# A List implementation backed by multiple small arrays rather than the traditional single large array.

Benedict Elliott Smith lists at laerad.com
Sat Apr 17 16:48:07 UTC 2010

```Hi Kevin,

As it happens I might have something useful still to contribute. As an
exercise in saving face I revisited the problem to see if I could achieve
the same complexity bounds as ChunkedArrayList but with a lower overhead. I
must admit I still didn't fully appreciate how the algorithm in
ChunkedArrayList worked until I tried to come up with an algorithm with
similar properties. What I have ended up with is almost identical except
adds I think a couple of incremental improvements, simply by redefining the
arrayIndex() method. I should note that I have not yet implemented more than
a prototype as it seems to me your implementation is excellent already, and
if it is decided to include my modifications the changes should be modest.

Firstly, (I hope that) what I have produced is a little more CPU pipe-line
friendly; there is less dependency on immediately preceding calculations at
each stage (i.e. so more operations should be able to proceed simultaneously
in the pipeline), and consists exclusively of shifts, addition/subtraction
and bit-wise (&)ands (except for the conditionals in
Integer.numberOfLeadingZeros(i)), although the total number of instructions
is approximately the same.

Secondly, I have modified the algorithm so that a "seed" size can be
specified (although I expect hard coding a suitable one will ultimately be
best). Whereas ChunkedArrayList currently requires that the pattern of array
allocation sizes be [1, 1, 2, 2, 2, 4(..*6), 8(..*12), 16(..*24)] we can now
support, for some "*s*", [*s*(..*2), 2*s*(..*3), 4*s*(..*6), 8*s*(..*12), 16
*s*(..*24)] etc. although when put in simple text like that it does appear
to trivialise the change. The benefit of this, though, is two fold: 1) for
small n the constant factor is reduced (both CPU and memory wise); and 2)
the sqrt(n) bounds are reached more quickly also.

As an illustration, consider setting *s* to 4, and assume the backing array
is size two and doubles in size with each growth; with ChunkedArrayList we
would resize at i=2, i=6, i=20, i=72; with *s* as 4 we would instead resize
at i=8,i=24,i=80,i=288; the cost at each would be some multiple of 2,4,8,16
respectively. As you can see the latter is much closer to the sqrt(n) cost -
both approach it eventually, but my suggestion is to reach it more quickly.
This is at the expense of more slowly reaching the sqrt(n) wasted memory
condition, but given the high constant factor cost wrt to memory at this
early stage, this seems a very sensible trade off. It seems likely this
should also have a positive impact on cache performance for smaller lists as
well.

Finally, after playing with this idea in my head I am confident I can extend
the core ideas of this data structure to hashing relatively easily, getting
the the same worst case O(sqrt(n)) insertion cost, and O(sqrt(n)) wasted
memory guarantees. I notice that this case hasn't been addressed yet,
although I see from Martin's recent mail that this was raised before. Unless
there are better suggestions for solving the hash table problem I will have
a go at it as it seems an interesting problem - that is, assuming there are
no objections?

I'm interested to hear your thoughts. I hope this time I've been a bit more
considered in what I've put forward, and hence less of a waste of time!

Code snippet for calculation of array index and item offset:

final int arraySizeShiftMinusSeed = ((31 -
Integer.numberOfLeadingZeros(index >>> seed)) >>> 1) ;
final int arraySizeShift = arraySizeShiftMinusSeed + seed ;
final int firstArrayOfThisSize = ((((1 << arraySizeShiftMinusSeed + 3) - (1
<< arraySizeShiftMinusSeed + 1))) >>> 1) - 1 ;
final int indexRemainder = index - ((1 << seed) << arraySizeShiftMinusSeed +
arraySizeShiftMinusSeed) ;
final int arrayOffset = indexRemainder >>> arraySizeShift ;

final int arrayIndex = firstArrayOfThisSize + arrayOffset ;
final int itemIndex = index & ((1 << arraySizeShift) - 1) ;

the first array size will be 1 << seed - 1 (i.e. seed is equal to *s* + 1);
seed only works for values for 2 or more at this moment, fyi

On 16 April 2010 00:18, Kevin L. Stern <kevin.l.stern at gmail.com> wrote:

> Oh no worries Benedict, thanks for your interest in the topic.  Let me know
> if you have any other questions or if you have any related ideas or
> concerns.
>
>
> On Thu, Apr 15, 2010 at 8:00 AM, Benedict Elliott Smith <lists at laerad.com>wrote:
>
>> Sorry Kevin - it sounds like I might be being of more hindrance than help.
>> that part of the discussion was clearly truncated by the time I had joined
>> the list - I haven't been able to find the history in the archives either...
>>
>> I was just wondering about the worst case cost of add() as described by
>> the worst case cost of an add is still sqrt(n), with a relatively high
>> constant factor. I had been thinking that once n passed a threshold the cost
>> of additions in this other structure would become a constant factor,
>> offering nice algorithmic complexity guarantees for large n; however since
>> sqrt(Integer.MAX_VALUE) is ~46,000, the maximum size of new array
>> allocations would have to be unrealistically small (assuming linear cost for
>> allocation) for this to be the case. It would still be nice to have a data
>> structure that avoids needing to copy data with each grow, whilst still
>> maintaining good memory performance.
