Routino : Algorithm
===================
This page describes the development of the algorithm that is used in
Routino for finding routes.
Simplest Algorithm
------------------
The algorithm to find a route is fundamentally simple: Start at the
beginning, follow all possible routes and keep going until you reach
the end.
While this method does work, it isn't fast. To be able to find a route
quickly needs a different algorithm, one that can find the correct
answer without wasting time on routes that lead nowhere.
Improved Algorithm
------------------
The simplest way to do this is to follow all possible segments from the
starting node to the next nearest node (an intermediate node in the
complete journey). For each node that is reached store the shortest
route from the starting node and the length of that route. The list of
intermediate nodes needs to be maintained in order of shortest overall
route on the assumption that there is a straight line route from here
to the end node.
At each point the intermediate node that has the shortest potential
overall journey time is processed before any other node. From the first
node in the list follow all possible segments and place the newly
discovered nodes into the same list ordered in the same way. This will
tend to constrain the list of nodes examined to be the ones that are
between the start and end nodes. If at any point you reach a node that
has already been reached by a longer route then you can discard that
route since the newly discovered route is shorter. Conversely if the
previously discovered route is shorter then discard the new route.
At some point the end node will be reached and then any routes with
potential lengths longer than this actual route can be immediately
discarded. The few remaining potential routes must be continued until
they are found to be shorter or have no possibility of being shorter.
The shortest possible route is then found.
At all times when looking at a node only those segments that are
possible by the chosen means of transport are followed. This allows the
type of transport to be handled easily. When finding the quickest route
the same rules apply except that the criterion for sorting is the
shortest potential route (assuming that from each node to the end is
the fastest possible type of highway).
This method also works, but again it isn't very fast. The problem is
that the complexity is proportional to the number of nodes or segments
in all routes examined between the start and end nodes. Maintaining the
list of intermediate nodes in order is the most complex part.
Final Algorithm
---------------
The final algorithm that is implemented in the router is basically the
one above but with an important difference. Instead of finding a long
route among a data set of 8,000,000 nodes (number of highway nodes in
UK at beginning of 2010) it finds one long route in a data set of
1,000,000 nodes and a few hundred very short routes in the full data
set. Since the time taken to find a route is proportional to the number
of nodes that need to be considered the main route takes 1/10th of the
time and the very short routes take almost no time at all.
The solution to making the algorithm fast is therefore to discard most
of the nodes and only keep the interesting ones. In this case a node is
deemed to be interesting if it is the junction of three or more
segments or the junction of two segments with different properties or
has a routing restriction different from the connecting segments. In
the algorithm and following description these are classed as
super-nodes. Starting at each super-node a super-segment is generated
that finishes on another super-node and contains the shortest path
along segments with identical properties (and these properties are
inherited by the super-segment). The point of choosing the shortest
route is that since all segments considered have identical properties
they will be treated identically when properties are taken into
account. This decision making process can be repeated until the only
the most important and interesting nodes remain.
To find a route between a start and finish point now comprises the
following steps (assuming a shortest route is required):
1. Find all shortest routes from the start point along normal segments
and stopping when super-nodes are reached.
2. Find all shortest routes from the end point backwards along normal
segments and stopping when super-nodes are reached.
3. Find the shortest route along super-segments from the set of
super-nodes in step 1 to the set of super-nodes in step 2 (taking
into account the lengths found in steps 1 and 2 between the
start/finish super-nodes and the ultimate start/finish point).
4. For each super-segment in step 3 find the shortest route between
the two end-point super-nodes.
This multi-step process is considerably quicker than using all nodes
but gives a result that still contains the full list of nodes that are
visited. There are some special cases though, for example very short
routes that do not pass through any super-nodes, or routes that start
or finish on a super-node. In these cases one or more of the steps
listed can be removed or simplified.
When the first route reaches the final node the length of that route is
retained as a benchmark. Any shorter complete route that is calculated
later would replace this benchmark. As routes are tested any partial
routes that are longer than the benchmark can be immediately discarded.
Other partial routes have the length of a perfect straight highway to
the final node added to them and if the total exceeds the benchmark
they can also be discarded. Very quickly the number of possible routes
is reduced until the absolute shortest is found.
For routes that do not start or finish on a node in the original data
set a fake node is added to an existing segment. This requires special
handling in the algorithm but it gives mode flexibility for the start,
finish and intermediate points in a route.
