Fall 2019

1

Can you fill a 𝟖×𝟖 board with the corners missing using dominoes?

Can you tile this?

With these?

Fall 2019

2

Can you fill a 𝟖×𝟖 board with the corners missing using dominoes?

Can you tile this?

With these?

Today’s Keywords

• Dynamic Programming• Gerrymandering

3

CLRS Readings

• Chapter 15• Chapter 16

4

Administrativa

• Midterm take-home due tonight at 11pm– Individual work!

• Homework 5 out tonight!– Due Thursday, October 24 at 11pm– Seam Carving!– Dynamic Programming (implementation)– Java or Python

5

Dynamic Programming

• Requires Optimal Substructure– Solution to larger problem contains the solutions to smaller ones

• Idea:1. Identify the recursive structure of the problem• What is the “last thing” done?

2. Save the solution to each subproblem in memory3. Select a good order for solving subproblems• “Top Down”: Solve each recursively• “Bottom Up”: Iteratively solve smallest to largest

6

Generic Top-Down Dynamic Programming Solnmem = {}def myDPalgo(problem):

if mem[problem] not blank:return mem[problem]

if baseCase(problem):solution = solve(problem)mem[problem] = solutionreturn solution

for subproblem of problem:subsolutions.append(myDPalgo(subproblem))

solution = OptimalSubstructure(subsolutions)mem[problem] = solutionreturn solution

7

DP Algorithms so far

• 2×𝑛 domino tiling (Fibonacci) • Log cutting• Matrix Chaining• Longest Common Subsequence• 4×𝑛 domino tiling (Homework 4 Problem)• Seam Carving (Homework 5 Problem)

8

Domino Tiling

9

Tile(n):Initialize Memory MM[0] = 0M[1] = 0for i = 0 to n:

M[i] = M[i-1] + M[i-2]return M[n]

M

0

1

2

3

4

5

6

Log Cutting

10

10987654321Length:

𝐶𝑢𝑡(𝑖): 0

0

Solve Smallest subproblem first

𝐶𝑢𝑡 4 = max

𝐶𝑢𝑡 3 + 𝑃[1]𝐶𝑢𝑡 2 + 𝑃 2𝐶𝑢𝑡 1 + 𝑃 3𝐶𝑢𝑡 0 + 𝑃[4]

4

Matrix Chaining

11

30

35

×𝑀9 35

15

×𝑀:15

5

×𝑀; 5

10

×𝑀<

10

20

×𝑀= 20

25

𝑀>

𝐵𝑒𝑠𝑡 𝑖, 𝑗 = minFG9

HI9𝐵𝑒𝑠𝑡 𝑖, 𝑘 + 𝐵𝑒𝑠𝑡 𝑘 + 1, 𝑗 + 𝑟L𝑟HM9𝑐F

𝐵𝑒𝑠𝑡 𝑖, 𝑖 = 0 0 15750 7875 9375 11875

0 2625 4375 7125 10500

0 750 2500 5375

35000 1000

50000

0

1 2 3 4 5 61

2

3

4

5

6

𝐵𝑒𝑠𝑡 1,6 = min

𝐵𝑒𝑠𝑡 1,1 + 𝐵𝑒𝑠𝑡 2, 6 + 𝑟9𝑟:𝑐>𝐵𝑒𝑠𝑡 1,2 + 𝐵𝑒𝑠𝑡 3, 6 + 𝑟9𝑟;𝑐>𝐵𝑒𝑠𝑡 1,3 + 𝐵𝑒𝑠𝑡 4, 6 + 𝑟9𝑟<𝑐>𝐵𝑒𝑠𝑡 1,4 + 𝐵𝑒𝑠𝑡 5, 6 + 𝑟9𝑟=𝑐>𝐵𝑒𝑠𝑡 1,5 + 𝐵𝑒𝑠𝑡 6, 6 + 𝑟9𝑟>𝑐>

