Efficient algorithm to find all the paths from A to Z?

Question

With a set of random inputs like this (20k lines):

A B
U Z
B A
A C
Z A
K Z
A Q
D A
U K
P U
U P
B Y
Y R
Y U
C R
R Q
A D
Q Z

Find all the pat

Answer 1

As I understand your question, Dijkstras algorithm cannot be applied as is, since shortest path problem per definition finds a single path in a set of all possible paths. Your task is to find all paths per-se.

Many optimizations on Dijkstras algorithm involve cutting off search trees with higher costs. You won't be able to cut off those parts in your search, as you need all findings.

And I assume you mean all paths excluding circles.

Algorithm:

• Pump network into a 2dim array 26x26 of boolean/integer. fromTo[i,j]. Set a 1/true for an existing link.

• Starting from the first node trace all following nodes (search links for 1/true).

• Keep visited nodes in a some structure (array/list). Since maximal depth seems to be 26, this should be possible via recursion.

• And as @soulcheck has written below, you may think about cutting of paths you have aleady visted. You may keep a list of paths towards the destination in each element of the array. Adjust the breaking condition accordingly.

• Break when

  • visiting the end node (store the result)
  • when visiting a node that has been visted before (circle)
  • visiting a node for which you have already found all paths to the destination and merge your current path with all the existing ones from that node.

Performance wise I'd vote against using hashmaps and lists and prefer static structures.

Hmm, while re-reading the question, I realized that the name of the nodes cannot be limited to A-Z. You are writing something about 20k lines, with 26 letters, a fully connected A-Z network would require far less links. Maybe you skip recursion and static structures :)

Ok, with valid names from AAA to ZZZ an array would become far too large. So you better create a dynamic structure for the network as well. Counter question: regarding performance, what is the best data structure for a less popuplate array as my algorithm would require? I' vote for an 2 dim ArrayList. Anyone?