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Facilities Planning. Objectives and Agenda: 1. Examples of Shortest Path Problem 2. Finding shortest paths: Dijkstra’s method 3. Other applications of Shortest path problem. Example (Shortest Path Problem). What is the shortest route from Point A to Point B ?.

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slide1

Facilities Planning

Objectives and Agenda:

1. Examples of Shortest Path Problem

2. Finding shortest paths: Dijkstra’s method

3. Other applications of Shortest path problem

slide2

Example (Shortest Path Problem)

What is the shortest route from Point A to Point B ?

What if some roads are specified as 1-way only ?

slide3

A Graph-model of the Shortest Route problem

Given a directed, weighted graph, G(V, E), start node s, end node v,

Find the minimum total weight path from s to v

Legend:

Edge  direct road

weight  road length

nodes  road intersections

ust = HKUST

tko = Tseung Kwan O

kt = Kowloon Tong

ch = Choi Hung

tkw = To Kwa Wan

pe = Prince Edward

hh = Hung Hom

slide4

Finding shortest paths: Dijkstra’s method

Strategy:

Find shortest path to one node;

[all other nodes remain]

Find shortest path to one of the remaining nodes;

Repeat … until done

How?

Upper bound on distance from s to u: d[u]

If Node k has the MIN upper bound 

d[k] is the shortest distance from s to k

Select node k to update upper bounds on remaining nodes

Key concept: Relaxation

slide5

Relaxation..

Two cases of relaxation

x

x

y

y

4

4

18

10

10

12

Relax( x, y)

Relax( x, y)

x

x

y

y

4

4

12

10

14

10

cyan edge => current immediate predecessor of y is x

slide6

Q = { }

Dijkstra’s method..

d[vi] =  for each vertex; d[s] = 0;

Make two lists: S = { }; Q = V

Find the node, u, in Q with minimum d[u]

Remove u from Q, and add u to S

For each edge (u, v),

Relax (u, v); update IP(v)

yes

no

DONE

slide7

Dijkstra’s method, Example

Find shortest path from ust to hh

slide8

Dijkstra’s method, Example..

S = set of nodes for which we know

the shortest distance from s

Q = remaining nodes in the graph

d[ust] = d*[ust] = 0

relax ust

slide9

Dijkstra’s method, Example...

mark IP’s of tko, ch

MIN d[u] in Q is tko

d*[tko] = 6

relax tko

slide10

Dijkstra’s method, Example….

MIN d[u] in Q is ch

d*[ch] = 7

relax ch

NOTE: what happens to IP of kt?

slide11

Dijkstra’s method, Example…..

MIN d[u] in Q is kt

d*[kt] = 9

relax kt

slide12

Dijkstra’s method, Example…...

MIN d[u] in Q is pe

d*[pe] = 10

relax pe

NOTE: what happens to IP of hh?

slide13

Dijkstra’s method, Example…....

MIN d[u] in Q is tkw

d*[tkw] = 11

relax tkw

slide14

Dijkstra’s method, Example……..

MIN d[u] in Q is hh

d*[hh] = 12

relax hh

DONE!

Shortest path: reverse(hh  tkw  ch  ust)

slide15

Dijkstra’s method, Proof

PROPERTIES OF RELAXATION

(i) d[u] is non-increasing: relaxation cannot increase d[u]

(ii) d[u] cannot go below the shortest distance from s to u.

connected graph  there is a shortest distance from s to u = d*[u]

d[u] ≥ d*[u]

(i) and (ii) => Once we have found the shortest path to node u,

d[u] will never change.

slide16

Q

Q

Dijkstra’s method, Proof..

What are such nodes ?

(1) The elements of set S

why?

(2) Next candidate: select min d[u] node from nodes in set Q

Let: d[v] be minimum among all nodes in Q

There are two possibilities for the shortest path from s to v

so … ?

slide17

x

x

y

y

4

4

4

Dijkstra’s method, concluding remarks

What if the graph is undirected ?

Replace each undirected edge with two directed edges

What if the some edge weights are negative?

slide18

Shortest path, Applications

- Telephone routes

- Which communication links to activate when a user makes

a phone call, e.g. from HK to New York, USA.

- Road systems design

Problem: how to determine the no. of lanes in each road?

Given: expected traffic between each pair of locations

Method: Estimate total traffic on each road link assuming

each passenger will use shortest path

- Many other applications, including:

- Finance (arbitrage),

- Assembly line inspection systems design, …

slide19

Prof Edsger Dijkstra [1930-2002]

next topic: transportation flow planning – max flow