15125

𝑗 == 𝑖

Generic Top-Down Dynamic Programming Solnmem = {}def myDPalgo(problem):

if mem[problem] not blank:return mem[problem]

if baseCase(problem):solution = solve(problem)mem[problem] = solutionreturn solution

for subproblem of problem:subsolutions.append(myDPalgo(subproblem))

solution = OptimalSubstructure(subsolutions)mem[problem] = solutionreturn solution

12

Generic Top-Down Dynamic Programming Soln

mem = [][]Matrix = [] #1-D array of all 2-D matricesdef matrixChain(i,j):

if mem[i][j] not blank:return mem[i][j]

if (j-i) == 0:mem[i][j] = 0return 0

if (j-i) == 1:solution = size(Matrix[i]) * size(Matrix[j]) * size(Matrix[j][0]) #cost to multiplymem[i][j] = solutionreturn solution

for k in i+1 .. j-1:subsolutions.append(matrixChain(i,k) + matrixChain(k+1,j) + cost(i,k,j))

solution = min(subsolutions) # optimal substructuremem[i][j] = solutionreturn solution 13

Longest Common Subsequence

14

𝐿𝐶𝑆 𝑖, 𝑗 =0 if 𝑖 = 0 or𝑗 = 0𝐿𝐶𝑆 𝑖 − 1, 𝑗 − 1 + 1 if 𝑋 𝑖 = 𝑌[𝑗]max(𝐿𝐶𝑆 𝑖, 𝑗 − 1 , 𝐿𝐶𝑆 𝑖 − 1, 𝑗 ) otherwise

0 0 0 0 0 0 0 0

0 0 1 1 1 1 1 1

0 0 1 1 1 2 2 2

0 0 1 2 2 2 2 2

0 1 1 2 2 2 3 3

0 1 2 2 3 3 3 4

0 1 2 2 3 3 4 4

𝑋 = 𝐴 𝑇 𝐶 𝑇𝑇 𝐺 𝐴1 2 3 74 5 60𝑌 =

0

1

3456

2𝑇

𝐶𝐴𝑇𝐴

𝐺

To fill in cell (𝑖, 𝑗) we need cells 𝑖 − 1, 𝑗 − 1 , 𝑖 − 1, 𝑗 , (𝑖, 𝑗 − 1)Fill from Top->Bottom, Left->Right (with any preference)

Seam Carving

15

S(n-1,k-1)

𝑝],H

S(n-1,k) S(n-1,k+1)

S(n,k)

𝑆 𝑛, 𝑘 = 𝑚𝑖𝑛 𝑆 𝑛 − 1, 𝑘 − 1 + 𝑒(𝑝],H)

𝑆 𝑛 − 1, 𝑘 + 𝑒(𝑝],H)

𝑆 𝑛 − 1, 𝑘 + 1 + 𝑒(𝑝],H)

16

17

Gerrymandering

• Manipulating electoral district boundaries to favor one political party over others

