How to find the shortest path of a weighted tree?

Question

I have a weighted tree which looks like (weights are in brackets)

          A1
        /   \
     B1(3)  B2(2)
     /   \  /  \
   C1(1) C2(3) C3(4)
   /   \ /  \  /  \
 D1(8) D2(7) D3(2) D4(5) 
    ......

So, each node has two children. And each node shares a child with a neighbour node. A depth of the tree can be very high.

3 + 1 + 8 = 12
3 + 1 + 7 = 11
3 + 3 + 7 = 13 ... and so on

What is the best way to find the shortest path? As a result I need not a sum of weights but a full path (lets say A1-B2-C3-D3).

I will be more than happy if you could reference me to the right algorithm.. Or provide java/pseudo code solution.

Thank you!

Update

I am looking for a full path from top to bottom

Answer 1

This may be a natural Dynamic Programming (DP) problem due to the child sharing property. I suggest using a bottom-up DP algorithm to solve this problem.

1. Define the state of each node as SP(n), which means shortest path from that node. We could notice that the SP(n) is only dependent on the SP(c), where c is child of n. And because of the child sharing property, the SP(n) may be reused by n's parents.

2. The state transformation equation is listed as below:

   SP(n) = min {for every c of n's children | SP(c) + weight(c)}

As for implementation, we scan bottom-up from leaves to compute the SP(n) until we reach the root. And the time cost is O(n) since we compute it in one run.

Answer 2

You may want to look at Alpha-beta-pruning . This algorithm basically removes part of the search tree as soon as they are known to be obsolete, ie a shorter path to the same position is already known.