Harbin, China2010 finals

ACM today… !

Knapsack

DP for NP!

Floyd-Warshall

DP for APSP!

ACM "abroad"

ACM progress…

Candy!

1800's

1950's

1970's

1920's

1940's

Candy!

1800's

1950's

1970's

1920's

1940's

Maximizing Candy!i = 1

i = 2

i = 3

i = 4

i = 5

Suppose you can consume 13 candywt units of candy.What choice will maximize your candy-value experience?

(1) if you can take fractional parts of candy packages…?

(2) if you can take any number of whole candy packages…?

(3) if you can take 0 or 1 of each whole candy package…?

w1 = 2

v1 = 100

w2 = 3

v2 = 120

w3 = 5

v3 = 230

w4 = 7

v4 = 560

w5 = 9

v5 = 675

Maximizing Candy!i = 1

i = 2

i = 3

i = 4

i = 5

Suppose you can consume 13 candywt units of candy.What choice will maximize your candy-value experience?

(1) if you can take fractional parts of candy packages…?

(2) if you can take any number of whole candy packages…?

(3) if you can take 0 or 1 of each whole candy package…?

GO WITH GREED!

w1 = 2

v1 = 100

vpw1 = 50

w2 = 3

v2 = 120

vpw2 = 40

w3 = 5

v3 = 230

vpw3 = 46

w4 = 7

v4 = 560

vpw4 = 80

w5 = 9

v5 = 675

vpw5 = 75

Knapsack problemi = 1

i = 2

i = 3

i = 4

i = 5

Suppose you can consume 13 candywt units of candy.

(2) if you can take any number of whole candy packages…?

"unbounded"

IDEA:

Consider all possible weights (integers) from 0 up to the weight you can carry -For each one choose the best from all N items

w1 = 2

v1 = 100

vpw1 = 50

w2 = 3

v2 = 120

vpw2 = 40

w3 = 5

v3 = 230

vpw3 = 46

w4 = 7

v4 = 560

vpw4 = 80

w5 = 9

v5 = 675

vpw5 = 75

Knapsack problemi = 1

i = 2

i = 3

i = 4

i = 5

Suppose you can consume 13 candywt units of candy.

(2) if you can take any number of whole candy packages…?

"unbounded"

0 1 2 3 4 5 6 7 8 9 10 11 12 13

TOTAL WEIGHT

max total value

V(w) =

w1 = 2

v1 = 100

vpw1 = 50

w2 = 3

v2 = 120

vpw2 = 40

w3 = 5

v3 = 230

vpw3 = 46

w4 = 7

v4 = 560

vpw4 = 80

w5 = 9

v5 = 675

vpw5 = 75

Knapsack problemi = 1

i = 2

i = 3

i = 4

i = 5

Suppose you can consume 13 candywt units of candy.

(3) if you can take 0 or 1 of each whole candy package…?

w1 = 2

v1 = 100

vpw1 = 50

w2 = 3

v2 = 120

vpw2 = 40

w3 = 5

v3 = 230

vpw3 = 46

w4 = 7

v4 = 560

vpw4 = 80

w5 = 9

"0-1 problem"

IDEA:

Do the same thing as before, but consider sublists of items that grow one-at-a-time

v5 = 675

vpw5 = 75

Knapsack problem w1 = 2

v1 = 100

i = 1

i = 2

i = 3

i = 4

i = 5

vpw1 = 50

w2 = 3

v2 = 120

vpw2 = 40

w3 = 5

v3 = 230

vpw3 = 46

w4 = 7

v4 = 560

vpw4 = 80

w5 = 9

Suppose you can consume 13 candywt units of candy.

"0-1 problem"

0 1 2 3 4 5 6 7 8 9 10 11 12 13

TOTAL WEIGHT

max total value

V(i,w) =

(3) if you can take 0 or 1 of each whole candy package…?

Total Items

0

01

012

0123

01234

012345v5 = 675

vpw5 = 75

Knapsack problem w1 = 2

v1 = 100

i = 1

i = 2

i = 3

i = 4

i = 5

vpw1 = 50

w2 = 3

v2 = 120

vpw2 = 40

w3 = 5

v3 = 230

vpw3 = 46

w4 = 7

v4 = 560

vpw4 = 80

w5 = 9

v5 = 675

vpw5 = 75

Suppose you can consume 13 candywt units of candy.

"0-1 problem"

0 1 2 3 4 5 6 7 8 9 10 11 12 13

TOTAL WEIGHT

max total value

V(i,w) =

(3) if you can take 0 or 1 of each whole candy package…?

