// shortest-path.h
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
//
// Author: allauzen@cs.nyu.edu (Cyril Allauzen)
//
// \file
// Functions to find shortest paths in an FST.
#ifndef FST_LIB_SHORTEST_PATH_H__
#define FST_LIB_SHORTEST_PATH_H__
#include <functional>
#include "fst/lib/cache.h"
#include "fst/lib/queue.h"
#include "fst/lib/shortest-distance.h"
#include "fst/lib/test-properties.h"
namespace fst {
template <class Arc, class Queue, class ArcFilter>
struct ShortestPathOptions
: public ShortestDistanceOptions<Arc, Queue, ArcFilter> {
typedef typename Arc::StateId StateId;
size_t nshortest; // return n-shortest paths
bool unique; // only return paths with distinct input strings
bool has_distance; // distance vector already contains the
// shortest distance from the initial state
ShortestPathOptions(Queue *q, ArcFilter filt, size_t n = 1, bool u = false,
bool hasdist = false, float d = kDelta)
: ShortestDistanceOptions<Arc, Queue, ArcFilter>(q, filt, kNoStateId, d),
nshortest(n), unique(u), has_distance(hasdist) {}
};
// Shortest-path algorithm: normally not called directly; prefer
// 'ShortestPath' below with n=1. 'ofst' contains the shortest path in
// 'ifst'. 'distance' returns the shortest distances from the source
// state to each state in 'ifst'. 'opts' is used to specify options
// such as the queue discipline, the arc filter and delta.
//
// The shortest path is the lowest weight path w.r.t. the natural
// semiring order.
//
// The weights need to be right distributive and have the path (kPath)
// property.
template<class Arc, class Queue, class ArcFilter>
void SingleShortestPath(const Fst<Arc> &ifst,
MutableFst<Arc> *ofst,
vector<typename Arc::Weight> *distance,
ShortestPathOptions<Arc, Queue, ArcFilter> &opts) {
typedef typename Arc::StateId StateId;
typedef typename Arc::Weight Weight;
ofst->DeleteStates();
ofst->SetInputSymbols(ifst.InputSymbols());
ofst->SetOutputSymbols(ifst.OutputSymbols());
if (ifst.Start() == kNoStateId)
return;
vector<Weight> rdistance;
vector<bool> enqueued;
vector<StateId> parent;
vector<Arc> arc_parent;
Queue *state_queue = opts.state_queue;
StateId source = opts.source == kNoStateId ? ifst.Start() : opts.source;
Weight f_distance = Weight::Zero();
StateId f_parent = kNoStateId;
distance->clear();
state_queue->Clear();
if (opts.nshortest != 1)
LOG(FATAL) << "SingleShortestPath: for nshortest > 1, use ShortestPath"
<< " instead";
if ((Weight::Properties() & (kPath | kRightSemiring))
!= (kPath | kRightSemiring))
LOG(FATAL) << "SingleShortestPath: Weight needs to have the path"
<< " property and be right distributive: " << Weight::Type();
while (distance->size() < source) {
distance->push_back(Weight::Zero());
enqueued.push_back(false);
parent.push_back(kNoStateId);
arc_parent.push_back(Arc(kNoLabel, kNoLabel, Weight::Zero(), kNoStateId));
}
distance->push_back(Weight::One());
parent.push_back(kNoStateId);
arc_parent.push_back(Arc(kNoLabel, kNoLabel, Weight::Zero(), kNoStateId));
state_queue->Enqueue(source);
enqueued.push_back(true);
while (!state_queue->Empty()) {
StateId s = state_queue->Head();
state_queue->Dequeue();
enqueued[s] = false;
Weight sd = (*distance)[s];
for (ArcIterator< Fst<Arc> > aiter(ifst, s);
!aiter.Done();
aiter.Next()) {
const Arc &arc = aiter.Value();
while (distance->size() <= arc.nextstate) {
distance->push_back(Weight::Zero());
enqueued.push_back(false);
parent.push_back(kNoStateId);
arc_parent.push_back(Arc(kNoLabel, kNoLabel, Weight::Zero(),
kNoStateId));
}
Weight &nd = (*distance)[arc.nextstate];
Weight w = Times(sd, arc.weight);
if (nd != Plus(nd, w)) {
nd = Plus(nd, w);
parent[arc.nextstate] = s;
arc_parent[arc.nextstate] = arc;
if (!enqueued[arc.nextstate]) {
state_queue->Enqueue(arc.nextstate);
enqueued[arc.nextstate] = true;
} else {
state_queue->Update(arc.nextstate);
}
}
}
if (ifst.Final(s) != Weight::Zero()) {
Weight w = Times(sd, ifst.Final(s));