• Coined in an 1812 Political cartoon• Governor Elbridge Gerry signed a

bill that redistricted Massachusetts to benefit his Democratic-Republican Party

18

The Gerrymander

According to the Supreme Court

• Gerrymandering cannot be used to:– Disadvantage racial/ethnic/religious groups

• It can be used to:– Disadvantage political parties

19

VA 5th District

20

VA 5th District

21

2018 Election

Gerrymandering Today

• Computers make it really effective

22

")4

")1

")3

")7

")2

Ivor

Carsley

Courthouse

Berlin

Sebrell

Precinct 3-1

Newville

Yale

Pons

Surry

Eastern

Orbit

Brandon

Chesapeake

Salem

Zuni

Little Zion

Piankatank

Templeton

Westville

Old Mill

Harris Grove

4-B

Bryan

Achilles

Harrison

Stony Creek

Roanes

Henry

Rushmere

Wakefield

Raynor

Precinct 3-1

Harcum

Enon

Bacon's Castle

New Market

Precinct 1-1

Bethel

Eltham

Cumberland

Courthouse

Saluda

Dendron

Claremont

Asbury

Sweet Hall

Precinct 2-1

Reams

Lee Hall

Wilton

Waller Mill

Whitlocks

Magruder

Bartlett

Watkins

Blackwater

Shackleford's

Courthouse

Blackwater

Courthouse

Rives

Chuckatuck

Providence Forge

Windsor

Stonehouse A

White Marsh

Bland

Stonehouse C

Powhatan A

Elko

Old Church

Tunstall

Carrollton

Little Mill

New Hope

Wall's Bridge

Courthouse

Nash

Roberts B

Driver

Seaford

Botetourt

Matoaca

Town Hall

Mars Hill

Rohoic

Dorey

Southern Branch

Church View

Edgehill

Timberneck

Walters

Airport

Thirty Nine

Battlefield

Black Creek

Dare

Cherry Hill

Eanes

Winfrees Store

Burbank

Ebenezer

Jamestown A

Union Branch

King's Fork

Yeates

Dunbar

609

Iron Bridge

Harmony Village

Nansemond River

Ettrick

Kentwood

HayesCourts Bldg

Sullivans

Antioch

Smithfield

4-A

Ecoff

806

Armstrong

Mehfoud

Clay

Spring Grove

Zion Grace

Stryker

Sandston

Bellwood

West Wakefield

Jefferson Park

Bird

Hilton

Yorktown

Gates

Elizabeth Scott

Precinct 5-1

Richard Bland

Deep Creek

Stonehouse B

Berkeley C

Bennetts Creek

413

Warwick

Chickahominy

Machen

Tabb

106

Cold Harbor

Quinton

Rural Point Precinct 1-1

811

Dinwiddie

Rolfe

West Point

812

Harrowgate

Chickahominy River

Drewryville

Manquin

Ocean View School

Greenwood

Downtown

Bland

Kiln Creek

Glen Lea

Ward 2

Phillips

Chesdin

Thirty-Six

903

Reed

410

Davis

River

Urbanna

Berkeley B Part 1

Wilder

Five Forks

Riverside

Bethel

Ten

Western

Jolliff One

Central

Village

Beulah

Donahoe

503

402

101

Wells

910

Ocean View Center

810

Drewry's Bluff

Fifth

Five

Laburnum

Southside

Dutch Gap

Sedley

404

Fourth

Totopotomoy

706

Kiln Creek

203

Bayview School

Palmer

Nelson

Kraft

Nine

703

Harwoods Mill

Atlee

Epes

Ward 4

Roberts C Part 1

Easton

Powhatan C

607

Titustown Center

Deer Park

Cooper

Carver

North Chester

602

Avalon

Lindsay

Lake Cohoon

Waverly

Edgehill

309

South Chester

505

610

Barron Black

412

Third

Queens Lake

Bethel

Smith

AzaleaBelmont

204

Sunray Ii

Second

Ratcliffe

814

Coventry

Willard

Riverview

Hermitage

Fairlawn

Thomas

Powhatan D

707

Watkins

Phenix

Cedar Fork

Aberdeen

Yates

Wythe

Thirty

Denbigh

Falling Creek

Masonic

510

Berkley

Newman

603

Stonewall Jackson

508

Camelot

Salem Church

909

501

Courthouse

301

Fellowship

Powhatan B

Montrose

306

Briarfield

GranbyWesley

One

802

104

Richneck

St. Julians

509

