Question: Is it possible to reconstruct a grid of polygones or polylines from the centroids?

I am looking for a arcGIS tool or script that will allow me to reconstruct a grid of polygons or polylines from their centroids.

![From centroids](https://i.stack.imgur.com/NTVLC.png)

![to polygons](https://i.stack.imgur.com/bLcSP.png)

Answer: 
apparently there is a tool that allows you to do that in mapinfo

<http://www.kxcad.net/mapinfo/mapinfo_professional/mapinfow-37-10.html>
calculating the shortest distance , there is a logic behind it.

When we ask the bing or google service to give a shortest distance betwen point A to point B , it gives us the route which is drivable , if there a one way from B to A which is short , it wont give that route.

So how you have to proceed it 

1. Calculate the route and distance from Point A to B
2. Calculate the route and distance from Point B to A.

take the Route which has shortest distance.