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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?

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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?

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IT always seems mysterious to me… Hooray! Not sure about these, however…

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Matrix of skillsets

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Ford-Fulkerson algorithm What's the maximum flow possible, from src to sink? s B E D C 13 t 16 104 9 12 14 7 20 4 source capacity Max Flow ! sink or target

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s B E D C 13 s B C D E FROM sBCDE t -1613-- --1012- -4--14 --9-- ---7- - - - 20 4 ------ t t 16 104 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 ?

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s B E D C 13 s B C D E FROM sBCDE t -1613-- --1012- -4--14 --9-- ---7- - - - 20 4 ------ t t 4/16 104 9 0/12 14 7 8/20 4 TO Old capacities 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 C D E FROM sBCDE -413-- 12-100- -4--14 -129-- ---7- - - - 8 4 --- -- t t TO Residual capacities. and the red edges? Backwards capacities! 12

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s B E D C 13 s B C D E FROM sBCDE t -1613-- --1012- -4--14 --9-- ---7- - - - 20 4 ------ t t 4 104 9 0 14 7 8 4 TO source sink (Step #1) Use depth- or breadth-first search to find any path from s to t. Max Flow s B C D E FROM sBCDE -413-- 12-100- -4--14 -129-- ---7- - - - 8 4 --- -- t t TO 12 (Step #2) Continue with the remaining capacities until no path exists! Old capacities Residual capacities. Backwards capacities. New capacities

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B E D C 12/13 11/16 0/101/4 0/9 12/12 11/14 7/7 19/20 4/4 max flow: 23 (Step #1) Use depth- or breadth-first search to find any path from s to t. Max Flow (Step #2) Continue with the remaining capacities until no path exists! s source t sink

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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…

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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_karp This is the algorithm.

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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

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

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

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

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

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

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

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

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Max flow thought experiment... s source t sink 1 1 1 1 1 1 all 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…

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

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

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general problems:max-flow problems: cowcarn feeding optimilk sandcas timecards tswift difficult-to-classify problems: This week's problems Try max flow! soda (last week) tour (last week)

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The challenge: is sometimes setting up the graph

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4 42 189 10 1000 50 1 3 20 1 3 8 2 3 1 4 How do we use the results? What is flowing? tswift There 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

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4 42 189 10 1000 50 1 3 20 1 3 8 2 3 1 4 source 42 189 10 1000 ingredient costs How do we use the results? What is flowing? tswift There are four ingredients available ~ at these costs hay-flavored coffee spam chocolate There are four smoothie recipes available ~ with these rewards Hay coffee Spam Choc. 4 ingredients (hay) & 4 smoothie recipes each recipe requires ingredients 1 2 3 4 ingredients

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4 42 189 10 1000 50 1 3 20 1 3 8 2 3 1 4 source sink 42 189 10 1000 ingredient costs recipe rewards 50 20 8 3 How do we use the results? What is flowing? tswift There are four ingredients available ~ at these costs hay-flavored coffee spam chocolate There are four smoothie recipes available ~ with these rewards Hay coffee Spam Choc. 4 ingredients (hay) & 4 smoothie recipes each recipe requires ingredients Hay- spam Spam- hay coffee Choco- hay 1 2 3 4 ingredients recipes

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4 42 189 10 1000 50 1 3 20 1 3 8 2 3 1 4 source sink 42 189 10 1000 ingredient costs recipe rewards 50 20 8 3 How do we use the results? What is flowing? tswift There are four ingredients available ~ at these costs hay-flavored coffee spam chocolate There are four smoothie recipes available ~ with these rewards Hay coffee Spam Choc. 4 ingredients (hay) & 4 smoothie recipes each recipe requires ingredients Hay- spam Spam- hay coffee Choco- hay 1 2 3 4 One purple edge is missing… Which? ingredients recipes What should these capacities be?

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general problems:max-flow problems: cowcarn feeding optimilk sandcas timecards tswift difficult-to-classify problems: This week's problems Try max flow! soda (last week) tour (last week)

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This code only looks obfuscated!

