Chapter 14 Solutions
1.
Shortest path algorithm has two main steps.
A labeling correcting step
and a determining the minimum of the corrected labels.
Label Correcting Step: If (Label(I)+Distance(I,J)<=Label(J)) then Label(J)
= label(I)+Distance(I,J) for all arcs out of node I.
Find Minimum of Corrected Labels:
Min = Large Number
For I = 1 to Number of Nodes
If (Label(I) is Not Permanent and Label(I) < Min)
Min = Label(I)
End For
Both the steps will be carried for n times for a problem with n nodes.
The first step will be conducted for all the arcs in the graph.
The total
number of possible arcs if all the n nodes are connected to each other.
That is number of arcs are (n1)^2. Thus the comparison will be done at
most (n1)^2. Hence the complexity is O(n2).
The finding of the minimum valued node will be carried out at most n
times, in which case the sink will be permanently labeled last.
Each for
execution will perform n operations.
Thus this step has also has a
complexity of O(n2).
Since both the steps are of order (n2) the algorithm
is of complexity O(n2).
2.
The greedy algorithm for finding minimal spanning tree first sorts the
arcs in ascending of arc cost.
Then it adds the arc in spanning tree if
there is not a path between from node and to node of the arc.
If all the
nodes are connected or (n1) arcs are added to tree the algorithm
terminates.
The sorting algorithm is of order nlog(n)).
After an arc is
added the checking whether there is a path needs at most n operations.
Since the number of arcs is at most n2 and the number of operations in
the path checking is also n*n.
Hence the algorithm is of order O(n2).
In the worst case scenario the arc chosen last can be the one that
completes tree.
Hence theoretically it is not possible for this algorithm
to be better than O(n2) in the worst case.
3.
For the algorithm described in the worst case scenario each of the n
numbers have to be compared with (n1) numbers before we can arrive
at an answer.
Since in each step requires (n1) comparison with n such
steps are possible, the algorithm is of order O(n2).
4.
Genetic Search:
Given the flow and the cost between each pair of nodes in a network
with n nodes, the pHub Location Problem (PHLP) turns a subset of
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View Full Documentthe n nodes into hubs and assigns each of the remaining np nodes to a
single hub at the minimum cost possible.The main components that
describe the implementation of GAs to the PHHP are given here:
(1) Representation
The data structure used to represent a population of individuals
(solutions) is the array. Each individual is defined as a special kind of
structure that has several pieces of information attached to it. The first
piece of information is a string denoted by IND[i]=b1b2b3.
....bn,
where the length of the string, n,
corresponds to the total number of
nodes in the network and bk represents node k. The value 1 indicates
that node k is a hub and the value 0 indicates it is not. Make sure any
random p positions are 1. Each string remains a feasible solution to the
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 Fall '07
 HOCHBAUM

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