if (f_distance != Plus(f_distance, w)) {
f_distance = Plus(f_distance, w);
f_parent = s;
}
}
}
(*distance)[source] = Weight::One();
parent[source] = kNoStateId;
StateId s_p = kNoStateId, d_p = kNoStateId;
for (StateId s = f_parent, d = kNoStateId;
s != kNoStateId;
d = s, s = parent[s]) {
enqueued[s] = true;
d_p = s_p;
s_p = ofst->AddState();
if (d == kNoStateId) {
ofst->SetFinal(s_p, ifst.Final(f_parent));
} else {
arc_parent[d].nextstate = d_p;
ofst->AddArc(s_p, arc_parent[d]);
}
}
ofst->SetStart(s_p);
}
template <class S, class W>
class ShortestPathCompare {
public:
typedef S StateId;
typedef W Weight;
typedef pair<StateId, Weight> Pair;
ShortestPathCompare(const vector<Pair>& pairs,
const vector<Weight>& distance,
StateId sfinal, float d)
: pairs_(pairs), distance_(distance), superfinal_(sfinal), delta_(d) {}
bool operator()(const StateId x, const StateId y) const {
const Pair &px = pairs_[x];
const Pair &py = pairs_[y];
Weight wx = Times(distance_[px.first], px.second);
Weight wy = Times(distance_[py.first], py.second);
// Penalize complete paths to ensure correct results with inexact weights.
// This forms a strict weak order so long as ApproxEqual(a, b) =>
// ApproxEqual(a, c) for all c s.t. less_(a, c) && less_(c, b).
if (px.first == superfinal_ && py.first != superfinal_) {
return less_(wy, wx) || ApproxEqual(wx, wy, delta_);
} else if (py.first == superfinal_ && px.first != superfinal_) {
return less_(wy, wx) && !ApproxEqual(wx, wy, delta_);
} else {
return less_(wy, wx);
}
}
private:
const vector<Pair> &pairs_;
const vector<Weight> &distance_;
StateId superfinal_;
float delta_;
NaturalLess<Weight> less_;
};
// N-Shortest-path algorithm: this version allow fine control
// via the otpions argument. See below for a simpler interface.
//
// 'ofst' contains the n-shortest paths in 'ifst'. 'distance' returns
// the shortest distances from the source state to each state in
// 'ifst'. 'opts' is used to specify options such as the number of
// paths to return, whether they need to have distinct input
// strings, the queue discipline, the arc filter and the convergence
// delta.
//
// The n-shortest paths are the n-lowest weight paths w.r.t. the
// natural semiring order. The single path that can be
// read from the ith of at most n transitions leaving the initial
// state of 'ofst' is the ith shortest path.
// The weights need to be right distributive and have the path (kPath)
// property. They need to be left distributive as well for nshortest
// > 1.
//
// The algorithm is from Mohri and Riley, "An Efficient Algorithm for
// the n-best-strings problem", ICSLP 2002. The algorithm relies on
// the shortest-distance algorithm. There are some issues with the
// pseudo-code as written in the paper (viz., line 11).
template<class Arc, class Queue, class ArcFilter>
void ShortestPath(const Fst<Arc> &ifst, MutableFst<Arc> *ofst,
vector<typename Arc::Weight> *distance,
ShortestPathOptions<Arc, Queue, ArcFilter> &opts) {
typedef typename Arc::StateId StateId;
typedef typename Arc::Weight Weight;
typedef pair<StateId, Weight> Pair;
typedef ReverseArc<Arc> ReverseArc;
typedef typename ReverseArc::Weight ReverseWeight;
size_t n = opts.nshortest;
if (n == 1) {
SingleShortestPath(ifst, ofst, distance, opts);
return;
}
ofst->DeleteStates();
ofst->SetInputSymbols(ifst.InputSymbols());
ofst->SetOutputSymbols(ifst.OutputSymbols());
if (n <= 0) return;
if ((Weight::Properties() & (kPath | kSemiring)) != (kPath | kSemiring))
LOG(FATAL) << "ShortestPath: n-shortest: Weight needs to have the "
<< "path property and be distributive: "
<< Weight::Type();
if (opts.unique)
LOG(FATAL) << "ShortestPath: n-shortest-string algorithm not "
<< "currently implemented";
// Algorithm works on the reverse of 'fst' : 'rfst' 'distance' is
// the distance to the final state in 'rfst' 'ofst' is built as the
// reverse of the tree of n-shortest path in 'rfst'.
