In this post, we are going to present an implementation of Dijkstra's algorithm in Cython. Dijkstra's algorithm is a shortest path algorithm. It was conceived by Edsger W. Dijkstra in 1956, and published in 1959 [1].
Credit: Wikimedia Commons (CC BYSA 3.0)
From a directed graph
There are many variants and evolutions of this algorithm but we focus here on this classical onetoall version. In the present implementation, we are not going to store the shortest paths, but only the cost associated with the shortest path at each vertex. However, Dijkstra's algorithm can be built using an array of predecessors: for each vertex
The use cases here are road networks, with rather sparse networks. We are going to run the SSSP algorithm on the DIMACS road networks that we downloaded in a previous post: Download some benchmark road networks for Shortest Paths algorithms .
New York road network from the DIMACS implementation challenge
Also, we are going to use the minprioriy queue, based on a binary heap, presented in another post: A Cython implementation of a priority queue. The heap elements correspond to graph vertices, with the key value being the travel time from the source.
SSSP algorithm
Here is a short description of the algorithm from Cormen et al. [2]:
Dijkstra's algorithm maintain a set
$S$ of vertices whose final shortestpath weights from the source$s$ have already been determined. The algorithm repeatedly selects the vertex$u \in VS$ with the minimum shortestpath estimate, adds$u$ to$S$ , and relaxes all edges leaving$u$ .
A the beginning of the algorithm, all vertices
However, there are two variants regarding the initialization of the heap. As described on the wikipedia page:
Instead of filling the priority queue with all nodes in the initialization phase, it is also possible to initialize it to contain only source.
We tested both versions. Let's start with the simplest version where the queue is filled with all vertices.
First approach: initialize the priority queue with all nodes
The algorithm can be decomposed in the following steps:

initialization:

$S=\emptyset$ .$S$ is the set of "scanned" vertices for which the shortest path has been evaluated and won't be modified anymore.  insert all vertices
$v$ but$s$ into the priority queue$Q$ with an infinite key value:$v.key=\infty$  insert the source vertex
$s$ into$Q$ with a 0 key value:$s.key=0$


loop:

while
$Q$ is not empty extract the element
$u$ from$Q$ with min priority  add
$u$ to$S$ 
for each outgoing edge
$(u, v) \in E $ :
if
$v \notin S$ and$v.key \gt u.key + w(u,v) $ : decrease key of
$v$ with key value v_key$u.key + w(u,v)$
 decrease key of

if
 extract the element

while
Second approach: initialize the priority queue with only the source
This time, only the source vertex

initialization:

$S=\emptyset$ .  insert the source vertex
$s$ into$Q$ with a 0 key value:$s.key=0$


loop:

while
$Q$ is not empty extract the element
$u$ from$Q$ with min priority  add
$u$ to$S$ 
for each outgoing edge
$(u, v) \in E$ :
if
$v \notin S$ :
if
$v \notin Q$ : insert
$v$ into$Q$ with key value:$u.key + w(u,v)$
 insert

else:

if
$v.key \gt u.key + w(u,v) $ : decrease key of
$v$ with key value v_key$u.key + w(u,v)$
 decrease key of

if

if

if
 extract the element

while
General idea
Without going into the details, the idea of this algorithm is quite simple: at each iteration, we consider the vertex
But how can we be sure that there is not another distinct path
In the following, we use an enum
state for each vertex

