1. ACM so far… Sep 6 Welcome! and DP problems ~ 5 problems
Sep Lab session ~ 4 problems
Sep Discussion session on graph problems ~ 4 problems
Sep Lab session on graph problems ~ 4 problems
Oct Discussion session on geometry problems ~ 4 problems
Oct Lab session on geometry problems ~ 4 problems
Oct Lab & local ACM qualifying contest ~ 6 problems
Oct No meeting
Nov Discussion session on Max Flow ~ 4 problems
Nov Lab session ~ 4 problems
Nov (Sat.) ACM Regional contest (in Riverside...)
Nov Final meeting (may be a make-up lab if we miss one)

2. This code looks obfuscated!

3. International Obfuscated C Coding Contest
Donut!
International Obfuscated C Coding Contest
donut.c

4. What's the maximum flow possible, from src to sink?
Max Flow ! B 12 D 16 20
sink or target t s 10 4 9 7
source 13 4 C E 14
capacity
What's the maximum flow possible, from src to sink?
Ford-Fulkerson algorithm

5. Max Flow B D t s C E What's left ? Capacity GraphTO
Max Flow
Capacity Graph s B C D E t s - 16 13 10 12 4 14 9 7 20 B
(Step #1) Use depth- or breadth-first search to find any path from s to t. C
FROM D E B 12 D t 16 20
sink t s 10 4 9 7
source 13 4 C E 14
What's left ?

6. Max Flow B D t s C E What's left… Old capacities- 16 13 10 12 4 14 9 7 20 B
(Step #1) Use depth- or breadth-first search to find any path from s to t. C
FROM D E B
0/12 D t
4/16 12
8/20
sink 12 12 t s 10 4 9 7
source 13 4 C E 14 TO s B C D E t
What's left… s - 4 13 12 10 14 9 7 8
Residual capacities. B C
FROM D
and the red edges? E
Backwards capacities! t

7. (Step #2) Continue with the remaining capacities until no path exists!TO
Max Flow
Old capacities s B C D E t s - 16 13 10 12 4 14 9 7 20 B
(Step #1) Use depth- or breadth-first search to find any path from s to t. C
FROM D E B D t 4 12 8
sink 12 12 t s 10 4 9 7
source 13 4 C E 14 TO s B C D E t
Residual capacities. s - 4 13 12 10 14 9 7 8
Backwards capacities. B C
FROM D
(Step #2) Continue with the remaining capacities until no path exists! E t

8. Max Flow max flow: 23 B D s t C E
(Step #1) Use depth- or breadth-first search to find any path from s to t. B
12/12 D
11/16
19/20
sink s
0/10
1/4 t
0/9
7/7
source
12/13
4/4 C E
11/14
(Step #2) Continue with the remaining capacities until no path exists!
max flow: 23

9. And the code needed to run it…
Setting up…
And the code needed to run it…
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
Linked at the ACM website by the slides…

10. Get into the flow! edmonds_karp def max_flow(C, source, sink):
A little bit of name contention…
This is the algorithm.
edmonds_karp
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

11. A brief BFS algorithm using the Capacity matrix
Useful alone, too
A brief BFS algorithm using the Capacity matrix
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

12. But is max flow good for anything?
that is, beyond solving "max flow" problems...

13. and some acceptable possibilities ...
Matching!
and some acceptable possibilities ...
we have four brides
and six grooms
a bipartite graph

14. and some acceptable possibilities ...
Matching!
and some acceptable possibilities ...
we have four brides
and six grooms
a maximal matching
== no more matchings without rearrangement

15. and some acceptable possibilities ...
Matching!
and some acceptable possibilities ...
we have four brides
and six grooms
a maximum matching
== no rearrangements will yield more matchings

16. connect a source to the left side...
Maximum matching is max flow...
connect a source to the left side...
all 1s s
source

17. connect a source to the left side...
Maximum matching is max flow...
connect a source to the left side...
make all capacities = 1 1 1
all 1s 1 s 1
source 1 1

18. Maximum matching is max flow...
connect a source to the left side...
make all capacities = 1
put a sink on the right 1 1
all 1s
all 1s 1
sink s 1 t
source 1 1
what do the source and sink constraints ensure?

19. Max flow thought experiment...
Suppose this is the flow so far (3 units): 1 1
all 1s
all 1s 1
sink s 1 t
source 1 1
Draw what happens in the next step of the max-flow algorithm!
how to get from maximal matching to maximum matching…

20. Max flow thought experiment...
... the path it finds ... 1 1
all 1s
all 1s 1
sink s 1 t
source 1 1
What's going on here?

21. Max flow thought experiment...
Done! 1 1
all 1s
all 1s 1
sink s 1 t
source 1 1
Maximum matching == 4

22. This week's problems… dinner dining hardware muddy optmilk
all can be done with maxflow…

23. is often setting up the graph
The challenge:
is often setting up the graph

24. can an assignment be made without putting teammates together?
Input
dinner
number of teams
number of tables
# of people in each team
4 5
0 0
Output 1
capacity of each table
again…
can an assignment be made without putting teammates together?
end…
tables with capacities 3
teams with sizes 5
seating assignments! 4 3
no teammates 6 5 2 4 5

25. fully connected with edge weights of 1
dinner's maxflow graph
How do these edge weights reflect the problem constraints?
Team
Table 2 4
Table 5 3
Team s 6
sink
Table 5
fully connected with edge weights of 1 t
source
Team 3
Table 5 4
Team
Table
How does the maxflow here relate to whether the seating is possible or not?

