Re: "set" data structure in Mathematica? (speeding up graph traversal function)

• To: mathgroup at smc.vnet.net
• Subject: [mg117595] Re: "set" data structure in Mathematica? (speeding up graph traversal function)
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Thu, 24 Mar 2011 06:27:28 -0500 (EST)

```Oops, I mangled the first timing, a bit:

Clear[next, path, p]
nodes = Union@Flatten[graph /. Rule -> List];
next[i_Integer] := next[i] = Cases[graph, Rule[i, j_] :> j]
path[i_Integer] := path@p@i
path[p[a___, last_]] :=
Module[{c = Complement[next@last, {a, last}]},
If[c == {},
p[a, last],
Flatten[path@p[a, last, #] & /@ c]
]
]
(paths = (path /@ nodes); Length@Flatten@paths) // Timing

{1.75351, 47165}

Bobby

On Wed, 23 Mar 2011 02:54:36 -0500, DrMajorBob <btreat1 at austin.rr.com>
wrote:

> Here's an improvement, I think.
>
> For the test graph you listed below, I get
>
> Clear[next, path, p, listPath]
> nodes = Union@Flatten[graph /. Rule -> List];
> next[i_Integer] := next[i] = Cases[graph, Rule[i, j_] :> j]
> path[i_Integer] := path@p@i
> path[p[a___, last_]] :=
>   Module[{c = Complement[next@last, {a, last}]},
>    If[c == {},
>     p[a, last],
>     Flatten[path@p[a, last, #] & /@ c]
>     ]
>    ]
> (paths = (path /@ nodes); Length@Flatten[pathways]) // Timing
>
> {1.73881, 405496}
>
> Omitting Flatten and using p -> List can get a structure like yours
> (mostly):
>
> Clear[next, path, p, listPath]
> nodes = Union@Flatten[graph /. Rule -> List];
> next[i_Integer] := next[i] = Cases[graph, Rule[i, j_] :> j]
> path[i_Integer] := path@p@i
> path[p[a___, last_]] :=
>   Module[{c = Complement[next@last, {a, last}]},
>    If[c == {},
>     p[a, last],
>     Flatten[path@p[a, last, #] & /@ c]
>     ]
>    ]
> ((paths = Flatten[path /@ nodes] /. p -> List) // Length) // Timing
>
> {1.85056, 47165}
>
> Compare the above with
>
> Clear[traverse]
> traverse[path_][node_] :=
>   With[{l = ReplaceList[node, graph], p = Append[path, node]},
>    If[l === {}, {p},
>     If[MemberQ[path, node], {}, Join @@ (traverse[p] /@ l)]]]
> pathways = traverse[{}] /@ nodes; // Timing
>
> {8.09828, Null}
>
> paths == Cases[pathways, {__Integer}, Infinity]
>
> True
>
> Bobby
>
> On Tue, 22 Mar 2011 05:06:19 -0500, Szabolcs Horv=E1t
> <szhorvat at gmail.com>
> wrote:
>
>> Dear MathGroup,
>>
>> I am trying to speed up a function.
>>
>> I'm looking for an efficient set-like data structure, to be used as
>> follows:
>>
>> I have a recursive function f, which works like this (omitting the
>> details):
>>
>> f[set_][value_] :=
>>    if value in set then
>>      return nothing
>>    otherwise
>>      newSet = insert value into set
>>      return f[newSet] /@ (list of computed values)
>>
>> In other words, the function calculates a list of values, then calls
>> itself recursively with each of them.  These values are collected into a
>> set during the recursive calls, and one of the stopping conditions is
>> that a value gets repeated.
>>
>> Note: of course there's another stopping condition as well, otherwise
>> the function wouldn't return anything, but that is irrelevant here.
>>
>> Also note: each branch of the recursion will have its own collection
>> (set) of values.
>>
>> The trivial solution would be just using a List of values, inserting new
>> values with Append, and testing whether they're already in the list with
>> MemberQ.  I found that the function spends most of its time in MemberQ,
>> so I am looking for a more efficient solution than a plain list that
>> needs to be iterated over to find elements in it.
>>
>> I tried using newSet = Union[set, {value}] instead of append, and
>> testing whether Length[newSet] === Length[set], but this turned out to
>> be even slower.
>>
>> I wonder if there's a better solution.