307

Mallory

Tarrallton

Crestview

111

Adams

Lakeside

Booker

Bassette

Old Dominion

Sanford

308

902

Glenside

Campostella

Hunterdale

102

105

Rosemont

Longan

Twenty

Beach

River Birch

Tuckahoe112

Shady Grove

302

Staples Mill

Jamestown B

Berkeley

Langley

Harbour View

Wilson

Sarah's Creek

911

First Ward First Precinct

Seventh Ward First Precinct

Phoebus

Berkeley A Part 1

Greendale

East Hampton

Roberts A Part 2

First

Reservoir

Pleasants

Nansemond

Tallwood

Charles

McIntosh

Jenkins

Ward 1

Manchester

Georgetown

Skipwith

Sandy Bottom

Poplar Halls

Second Ward First Precinct

Fourteen

Meadowbrook

604

Precinct 4-1

Churchland

305

Ward 3

Ward 7

Saunders

Seven

Hungary

Taylor Road

Moody

504

304

United Way

Syms

Forrest

Berkeley B Part 2

Johnson

South Norfolk Recreation

Beaufont 908

Fairfield

Maplewood606

Carver

Indian River

Oaklette

Third Ward First Precinct

East Ocean View

City Hall

Crestwood

LongdaleYellow Tavern

Thirty-Four Ingleside

Ward 5Ward 6

Silverwood

Chippenham

213

Bailey Creek

Thirty-Five

Reon

701

Stuart

Tucker Capps

Thirty-Two

Bayside

Hidenwood

Providence

Seventeen

114

Tanglewood

Chamberlayne

206

Boulevard

Nineteen

MechanicsvilleHighland Gardens

211

Thirty-One

Belmont

Sherry Park

Sixteen

Carver School

Bland

Laurel Meadow

Wellesley

Providence

Thirty Seven

Westwood

Sunray I

Brambleton

Thirty Eight

E. W. Chittum School

Northside

Twenty-Five

Dumbarton

Joliff Middle School

Sedgefield

Larrymore

Holllybrook

Chesterfield

705

Roberts A Part 1

Central Gardens

Pebble Creek

Crossroads

702

Taylor Elementary SchoolMaury

Jones

Newmarket

Twenty-Eight

Fourth Ward First Precinct

113

Beaverdam Creek

Park PlaceBallentine

Twenty-Two

Hanover Grove

Thirteen

Johnson Park

Little Creek

303

Three Chopt

Twenty-Seven

Geneva Park

Windsor

Chrysler Museum

Kecoughtan

Bowling Park

Suburban Park

Maude Trevvett

Highland Springs

Eleven

Sixth Ward First Precinct

Tyler

Young Park

Jefferson

Thirty-Three

207

Rollingwood

Sherwood Rec CenterLafayette

Twenty-Three

Summit CourtHilliard

208

Greenwood

Larchmont Library

Fifth Ward First Precinct

Twenty-One

Third Presbyterian

Azalea Gardens

Twenty-Four

Shell

Twenty-Six

Twenty-Nine

South Morrison

Brookland

212Nine Mile

Arrowhead

Marshall

College Park

South Norfolk

Union Chapel

Oakview

Lafayette-Winona

Hampton Library

Oceanair

Tucker

Norview Methodist

Tanner's Creek

Roberts C Part 2

Lambert's Point

Norfolk Highlands

Newsome Park

Norview Middle School

Magruder

Hunton Y

Jacobs

Larchmont Recreation Center

Monument Hills

Oyster Point

Centerville

Oscar Smith School

Coleman Place SchoolGhent Square

Courtland

Canterbury

Stratford Hall

Tucker House

ReamsHuguenot

Bon Air

4-C Elephants Fork/Westhaven

StudleyStudley

Sussex

Surry

York

Gloucester

Isle of Wight

Henrico

Mathews

New Kent

Prince George

Charles City

James City

Suffolk

Dinwiddie

Chesterfield

Middlesex

Southampton

Hampton

Norfolk

Hanover

Newport News

King William

Lancaster

King and Queen

Poquoson

Portsmouth

Greensville

Richmond city

Chesapeake

Petersburg

Hopewell

WilliamsburgColonial Heights

HB 251 (2012)Congressional District 3

Gerrymandering Today

• Computers make it really effective

23

Gerrymandering Today

24

How does it work?