26. 4 tools & 4 tasks
4 4 42
189 10
1000
50 1 3
20 1 3
8 2
3 1 4
hardware
hammer
There are four tools available ~ at these costs TV
coffee PC E4
each task requires some tools
hammer 50 42
waking folks in east 20 TV
There are four tasks available ~ with these rewards
189
sink
source 8
PERL stories
- use in ACM
- creating scripts
- LTS, BOT 10
coffee
sleep
1000 3
task rewards
tool costs PC
coding
What is flowing?
Tools
Jobs
How do we use the results?

27. Try max flow!

28. Try max flow! Sophs JRs SRs Elderly Pomona slate 1 slate 3 slate 2
flair 0
flair 0
flair 1
flair 2
flair 2
stems 1
stems 3
stems 1
stems 2
stems 2
loser 1
loser 2
loser 3
loser 2
loser 3
stone 1
stone 3
stone 2
stone 1
stone 2
guppy 2
PERL stories
- use in ACM
- creating scripts
- LTS, BOT
guppy 1
guppy 0
guppy 1
guppy 0
lasso 1
lasso 1
lasso 3
lasso 1
lasso 2
pluot 1
pluot 1
pluot 2
pluot 1
pluot 0
This term's first class to guess another's word earns 1 problem...
This term's last class to have its word guessed earns 1 problem...

29. old years…

30. hardware
4 4 42
189 10
1000
50 1 3
20 1 3
8 0
3 1 4
Tools
Jobs 50 42 20
189
sink
source 8
PERL stories
- use in ACM
- creating scripts
- LTS, BOT 10 3
1000
job rewards
tool costs
What is flowing?
How can max flow help us here?

31. hardware
4 4 42
189 10
1000
50 1 3
20 1 3
8 0
3 1 4
Tools
Jobs 50 42 20
189
sink
source
PERL stories
- use in ACM
- creating scripts
- LTS, BOT 8 10 3
1000
job rewards
tool costs
What is flowing?
How can max flow help us here?

32. number of cows
dining
Input
total # of foods
total # of drinks
4 3 3
1 2
2 3
1 3
Likes
3 1
1 2 3
# of foods cow[i] likes
foods
drinks 1
# of drinks cow[i] likes 2
Output 3
What is a cow-satisfying assignment here? 3
# of cows that can receive both a food and a drink they like…
foods
drinks
each can be used only once

33. Jotto! Frosh Sophs Jrs Srs audio 2 audio 1 audio 2 audio 1 graze 2
alloy 2
alloy 1
alloy 1
alloy 1
fresh 1
fresh 2
fresh 2
fresh 2
armor 2
armor 2
armor 2
armor 1
brave 2
brave 3
brave 1
brave 1
wreak 2
wreak 3
wreak 1
wreak 2
PERL stories
- use in ACM
- creating scripts
- LTS, BOT
fjord 1
fjord 2
fjord 1
fjord 5
This term's first class to guess another's word earns 1 problem...
This term's last class to have its word guessed earns 1 problem...

34. Stake
Height and width of the field
Input 4
the pattern
Output 3
Maximum number of cows such that no two share a column and no two share a row.

35. 4 tools & 4 tasks
4 4 42
189 10
1000
50 1 3
20 1 3
8 2
3 1 4
hardware
hammer
There are four tools available ~ at these costs TV
coffee PC
each task requires some tools
hammer E4 42 50
189
waking folks in east TV
There are four tasks available ~ with these rewards 20
sink
source 10
PERL stories
- use in ACM
- creating scripts
- LTS, BOT 8
sleep
coffee 3
1000
tool costs
task rewards
coding PC
What is flowing?
Tools
Jobs
How do we use mf to maximize our profit?

36. Stake as matching who are the brides? and the grooms?
and the constraints?

37. Try one or more of this week's problems!
this one is from last week... ?
Try one or more of this week's problems!

38. Dijkstra, for single-source shortest paths
shortest dist from S
For all nk , track <nk, Inf>
Put <S,0> into your queue Q.
While Q not empty:
Remove Q's nearest node <nc,dc>
For each edge [nc, nk, dc2k]:
Let dk be nk's distance: <nk,dk>
If dc + dc2k < dk: set dk = dc + dc2k
Put <nk,dk> into Q...

39. Dijkstra, for single-source shortest paths
shortest dist from S
For all nk , track <nk, Inf>
Put <S,0> into your queue Q.
While Q not empty:
Remove Q's nearest node <nc,dc>
For each edge [nc, nk, dc2k]:
Let dk be nk's distance: <nk,dk>
If dc + dc2k < dk: set dk = dc + dc2k
Put <nk,dk> into Q...

40. Jotto! Frosh Sophs Jrs Srs audio 2 audio 1 audio 2 audio 1 graze 2
alloy 2
alloy 1
alloy 1
alloy 1
fresh 1
fresh 2
fresh 2
fresh 2
armor 2
armor 2
armor 2
armor 1
brave 2
brave 3
brave 1
brave 1
wreak 2
wreak 3
wreak 1
wreak 2
PERL stories
- use in ACM
- creating scripts
- LTS, BOT
fjord 1
fjord 2
fjord 1
fjord 5
taper 1
taper 2
taper 4
taper 1
tater 1
tater 2
tater 4
tater 1
This term's first class to guess another's word earns 1 problem...
This term's last class to have its word guessed earns 1 problem...