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Chapter 4 Network Algorithms

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Title: Chapter 4 Network Algorithms


1
Chapter 4 Network Algorithms
  • Section 4.1 Shortest Paths
  • Colleen Raimondi

2
Section 4.1 Shortest Paths
In this chapter we present algorithms for the
solution of two important network optimization
problems.
Network a graph with a positive integer k(e)
assigned to each edge e, it will typically
represent the length of an edge, in units such
as miles, or represent capacity of an edge, in
units such as megawatts or gallons per minute.
Note Edge (a,b) has a capacity of 5, so
k(a,b)5.
3
Section 4.1 Shortest Paths
We begin with an algorithm for a relatively
simple problem, finding a shortest path in a
network from point a to point z. We say a
shortest path because, in general, there may be
more than one shortest path from a to z.
So when we find a shortest path, we must be able
to prove it is shortest without explicitly
comparing it with all over a-z paths. Although
the problem is now starting to sound difficult,
there is still a straightforward algorithmic
solution.
4
Section 4.1 Shortest Path
Dijkstras algorithm
Let variable m be a distance counter. For
increasing values of m, label vertices whose
minimal distance from a vertex a is m. The
first label of a vertex will be the previous
vertex on the shortest path from a to x. The
second label of x will be the length of the
shortest path from a to x.
5
Section 4.1 Shortest Path
  • Shortest Path Algorithm
  • Set m1 and label vertex a with (-,0) (the -
    represents a blank).
  • Check each edge e(p,q) from some labeled vertex
    p to some unlabeled vertex q. Suppose ps labels
    are r,d(p). If d(p)k(e)m, label q with (p,m).
  • If all vertices are not yet labeled, increment m
    by one and go to Step 2. Otherwise go to Step 4.
    If we are only interested in a shortest path to
    z, then we go to Step 4 when z is labeled.
  • For any vertex y, a shortest path from a to y has
    length d(y), the second label of y. Such a path
    may be found by backtracking from y (using the
    first labels).

6
Section 4.1 Shortest Path
We want to find the shortest path from a to c.
a(-,0)
m1 m3 m5 The shortest path would be a-d-c.
e(d,3)
c(d,5)
d(a,1)
7
Section 4.1 Shortest Paths
Note The algorithm given previously has one
significant inefficiency if all sums d(p)k(e)
in Step 2 have values of at least mgtm, then the
distance counter m should be increased
immediately to m.
8
Section 4.1 Example
Example 1 Pg. 134
m1 m3 m4 m5 m6 m7
a(-,0)
The shortest path is a-i-e-g-h.
We want to find a shortest path from point a to
point h.
9
Section 4.1 Class Work As seen on page 135 1
Use the shortest path algorithm to find the
shortest path between vertex c and vertex m.
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