acm so far… sep 11 welcome! and dp problems ~ 6 problems sep 18 lab session ~ 6 problems sep 25...

ACM so far…

Sep 11 Welcome! and DP problems ~ 6 problems

Sep 18 Lab session ~ 6 problems

Sep 25 Discussion session on graph problems ~ 6 problems

Oct 2 Lab session on graph problems ~ 6 problems

Oct 9 Scott Ellsworth on Google Irvine ~ 6 problems

Oct 16 Discussion session on maxflow problems ~ 6 problems

Oct 23 (9pm) Lab & local ACM qualifying contest ~ 6 problems

Oct 30 Discussion session on geometry problems ~ 6 problems

Nov 6 Lab session on geometry problems ~ 6 problems

Nov 10 (Sat.) ACM Regional contest (in Riverside...)

Nov 13 Final meeting

Job-fair thoughts?

Our "sources"• Aaron Gable, Amber Yust, and other alums…

• Cory Simmonsen

Web technologies

Build systems

Distributed systems (scaling up)

Testing!

Working with a large existing codebase…

What should be improved?

IT always seems mysterious to me…

Hooray!

Not sure about these, however…

Matrix of skillsets

Ford-Fulkerson algorithm

What's the maximum flow possible, from src to sink?

s

B

E

D

C13

t

16

10 4 9

12

14

7

20

4source

capacity

Max Flow !

sink or target

s

B

E

D

C13

sB

C

D

E

FROM

s B C D E

t

- 16 13 - -- - 10 12 -

- 4 - - 14

- - 9 - -

- - - 7 -

--

-

20

4

- - - - - -t

t

16

10 4 9

12

14

7

20

4

TO

Capacity Graph

source

sink

(Step #1) Use depth- or breadth-first search to find any path from s to t.

Max Flow

What's left ?

s

B

E

D

C13

sB

C

D

E

FROM

s B C D E

t

- 16 13 - -- - 10 12 -

- 4 - - 14

- - 9 - -

- - - 7 -

--

-

20

4

- - - - - -t

t

4/16

10 4 9

0/12

14

7

8/20

4

TO

Old capacities

source

sink


Max Flow

What's left… sB

C

D

E

FROM

s B C D E

- 4 13 - -12 - 10 0 -

- 4 - - 14

- 12 9 - -

- - - 7 -

--

-

8

4

- - - 12 - -t

t

TO

Residual capacities.

and the red edges?Backwards capacities!

12 12

12

s

B

E

D

C13

sB

C

D

E

FROM

s B C D E

t

- 16 13 - -- - 10 12 -

- 4 - - 14

- - 9 - -

- - - 7 -

--

-

20

4

- - - - - -t

t

4

10 4 9

0

14

7

8

4

TO

source

sink


Max Flow

sB

C

D

E

FROM

s B C D E

- 4 13 - -12 - 10 0 -

- 4 - - 14

- 12 9 - -

- - - 7 -

--

-

8

4

- - - 12 - -t

t

TO

12 12

12

(Step #2) Continue with the remaining capacities until no path exists!

Old capacities

Residual capacities.

Backwards capacities.

New capacities

B

E

D

C12/13

11/16

0/10 1/4 0/9

12/12

11/14

7/7

19/20

4/4

max flow: 23


Max Flow

(Step #2) Continue with the remaining capacities until no path exists!

ssource

t

sink

Setting up…

if __name__ == "__main__":

# make a capacity graph # node A B C D E F C = [ [ 00, 16, 13, 00, 00, 00 ], # A [ 00, 00, 10, 12, 00, 00 ], # B [ 00, 04, 00, 00, 14, 00 ], # C [ 00, 00, 9, 00, 00, 20 ], # D [ 00, 00, 00, 7, 00, 4 ], # E [ 00, 00, 00, 00, 00, 00 ] ] # F

print "C is", C source = 0 # A sink = 5 # F

max_flow_value = max_flow( C, source, sink ) print "max_flow_value is", max_flow_value

And the code needed to run it…

Linked at the ACM website by the slides…

Get into the flow!

def max_flow(C, source, sink): n = len(C) # C is the capacity matrix F = [[0] * n for i in range(n)] # F is the flow matrix # residual capacity from u to v is C[u][v] - F[u][v]

while True: path = BFS(C, F, source, sink) if not path: break # no path - we're done!

