ordering : S× S→ false, true

I Reflexive: a aI Antisymmetric: (a b), (b a)⇒ (a = b)I Transitive: (a b), (b c)⇒ (a c)I Total: if not (a b) then (b a)

I Given a sequence of N items A1,A2, . . . ,AN , each from the set SI Find a permutation P = [p1, p2, . . . , pN ], such that:

I for all i ∈ [1;N − 1] it holds that Api Api+1

I In other words, construct a sequence B1,B2, . . .BN , such that:I Every Bi has exactly one corresponding Aj , and vice versaI Bi Bi+1

I Note:I Strict ordering: A ≺ B := A B and A 6= BI For numbers (S = Z,R, . . .), is often ≤I We will denote as ≤, and ≺ as <, in the rest of the week materials

4 / 8


I Given a set S with total ordering : S× S→ false, trueI Reflexive: a aI Antisymmetric: (a b), (b a)⇒ (a = b)I Transitive: (a b), (b c)⇒ (a c)I Total: if not (a b) then (b a)





4 / 8



I Given a sequence of N items A1,A2, . . . ,AN , each from the set S

I Find a permutation P = [p1, p2, . . . , pN ], such that:I for all i ∈ [1;N − 1] it holds that Api Api+1



4 / 8







4 / 8






I Note:I Strict ordering: A ≺ B := A B and A 6= B

I For numbers (S = Z,R, . . .), is often ≤I We will denote as ≤, and ≺ as <, in the rest of the week materials

4 / 8






I Note:I Strict ordering: A ≺ B := A B and A 6= BI For numbers (S = Z,R, . . .), is often ≤

I We will denote as ≤, and ≺ as <, in the rest of the week materials

4 / 8







4 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering1. Does this book exist?

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering1. Does this book exist? Gamma

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering1. Does this book exist? Gamma → YES

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

8G

am

ma

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering1. Does this book exist? Kant

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering1. Does this book exist? Kant → NO

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

11

Ho

are

12

Ker

nig

ha

n

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering2. How many books of Patashnik is there?

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering2. How many books of Patashnik is there? Two

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

18

Pa

tash

nik

19

Pa

tash

nik

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering3. How many books have a name smaller than Ritchie?

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

20

Rit

chie

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast query answering3. How many books have a name smaller than Ritchie? 19 (the index minus 1)

1A

bel

son

2A

ho

3B

roo

ks

4C

orm

en

5D

ijk

stra

6E

ckel

7F

oris

ek

8G

am

ma

9G

od

el

10

Gra

ha

m

11

Ho

are

12

Ker

nig

ha

n

13

Kn

uth

14

La

mp

ort

15

McC

on

nel

l

16

Mey

er

17

Nor

vig

18

Pa

tash

nik

19

Pa

tash

nik

21

Ru

ssel

22

Set

hi

23

Sed

gew

ick

24

Sk

ien

a

25

Sp

ols

ky

26

Str

ou

stru

p

27

Su

ssm

an

28

Ta

nen

ba

um

29

Tu

rin

g

30

Wir

th

20

Rit

chie

5 / 8

Why sorting?

Why do we need sorting? – Prepare to fast data processing

Given segment endpoints where paint was applied. Find which length was painted

6 / 8

Why sorting?

Why do we need sorting? – Prepare to fast data processingGiven segment endpoints where paint was applied.

Find which length was painted

1 4

2 5

8 9

7 11

10 12

6 / 8

Why sorting?

Why do we need sorting? – Prepare to fast data processingGiven segment endpoints where paint was applied. Find which length was painted

1 4

2 5

8 9

7 11

10 12

6 / 8

Why sorting?


1 4

2 5

8 9

7 11

10 12

Solution:

I Sort segments by the left coordinateI Traverse segments in the sorted order

I Track the endpoint of the current segment clusterI Finish the cluster when the next segment is beyond this point

6 / 8

Why sorting?


1 4

2 5

8 9

7 11

10 12

Solution:

I Sort segments by the left coordinate

I Traverse segments in the sorted order


6 / 8

Why sorting?


1 4

2 5

8 9

7 11

10 12

Solution:

I Sort segments by the left coordinateI Traverse segments in the sorted order


6 / 8

Why sorting?

Why do we need sorting? – Construct an optimal solution

7 1 4 6 8 2 9 3 1 4 3 5 9 8 2

3 1 2 8 6 5 4 7 4 9 4 5 1 3 8

7 / 8

Why sorting?

Why do we need sorting? – Construct an optimal solutionExample:

I Given two sequences A = [A1, . . .AN ] and B = [B1, . . .BN ]

I Find permutations P and Q such that∑N

i=1 APi· BQi

is maximum possible

7 1 4 6 8 2 9 3 1 4 3 5 9 8 2

3 1 2 8 6 5 4 7 4 9 4 5 1 3 8

7 / 8

Why sorting?




i=1 APi· BQi

is maximum possible

Solution:

I Sort A in non-decreasing order

I Sort B in non-decreasing order

I These facts will be proven in a this week’s video

7 1 4 6 8 2 9 3 1 4 3 5 9 8 2

3 1 2 8 6 5 4 7 4 9 4 5 1 3 8

7 / 8

Why sorting?




i=1 APi· BQi

is maximum possible

Solution:




1 1 2 2 3 3 4 4 5 6 7 8 8 9 9

3 1 2 8 6 5 4 7 4 9 4 5 1 3 8

7 / 8

Why sorting?




i=1 APi· BQi

is maximum possible

Solution:




1 1 2 2 3 3 4 4 5 6 7 8 8 9 9

1 1 2 3 3 4 4 4 5 5 6 7 8 8 9

7 / 8

Why sorting?




i=1 APi· BQi

is minimum possible

Solution:


I Sort B in non-increasing order


1 1 2 2 3 3 4 4 5 6 7 8 8 9 9

9 8 8 7 6 5 5 4 4 4 3 3 2 1 1

7 / 8

What will we learn this week for sorting?

I Insertion sort: a simple sorting algorithm

I If a sorted sequence yields an optimal result, how to prove it?

I Quick sort: a very fast algorithm

I Merge sort: can never be too slow

I Stable and unstable sorting algorithms

I How to compare various objects

I An Ω(n log n) bound on the complexity of comparison-based algorithms

I Bucket sort and radix sort: Linear non-comparison sorting algorithms

I Sorting algorithms in standard libraries

8 / 8

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