[Arjen Markus] (15 august 2005) Here is a very simple algorithm to find the shortest paths in a graph from any node to any other node. The computation is done using "Floyd's algorithm" and it consists of two steps:
* Compute a matrix of indices (encodings of the shortest paths)
* Use that to construct the path from one node to the next
It uses Tcllib's struct::graph module to store the graph in a convenient way.
Of course there are more efficient algorithms, but this one is delightfully simple.
----
# shortest_path.tcl --
# Find the shortest path in a graph, using
# Floyd's algorithm
#
package require struct
# mkMatrix --
# Make a square matrix with uniform entries
# Arguments:
# size Size (number of columns/rows) of the matrix
# value Default value to use
# Result:
# A list of lists that represents the matrix
#
proc mkMatrix {size value} {
set row {}
for { set i 0 } { $i < $size } { incr i } {
lappend row $value
}
set matrix {}
for { set i 0 } { $i < $size } { incr i } {
lappend matrix $row
}
return $matrix
}
# mkPath --
# Use the resulting matrix to print the shortest path
# Arguments:
# indices Matrix of indices
# names Names of the nodes
# from The name of the node to start with
# to The name of the node to go to
# Result:
# A list of intermediate nodes along the path
#
proc mkPath {indices names from to} {
set f [lsearch $names $from]
set t [lsearch $names $to]
set ipath [IntermediatePath $indices $f $t]
set path [list $from]
foreach node $ipath {
lappend path [lindex $names $node]
}
lappend path $to
return $path
}
# IntermediatekPath --
# Construct the intermediate path
# Arguments:
# indices Matrix of indices
# from The node to start with
# to The node to go to
# Result:
# A list of intermediate nodes along the path
#
proc IntermediatePath {indices from to} {
set path {}
set next [lindex $indices $from $to]
if { $next >= 0 } {
set path [concat $path [IntermediatePath $indices $from $next]]
lappend path $next
set path [concat $path [IntermediatePath $indices $next $to]]
}
return $path
}
# floydPaths --
# Construct the matrix that encodes the shortest paths,
# via Floyd's algorithm
# Arguments:
# distances Matrix of distances
# lmatrix (Optional) the name of a variable to hold the
# shortest path lengths as a matrix
# Result:
# A matrix encoding the shortest paths
#
proc floydPaths {distances {lmatrix {}}} {
if { $lmatrix != {} } {
upvar 1 $lmatrix lengths
}
set size [llength $distances]
set indices [mkMatrix $size -1]
set lengths $distances
for { set k 0 } { $k < $size } { incr k } {
for { set i 0 } { $i < $size } { incr i } {
for { set j 0 } { $j < $size } { incr j } {
set dik [lindex $lengths $i $k]
set dij [lindex $lengths $i $j]
set dkj [lindex $lengths $k $j]
if { $dik == {} || $dkj == {} } {
continue ;# No connection - distance infinite
}
if { $dij == {} || $dik+$dkj < $dij } {
lset indices $i $j $k
lset lengths $i $j [expr {$dik+$dkj}]
}
}
}
}
return $indices
}
# determinePaths --
# Construct the matrix that encodes the shortest paths from
# the given graph
# Arguments:
# graph Graph to be examined
# key Name of the (non-negative) attribute) holding the
# length of the arcs (defaults to "distance")
# lmatrix (Optional) the name of a variable to hold the
# shortest path lengths as a matrix
# Result:
# A matrix encoding the shortest paths
#
proc determinePaths {graph {key distance} {lmatrix {}} } {
if { $lmatrix != {} } {
upvar 1 $lmatrix lengths
}
set names [$graph nodes]
set distances [mkMatrix [llength $names] {}]
for { set i 0 } { $i < [llength $names] } { incr i } {
lset distances $i $i 0 ;# Distance of a node to itself is 0
}
foreach arc [$graph arcs $key] {
set from [lsearch $names [$graph arc source $arc]]
set to [lsearch $names [$graph arc target $arc]]
set d [$graph arc get $arc $key]
if { $from != $to } {
lset distances $from $to $d
}
}
puts $distances
return [floydPaths $distances lengths]
}
# Small test --
# Construct a graph, make a matrix of distances out of it
# and query a few shortest paths. Note: the graph is undirected,
# so the arrows are doubled.
#
set names {A B C D E F G}
set distances {
{ 0 7 3 {} {} {} {}}
{ 7 0 {} 8 {} {} 40}
{ 3 {} 0 12 4 {} {}}
{{} 8 12 0 {} {} {}}
{{} {} 4 {} 0 10 7}
{{} {} {} {} 10 0 8}
{{} 40 {} {} 7 8 0}}
# Construct the graph:
#
set graph [::struct::graph]
set names {A B C D E F G}
set arcs {
A B 7
A C 3
B D 8
B G 40
C D 12
C E 4
E F 10
E G 7
F G 8
}
#
#
foreach n $names {
$graph node insert $n
}
foreach {from to distance} $arcs {
set arc [$graph arc insert $from $to]
$graph arc append $arc distance $distance
set arc [$graph arc insert $to $from]
$graph arc append $arc distance $distance
}
#
# Now that we have our graph, examine some shortest paths
#
# Note: the ordering of the nodes in the graph is not the
# same as the order in which they were created! Hence the
# call to [$graph nodes].
set indices [determinePaths $graph "distance" lengths]
puts $indices
puts [mkPath $indices [$graph nodes] A B]
puts [mkPath $indices [$graph nodes] B G]
See also: [A-star]
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%|[Category Graph Theory]|[Category Mathematics]|%
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