>>
>> ascertain the behaviour of this class, as I couldn't locate a free copy
>> of [Brodnik99resizablearrays], and it seems this was a bad idea; as the
>> sqrt(n) cost appears to be associated with growing the backing array, rather
>> than with what I assumed to be copying data between arraylets, and it seems
>> this cost is pretty optimal. That will teach me to post to a list without
>> getting my facts straight first. The interesting thing is simply that the
>> constant factor for this implementation still seems to be quite high,
>> although perhaps that is simply because I was not benchmarking sufficiently
>> large values of n.
>>
>>
>>
>> On 15 April 2010 12:12, Kevin L. Stern <kevin.l.stern at gmail.com> wrote:
>>
>>> Hi Benedict,
>>>
>>> Unless I am misreading your post, this now has a very similar feel to the
>>> first data structure that I posted to the list.  Martin Buchholz then
>>> pointed out that we can incorporate the ideas from
>>> [Brodnik99resizablearrays] and reap additional benefits.
>>>
>>> Regards,
>>>
>>> Kevin
>>>
>>>
>>> On Thu, Apr 15, 2010 at 4:07 AM, Benedict Elliott Smith <
>>>
>>>> Hi Kevin,
>>>>
>>>> Yes, as I was going to bed last night I realised I had not fully
>>>> addressed the problem that was originally being visited; only reduced the
>>>> constant factor for addition to the end of the list. A trivial modification
>>>> fixes that, however; same scheme but up to some maximum arraylet size (of a
>>>> power of 2), after which the array is increased in size linearly.
>>>> Performance doesn't seem to have been affected appreciably, although not
>>>> been exhaustive in the benchmarking:
>>>>
>>>> 10 items inserts versus ArrayList: Chunked=1.15, ExpArray=1.16
>>>> 10 items inserts Chunked / ExpArray = 0.99
>>>> 10 items get versus ArrayList: Chunked=1.15, ExpArray=1.16
>>>> 10 items get Chunked / ExpArray = 0.99
>>>> 100 items inserts versus ArrayList: Chunked=1.24, ExpArray=1.01
>>>> 100 items inserts Chunked / ExpArray = 1.23
>>>> 100 items get versus ArrayList: Chunked=1.24, ExpArray=1.01
>>>> 100 items get Chunked / ExpArray = 1.23
>>>> 1000 items inserts versus ArrayList: Chunked=1.22, ExpArray=1.03
>>>> 1000 items inserts Chunked / ExpArray = 1.19
>>>> 1000 items get versus ArrayList: Chunked=1.22, ExpArray=1.03
>>>> 1000 items get Chunked / ExpArray = 1.19
>>>> 10000 items inserts versus ArrayList: Chunked=1.22, ExpArray=1.03
>>>> 10000 items inserts Chunked / ExpArray = 1.18
>>>> 10000 items get versus ArrayList: Chunked=1.22, ExpArray=1.03
>>>> 10000 items get Chunked / ExpArray = 1.18
>>>> 100000 items inserts versus ArrayList: Chunked=0.82, ExpArray=0.75
>>>> 100000 items inserts Chunked / ExpArray = 1.09
>>>> 100000 items get versus ArrayList: Chunked=0.82, ExpArray=0.75
>>>> 100000 items get Chunked / ExpArray = 1.09
>>>>
>>>> user configurable. Even with a low setting as above (65K) performance seems
>>>> pretty consistent.
>>>>
>>>> Code for calculating index and array offset are pretty straight forward;
>>>> haven't given much thought to optimisations just yet:
>>>>
>>>> private final int indexFor(int a, int i) {
>>>> return 1 + i - (a > maxArrayIndex ? (1 + a - maxArrayIndex) <<
>>>> maxArraySizeShift : 1 << a) ;
>>>>  }
>>>> private final int arrayFor(int i) {
>>>> return i >= (maxArraySize << 1) ? (i + 1 >>> maxArraySizeShift) +
>>>> maxArrayIndex - 1 : 31 - Integer.numberOfLeadingZeros(i + 1) ;
>>>>  }
>>>>
>>>> Regarding the double list idea - yes, I agree, I certainly didn't think
>>>> that one through fully!
>>>>
>>>>
>>>>
>>>> On 15 April 2010 02:44, Kevin L. Stern <kevin.l.stern at gmail.com> wrote:
>>>>
>>>>> Hi Benedict,
>>>>>
>>>>> Like you, I am relatively new to this mailing list; I am also trying to
>>>>> tread lightly so as not to step on any toes.  That being said, I think that
>>>>> I can offer a response to your inquiry.
>>>>>
>>>>> Regarding:  "The idea is to simply double the new array size each time
>>>>> a new array needs to be allocated"
>>>>>
>>>>> It seems this would not address the desire of offering an alternative
>>>>> to the allocation of a large backing array for ArrayList (your largest
>>>>> backing array could still reach a size of 1/2 * Integer.MAX_VALUE) and would
>>>>> not address the desire of wasting the (asymptotically) minimum amount of
>>>>> memory in the worst case while maintaining O(1) amortized time bounds.  The
>>>>> data structure described in [Brodnik99resizablearrays] has a maximum backing
>>>>> array size of sqrt(n) and caps wasted memory at sqrt(n).  What advantage
>>>>> over ArrayList do you see in your data structure?
