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ADVANCED ALGORITHMS
Lecture 12: Flows/cuts
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ANNOUNCEMENTS
➤ HW 1 grades out, see solutions/hints ➤ HW 2 due TOMORROW, 11:59 PM
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ADVANCED ALGORITHMS Lecture 12: Flows/cuts 1 ANNOUNCEMENTS HW 1 - - PowerPoint PPT Presentation
ADVANCED ALGORITHMS Lecture 12: Flows/cuts 1 ANNOUNCEMENTS HW 1 - - PowerPoint PPT Presentation
ADVANCED ALGORITHMS Lecture 12: Flows/cuts 1 ANNOUNCEMENTS HW 1 grades out, see solutions/hints HW 2 due TOMORROW , 11:59 PM 2 TODAYS PLAN Flows and cuts in graphs Some applications 3 MIN S-T CUT Problem:
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TODAY’S PLAN
➤ Flows and cuts in graphs ➤ Some applications
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MIN S-T CUT
➤ Problem: given a directed graph with edge weights, find a small set of
(weighted) edges to “cut” so that there is no longer a directed path from s to t.
➤ Note: cut allowed to leave paths from t to s
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MAXIMUM FLOW
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➤ Problem: given a directed graph with capacities ce on the edges, and a
source s, sink t, send as much flow as possible without violating capacities.
➤ Flow defined intuitively — using a collection of paths ➤ Motivation: max throughput in network routing ➤ (Many others — will see today)
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EXAMPLE
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MAXIMUM FLOW IS ALWAYS <= MIN CUT
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➤ Saw last class: ➤ Take any flow — collection of paths P, flow on path p defined as fp ➤ Take any cut — set of edges whose removal leaves no s —> t path ➤ Obs: every path in our flow f uses some edge of cut ➤ Imagine we go through edges in cut, for each e, remove all paths
that goes through e
➤ Flow removed each time <= weight of edge
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MAXIMUM FLOW VS MIN CUT
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ALGORITHMS FOR FLOW — ADDING PATHS ITERATIVELY
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20 20 20 10 10 Algorithm:
- Find some s-t path, assign flow
= min edge capacity
- Decrease capacities
- Repeat
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ITERATIVE ADDITION — BAD ORDER
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FORD-FULKERSON ALGORITHM
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➤ Main idea: can we repair bad choices? (a la backtracking)
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RESIDUAL GRAPH
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RESIDUAL GRAPH
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➤ Idea: build network that allows “cancelling” some of current flow ➤ Formally, given flow f, add reverse edges of weight = total flow on edge
‘e’, reduce capacity of forward edge Key insight:
- A flow f’ in the residual graph leads to a valid flow f + f’ in G
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FORD-FULKERSON ALGORITHM
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➤ Main idea: can we repair bad choices? (a la backtracking) ➤ Algorithm: (initialize f = 0) ➤ Build residual Gf ➤ Find “augmenting path” f’ in Gf ; set f = f + f’ ➤ Repeat
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CORRECTNESS
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➤ High level idea: if flow is not optimal, there is always a path of min
capacity >0 in the residual.
➤ T
wo ways of proof. First via looking at optimal flow. (Tends to get messy)
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CORRECTNESS
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➤ High level idea: if flow is not optimal, there is always a path of min
capacity >0 in the residual.
➤ Another proof: if there is no augmenting path, then there exists cut in
G of same value as current flow
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CORRECTNESS
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MAX FLOW - MIN CUT THEOREM
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RECAP
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APPLICATIONS — CONNECTIVITY IN NETWORKS
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APPLICATIONS — IMAGE SEGMENTATION
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APPLICATIONS — MAX MATCHING
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APPLICATIONS — BASEBALL ELIMINATION
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