Algorithm Design

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Chapter 4 Greedy Algorithms
Figttre 4.18 A directed graph can have many different arborescences. Parts (b) and (c)
depict two different aborescences, both rooted at node r, for the graph in part (a).
(4.t4) A subgraph T = (V, F) of G is an arborescence with respect to root r if
and only if T has no cycles, and for each node v ~ r, there is exactly one edge
in F that enters v.
Proof. If T is an arborescence with root r, then indeed every other node v
has exactly one edge entering it: this is simply the last edge on the unique r-v
path.
Conversely, suppose T has no cycles, and each node v # r has exactly
one entering edge. In order to establish that T is an arborescence, we need
only show that there is a directed path from r to each other node v. Here is
how to construct such a path. We start at v and repeatedly follow edges in
the backward direction. Since T has no cycles, we can never return tO a node
we’ve previously visited, and thus this process must terminate. But r is the
only node without incoming edges, and so the process must in fact terminate
by reaching r; the sequence of nodes thus visited yields a path (in the reverse
direction) from r to v. m
It is easy to see that, just as every connected graph has a spanning tree, a
directed graph has an arborescence rooted at r provided that r can reach every
node. Indeed, in this case, the edges in a breadth-first search tree rooted at r
will form an arborescence.
(4.t5) A directed graph G has an arborescence rooted at r if and only if the¢e
_
4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm
The basic problem we consider here is the following. We are given a
directed graph G = (V, E), with a distinguished root node r and with a nonnegative
cost ce >_ 0 on each edge, and we wish to compute an arborescence
rooted at r of minimum total cost. (We will refer to this as an optimal arborescence.)
We will assume throughout that G at least has an arborescence rooted
at r; by (4.35), this can be easily checked at the outset.
L4~ Designing the Algorithm
Given the relationship between arborescences and trees, the minimum-cost
arborescence problem certainlyhas a strong initial resemblance to the Minimum
Spanning Tree Problem for undirected graphs. Thus it’s natural to start
by asking whether the ideas we developed for that problem can be carried
over directly to this setting. For example, must the minimum-cost arborescence
contain the cheapest edge in the whole graph? Can we safely delete the
most expensive edge on a cycle, confident that it cannot be in the optimal
arborescence?
Clearly the cheapest edge e in G will not belong to the optimal arborescence
if e enters the root, since the arborescence we’re seeking is not supposed to
have any edges entering the root. But even if the cheapest edge in G belongs
to some arborescence rooted at r, it need not belong to the optimal one, as
the example of Figure 4.19 shows. Indeed, including the edge of cost 1 in
Figure 4.!9 would prevent us from including the edge of cost 2 out of the
root r (since there can only be one entering edge per node); and this in turn
would force us to incur an unacceptable cost of 10 when we included one of
10 10
(a)
Figure 4.19 (a) A directed graph with costs onits edges, and (b) an optimal arborescence
rooted at r for this graph.
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