>>>>>
>>>>> Regarding:  "Also, with regard to a Deque implementation, it seems that
>>>>> the simplest solution would be to simply have two lists, with one accepting
>>>>> inserts for near the beginning and being ordered in reverse whilst the other
>>>>> accepted inserts for near to the end."
>>>>>
>>>>> What happens with your structure when you add n elements and then
>>>>> remove element 0 n times?  I think that once you work out all the kinks
>>>>> you'll end up with the two stacks approach, which is mentioned in
>>>>> [Brodnik99resizablearrays] and which I mentioned in an earlier email, or
>>>>> you'll end up with the circular list approach, which is not friendly to O(1)
>>>>> amortized time bounds in a data structure that resizes more often than O(n)
>>>>> due to the 'unshift' to the front = 0 position.  I think the best approach
>>>>> is the one mentioned in [Brodnik99resizablearrays], which is the approach
>>>>> that I am currently working on.  Incidentally, this approach also provides
>>>>> for a much improved index unpacking procedure using only bit shifts and bit
>>>>> masks, although it is at the expense of (O(1)) additional work during
>>>>> resize.
>>>>>
>>>>> Regards,
>>>>>
>>>>> Kevin
>>>>>
>>>>>
>>>>>
>>>>> On Wed, Apr 14, 2010 at 4:42 PM, Benedict Elliott Smith <
>>>>>
>>>>>> Hi,
>>>>>>
>>>>>> I hope you don't consider it rude to involve myself in this
>>>>>> conversation towards the end - I joined the mailing list only recently.
>>>>>>
>>>>>> I'm not sure if this offers a huge amount to the discussion, but I
>>>>>> have tinkered with a "chunked" array list which seems to offer better time
>>>>>> performance in general at the cost of greater (worst case) memory
>>>>>> utilisation. It is easier to understand IMHO as well, although this is not
>>>>>> necessarily a great benefit here. It turns out the idea is very similar to
>>>>>> the one implemented already by Kevin, though; but perhaps simpler. The idea
>>>>>> is to simply double the new array size each time a new array needs to be
>>>>>> allocated, or in effect allocate an array that is the size of all existing
>>>>>> arrays put together. With this scheme the calculation for array and offset
>>>>>> are really very straight forward ( floor(log(i)) and 1 + i -
>>>>>> 2^floor(log(i))) ). Memory utilisation is the same as for ArrayList, but
>>>>>> obviously inserts at the end are much quicker.
>>>>>>
>>>>>> I have prototyped the data structure this evening and benchmarked
>>>>>> additions at the end of the list, for which the performance is pretty
>>>>>> impressive.
>>>>>>
>>>>>> Some random statistics for addition only on the client JVM (I have
>>>>>> quickly dubbed my implementation ExpArrayList)
>>>>>> All statistics were run in two rounds with ~1000 runs per round per
>>>>>> statistic per list, and the second round results were used.
>>>>>>
>>>>>> 10 items versus ArrayList: Chunked=1.14, ExpArray=1.02
>>>>>> 10 items Chunked / ExpArray = 1.12
>>>>>> 100 items versus ArrayList: Chunked=1.20, ExpArray=0.82
>>>>>> 100 items Chunked / ExpArray = 1.45
>>>>>> 1000 items versus ArrayList: Chunked=1.03, ExpArray=0.51
>>>>>> 1000 items Chunked / ExpArray = 2.02
>>>>>> 10000 items versus ArrayList: Chunked=0.88, ExpArray=0.49
>>>>>> 10000 items Chunked / ExpArray = 1.79
>>>>>> 100000 items versus ArrayList: Chunked=0.32, ExpArray=0.20
>>>>>> 100000 items Chunked / ExpArray = 1.64
>>>>>>
>>>>>> and server JVM:
>>>>>> 10 items versus ArrayList: Chunked=1.00, ExpArray=1.16
>>>>>> 10 items Chunked / ExpArray = 0.86
>>>>>> 100 items versus ArrayList: Chunked=1.29, ExpArray=0.96
>>>>>> 100 items Chunked / ExpArray = 1.34
>>>>>> 1000 items versus ArrayList: Chunked=1.16, ExpArray=0.92
>>>>>> 1000 items Chunked / ExpArray = 1.27
>>>>>> 10000 items versus ArrayList: Chunked=0.93, ExpArray=0.84
>>>>>> 10000 items Chunked / ExpArray = 1.12
>>>>>> 100000 items versus ArrayList: Chunked=0.71, ExpArray=0.65
>>>>>> 100000 items Chunked / ExpArray = 1.10
>>>>>>
>>>>>> Interestingly insertion at the beginning of the list appears to be
>>>>>> quicker with ExpArrayList, at least on the server JVM, whereas I would have
>>>>>> expected them to be fairly close.
