627
628
629
630
631
We will start with the properties of any shortest route which may exist between the points of the shortest route. Any such general shortest route must exhibit two necessary and sufficient properties to be called the shortest route.
Property 1: All the points should be joined to the nearest branch to it.
Property 2: Each segment must be joined to the nearest segment to it.
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
We would explain these two points in detail. For illustration only let us consider the following hypothetical route of [Fig.7] supposed to be the shortest route between these points.
Note: The route shown is topological and hypothetical and their may be other shortest route between these points. The main point is not the position of points but the property each point exhibits.
Now each point on the network is joined to a branch for which ‘a+b-c’ is minimum possible. If it has been joined to any other branch (say) for which the ‘a+b-c’ value is larger, that route would not be the shortest route. Also every segment (It may be any segment not only the adjoining segment) is joined to its nearest segment via ‘a+b-c-d’ rule. The reason for above is same as of property one.
\ If every point is joined to the nearest branches respectively and no further alteration in the position and order produces further shortening of route, we can safely conclude that the route indeed the shortest. Our method produces the shortest route because it is based on the properties which make the shortest route the shortest. We can end by continuous application of properties 1 and 2 only at the shortest route.
648
649
650
651
652
653
654
655
656
657
658
659
This condition is reached when conditions (properties) 1 and 2 are satisfied.
Hence these two properties are the necessary and sufficient conditions for a shortest route to be the shortest.
The method which we have used gives after each step a decreasing sequence of shorter routes and in the end this decreasing sequence terminates<