# find the path's flow, that is, the "bottleneck" edges = [C[u][v]-F[u][v] for u,v in path] path_flow = min( edges ) print "Augmenting by", path_flow for u,v in path: # traverse path to update flow F[u][v] += path_flow # forward edge up F[v][u] -= path_flow # backward edge down

return sum([F[source][i] for i in range(n)]) # out from source

A little bit of name contention…

edmonds_karpThis is the algorithm.

Useful alone, too

def BFS(C, F, source, sink): queue = [source] # the BFS queue paths = {source: []} # stores 1 path per graph node while queue:

u = queue.pop(0) # next node to explore (expand) for v in range(len(C)): # for each possible next node

# path from u to v? and not yet at v? if C[u][v] - F[u][v] > 0 and v not in paths: paths[v] = paths[u] + [(u,v)] if v == sink: return paths[v]

queue.append(v) # go from v in the future return None

A brief BFS algorithm using the Capacity matrix

But is max flow good for anything?

that is, beyond solving "max flow" problems...

we have four brides and six grooms

Matching! and some acceptable possibilities ...

a bipartite graph



a maximal matching == no more matchings without rearrangement



a maximum matching == no rearrangements will yield more matchings

Maximum matching is max flow...

ssource

connect a source to the left side...

all 1s


ssource


make all capacities = 1

1

1

11

1

1

all 1s


ssource

t

sink


put a sink on the right

make all capacities = 1

1

1

11

1

1

all 1sall 1s

what do the source and sink constraints ensure?

Max flow thought experiment...

ssource

t

sink

1

1

11

1

1

all 1sall 1s

Suppose this is the flow so far (3 units):

Draw what happens in the next step of the max-flow algorithm!how to get from maximal matching to maximum matching…


ssource

t

sink

1

1

11

1

1

all 1sall 1s

... the path it finds ...

What's going on here?


ssource

t

sink

1

1

11

1

1

all 1sall 1s

Done!

Maximum matching == 4

general problems: max-flow problems:

cowcarn

feeding

optimilk

sandcastimecards

tswift

difficult-to-classify problems:

This week's problemsTry max flow!

soda (last week)tour (last week)

The challenge:

is sometimes setting up the graph

4 44218910100050 1 320 1 3 8 2 3 1 4

How do we use the results?

What is flowing?

tswiftThere are four ingredients available ~ at these costs

hay-flavored

coffee

spam

chocolate

There are four smoothie recipes available ~ with these rewards

4 ingredients (hay) & 4 smoothie recipes

each recipe requires ingredients

1

2

3

4

4 44218910100050 1 320 1 3 8 2 3 1 4

source

42

189

10

1000ingredient

costs


What is flowing?


hay-flavored

coffee

spam

chocolate


Hay

coffee

Spam

Choc.



1

2

3

4ingredients

4 44218910100050 1 320 1 3 8 2 3 1 4

sourcesink

42

189

10

1000ingredient

costs

recipe rewards

50

20

8

3


What is flowing?


hay-flavored

coffee

spam

chocolate


Hay

coffee

Spam

Choc.



Hay-spam

Spam-hay

coffee

Choco-hay

1

2

3

4ingredients recipes

4 44218910100050 1 320 1 3 8 2 3 1 4

sourcesink

42

189

10

1000ingredient

costs

recipe rewards

50

20

8

3


What is flowing?


hay-flavored

coffee

spam

chocolate


Hay

coffee

Spam

Choc.



Hay-spam

Spam-hay

coffee

Choco-hay

1

2

3

4

One purple edge is missing… Which?

ingredients recipes

What should these capacities be?

general problems: max-flow problems:

cowcarn

feeding

optimilk

sandcastimecards

tswift

difficult-to-classify problems:

This week's problemsTry max flow!

soda (last week)tour (last week)

This code only looks

obfuscated!

Donut!

http://www.ioccc.org/

donut.c

International Obfuscated C Coding Contest

This code

IS

obfuscated!

2011's winner…

http://www.ioccc.org/

donut.c

International Obfuscated C Coding Contest

This code

IS

obfuscated!

Tools

4 44218910100050 1 320 1 3 8 2 3 1 4

source

Jobs

sink

42

189

10

1000tool costs

task rewards

50

20

8

3


What is flowing?

hardwareThere are four tools available ~ at these costs

hammer

TV

coffee

PC

There are four tasks available ~ with these rewards

hammer

TV

coffee

PC

4 tools & 4 tasks

each task requires some tools

E4

waking folks in

east

sleep

coding

dinner

4 54 5 3 53 5 2 6 44 54 5 3 53 5 2 6 30 0

number of teams Input

Output

number of tables

# of people in each team

can an assignment be made without putting teammates together?