>>
>> One way would be using "function definitions", i.e. "inserting" 'value'
>> into 'set' as set[value] = True, and testing membership simply as
>> TrueQ[set[value]], but this object can't easily be passed around in a
>> recursive function because it is essentially a global variable.  Note
>> again that the recursion has several branches as the function is mapped
>> over a whole list of values.
>>
>> Does anyone have an idea how to solve this problem?
>>
>>   -- Szabolcs
>>
>> P.S.  The problem I'm trying to solve is to find (actually just count)
>> ALL acyclic paths in a directed graph, starting from a given node.
>> Starting from a node, I descend along the connections until there's no
>> way to go, or until I encounter a node that I have already visited.
>>
>> Perhaps it's better if I post the function ...
>>
>> traverse[graph_][path_][node_] :=
>>   With[
>>    {l = ReplaceList[node, graph],
>>     p = Append[path, node]},
>>    If[l === {},
>>     {p},
>>     If[
>>      MemberQ[path, node],
>>      {},
>>      Join @@ (traverse[graph][p] /@ l)
>>      ]
>>     ]
>>    ]
>>
>> graph is a list of rules representing a simple directed graph, path is
>> the list of already visited nodes (initially {}), node is the starting
>> node.  It's generally a good idea to pass the graph as a dispatch table
>> (Dispatch[]), but then MemberQ[] takes even longer to evaluate than
>> ReplaceList[].
>>
>> If there are cycles in the graph, there will be a large number of paths,
>> and the function will be slow.
>>
>> P.P.S
>>
>> Here's a small graph with some loops to test with:
>>
>> {1 -> 2, 3 -> 4, 5 -> 1, 5 -> 2, 6 -> 2, 6 -> 5, 7 -> 2, 7 -> 6,
>>   8 -> 1, 8 -> 2, 8 -> 5, 8 -> 6, 9 -> 2, 9 -> 5, 9 -> 6, 9 -> 7,
>>   10 -> 2, 10 -> 6, 10 -> 7, 11 -> 6, 11 -> 7, 12 -> 6, 12 -> 7,
>>   13 -> 6, 13 -> 7, 14 -> 2, 14 -> 5, 14 -> 6, 14 -> 7, 15 -> 3,
>>   15 -> 6, 15 -> 7, 15 -> 8, 16 -> 3, 16 -> 6, 16 -> 7, 16 -> 13,
>>   17 -> 3, 17 -> 6, 17 -> 7, 17 -> 9, 17 -> 11, 17 -> 13, 18 -> 3,
>>   18 -> 6, 18 -> 7, 18 -> 9, 18 -> 11, 18 -> 12, 18 -> 13, 18 -> 14,
>>   18 -> 15, 18 -> 19, 18 -> 20, 19 -> 6, 19 -> 7, 19 -> 9, 19 -> 10,
>>   19 -> 11, 19 -> 12, 19 -> 13, 19 -> 14, 19 -> 16, 19 -> 17, 19 -> 18,
>>    19 -> 20, 20 -> 3, 20 -> 7, 20 -> 9, 20 -> 10, 20 -> 11, 20 -> 12,
>>   20 -> 14, 20 -> 16, 20 -> 18, 20 -> 19, 21 -> 3, 21 -> 9, 21 -> 10,
>>   21 -> 11, 21 -> 12, 21 -> 13, 21 -> 14, 21 -> 15, 21 -> 16, 21 -> 17,
>>    21 -> 19, 21 -> 20, 22 -> 3, 22 -> 7, 22 -> 9, 22 -> 10, 22 -> 11,
>>   22 -> 12, 22 -> 13, 22 -> 14, 22 -> 16, 22 -> 19, 22 -> 20, 23 -> 3,
>>   23 -> 7, 23 -> 9, 23 -> 10, 23 -> 14, 23 -> 16, 23 -> 17, 23 -> 18,
>>   23 -> 19, 23 -> 20, 24 -> 7, 24 -> 9, 24 -> 10, 24 -> 11, 24 -> 14,
>>   24 -> 15, 24 -> 16, 24 -> 17, 24 -> 18, 24 -> 20, 25 -> 3, 25 -> 9,
>>   25 -> 10, 25 -> 11, 25 -> 14, 25 -> 15, 25 -> 16, 25 -> 18, 25 -> 19,
>>    25 -> 20, 26 -> 7, 26 -> 9, 26 -> 10, 26 -> 11, 26 -> 13, 26 -> 14,
>>   26 -> 15, 26 -> 16, 26 -> 19, 26 -> 20, 27 -> 3, 27 -> 6, 27 -> 7,
>>   27 -> 9, 27 -> 10, 27 -> 11, 27 -> 12, 27 -> 13, 27 -> 14, 27 -> 15,
>>   27 -> 16, 27 -> 17, 27 -> 18, 27 -> 19, 27 -> 20, 27 -> 22, 27 -> 28,
>>    28 -> 3, 28 -> 9, 28 -> 10, 28 -> 11, 28 -> 12, 28 -> 13, 28 -> 14,
>>   28 -> 15, 28 -> 16, 28 -> 17, 28 -> 18, 28 -> 19, 28 -> 20, 28 -> 22,
>>    28 -> 27, 28 -> 29, 29 -> 3, 29 -> 6, 29 -> 7, 29 -> 8, 29 -> 9,
>>   29 -> 10, 29 -> 11, 29 -> 12, 29 -> 13, 29 -> 14, 29 -> 15, 29 -> 16,
>>    29 -> 17, 29 -> 18, 29 -> 19, 29 -> 20, 29 -> 21, 29 -> 22,
>>   29 -> 23, 29 -> 24, 29 -> 25, 29 -> 26, 29 -> 27, 29 -> 28}
>>
>
>
> --
> DrMajorBob at yahoo.com
>

--
DrMajorBob at yahoo.com

```

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