>>>>>> Amazingly ExpArrayList is faster even than ArrayList for insertion at
>>>>>> the beginning of large lists, which I haven't yet tried to understand.
>>>>>> Insertion in the middle is similar.
>>>>>>
>>>>>> 10 items versus ArrayList: Chunked=9.82, ExpArray=3.80
>>>>>> 10 items Chunked / ExpArray = 2.59
>>>>>> 100 items versus ArrayList: Chunked=7.30, ExpArray=3.41
>>>>>> 100 items Chunked / ExpArray = 2.14
>>>>>> 1000 items versus ArrayList: Chunked=2.83, ExpArray=1.09
>>>>>> 1000 items Chunked / ExpArray = 2.59
>>>>>> 10000 items versus ArrayList: Chunked=1.56, ExpArray=0.72
>>>>>> 10000 items Chunked / ExpArray = 2.16
>>>>>>
>>>>>> Finally, there are promising results for get() from the ExpArrayList
>>>>>> as well (server JVM), again somehow beating ArrayList for larger lists:
>>>>>> 10 items get versus ArrayList: Chunked=1.27, ExpArray=1.16
>>>>>> 10 items get Chunked / ExpArray = 1.10
>>>>>> 100 items get versus ArrayList: Chunked=1.45, ExpArray=1.17
>>>>>> 100 items get Chunked / ExpArray = 1.25
>>>>>> 1000 items get versus ArrayList: Chunked=1.42, ExpArray=1.07
>>>>>> 1000 items get Chunked / ExpArray = 1.33
>>>>>> 10000 items get versus ArrayList: Chunked=1.26, ExpArray=1.02
>>>>>> 10000 items get Chunked / ExpArray = 1.24
>>>>>> 100000 items get versus ArrayList: Chunked=1.05, ExpArray=0.86
>>>>>> 100000 items get Chunked / ExpArray = 1.22
>>>>>>
>>>>>>
>>>>>> I'm willing to explore this further but I'm not sure how desirable
>>>>>> that is, given that Kevin's data structure appears to perform pretty well
>>>>>> already wrt to CPU time, and better wrt to memory utilisation, and in effect
>>>>>> this mostly changes only the function to determine which array to use, not
>>>>>> the body of the implementation. Let me know if you would like a copy of the
>>>>>> source code and I will find somewhere to upload it.
>>>>>>
>>>>>> Also, with regard to a Deque implementation, it seems that the
>>>>>> simplest solution would be to simply have two lists, with one accepting
>>>>>> inserts for near the beginning and being ordered in reverse whilst the other
>>>>>> accepted inserts for near to the end. The only trick would be having the
>>>>>> list at the beginning support iteration in reverse order cheaply, but this
>>>>>> could easily be achieved by creating an extension of List with a
>>>>>> reverseIterator() method.
>>>>>>
>>>>>>
>>>>>> Anyway, not sure if this helped at all but fancied joining in...
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> On 14 April 2010 12:25, Joe Kearney <joe.j.kearney at googlemail.com>wrote:
>>>>>>
>>>>>>> Hi Kevin,
>>>>>>>
>>>>>>> It implements List, as well as Deque. It is indeed based on
>>>>>>> ArrayDeque, with the added operations to implement list. It does so
>>>>>>> reasonably efficiently, moving the fewest elements possible on each
>>>>>>> operation, that is zero for the queue operations, at most n/2 for the rest
>>>>>>> and all of them for a backing array resize.
>>>>>>>
>>>>>>> The idea is to get a replacement for arraylist that performs like
>>>>>>> arraydeque on remove(0). As a side effect, we should be able to get better
>>>>>>> performance on other operations by requiring fewer elements to be moved.
>>>>>>>
>>>>>>> Thanks,
>>>>>>> Joe
>>>>>>>
>>>>>>> 2010/4/14 Kevin L. Stern <kevin.l.stern at gmail.com>
>>>>>>>
>>>>>>> Hi Joe,
>>>>>>>>
>>>>>>>> I was referring to the ChunkedArrayList when I stated that add does
>>>>>>>> not amortize to constant time when the data structure employs the circular
>>>>>>>> list trick to achieve deque behavior; ChunkedArrayList potentially resizes
>>>>>>>> every n^(1/2) operations.
>>>>>>>>
>>>>>>>> Regarding your CircularArrayList, does it differ from Java's
>>>>>>>> ArrayDeque?  I took only a cursory look at it, so please understand if I
>>>>>>>> have missed your reason for creating CircularArrayList altogether.
>>>>>>>>
>>>>>>>> Regards,
>>>>>>>>
>>>>>>>> Kevin
>>>>>>>>
>>>>>>>>
>>>>>>>> On Tue, Apr 13, 2010 at 6:52 AM, Joe Kearney <
>>>>>>>>
>>>>>>>>> Hi Kevin, Martin,
>>>>>>>>>
>>>>>>>>> To add another discussion point, I've been writing a
>>>>>>>>> draft/proof-of-concept of retrofitting the List interface onto ArrayDeque.