Algorithm Evolution
- - - - - - - - - -
In Routino versions 1.0 to 1.4 the algorithm used to select a
super-node was the same as above except that node properties were not
included. Routino versions 1.4.1 to 1.5.1 used a slightly different
algorithm which only chose nodes that were junctions between segments
with different properties (or has a routing restriction that is
different from connecting segments in versions 1.5 and 1.5.1). The
addition of turn restrictions (described in more detail below) requires
the original algorithm since the super-segments more accurately reflect
the underlying topology.
Algorithm Implementation
- - - - - - - - - - - -
The algorithm that is used for finding the route between the
super-nodes using super-segments is the A* algorithm (or a slight
variation of it). This was not a deliberate design decision, but
evolved into it during development. This algorithm relies on
calculating the lowest score (shortest distance or quickest time) to
each node from the starting node. The remaining score for the path to
the destination node is estimated (based on a straight line using the
fastest type of highway) and added to the current score and the result
recorded. At each step the unvisited node that has the lowest current
score is examined and all nodes connected to it have their scores
calculated. When the destination node has been reached all remaining
unvisited nodes with scores higher than the destination node's score
can be discarded and the few remaining nodes examined.
The algorithm used to find the route between super-nodes using normal
segments is Dijkstra's algorithm (although it is implemented as the
same algorithm as above but with no estimated cost). Since these routes
tend to be short and the CPU time for calculating the heuristic cost
function is relatively large this tends to give a quicker solution.
Routing Preferences
-------------------
One of the important features of Routino is the ability to select a
route that is optimum for a set of criteria such as preferences for
each type of highway, speed limits and other restrictions and highway
properties.
All of these features are handled by assigning a score to each segment
while calculating the route and trying to minimise the score rather
than simply minimising the length.
Segment length
When calculating the shortest route the length of the segment is
the starting point for the score.
Speed preference
When calculating the quickest route the time taken calculated
from the length of the segment and the lower of the highway's
own speed limit and the user's speed preference for the type of
highway is the starting point for the score.
One-way restriction
If a highway has the one-way property in the opposite direction
to the desired travel and the user's preference is to obey
one-way restrictions then the segment is ignored.
Weight, height, width & length limits
If a highway has one of these limits and its value is less than
the user's specified requirement then the segment is ignored.
Highway preference
The highway preference specified by the user is a percentage,
these are scaled so that the most preferred highway type has a
weighted preference of 1.0 (0% always has a weighted preference
of 0.0). The calculated score for a segment is divided by this
weighted preference.
Highway properties
The other highway properties are specified by the user as a
percentage and each highway either has that property or not. The
user's property preference is scaled into the range 0.0 (for 0%)
to 1.0 (for 100%) to give a weighted preference, a second
"non-property" weighted preference is calculated in the same way
after subtracting the user's preference from 100%. If a segment
has a particular property then the calculated score is divided
by the weighted preference for that property, if not then it is
divided by the non-property weighted preference. A non-linear
transformation is applied so that changing property preferences
close to 50% do not cause large variations in routes.
Data Pruning
------------
From version 2.2 there are options to "prune" nodes and segments from
the input data which means to remove nodes and/or segments without
significantly changing the routing results.
The pruning options must meet a number of conditions to be useful:
* The topology relevant to routing must remain unchanged. The
instructions that are produced from the reduced set of nodes and
segments must be sufficiently accurate for anybody trying to follow
them on the ground.
* Any restrictions belonging to nodes or segments that stop certain
types of traffic from following a particular highway must be
preserved.
* The total length must be calculated using the original data and not
the simplified data which by its nature will typically be shorter.
* The location of the remaining nodes and segments must be a good
representation of the original nodes and segments. Since the
calculated route may be displayed on a map the remaining nodes and
segments must clearly indicate the route to take.
The prune options all have user-controllable parameters which allow the
geographical accuracy to be controlled. This means that although the
topology is the same the geographical accuracy can be sacrificed
slightly to minimise the number of nodes and segments.
The pruning options that are available are:
* Removing the access permissions for a transport type from segments
if it is not possible to route that transport type from those
segments to a significant number of other places. The limit on the
pruning is set by the total length of the isolated group of
segments. This significantly increases the chance that a route will
be found by not putting waypoints in inaccessible places.
* Removing short segments, the limit is set by the length of the
segment. This removes a number of redundant segments (and
associated nodes) but rules are applied to ensure that removing the
segments does not alter junction topology or remove node access
permissions or changes in way properties.
* Removing nodes from almost straight highways, the limit is set by
the distance between the remaining segments and the original nodes.