Total Items

0

01

012

0123

01234

012345

DP for APSP…

Directed graph as adjacency matrix:

1

2

34

14

14

14

10

50

100

Directed graph:

src

1

2

3

4

1 2 3 4

"from"

dst"to"

100inf140

0

0

0

inf 5014

infinf

infinf

14

10

0 intermediate nodes

Floyd-Warshall!

src

1

2

3

4

1 2 3 4

"from"

dst"to"

100inf140

0

0

0

inf 5014

infinf

infinf

14

10

0 intermediate nodes

src

1

2

3

4

1 2 3 4

"from"

dst"to"

100inf140

0

0

0

inf 5014

infinf

inf24

14

10

1 intermediate node(s)!1

Floyd-Warshall!

src

1

2

3

4

1 2 3 4

"from"

dst"to"

100inf140

0

0

0

inf 5014

infinf

inf24

14

10

src

1

2

3

4

1 2 3 4

"from"

dst"to"

64inf140

0

0

0

inf 5014

infinf

inf24

14

10

2 intermediate node(s)!1

1 intermediate node(s)1 2

Floyd-Warshall!

src

1

2

3

4

1 2 3 4

"from"

dst"to"

4228140

0

0

0

inf 2814

infinf

3824

14

10

src

1

2

3

4

1 2 3 4

"from"

dst"to"

4228140

0

0

0

38

2814

38

24

3824

14

10

4 intermediate node(s) ~ done!1

3 intermediate node(s)1 22 3 3 4

All-pairs shortest paths:

Wikipedia's code

Longest paths? What has to be true about the

graph?

My hope is for FW to become a

Wikipedia pseudocode primitivenot yet!

Tracking the paths…Floyd Warshall…

Dynamic programming

When a seemingly intractable problem has lots of repeated substructure, go DP!

Build a table of partial results.

Replace computation with table look-up when possible

notree

Input

Output

4 50 1 1 0 92 5 5 1 331 5 5 1 100 1 1 0 99

dimensions grid of costs for clearing trees

6

minimum cost

4 50 1 1 0 92 5 5 1 331 5 5 1 100 1 1 0 99

Cows helicopter in and clear trees in order to "connect

the dots"

notreeinline bool valid (int r, int c){ return r >= 0 && r < R && c >= 0 && c < C;}

void init (){ memset (dist, 63, sizeof (dist)); int dr [] = {-1, 0, 1, 0}, dc [] = {0, 1, 0, -1};

for (int r = 0; r < R; r++) for (int c = 0; c < C; c++) for (int d = 0; d < 4; d++) { int nr = r + dr [d], nc = c + dc [d];

if (valid (nr, nc)) dist [r][c][nr][nc] = cost [nr][nc]; }

for (int a = 0; a < R; a++) for (int b = 0; b < C; b++) for (int c = 0; c < R; c++) for (int d = 0; d < C; d++) for (int e = 0; e < R; e++) for (int f = 0; f < C; f++) dist [c][d][e][f] = min (dist [c][d][e][f], dist [c][d][a][b] + dist [a][b][e][f]);}

Where's Floyd Warshall here?

4 50 1 1 0 92 5 5 1 331 5 5 1 100 1 1 0 99

cost [nr][nc]

notree

Build a table! All subsets of treats…000 001 010 011 100 101 110 111

All starting points (r,c)

…

(0,0)

(0,1)

(0,9)

(1,0)

(1,1)

(1,9)

(9,0)

(9,1)

(9,9)

…

…

…

…

Each cell holds the minimum cost, starting from (r,c) of reaching all the treats in the

subset.

Which squares are easy?

notree

Build a table! All subsets of treats…000 001 010 011 100 101 110 111

All starting points (r,c)

…

(0,0)

(0,1)

(0,9)

(1,0)

(1,1)

(1,9)

(9,0)

(9,1)

(9,9)

…

…

…

…

Each cell holds the minimum cost, starting from (r,c) of reaching all the treats in the

subset.

Which squares are easy?

T[ss][r,c] = min T[s1][r,c] + T[s2][r,c]

s1 & s2 = s

T[ss][r,c] = min T[ss][a,b] + dist[r,c][a,b]

(a,b)

notree

Build a table! All subsets of treats…000 001 010 011 100 101 110 111

All starting points (r,c)

…

(0,0)

(0,1)

(0,9)

(1,0)

(1,1)

(1,9)

(9,0)

(9,1)

(9,9)

…

…

…

…

Each cell holds the minimum cost, starting from (r,c) of reaching all the treats in the

subset.

T[ss][r,c] = min T[s1][r,c] + T[s2][r,c]

s1 & s2 = s

T[ss][r,c] = min T[ss][a,b] + dist[r,c][a,b]

(a,b)

for (int s = ss; s != 0; s = s - 1 & ss) for (int i = 0; i < R; i++) for (int j = 0; j < C; j++) T[ss][i][j] = min(T[ss][i][j], T[s][i][j] + T[ss^s][i][j]);Crazy!

ACM contest == CS60?

Greg Rae, HMC trustee

Most venerable problem in cs60? (and

cs42?)

these lines are from CS

60, Fall 1999

ACM regionals ~

CS 60?

Tautology checking (and parsing!)

( no Prolog permitted )

ACM regionals ~

CS 60?

20 questions: given objects, given questions, and given the answers to the questions (!),

Identify the object in the specified number of questions…

ACM regionals ~

CS 60?

from top left to bottom right as cheaply as possible…

"Quiz"

Read over this week's problems…

Which is shortest paths?

Which is knapsack?

What else is there?

Finished? Jotto!

Jotto!

Sophs Jrs Srs “Other”

A word-guessing game similar to mastermind…

icily 0 icily 0 icily 1 icily 1

Next Tuesday: Labbring a laptop, if you

have one…

strep 2 strep 2 strep 2 strep 1

spork 1 spork 3 spork 0 spork 0