>>>>>>>>> This works over the raw array, it doesn't use the fancier structures being
>>>>>>>>> discussed elsewhere on this list that deal with splitting huge arrays into
>>>>>>>>> arraylets, or that provide for O(1) insert in the middle.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> I'd be interested if you have any comments in the context of this
>>>>>>>>> discussion. The code is not entirely ready yet, a couple of tests fail
>>>>>>>>> (6/789) because of a corner case I haven't nailed yet, but the idea is there
>>>>>>>>> at least. I'd like to add array shrinking later, when the size dips below
>>>>>>>>> capacity*0.4 perhaps, to avoid flickering up and down around...
>>>>>>>>>
>>>>>>>>> Tests show performance to be close to ArrayList for the O(1)
>>>>>>>>> operations. Timings for indexed reads and writes showed
>>>>>>>>> no discernible difference between implementations last time I ran the
>>>>>>>>> tests. I don't understand at the moment why the iterator add at index
>>>>>>>>> size/3, size/2 perform 30% slower than ArrayList on smaller lists, nor the
>>>>>>>>> dodgy numbers for ArrayList.insert(5), I'll look at this soon.  Those
>>>>>>>>> operations that become O(1) in a circular implementation (that are
>>>>>>>>> implemented and tested here) are faster than in ArrayList. Insert/remove in
>>>>>>>>> the middle are somewhat faster than ArrayList because we only have to copy
>>>>>>>>> at most half of the elements, except when resizing the array.
>>>>>>>>>
>>>>>>>>> Kevin, I don't fully understand your point about not amortizing to
>>>>>>>>> O(1). Certainly that's true for insert not at head or tail. Otherwise this
>>>>>>>>> implementation only moves array elements to the front on an array resize
>>>>>>>>> operation which happens every O(ln n) operations at most, if we do lots of
>>>>>>>>> adds, maybe a little more if we add array shrinking too.  This is the same
>>>>>>>>> as ArrayList. Are you just referring to the add-in-the-middle case?
>>>>>>>>>
>>>>>>>>> Some performance results below, code for these is in the repository
>>>>>>>>> above too. This was the second run, after a warmup.
>>>>>>>>>
>>>>>>>>> Thanks,
>>>>>>>>> Joe
>>>>>>>>>
>>>>>>>>> ------------------------------------------------ CircularArrayList
>>>>>>>>> ------------------------------------------------
>>>>>>>>>   insert(5)   removeRnd   removeMid   remove(0)
>>>>>>>>>      10          20          67          70         125
>>>>>>>>> 102          90         240         191         138
>>>>>>>>>    100          19          67          70         166         138
>>>>>>>>>          94         230         194         118
>>>>>>>>>   1000          28          64          67         681         538
>>>>>>>>>          91         324         382         119
>>>>>>>>>   10000          30          65          67        5884
>>>>>>>>>  4425          94        1296        2330         124
>>>>>>>>> ---------------------------------------------------- ArrayList
>>>>>>>>> ----------------------------------------------------
>>>>>>>>>   insert(5)   removeRnd   removeMid   remove(0)
>>>>>>>>>      10          23          68          70         100
>>>>>>>>>  69       32913         162         130         105
>>>>>>>>>    100          20          67          70         129         104
>>>>>>>>>       21944         169         134         135
>>>>>>>>>   1000          29          63          67         651         506
>>>>>>>>>        9602         364         333         526
>>>>>>>>>   10000          30          63          66        5878
>>>>>>>>>  4414        9947        2312        2280        4437
>>>>>>>>>
>>>>>>>>> 2010/4/13 Kevin L. Stern <kevin.l.stern at gmail.com>
>>>>>>>>>
>>>>>>>>> Hi Martin,
>>>>>>>>>>
>>>>>>>>>> operations in the previous email.  The approach to achieving this suggested
>>>>>>>>>> in [Brodnik99resizablearrays] is tantamount to making ArrayList operations
>>>>>>>>>> absolute O(1) by keeping around an array of size (3/2)*n and filling it with
>>>>>>>>>> a constant number of entries from the main array each time add is called.
>>>>>>>>>> Although this distributes the work done during a resize across the n
>>>>>>>>>> operations required to enter a resize-required state, it is at the expense
>>>>>>>>>> of additional memory usage and slower add operations.  My thought is that
>>>>>>>>>> this would be a fine approach for a real-time application that requires hard
>>>>>>>>>> guarantees on performance but would be a liability in so many Java
>>>>>>>>>> applications that do not require these hard guarantees.  I look forward to
>>>>>>>>>> hearing your thoughts on the matter, though.
>>>>>>>>>>
>>>>>>>>>> Kevin
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On Tue, Apr 13, 2010 at 6:18 AM, Kevin L. Stern <
>>>>>>>>>> kevin.l.stern at gmail.com> wrote:
>>>>>>>>>>
>>>>>>>>>>> Hi Martin,
>>>>>>>>>>>
>>>>>>>>>>> It's interesting to note that the old circular list trick will
>>>>>>>>>>> not suffice to turn this data structure into a deque since we might be
>>>>>>>>>>> copying all n elements back to the front = 0 position every n^(1/2)
>>>>>>>>>>> operations (add wouldn't amortize to O(1)).  We could use the old two stacks
>>>>>>>>>>> trick (push elements onto one stack, flip (the bottom) half (of) the
>>>>>>>>>>> elements to the 'other' stack when the 'other' stack becomes empty),
>>>>>>>>>>> mentioned in [Brodnik99resizablearrays], but I find this to be a bit CS
>>>>>>>>>>> 101.  In [Brodnik99resizablearrays] the authors suggest a method for making
>>>>>>>>>>> all blocks roughly the same size, allowing us to expand/shrink capacity at
>>>>>>>>>>> the beginning or the end; this is the approach that I will take to create a
>>>>>>>>>>> deque.