• States are broken into precincts• All precincts have the same size• We know voting preferences of each precinct• Group precincts into districts to maximize the number of districts

won by my party

25

R:65D:35

R:45D:55

R:47D:53

R:60D:40

Overall: R:217 D:183

vs

How does it work?

• States are broken into precincts• All precincts have the same size• We know voting preferences of each precinct• Group precincts into districts to maximize the number of districts

won by my party

26

R:65D:35

R:45D:55

R:47D:53

R:60D:40

Overall: R:217 D:183

R:65D:35

R:45D:55

R:47D:53

R:60D:40

R:125 R:92

R:65D:35

R:45D:55

R:47D:53

R:60D:40

R:112 R:105

Gerrymandering Problem Statement

• Given:– A list of precincts: 𝑝9, 𝑝:, … , 𝑝]– Each containing 𝑚 voters

• Output:– Districts 𝐷9, 𝐷: ⊂ {𝑝9, 𝑝:, … , 𝑝]}– Where 𝐷9 = |𝐷:|– 𝑅 𝐷9 > g]

<and 𝑅 𝐷: > g]

<• 𝑅(𝐷L) gives number of “Regular Party” voters in 𝐷L• 𝑅 𝐷L > ij

<means 𝐷L is majority “Regular Party”

– “failure” if no such solution is possible

27

𝑚 ⋅𝑛2 ⋅12

Valid Gerrymandering!

Dynamic Programming

• Requires Optimal Substructure– Solution to larger problem contains the solutions to smaller ones

• Idea:1. Identify the recursive structure of the problem• What is the “last thing” done?

2. Save the solution to each subproblem in memory3. Select a good order for solving subproblems• “Top Down”: Solve each recursively• “Bottom Up”: Iteratively solve smallest to largest

28

Dynamic Programming

• Requires Optimal Substructure– Solution to larger problem contains the solutions to smaller ones

• Idea:1. Identify the recursive structure of the problem• What is the “last thing” done?

2. Save the solution to each subproblem in memory3. Select a good order for solving subproblems• “Top Down”: Solve each recursively• “Bottom Up”: Iteratively solve smallest to largest

29

World Two

World One

Consider the last precinct

30

𝐷9𝑘 precincts𝑥 voters for R

𝐷:𝑛 − 𝑘 − 1 precincts𝑦 voters for R

𝑝]

𝐷9𝑘 + 1 precincts𝑥 + 𝑅(𝑝]) voters for R

𝐷:𝑛 − 𝑘 precincts𝑦 + 𝑅(𝑝]) voters for R

If we assign 𝑝]to 𝐷9

If we assign 𝑝]to 𝐷:

After assigning the first 𝑛 − 1 precincts

𝑝9, 𝑝:, … , 𝑝]G9

Valid gerrymandering if: 𝑘 + 1 = ]

:,

𝑥 + 𝑅 𝑝] , 𝑦 >g]<

Valid gerrymandering if:n − 𝑘 = ]

:,

𝑥, 𝑦 + 𝑅 𝑝] > g]<

𝐷9𝑘 + 1 precincts𝑥 + 𝑅(𝑝]) voters for R

𝐷9𝑘 precincts𝑥 voters for R

𝐷:𝑛 − 𝑘 − 1 precincts𝑦 voters for R

𝐷:𝑛 − 𝑘 precincts𝑦 + 𝑅(𝑝]) voters for R

Define Recursive Structure

𝑆 𝑗, 𝑘, 𝑥, 𝑦 =

31

True if from among the first 𝒋 precincts:𝒌 are assigned to 𝐷9exactly 𝒙 vote for R in 𝐷9exactly 𝒚 vote for R in 𝐷:

4D Dynamic Programming!!!

