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Subject: Re: [boost] space complexity of push_relabel_max_flow
From: Dan Jiang (danjiang_at_[hidden])
Date: 2010-04-14 18:15:35

> Date: Tue, 13 Apr 2010 22:31:17 -0400
> From: jewillco_at_[hidden]
> To: boost_at_[hidden]
> Subject: Re: [boost] space complexity of push_relabel_max_flow
> On Tue, 13 Apr 2010, Dan Jiang wrote:
> >
> > Thanks for the reply. Please see my reply inline.
> >
> >> Date: Tue, 13 Apr 2010 12:53:45 -0400
> >> From: jewillco_at_[hidden]
> >> To: boost_at_[hidden]
> >> Subject: Re: [boost] space complexity of push_relabel_max_flow
> >>
> >> On Tue, 13 Apr 2010, Dan Jiang wrote:
> >>
> >>> Hi All, I am a new BOOST user and like it very much. I have a question
> >>> regarding space complexity of push_relabel_max_flow. The
> >>> push_relabel_max_flow is very efficient compared to Edmond's and Karp.
> >>> In my application, I need to deal with half a million nodes and half a
> >>> billion arcs. It requires lots of ram (>= 6gb). Two questions:
> >>
> >>> 1) How do I find out space used per node and per arc?
> >>
> >> I do not know off hand for adjacency_list; it is probably reasonable (one
> >> integer per edge, some per vertex, plus properties). However, remember
> >> that dynamically allocated vectors can have up to a 100% space overhead.
> >> I would recommend using compressed_sparse_row_graph and then changing the
> >> EdgeIndex and Vertex types to uint32_t (since you have fewer than 2^32 of
> >> each). That will use 4 * (nedges + nvertices + 1) bytes of space, plus
> >> properties; accesses will also likely be faster than for adjacency_list.
> >
> > I'll try that later on.
> OK. I saw your other email about that -- I'll answer it soon.
> >>> 2) My network does not need any reverse edge (currently I set the
> >>> capacity of reverse edge to 0), is there anyway to not supply reverse
> >>> edges to save space?
> >>
> >> It does not appear so, because of the residual capacities that need to be
> >> stored. If you can find an algorithm description that doesn't need
> >> reverse edges, though, it might be possible to change the algorithm to
> >> remove them.
> >
> > I just read the push_relabel_max_flow algorithm, it requires reverse edge list and their capacities.
> It actually doesn't need the edges AFAICT. It just uses them to look
> elements up in property maps, so it might be possible to work around that
> requirement. However, there are other memory uses in the algorithm that
> are likely to be bigger.
> >>> 3) Does push_relabel_algorithm dynamically allocates lots of memory
> >>> (besides the ram required by the input graph, i.e., adjacency_list)?
> >>
> >> It looks like there is a bunch of that. I see six vectors and a queue in
> >> there, and one of those vectors is a vector of std::lists.
> >>
> >>> 4) Is it possible to implement custom file-based edge lists so that I
> >>> could explicitly page to disk when not enough ram is available? How
> >>> slower would it be?
> >>
> >> It would probably be quite a bit slower. If you need to go beyond your
> >> system's memory, why not let the OS's swap mechanism do the disk accesses?
> >> That would take care of all of the caching and such that you would need,
> >> and I'm not sure writing your own would provide much benefit.
> > I tried OS paging on 64-bit OS using my application. The performance
> > decreases from 100 to 1000times and plus the paging happens only when
> > available physical ram runs out. I'd like to be able to do partial
> > paging on demand. I am looking into using STLdb to hold capacity
> > properties, one for forward edge, another for reverse edge. But saving
> > is just 2*4*NumOfArcs (I am using float for capacity), which might not
> > be dramatic saving...
> I'm surprised OS paging was so bad, but that can happen and the algorithm
> doesn't seem to use a lot of locality between related data structures.
> The 2*4*nedges thing is probably not worth doing too much work to get yet.
> >> How much memory do you have, and what type are you using for your edge
> >> capacities? I have some ideas for how to shrink things, but I would first
> >> like to see a baseline of how bad things are now. If you switch to
> >> compressed_sparse_row_graph and try to run the algorithm (possibly on
> >> scaled-down graphs), how much memory does it use? Running that for a few
> >> data sizes might give an idea of how much actual space per vertex and edge
> >> the algorithm uses.
> > I have 4gb physical ram, usually with 3.2gb free. I use float for edge
> > capacity. I'll look into compressed_sparse_row_graph and see if there is
> > much saving.
> It will help you with the graph structure information and maybe the
> properties. Let's see what the actual usage is now and how that compares
> to what you have space for.
> >> BTW, is there any regular structure in your graph? For example, is it a
> >> grid graph like some computer vision algorithms use?
> >
> > My application is for mining. I construct a network from voxel model
> > which is typically of size 200 * 200 * 125. For each "chosen" voxel (a
> > voxel deemed to be used as a node in a graph), I create about 100 arcs
> > to other relevant voxels. I my test case, I have 0.5million voxels and
> > hence 500mil arcs. I need to find max flow from my source voxel to
> > destination voxel. If I use STLdb, I can decide at run time when to page
> > all property maps used in the push_relabel_max_flow algorithm. vertex
> > and edge lists are not to be paged as STDdb supports only maps.
> How do you define "relevant"? Are they local in terms of the grid of
> voxels? Is there any regular structure at all, such as linking to all
> neighbors within a certain distance that have a certain weight or
> something? What are you able to say about the conditions for which edges
> are put in? Can any of the data used to create edges be reused for
> computing capacities?
"relevant" is parameterized on a formula which means, # of arcs from a node is a variable whose value ranges from 5 to 150. So there is really no patten because the parameter used to compute "relavency" is user-defined.
I think CSR would work better in terms of memory requirement. However, I do not know how much as push_relabel_max_flow (or anall max flow algorithms in BOOST) requires "capacity" map and "reverse_edge" map which seem not provided by the CSR graph (in fact, I do not even know how to get these two maps from a CSR if possible at all, see my reply in another email).
Dan Jiang
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