This operation computes the shortest distance from the initial state to every state (when `reverse`

is `false`

) or
from every state to the final states (when `reverse`

is `true`

). The *shortest distance* from *p* to *q* is the
⊕-sum of the weights of all the paths between *p* and *q*.

The weights must must be right (left) distributive if `reverse`

is `false`

(`true`

)
and *k* -closed (i.e., 1 ⊕ *x* ⊕ *x* ^{2} ⊕ ... ⊕ *x* ^{ k +1} = 1 ⊕ *x* ⊕ *x* ^{2} ⊕ ... ⊕ *x* ^{ k }) (valid for `TropicalWeight`

).

`A`

, over the tropical semiring:

(TropicalWeight)

State | Distance |
---|---|

`0` |
`0` |

`1` |
`3` |

`2` |
`5` |

`3` |
`7` |

ShortestDistance(A, &distance); fstshortestdistance a.fst

State | Distance |
---|---|

`0` |
`10` |

`1` |
`7` |

`2` |
`7` |

`3` |
`3` |

ShortestDistance(A, &distance, true); fstshortestdistance --reverse A.fst

`ShortestDistance:`

- TIme:
- Acyclic:
*O(V + E)* - Tropical semiring:
*O(V log V + E)* - General:
*exponential*

- Acyclic:
- Space:
*O(V)*

See here for a discussion on efficient usage.

- Mehryar Mohri. Semiring Framework and Algorithms for Shortest-Distance Problems,
*Journal of Automata, Languages and Combinatorics*, 7(3):321-350, 2002.

-- CyrilAllauzen - 05 Jul 2007

I | Attachment | History | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|---|

jpg | shortestdistance.jpg | r1 | manage | 9.4 K | 2007-07-09 - 20:31 | CyrilAllauzen | Shortest distance input example |

Topic revision: r7 - 2014-04-23 - MichaelRiley

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