>>>>>>>>>>>
>>>>>>>>>>> The FAQ for the Sun Contributor Agreement Q3 (
>>>>>>>>>>> http://www.sun.com/software/opensource/contributor_agreement.jsp#sa_3)
>>>>>>>>>>> indicates that one should check with the project to determine where the SCA
>>>>>>>>>>> should be sent.  Do you know where I would find this information?
>>>>>>>>>>>
>>>>>>>>>>> Kevin
>>>>>>>>>>>
>>>>>>>>>>> @MISC{Brodnik99resizablearrays,
>>>>>>>>>>>     author = {Andrej Brodnik and Svante Carlsson and Erik D.
>>>>>>>>>>> Demaine and J. Ian Munro and Robert Sedgewick},
>>>>>>>>>>>     title = {Resizable Arrays in Optimal Time and Space},
>>>>>>>>>>>     year = {1999}
>>>>>>>>>>>
>>>>>>>>>>> }
>>>>>>>>>>>
>>>>>>>>>>> On Sun, Apr 11, 2010 at 4:17 PM, Martin Buchholz <
>>>>>>>>>>>
>>>>>>>>>>>> Hi Kevin,
>>>>>>>>>>>>
>>>>>>>>>>>> Thanks for your continuing work on this.
>>>>>>>>>>>>
>>>>>>>>>>>> I like the test results, and agree with your analysis.
>>>>>>>>>>>> I'm especially happy that you're beating
>>>>>>>>>>>> ArrayList at some operations.
>>>>>>>>>>>>
>>>>>>>>>>>> I'd like to see O(1) addition at the beginning,
>>>>>>>>>>>> implement both List and Deque (I regret
>>>>>>>>>>>> our not having done this with ArrayDeque).
>>>>>>>>>>>>
>>>>>>>>>>>> An additional property that would be nice to
>>>>>>>>>>>> have (but don't try too hard)
>>>>>>>>>>>> is to provide some kind of real-time
>>>>>>>>>>>> guarantees on the cost of an individual operation,
>>>>>>>>>>>> not just amortized time.  E.g. ArrayList.add
>>>>>>>>>>>> is worst-case O(n), making it unsuitable for use
>>>>>>>>>>>> in some real-time applications.
>>>>>>>>>>>>
>>>>>>>>>>>> I will help get your changes into the obvious
>>>>>>>>>>>> software distributions.  I assume you're happy
>>>>>>>>>>>> with having this class included in any of
>>>>>>>>>>>> Doug Lea's jsr166, guava-libraries, or the JDK itself.
>>>>>>>>>>>> You should sign a Sun contributor agreement,
>>>>>>>>>>>> or whatever the Oracle equivalent is,
>>>>>>>>>>>> if you have not done so yet.
>>>>>>>>>>>>
>>>>>>>>>>>> Doug Lea likes public domain,
>>>>>>>>>>>> guava-libraries likes the Apache license.
>>>>>>>>>>>>
>>>>>>>>>>>> We should get various people a chance to give
>>>>>>>>>>>> a thumbs up on the design of this class -
>>>>>>>>>>>> Doug Lea, Josh Bloch.
>>>>>>>>>>>>
>>>>>>>>>>>> Martin
>>>>>>>>>>>>
>>>>>>>>>>>> On Sun, Apr 11, 2010 at 09:32, Kevin L. Stern <
>>>>>>>>>>>> kevin.l.stern at gmail.com> wrote:
>>>>>>>>>>>> > Hello Martin,
>>>>>>>>>>>> >
>>>>>>>>>>>> > I spent some time this weekend trying to bring out bugs in the
>>>>>>>>>>>> > implementation; I believe the latest version to be in decent
>>>>>>>>>>>> shape.  I have
>>>>>>>>>>>> > also gathered some data on the performance of ChunkedArrayList
>>>>>>>>>>>> over
>>>>>>>>>>>> > ArrayList using the latest 1.6 JDK, which I've included below
>>>>>>>>>>>> (note that the
>>>>>>>>>>>> > numbers represent the time spent performing the specified
>>>>>>>>>>>> operation with
>>>>>>>>>>>> > ChunkedArrayList over the time spent with ArrayList, so 1.00
>>>>>>>>>>>> indicates
>>>>>>>>>>>> > equivalent performance, < 1.00 indicates that ChunkedArrayList
>>>>>>>>>>>> is less
>>>>>>>>>>>> > costly and > 1.00 indicates that ArrayList is less costly).