SCANNED
:$v \in S$ 
IN_HEAP
:$v \in Q$ 
NOT_IN_HEAP
:$v \notin S$ and$v \notin Q$
Cython implementation
The Cython implementation makes use of two important components presented in previous posts:
 the Cython priority queue : A Cython implementation of a priority queue
 the forward star representation of the graph, in NumPy arrays: Forward and reverse stars in Cython
The forward star representation is allowing an efficient access to the outgoing edges from a given node. We would use the reverse star to access the incoming edges, in the case of a Single Target Shortest Path algorithm, to compute the shortest paths from any node in the graph to a target node.
The Cython code for the priority queue (pq_bin_heap_basic
) and the forward star representation have been placed into Cython modules. The code is taken straightly from the indicated posts. The following implementation corresponds to the second approach, in which only the source vertex is inserted in the queue at the beginning.
import numpy as np
cimport numpy as cnp
cimport pq_bin_heap_basic as bhb
DTYPE = np.float64
ctypedef cnp.float64_t DTYPE_t
cpdef cnp.ndarray path_length_from_bin_basic(
cnp.uint32_t[::1] csr_indices,
cnp.uint32_t[::1] csr_indptr,
DTYPE_t[::1] csr_data,
int origin_vert_in,
int vertex_count):
""" Compute singlesource shortest path (onetoall)
using a priority queue based on a binary heap.
"""
cdef:
size_t tail_vert_idx, head_vert_idx, idx # indices
DTYPE_t tail_vert_val, head_vert_val # vertex travel times
bhb.PriorityQueue pqueue
bhb.ElementState vert_state # vertex state
size_t origin_vert = <size_t>origin_vert_in
with nogil:
# initialization of the heap elements
# all nodes have INFINITY key and NOT_IN_HEAP state
bhb.init_pqueue(&pqueue, <size_t>vertex_count)
# the key is set to zero for the origin vertex,
# which is inserted into the heap
bhb.insert(&pqueue, origin_vert, 0.0)
# main loop
while pqueue.size > 0:
tail_vert_idx = bhb.extract_min(&pqueue)
tail_vert_val = pqueue.Elements[tail_vert_idx].key
# loop on outgoing edges
for idx in range(<size_t>csr_indptr[tail_vert_idx], <size_t>csr_indptr[tail_vert_idx + 1]):
head_vert_idx = <size_t>csr_indices[idx]
vert_state = pqueue.Elements[head_vert_idx].state
if vert_state != bhb.SCANNED:
head_vert_val = tail_vert_val + csr_data[idx]
if vert_state == bhb.NOT_IN_HEAP:
bhb.insert(&pqueue, head_vert_idx, head_vert_val)
elif pqueue.Elements[head_vert_idx].key > head_vert_val:
bhb.decrease_key(&pqueue, head_vert_idx, head_vert_val)
# copy the results into a numpy array
path_lengths = cnp.ndarray(vertex_count, dtype=DTYPE)
cdef:
DTYPE_t[::1] path_lengths_view = path_lengths
with nogil:
for i in range(<size_t>vertex_count):
path_lengths_view[i] = pqueue.Elements[i].key
# cleanup
bhb.free_pqueue(&pqueue)
return path_lengths
Because we do not want the post to be loaded with too many lines of code, we do not show here the Python code to load the graphs into dataframes, convert them into the forward star representation (csr_indices
, csr_indptr
and csr_data
). For the same reason, we do now show the implementation of the first approach neither [initialize the priority queue with all nodes].
A Visualization of the algorithm
The following animated gif has been made in two steps. Some printf
statements have been added to the above code to print the vertex indices [added to and removed from the heap] at each step of the iteration. Then using this "trace" text file and the vertex coordinates, some figures have been generated every 1000 steps. Vertices in the heap are colored in red while those that have been scanned are in blue. We can observe the front propagation process of the algorithm.
Animation of Dijkstra's algorithm [second approach] applied to the New York road network
This New York network has 3730100 edges and 264346 vertices. It is interesting to observe that in this second approach, the heap size remains relatively small compared to the number of vertices.
The heap size figure and the animated gif corresponds to the same run [New York network], with source node index idx_from = 1000
.
Validation and infinite travel time
Results from our path_length
functions have been checked against SciPy (scipy.sparse.csgraph.dijkstra
):
path_lengths_ref = dijkstra(
csgraph=graph_csr, directed=True, indices=idx_from, return_predecessors=False
)
When a node cannot be reached from the source vertex DTYPE
data type, i.e.:
DTYPE_INF_PY = np.finfo(dtype=np.float64).max
However, the SciPy dijkstra
function returns np.inf
values for these nodes, so we need do replace these DTYPE_INF_PY
values with np.inf
ones in order to get the same output:
# deal with infinity
path_lengths = np.where(
path_lengths == DTYPE_INF_PY, np.inf, path_lengths
)
Then we can compare the different results with the following command:
assert np.allclose(
path_lengths, path_lengths_ref, rtol=1e05, atol=1e08, equal_nan=True
)
Execution timings
Let's compare the two approaches on the USA network, 57708624 edges and 23947347 vertices. These algoritms have two distinct phases: setup and run. During the setup phase, the graph data structured are prepared for the algorithm to run. This setup phase only needs to be run once for any number of calls to the path_length
functions. In the following, we only measure the execution time of the run
phase. We use the best time over 3 runs.
We can see that the second strategy is far more efficient. This may be due to the fact that the heap size remains smaller in the second approach, and that the decrease_key
operation is expensive as compared to the insert
one.
In the following posts, we will study various priority queue versions, and compare the resulting implementation with some shortest path libraries available in Python.
References
[1] Dijkstra, E. W., A note on two problems in connexion with graphs, Numerische Mathematik. 1: 269–271 (1959), doi:10.1007/BF01386390.
[2] Cormen et al., Introduction to Algorithms, MIT Press and McGrawHill, coll. « third », 2009.