01

capacity of each table

again…

end…

35

2 6 4

tables with capacities

teams with sizes

5

34

5

seating assignments!

no teammates

dinner's maxflow graph

ssource t

sink

How does the maxflow here relate to whether the seating is possible or not?

Team

Team

Team

Team

Table

Table

Table

Table

Table

4

3

6

5

2

5

5

3

4

fully connected with edge

weights of 1

How do these edge weights reflect the problem constraints?

JRs SRs Elderly

slate 3

This term's first class to guess another's word earns 1 problem...

slate 2 slate 1

This term's last class to have its word guessed earns 1 problem...

Sophs

slate 1

flair 0 flair 1 flair 2 flair 0

Pomona

slate 3

flair 2

stems 3 stems 1 stems 2 stems 1 stems 2

loser 2 loser 3 loser 2loser 1 loser 3

stone 3 stone 2 stone 1 stone 1 stone 2

guppy 1 guppy 0 guppy 1 guppy 2 guppy 0

Try max flow!

lasso 1 lasso 3 lasso 1 lasso 1 lasso 2

pluot 1 pluot 2 pluot 1 pluot 1 pluot 0

old years…

hardware

Tools

4 44218910100050 1 320 1 3 8 0 3 1 4

source

Jobs

sink

42

189

10

1000

tool costs

job rewards

50

20

8

3

How can max flow help us here?

What is flowing?

4 3 32 2 1 2 3 12 2 2 3 1 22 2 1 3 1 22 1 1 3 3

number of cows

Inputtotal # of foods

total # of drinks

# of foods cow[i] likes

# of drinks cow[i] likesfoods drinks

0

Output

# of cows that can receive both a food and

a drink they like…

3

each can be used only once

Likes

foods drinks

1

2

3

1 2

2 3

1 3

1 3

3 1

1 2

1 2

3What is a cow-satisfying assignment here?

dining

Jotto! Sophs Jrs Srs

audio 1 audio 2 audio 1

Frosh

audio 2 graze 3 graze 1 graze 1 graze 2

alloy 1 alloy 1 alloy 1 alloy 2 fresh 2 fresh 2 fresh 2 fresh 1



armor 2 armor 2 armor 1 armor 2 brave 3 brave 1 brave 1 brave 2 wreak 3 wreak 1 wreak 2 wreak 2 fjord 1 fjord 5 fjord 1 fjord 2

Stake

4 1 0 1 1 0 1 0 0 0 0 0 1 0 1 0 1

Input

Output3

Height and width of the field

the

patt

ern

Maximum number of cows such that no two share a column and no two share a row.

Tools

4 44218910100050 1 320 1 3 8 2 3 1 4

source

Jobs

sink

50

20

8

3task rewards

tool costs

42

189

10

1000

How do we use mf to maximize our profit?

What is flowing?

hardwareThere are four tools available ~ at these costs

hammer

TV

coffee

PC

There are four tasks available ~ with these rewards

hammer

TV

coffee

PC

4 tools & 4 tasks

each task requires some tools

E4

waking folks in

east

sleep

coding

Stake as matchingwho are the brides? and the grooms?

and the constraints?

This week's problems…

Try one or more of this week's problems!

this one is from last week... ?

Dijkstra, for single-source shortest paths

Put <S,0> into your queue Q.

While Q not empty:

Remove Q's nearest node <nc,dc>

For each edge [nc, nk, dc2k]:

For all nk , track <nk, Inf>

Let dk be nk's distance: <nk,dk>

If dc + dc2k < dk: set dk = dc + dc2k

Put <nk,dk> into Q...

S

shortest dist from S

Jotto! Sophs Jrs Srs

audio 1 audio 2 audio 1

Frosh

audio 2 graze 3 graze 1 graze 1 graze 2

alloy 1 alloy 1 alloy 1 alloy 2 fresh 2 fresh 2 fresh 2 fresh 1



armor 2 armor 2 armor 1 armor 2 brave 3 brave 1 brave 1 brave 2 wreak 3 wreak 1 wreak 2 wreak 2 fjord 1 fjord 5 fjord 1 fjord 2 taper 4 taper 1 taper 1 taper 2 tater 4 tater 1 tater 1 tater 2

acm so far… sep 11 welcome! and dp problems ~ 6 problems sep 18 lab session ~ 6 problems sep 25...

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