>>>>>>>>>>>> I've noticed
>>>>>>>>>>>> > relatively significant variability in a few of the numbers
>>>>>>>>>>>> when I switch
>>>>>>>>>>>> > hardware; though, these data do seem to represent rough
>>>>>>>>>>>> performance
>>>>>>>>>>>> > expectations.  For my test I generated x elements and then
>>>>>>>>>>>> timed the process
>>>>>>>>>>>> > of adding them to ArrayList/ChunkedArrayList, then I performed
>>>>>>>>>>>> a get
>>>>>>>>>>>> > operation on each for indices 0 through x-1 and finally I used
>>>>>>>>>>>> the iterator
>>>>>>>>>>>> > mechanism to retrieve the first through xth element (of
>>>>>>>>>>>> course, I performed
>>>>>>>>>>>> > each of these operations multiple times throwing away the
>>>>>>>>>>>> timing for the
>>>>>>>>>>>> > first few iterations to warm up the JVM).
>>>>>>>>>>>> >
>>>>>>>>>>>> > Regarding the question of whether or not this belongs in
>>>>>>>>>>>> java.util, I would
>>>>>>>>>>>> > suggest that if it is desirable from a GC point of view to
>>>>>>>>>>>> eliminate the
>>>>>>>>>>>> > large backing array from ArrayList then your suggestion of
>>>>>>>>>>>> achieving this by
>>>>>>>>>>>> > way of a data structure that is both time and space optimal is
>>>>>>>>>>>> a
>>>>>>>>>>>> > particularly elegant solution as it not only guarantees that
>>>>>>>>>>>> no backing
>>>>>>>>>>>> > array will be larger than sqrt(n) elements but it also
>>>>>>>>>>>> provides dynamic
>>>>>>>>>>>> > shrinking behavior, has less maximum memory overhead than
>>>>>>>>>>>> ArrayList, and
>>>>>>>>>>>> > copies (asymptotically) fewer elements during a resize than
>>>>>>>>>>>> ArrayList.  Of
>>>>>>>>>>>> > course, this data structure does not do everything better than
>>>>>>>>>>>> ArrayList; in
>>>>>>>>>>>> > particular, indexed access is more costly, due to the required
>>>>>>>>>>>> decomposition
>>>>>>>>>>>> > of the index into backing array index and offset and the
>>>>>>>>>>>> > indirection, and insertion-at-an-index is more costly, due to
>>>>>>>>>>>> the multiple
>>>>>>>>>>>> > array copies necessary to complete the shift.  That being
>>>>>>>>>>>> said, I think that
>>>>>>>>>>>> > the additional cost of indexed access is partially mitigated
>>>>>>>>>>>> by the
>>>>>>>>>>>> > availability of iterator and listIterator, whose
>>>>>>>>>>>> implementations do not use
>>>>>>>>>>>> > the index decomposition procedure, and the additional cost of
>>>>>>>>>>>> > insertion-at-an-index is partially mitigated by the fact that
>>>>>>>>>>>> > insertion-at-an-index is already an undesirable operation on
>>>>>>>>>>>> ArrayList due
>>>>>>>>>>>> > to its linear time complexity.
>>>>>>>>>>>> >
>>>>>>>>>>>> > Kevin
>>>>>>>>>>>> >
>>>>>>>>>>>> > 1000000 elements:
>>>>>>>>>>>> > Client JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 1.30
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 1.80
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 0.52
>>>>>>>>>>>> >
>>>>>>>>>>>> > Server JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 0.81
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 2.87
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 1.31
>>>>>>>>>>>> >
>>>>>>>>>>>> > 100000 elements:
>>>>>>>>>>>> > Client JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 0.96
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 1.86
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 0.48
>>>>>>>>>>>> >
>>>>>>>>>>>> > Server JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 0.96
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 1.89
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 2.68
>>>>>>>>>>>> >
>>>>>>>>>>>> > 10000 elements:
>>>>>>>>>>>> > Client JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 1.04
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 2.33
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 0.53
>>>>>>>>>>>> >
>>>>>>>>>>>> > Server JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 0.97
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 2.45
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 2.52
>>>>>>>>>>>> >
>>>>>>>>>>>> > 1000 elements:
>>>>>>>>>>>> > Client JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 0.99
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 2.27
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 0.54
>>>>>>>>>>>> >
>>>>>>>>>>>> > Server JVM:
>>>>>>>>>>>> > Add to ChunkedArrayList over ArrayList: 0.84
>>>>>>>>>>>> > Indexed access ChunkedArrayList over ArrayList: 1.23
>>>>>>>>>>>> > Iterator ChunkedArrayList over ArrayList: 1.11
>>>>>>>>>>>> >
>>>>>>>>>>>> >
>>>>>>>>>>>> > On Fri, Apr 9, 2010 at 7:42 PM, Martin Buchholz <
>>>>>>>>>>>> >>
>>>>>>>>>>>> >> My feeling on whether to support O(1) at both ends
>>>>>>>>>>>> >> is that any flavor of this that ends up in the JDK eventually
>>>>>>>>>>>> >> should really do this.  My idea is that we can
>>>>>>>>>>>> >> wholeheartedly recommend this collection class
>>>>>>>>>>>> >> for overall good behavior without any of the surprising
>>>>>>>>>>>> >> performance traps of existing collection classes.