This removes a large number of redundant nodes (and therefore
segments) but again care is taken not to remove node access
permissions or changes in way properties.
Turn Restrictions
-----------------
The addition of turn restrictions in version 2.0 adds a set of further
complications because it introduces a set of constraints that are far
more complex than one-way streets.
A turn restriction in the simplest case is a combination of a segment,
node and segment such that routes are not allowed to go from the first
segment to the second one through the specified node. Exceptions for
certain types of traffic can also be specified. Currently only this
simplest type of turn restriction is handled by the algorithm.
The first complication of turn restrictions is that the algorithm above
requires that super-segments are composed of segments with identical
properties. A turn restriction is not the same in both directions so a
super-segment cannot include any route through that turn restriction.
The node at the centre of the turn restriction must therefore be a
super-node to avoid this. In addition to this all nodes connected to
the turn restriction node by a single segment must also be super-nodes
to avoid any long-distance super-segments starting at the restricted
node.
The second complication of a turn restriction is that the optimum route
may require passing through the same node more than once. This can
happen where the route needs to work around a turn restriction by
driving past it, turning round (on a roundabout perhaps) and coming
back along the same highway. Without turn restrictions a route could be
defined purely by the set of nodes that were passed; no node would
exist more than once along a route between two points. With turn
restrictions the route is defined by a node and the segment used to get
there; no route between two points will ever need to follow the same
segment in the same direction more than once. This means that the
optimisation algorithm calculates scores for directed segments (indexed
by segment and end node) rather than for nodes.
A side-effect of this is that a route that works around a turn
restriction must be calculable using the super-segments that are stored
in the database. This puts a limit on the amount of database
optimisation that can be performed because if too many super-segments
are removed the optimum work-around may also be removed. The solution
to this is to ensure that the database preserves all loops that can be
used to turn around and reverse direction, previously super-segments
that started and finished on the same super-node were disallowed.
Another side-effect of having the route composed of a set of locations
(nodes) as well as the direction of travel (segments used to reach
them) is that via points in the route can be forced to continue in the
original direction. If the chosen method of transport obeys turn
restrictions then it will not reverse direction at a via point but will
find an optimum route continuing in the same direction. The only
exception to this is when the route ahead at a waypoint is into a
dead-end and an immediate U-turn is allowed.
A side-effect of having the starting direction at a via point defined
by the previous part of the route is that overall non-optimal routes
may be found even though each section between via points is optimal.
For a route with a start, middle and end point defined it can be the
case that the shortest route from the start to the middle arrives in
the opposite direction to that required for the optimal route from the
middle to the end. The calculation of the route in separate sections
therefore may give a non-optimum result even though each section is
itself optimum based on the start conditions.
Overall the presence of turn restrictions in the database makes the
routing slower even for regions of the map that have no turn
restrictions.
Data Implementation
-------------------
The hardest part of implementing this router is the data organisation.
The arrangement of the data to minimise the number of operations
required to follow a route from one node to another is much harder than
designing the algorithm itself.
The final implementation uses a separate table for nodes, segments and
ways. Each table individually is implemented as a C-language data
structure that is written to disk by a program which parses the
OpenStreetMap XML data file. In the router these data structures are
memory mapped so that the operating system handles the problems of
loading the needed data blocks from disk.
Each node contains a latitude and longitude and they are sorted
geographically so that converting a latitude and longitude coordinate
to a node is fast as well as looking up the coordinate of a node. The
node also contains the location in the array of segments for the first
segment that uses that node.
Each segment contains the location of the two nodes as well as the way
that the segment came from. The location of the next segment that uses
one of the two nodes is also stored; the next segment for the other
node is the following one in the array. The length of the segment is
also pre-computed and stored.
Each way has a name, a highway type, a list of allowed types of
traffic, a speed limit, any weight, height, width or length
restrictions and the highway properties.
The super-nodes are mixed in with the nodes and the super-segments are
mixed in with the segments. For the nodes they are the same as the
normal nodes, so just a flag is needed to indicate that they are super.
The super-segments are in addition to the normal segments so they
increase the database size (by about 10%) and are also marked with a
flag. Some segments are therefore flagged as both normal segments and
super-segments if they both have the same end nodes.
The relations are stored separately from the nodes, segments and ways.
For the turn restriction relations the initial and final segments are
stored along with the restricted node itself. Each node that has a turn
restriction is marked in the main node storage with a flag to indicate
this information.
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Copyright 2008-2013 Andrew M. Bishop.