if (!opts.has_distance)
ShortestDistance(ifst, distance, opts);
VectorFst<ReverseArc> rfst;
Reverse(ifst, &rfst);
distance->insert(distance->begin(), Weight::One());
while (distance->size() < rfst.NumStates())
distance->push_back(Weight::Zero());
// Each state in 'ofst' corresponds to a path with weight w from the
// initial state of 'rfst' to a state s in 'rfst', that can be
// characterized by a pair (s,w). The vector 'pairs' maps each
// state in 'ofst' to the corresponding pair maps states in OFST to
// the corresponding pair (s,w).
vector<Pair> pairs;
// 'r[s]', 's' state in 'fst', is the number of states in 'ofst'
// which corresponding pair contains 's' ,i.e. , it is number of
// paths computed so far to 's'.
StateId superfinal = distance->size(); // superfinal must be handled
distance->push_back(Weight::One()); // differently when unique=true
ShortestPathCompare<StateId, Weight>
compare(pairs, *distance, superfinal, opts.delta);
vector<StateId> heap;
vector<int> r;
while (r.size() < distance->size())
r.push_back(0);
ofst->SetStart(ofst->AddState());
StateId final = ofst->AddState();
ofst->SetFinal(final, Weight::One());
while (pairs.size() <= final)
pairs.push_back(Pair(kNoStateId, Weight::Zero()));
pairs[final] = Pair(rfst.Start(), Weight::One());
heap.push_back(final);
while (!heap.empty()) {
pop_heap(heap.begin(), heap.end(), compare);
StateId state = heap.back();
Pair p = pairs[state];
heap.pop_back();
++r[p.first];
if (p.first == superfinal)
ofst->AddArc(ofst->Start(), Arc(0, 0, Weight::One(), state));
if ((p.first == superfinal) && (r[p.first] == n)) break;
if (r[p.first] > n) continue;
if (p.first == superfinal)
continue;
for (ArcIterator< Fst<ReverseArc> > aiter(rfst, p.first);
!aiter.Done();
aiter.Next()) {
const ReverseArc &rarc = aiter.Value();
Arc arc(rarc.ilabel, rarc.olabel, rarc.weight.Reverse(), rarc.nextstate);
Weight w = Times(p.second, arc.weight);
StateId next = ofst->AddState();
pairs.push_back(Pair(arc.nextstate, w));
arc.nextstate = state;
ofst->AddArc(next, arc);
heap.push_back(next);
push_heap(heap.begin(), heap.end(), compare);
}
Weight finalw = rfst.Final(p.first).Reverse();
if (finalw != Weight::Zero()) {
Weight w = Times(p.second, finalw);
StateId next = ofst->AddState();
pairs.push_back(Pair(superfinal, w));
ofst->AddArc(next, Arc(0, 0, finalw, state));
heap.push_back(next);
push_heap(heap.begin(), heap.end(), compare);
}
}
Connect(ofst);
distance->erase(distance->begin());
distance->pop_back();
}
// Shortest-path algorithm: simplified interface. See above for a
// version that allows finer control.
// 'ofst' contains the 'n'-shortest paths in 'ifst'. The queue
// discipline is automatically selected. When 'unique' == true, only
// paths with distinct input labels are returned.
//
// The n-shortest paths are the n-lowest weight paths w.r.t. the
// natural semiring order. The single path that can be read from the
// ith of at most n transitions leaving the initial state of 'ofst' is
// the ith best path.
//
// The weights need to be right distributive and have the path
// (kPath) property.
template<class Arc>
void ShortestPath(const Fst<Arc> &ifst, MutableFst<Arc> *ofst,
size_t n = 1, bool unique = false) {
vector<typename Arc::Weight> distance;
AnyArcFilter<Arc> arc_filter;
AutoQueue<typename Arc::StateId> state_queue(ifst, &distance, arc_filter);
ShortestPathOptions< Arc, AutoQueue<typename Arc::StateId>,
AnyArcFilter<Arc> > opts(&state_queue, arc_filter, n, unique);
ShortestPath(ifst, ofst, &distance, opts);
}
} // namespace fst
#endif // FST_LIB_SHORTEST_PATH_H__
