International A LevelMaths: Decision 1EdexcelRevision NotesAlgorithms on GraphsThe Route Inspection AlgorithmRoute Inspection & Repeated Routes

Route Inspection & Repeated Routes (Edexcel International A Level Maths: Decision 1)

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Naomi C

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Naomi C

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Networks with more than 2 Odd Vertices

How do I solve the Chinese postman problem when a network has more than 2 odd nodes?

  • In any graph there will always be an even number of odd vertices (nodes)
    • The total sum of the degrees of the nodes is double the number of edges
  • If there are more than two odd vertices (nodes)
    • The shortest path between each possible pairing of the odd nodes must be considered
    • The shortest of these pairings will be the minimum weight of the paths that need to be repeated
  • In cases of four odd nodes, there will be three pairings of odd nodes
    • For example in a graph where the four odd nodes are P, Q, R and S the pairings would be
      • PQ and RS
      • PR and QS
      • PS and QR
    • In an exam question there will be a maximum of 4 odd nodes

What are the steps of the Chinese postman algorithm for networks with more than 2 odd nodes?

  • STEP 1
    Inspect the degree of all nodes and identify any odd nodes
     
  • STEP 2
    Find all possible pairings between the odd nodes
     
  • STEP 3
    For each possible pairing of odd nodes, find (by inspection) the shortest path between each pair of nodes
    The pairing with the smallest total of shortest paths will be repeated
    Add     any repeated edges required to the network
     
  • STEP 4
    Write down an Eulerian circuit of the adjusted network to find a possible route
    Find the sum of the edges traversed to find the total weight

What variations may there be on the Chinese postman algorithm?

  • A variation on the 4 odd nodes problem is that the start and end nodes can be any two of the odd nodes
    • The problem is to find the shortest route and the corresponding start and end nodes
    • To solve this problem
      • Find the length of the shortest paths for all possible pairings of the odd nodes
      • Choose the shortest of the shortest paths between any 2 of them to be repeated
      • The other two odd vertices will be your start and finish points
  • Another variation is that the weighting of an edge between a pair of nodes may change
    • This could depend on whether it is the first time it is being traversed or whether it is being repeated
    • For example, if an inspector was checking a pipeline for defects
      • The first time going along a section of pipeline could take longer during inspection
      • Walking along an already inspected section to get from one node to another may take less time

Worked example

The network shown below displays the distances, in kilometres, of the main roads between towns A, B, C, D and E.
Each road is to be inspected for potholes.

chinese-postman-example-2

a)

Explain why the network does not contain an Eulerian circuit.

Inspect the degree of each node

A: 3 (odd)
B: 3 (odd)
C: 3 (odd)
D: 3 (odd)
E: 2 (even) 

The graph does not contain an Eulerian circuit as some of the vertices are odd

b)

Find the shortest route that starts and finishes at town A and allows for each road to be inspected.

  • STEP 1
    There are 4 odd nodes, A, B, C and D
     
  • STEP 2
    The possible pairings between the odd nodes are AB and CD, AC and BD, and AD and BC
     
  • STEP 3
    By inspection, find the shortest paths (and the edges involved) for each pairing 
AB + CD = (ABD) + (CD) = 12 + 10 = 22
 
AC + BD = (ADC) + (BD) = 18 + 4 = 22
 
AD + BC = (AD) + (BC) = 8 + 9 = 17  leftwards arrow for blank of   shortest 
The shortest total of paths to be repeated is for AD and BC, so add the edges AD and BC to the graph
 chinese-postman-example-2-2
  • STEP 4
    Write down an Eulerian circuit (from the adjusted graph) starting at vertex A

Eulerian circuit: ADABDCBCEA

There are other possible Eulerian circuits that you could find

c)

State the total length of the shortest route.

Add the length of each edge in the graph, then add the weight of the repeated edges

15 + 17 + 8 + 4 + 9 + 10 + 6 = 69 (edges in the original graph)

17 (repeated edges)

Total length of the shortest route: 86 km

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    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.