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6.889: Algorithms for Planar Graphs and Beyond
Problem Set 10  Solutions
In this problem set you will develop an algorithm for canceling ﬂow cycles in a given ﬂow assignment.
In general graphs this can be done in
O
(
m
log
n
) time using Sleator’s and Tarjan’s dynamic trees.
You will use the relation between dual shortest paths and circulations to give a linear time algorithm
in planar graphs.
1. Let
G
be a planar graph with nonnegative capacities on its arcs. Let
φ
be shortest path
distances from
f
∞
in
G
*
. Let
θ
be the circulation induced by
φ
. That is,
θ
(
d
) =
φ
(head(
d
))

φ
(tail(
d
))
.
Recall that a residual path is a path whose darts all have strictly positive capacities. Show
that there are no counterclockwise residual cycles in the residual graph
G
θ
.
Solution:
Let
T
*
be a shortest path tree in
G
*
rooted at
f
∞
, and let
C
be a counterclockwise
cycle. Since
C
encloses at least one face (and by deﬁnition does not enclose
f
∞
), and since
T
*
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This note was uploaded on 01/20/2012 for the course CS 6.889 taught by Professor Erikdemaine during the Fall '11 term at MIT.
 Fall '11
 ErikDemaine
 Algorithms

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