𝑛 × 𝑛 ×𝑚𝑛 ×𝑚𝑛

32

Two ways to satisfy 𝑆 𝑗, 𝑘, 𝑥, 𝑦 :𝑆 𝑗, 𝑘, 𝑥, 𝑦 = True if:

from among the first 𝑗 precincts𝑘 are assigned to 𝐷9exactly 𝑥 vote for R in 𝐷9exactly 𝑦 vote for R in 𝐷:

𝐷9𝑘 precincts𝑥 voters for R

𝐷:𝑗 − 𝑘 precincts𝑦 voters for R

𝐷9𝑘 − 1 precincts𝑥 − 𝑅(𝑝F) voters for R

𝐷:𝑗 − 𝑘 precincts𝑦 voters for R

𝐷9𝑘 precincts𝑥 voters for R

𝐷:𝑗 − 1 − 𝑘 precincts𝑦 − 𝑅(𝑝F) voters for R

𝑝F

Then assign 𝑝Fto 𝐷9

Then assign 𝑝Fto 𝐷:

𝑝F

OR

𝑆 𝑗, 𝑘, 𝑥, 𝑦 = 𝑆 𝑗 − 1, 𝑘 − 1, 𝑥 − 𝑅 𝑝F , 𝑦 ∨ 𝑆 𝑗 − 1, 𝑘, 𝑥, 𝑦 − 𝑅 𝑝F

Final Algorithm

Initialize 𝑆(0,0,0,0) = Truefor 𝑗 = 1,… , 𝑛:

for 𝑘 = 1,… ,min(𝑗, ]:):

for 𝑥 = 0,… , 𝑗𝑚:for 𝑦 = 0,… , 𝑗𝑚:𝑆 𝑗, 𝑘, 𝑥, 𝑦 =

𝑆 𝑗 − 1, 𝑘 − 1, 𝑥 − 𝑅 𝑝F , 𝑦 ∨ 𝑆 𝑗 − 1, 𝑘, 𝑥, 𝑦 − 𝑅 𝑝FSearch for True entry at 𝑆(𝑛, ]

:, > g]

<, > g]

<)

33

𝑆 𝑗, 𝑘, 𝑥, 𝑦 = 𝑆 𝑗 − 1, 𝑘 − 1, 𝑥 − 𝑅 𝑝F , 𝑦 ∨ 𝑆 𝑗 − 1, 𝑘, 𝑥, 𝑦 − 𝑅 𝑝F

𝑆 𝑗, 𝑘, 𝑥, 𝑦 = True if:from among the first 𝑗 precincts𝑘 are assigned to 𝐷9exactly 𝑥 vote for R in 𝐷9exactly 𝑦 vote for R in 𝐷:

Initialize 𝑆(0,0,0,0) = Truefor 𝑗 = 1,… , 𝑛:

for 𝑘 = 1,… ,min(𝑗, ]:):

for 𝑥 = 0,… , 𝑗𝑚:for 𝑦 = 0,… , 𝑗𝑚:𝑆 𝑗, 𝑘, 𝑥, 𝑦 =

𝑆 𝑗 − 1, 𝑘 − 1, 𝑥 − 𝑅 𝑝F , 𝑦 ∨ 𝑆 𝑗 − 1, 𝑘, 𝑥, 𝑦 − 𝑅 𝑝FSearch for True entry at 𝑆(𝑛, ]

:, > g]

<, > g]

<)

Run Time

34

𝑛𝑛2𝑛𝑚𝑛𝑚

Θ(𝑛<𝑚:)

𝑆 𝑗, 𝑘, 𝑥, 𝑦 = 𝑆 𝑗 − 1, 𝑘 − 1, 𝑥 − 𝑅 𝑝F , 𝑦 ∨ 𝑆 𝑗 − 1, 𝑘, 𝑥, 𝑦 − 𝑅 𝑝F

Θ(𝑛<𝑚:)

• Input: list of precincts (size 𝑛), number of voters (integer 𝑚)• Runtime depends on the value of 𝑚, not size of 𝑚– Run time is exponential in size of input– Input size is 𝑛 + 𝑚 = 𝑛 + log𝑚

• Note: Gerrymandering is NP-Complete

35