Algorithm Design

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Chapter 7 Network Flow
with capacity 1; in this way, a unit of flow can traverse (ui, vi) and then move
directly to (uj, vj). Such a construction of edges is shown in Figure 7.17(b).
We extend this to a flow network by including a source and sink; we now
give the full construction in detail. The node set of the underlying graph G is
defined as follows.
o For each flight i, the graph G will have the two nodes ui and vi.
o G will also have a distinct source node s and sink node t.
The edge set of G is defined as fo!lows.
o For each i, there is an edge (ui, Vi) with a lower bound of ! and a capacity
of 1. (Each flight on the list mast be served.)
o For each i and j so that flight j is reachable from flight i, there is an edge
(vi, uj) with a lower bound of 0 and a capacity of 1. (The same plane can
perform flights i and j.)
o For each i, there is an edge (s, ai) with a lower bound of 0 and a capacity
of 1. (Any plane can begin the day with flight i.)
o For eachj, there is an edge (vp t) with a lower bound of 0 and a capacity
of 1. (Any plane can end the day with flight j.)
o There is an edge (s, t) with lower bound 0 and capacity k. (If we have
extra planes, we don’t need to use them for any of the flights.)
Finally, the node s will have a demand of -k, and the node t will have a
demand of k. All other nodes will have a demand of 0.
Our algorithm is to construct the network G and search for a feasible
circulation in it. We now prove the correctness of this algorithm.
/~ Analyzing the Algorithm
(7.54) There is a way to perform all flights using at most k planes if and only
if there is a feasible circulation in the network G.
Proof. First, suppose there is a way to perform all flights using k’