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Donut! http://www.ioccc.org/ donut.c International Obfuscated C Coding Contest This code IS obfuscated!

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2011's winner… http://www.ioccc.org/ donut.c International Obfuscated C Coding Contest This code IS obfuscated!

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Tools 4 42 189 10 1000 50 1 3 20 1 3 8 2 3 1 4 source Jobs sink 42 189 10 1000 tool costs task rewards 50 20 8 3 How do we use the results? What is flowing? hardware There 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

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dinner 4 5 4 5 3 5 3 5 2 6 4 4 5 4 5 3 5 3 5 2 6 3 0 number of teams Input Output number of tables # of people in each team can an assignment be made without putting teammates together? 0101 capacity of each table again… end… 3 5 2 6 4 tables with capacities teams with sizes 5 3 4 5 seating assignments! no teammates

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dinner's maxflow graph s source t sink How does the maxflow here relate to whether the seating is possible or not? Team 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?

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JRsSRsElderly slate 3 This term's first class to guess another's word earns 1 problem... slate 2slate 1 This term's last class to have its word guessed earns 1 problem... Sophs slate 1 flair 0flair 1flair 2flair 0 Pomona slate 3 flair 2 stems 3stems 1stems 2stems 1stems 2 loser 2loser 3loser 2loser 1 loser 3 stone 3stone 2stone 1 stone 2 guppy 1guppy 0guppy 1guppy 2guppy 0 Try max flow! lasso 1lasso 3lasso 1 lasso 2 pluot 1pluot 2pluot 1 pluot 0

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old years…

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hardware Tools 4 42 189 10 1000 50 1 3 20 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?

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hardware Tools 4 42 189 10 1000 50 1 3 20 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?

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4 3 3 2 2 1 2 3 1 2 2 2 3 1 2 2 2 1 3 1 2 2 1 1 3 3 number of cows Input total # of foods total # of drinks # of foods cow[i] likes # of drinks cow[i] likes foodsdrinks 0 Output # of cows that can receive both a food and a drink they like… 3 each can be used only once Likes foodsdrinks 1 2 3 1 2 2 3 1 3 3 1 1 2 3 What is a cow-satisfying assignment here? dining

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Jotto! SophsJrsSrs audio 1audio 2audio 1 Frosh audio 2 graze 3graze 1 graze 2 alloy 1 alloy 2 fresh 2 fresh 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... armor 2 armor 1armor 2 brave 3brave 1 brave 2 wreak 3wreak 1wreak 2 fjord 1fjord 5 fjord 1 fjord 2

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Stake 4 1 0 1 1 0 1 0 0 0 0 0 1 0 1 Input Output 3 Height and width of the field the pattern Maximum number of cows such that no two share a column and no two share a row.

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Tools 4 42 189 10 1000 50 1 3 20 1 3 8 2 3 1 4 source Jobs sink 50 20 8 3 task rewards tool costs 42 189 10 1000 How do we use mf to maximize our profit? What is flowing? hardware There 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

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

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

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Dijkstra, for single-source shortest paths Put into your queue Q. While Q not empty: Remove Q's nearest node For each edge [n c, n k, d c2k ]: For all n k, track Let d k be n k 's distance: If d c + d c2k < d k : set d k = d c + d c2k Put into Q... S shortest dist from S

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Dijkstra, for single-source shortest paths Put into your queue Q. While Q not empty: Remove Q's nearest node For each edge [n c, n k, d c2k ]: For all n k, track Let d k be n k 's distance: If d c + d c2k < d k : set d k = d c + d c2k Put into Q... S shortest dist from S

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Jotto! SophsJrsSrs audio 1audio 2audio 1 Frosh audio 2 graze 3graze 1 graze 2 alloy 1 alloy 2 fresh 2 fresh 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... armor 2 armor 1armor 2 brave 3brave 1 brave 2 wreak 3wreak 1wreak 2 fjord 1fjord 5 fjord 1 fjord 2 taper 4taper 1 taper 2 tater 4tater 1 tater 2

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