# Working with Shortest Paths¶

Storm can enumerate the most probable paths of a model, leading from the initial state to a defined set of target states, which we refer to as shortest paths.
In particular, the model states visited along those paths are available as sets and can be accumulated, yielding a *sub-model*.

## Background¶

The underlying implementation uses the *recursive enumeration algorithm* [JM1999], substituting distance for probability – which is why we refer to the most probable paths as the *shortest* paths.

This algorithm computes the shortest paths recursively and in order, i.e., to find the 7th shortest path, the 1st through 6th shortest paths are computed as precursors. The next (i.e., 8th shortest) path can then be computed efficiently.

It is crucial to note that *any* path is eligible, including those that (repeatedly) traverse loops (i.e., *non-simple* paths). This is a common case in practice: Often a large number of similar paths that differ only in the order and number of loop traversals occur successively in the sequence of shortest paths. (For applications that are only interested in simple paths, this is rather unfortunate.)

## Examining Shortest Paths¶

See also

As in Getting Started, we import some required modules and build a model from the example files:

```
>>> import stormpy.examples
>>> import stormpy.examples.files
>>> path = stormpy.examples.files.prism_dtmc_die
>>> prism_program = stormpy.parse_prism_program(path)
>>> model = stormpy.build_model(prism_program)
```

We also import the `ShortestPathsGenerator`

:

```
>>> from stormpy.utility import ShortestPathsGenerator
```

and choose a target state (by its ID) to which we want to compute the shortest paths:

```
>>> state_id = 8
```

It is also possible to specify a set of target states (as a list, e.g., `[8, 10, 11]`

) or a label in the model if applicable (e.g., `"observe0Greater1"`

).
For simplicity, we will stick to using a single state for now.

We initialize a `ShortestPathsGenerator`

instance:

```
>>> spg = ShortestPathsGenerator(model, state_id)
```

Now we can query the k-shortest path by index. Note that 1-based indices are used, so that the 3rd shortest path indeed corresponds to index `k=3`

.
Let us inspect the first three shortest paths:

```
>>> for k in range(1, 4):
... path = spg.get_path_as_list(k)
... distance = spg.get_distance(k)
... print("{}-shortest path to state #{}: {}, with distance {}".format(k, state_id, path, distance))
1-shortest path to state #8: [8, 4, 1, 0], with distance 0.125
2-shortest path to state #8: [8, 4, 1, 3, 1, 0], with distance 0.03125
3-shortest path to state #8: [8, 4, 1, 3, 1, 3, 1, 0], with distance 0.0078125
```

As you can see, the distance (i.e., probability of the path) is also available. Note that the paths are displayed as a backward-traversal from the target to the initial state.

- JM1999
Víctor M. Jiménez, Andrés Marzal. Computing the K Shortest Paths: A New Algorithm and an Experimental Comparison. International Workshop on Algorithm Engineering, 1999