Examining the VM splay tree effectiveness

[Andre Oppermann]

On 30.09.2010 19:15, Matthew Fleming wrote:
> On Thu, Sep 30, 2010 at 9:37 AM, Andre Oppermann<andre at freebsd.org>  wrote:
>> Just for the kick of it I decided to take a closer look at the use of
>> splay trees (inherited from Mach if I read the history correctly) in
>> the FreeBSD VM system suspecting an interesting journey.
>>
>> The VM system has two major structures:
>>   1) the VM map which is per process and manages its address space
>>     by tracking all regions that are allocated and their permissions.
>>   2) the VM page table which is per VM map and provides the backing
>>     memory pages to the regions and interfaces with the pmap layer
>>     (virtual->physical).
>>
>> Both of these are very frequently accessed through memory allocations
>> from userspace and page faults from the pmap layer.  Their efficiency
>> and SMP scalability is crucial for many operations beginning with fork/
>> exec, going through malloc/mmap and ending with free/exit of a process.
>>
>> Both the vmmap and page table make use of splay trees to manage the
>> entries and to speed up lookups compared to long to traverse linked
>> lists or more memory expensive hash tables.  Some structures though
>> do have an additional linked list to simplify ordered traversals.
>>
>> A splay tree is an interesting binary search tree with insertion,
>> lookup and removal performed in O(log n) *amortized* time.  With
>> the *amortized* time being the crucial difference to other binary trees.
>> On every access *including* lookup it rotates the tree to make the
>> just found element the new root node.  For all gory details see:
>>   http://en.wikipedia.org/wiki/Splay_tree
>>
>> This behavior has a few important implications:
>>   1) the root node and constant rotation works like a cache with
>>     least recent looked up node at the top and less recently ones
>>     close to the top;
>>   2) every lookup that doesn't match the root node, ie. a cache miss,
>>     causes a rotation of the tree to make the newly found node the new
>>     root;
>>   3) during the rotate it has to re-arrange and touch possibly a large
>>     number of other nodes;
>>   4) in worst case the tree may resemble a linked list.  A splay tree
>>     makes no guarantee that it is balanced.
>>
>> For the kernel and the splay tree usage in the VM map and page table
>> some more implications come up:
>>   a) the amortization effect/cost only balance if the majority of
>>     lookups are root node (cache) hits, otherwise the cost of
>>     rebalancing destroys any advantage and quickly turns into a
>>     large overhead.
>>   b) the overhead becomes even worse due to touching many nodes and
>>     causing a lot of CPU cache line dirtying.  For processes with
>>     shared memory or threads across CPU's this overhead becomes
>>     almost excessive because the CPU's stomp on each others cached
>>     nodes and get their own caches invalidated.  The penalty is a
>>     high-latency memory refetch not only for the root node lookup
>>     but every hop in the following rebalancing operation =>  thrashing.
>>
>> To quantify the positive and negative effects of the splay tree I
>> instrumented the code and added counters to track the behavior of
>> the splay tree in the vmmap and page table.
>>
>> The test case is an AMD64 kernel build to get a first overview.
>> Other system usages may have different results depending on their
>> fork and memory usage patters.
>>
>> The results are very interesting:
>>
>>   The page table shows a *very* poor root node hit rate and an excessive
>>   amount of rotational node modifications representing almost the
>>   worst case for splay trees.
>>
>> http://chart.apis.google.com/chart?chs=700x300&chbh=a,23&chds=0,127484769&chd=t:16946915,16719791,48872230,131057,74858589,105206121&cht=bvg&chl=insert|remove|lookup|cachehit|rotates|rotatemodifies
>>
>>   The vmmap shows a high root node hit rate but still a significant
>>   amount of rotational node modifications.
>>
>> http://chart.apis.google.com/chart?chs=700x300&chbh=a,23&chds=0,21881583&chd=t:724293,723701,20881583,19044544,3719582,4553350&cht=bvg&chl=insert|remove|lookup|cachehit|rotates|rotatemodifies
>>
>>  From these observations I come to the conclusion that a splay tree
>> is suboptimal for the normal case and quickly horrible even on small
>> deviations from the normal case in the vmmap.  For the page table it
>> is always horrible.  Not to mention the locking complications due to
>> modifying the tree on lookups.
>>
>> Based on an examination of other binary trees I put forward the
>> hypothesis that a Red-Black tree is a much better, if not the optimal,
>> fit for the vmmap and page table.  RB trees are balanced binary trees
>> and O(log n) in all cases.  The big advantage in this context is that
>> lookups are pure reads and do not cause CPU cache invalidations on
>> other CPU's and always only require a read lock without the worst
>> case properties of the unbalanced splay tree.  The high cache locality
>> of the vmmap lookups can be used with the RB tree as well by simply
>> adding a pointer to the least recently found node.  To prevent write locking
>> this can be done lazily.  More profiling is required to make
>> a non-speculative statement on this though.  In addition a few of
>> the additional linked lists that currently accompany the vmmap and
>> page structures are no longer necessary as they easily can be done
>> with standard RB tree accessors.  Our standard userspace jemalloc
>> also uses RB trees for its internal housekeeping.  RB tree details:
>>   http://en.wikipedia.org/wiki/Red-black_tree
>>
>> I say hypothesis because I haven't measured the difference to an
>> RB tree implementation yet.  I've hacked up a crude and somewhat
>> mechanical patch to convert the vmmap and page VM structures to
>> use RB trees, the vmmap part is not stable yet.  The page part
>> seems to work fine though.
>>
>> This is what I've hacked together so far:
>>   http://people.freebsd.org/~andre/vmmap_vmpage_stats-20100930.diff
>>   http://people.freebsd.org/~andre/vmmap_vmpage_rbtree-20100930.diff
>>
>> The diffs are still in their early stages and do not make use of
>> any code simplifications becoming possible by using RB trees instead
>> of the current splay trees.
>>
>> Comments on the VM issue and splay vs. RB tree hypothesis welcome.
>
> Do you have actual performance numbers from a real workload?

No, not yet.  I obviously would have posted those numbers as well.
What do you consider a representative real workload?

> I implemented an AA tree locally (basically a 2,3 tree that uses
> coloring to represent the 3 nodes) and the performance was much worse
> than the splay tree.  I assume this is due to the overhead of keeping
> the tree balanced; I didn't instrument either the splay or the AA
> code.  I suspect that the overhead of an RB tree would similarly wash
> out the benefits of O(log_2 n) time, but this is theory.  The facts
> would be measuring several different workloads an looking at the
> system/real times for them between the two solutions.

I think there are two pronounced effects to consider:
  a) the algorithmic complexity
  b) the real world implications of said complexity and operations
     on today's CPU's multi-hierarchy caches and associated SMP side
     effects.

> We've talked internally at $work about using a B+-tree with maybe
> branching factor 5-7; whatever makes sense for the size of a cache
> line.  This seems likely to be better for performance than an RB-tree
> but does require a lot more changes, since separate memory is needed
> for the tree's nodes outside the vm_page structure.  There just hasn't
> been time to implement it and try it out.
 >
> Unfortunately I won't have the time to experiment at $work for a few
> months on this problem.

-- 
Andre