>>>>>>>>>>>> >>
>>>>>>>>>>>> >> But for the preliminary version, it makes sense to
>>>>>>>>>>>> >> support only O(1) at one end, if it simplifies the
>>>>>>>>>>>> >> implementation.  Random access will of course
>>>>>>>>>>>> >> be worse than ArrayList, but by how much?
>>>>>>>>>>>> >> We can do some benchmarking and look for
>>>>>>>>>>>> >> micro-optimizations now.
>>>>>>>>>>>> >>
>>>>>>>>>>>> >> Kevin, what is you own personal feeling?
>>>>>>>>>>>> >> Is the algorithm correct, and efficient enough?
>>>>>>>>>>>> >> Do you think your new collection belongs in java.util?
>>>>>>>>>>>> >>
>>>>>>>>>>>> >> Martin
>>>>>>>>>>>> >>
>>>>>>>>>>>> >> On Sun, Apr 4, 2010 at 04:12, Kevin L. Stern <
>>>>>>>>>>>> kevin.l.stern at gmail.com>
>>>>>>>>>>>> >> wrote:
>>>>>>>>>>>> >> > The data structure is available at the second link that I
>>>>>>>>>>>> originally
>>>>>>>>>>>> >> > provided (once again, it is
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> >
>>>>>>>>>>>> ).
>>>>>>>>>>>> >> > This does not have O(1) time insertion at the front as yet
>>>>>>>>>>>> as it was
>>>>>>>>>>>> >> > unclear
>>>>>>>>>>>> >> > to me whether or not it was agreed upon:
>>>>>>>>>>>> >> > _________________
>>>>>>>>>>>> >> > From: Osvaldo Doederlein <opinali at gmail.com>
>>>>>>>>>>>> >> > Date: Mon, Mar 29, 2010 at 10:08 AM
>>>>>>>>>>>> >> > Subject: Re: A List implementation backed by multiple small
>>>>>>>>>>>> arrays
>>>>>>>>>>>> >> > rather
>>>>>>>>>>>> >> > than the traditional single large array.
>>>>>>>>>>>> >> > To: Martin Buchholz <martinrb at google.com>
>>>>>>>>>>>> >> > Cc: "Kevin L. Stern" <kevin.l.stern at gmail.com>,
>>>>>>>>>>>> >> > core-libs-dev at openjdk.java.net
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> > Initially, it would be good enough to replace only
>>>>>>>>>>>> java.util.ArrayList
>>>>>>>>>>>> >> > with
>>>>>>>>>>>> >> > minimal overhead. ArrayList does not support efficient
>>>>>>>>>>>> >> > other
>>>>>>>>>>>> >> > enhancements of ArrayDeque; but ArrayList is still a much
>>>>>>>>>>>> more important
>>>>>>>>>>>> >> > and
>>>>>>>>>>>> >> > popular collection, it's the primary "straight replacement
>>>>>>>>>>>> for primitive
>>>>>>>>>>>> >> > arrrays" and I guess it should continue with that role.
>>>>>>>>>>>> >> > _________________
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> > As a disclaimer, I'm still tinkering with this so I'll be
>>>>>>>>>>>> updating the
>>>>>>>>>>>> >> > document at the provided link as I find improvements.
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> > Thoughts?
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> > Thanks,
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> > Kevin
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> > On Thu, Apr 1, 2010 at 10:28 PM, Martin Buchholz <
>>>>>>>>>>>> >> > wrote:
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >> Hi Kevin,
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >> You're probably the only one on this list who has
>>>>>>>>>>>> >> >> seriously read the paper.  It is not surprising that
>>>>>>>>>>>> >> >> taking a research paper into production would
>>>>>>>>>>>> >> >> discover bugs - the research never had to undergo
>>>>>>>>>>>> >> >> rigorous testing.  (I like the Java culture of
>>>>>>>>>>>> >> >> combining spec + implementation + test suite)
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >> I suggest you ask the authors directly about the bug.
>>>>>>>>>>>> >> >> They would probably also be interested to hear
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >> Are you aware of Integer.numberOfLeadingZeros?
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >> Martin
>>>>>>>>>>>> >> >>
>>>>>>>>>>>> >> >> On Wed, Mar 31, 2010 at 19:34, Kevin L. Stern <
>>>>>>>>>>>> kevin.l.stern at gmail.com>
>>>>>>>>>>>> >> >> wrote:
>>>>>>>>>>>> >> >> > I'm almost convinced now that the paper is incorrect.
>>>>>>>>>>>> The code below
>>>>>>>>>>>> >> >> > gives
>>>>>>>>>>>> >> >> > me the appropriate index into the index array and the
>>>>>>>>>>>> offset into the
>>>>>>>>>>>> >> >> > data
>>>>>>>>>>>> >> >> > block.  That being said, remember when I mentioned that
>>>>>>>>>>>> this will
>>>>>>>>>>>> >> >> > include a
>>>>>>>>>>>> >> >> > bit more work to access an element than a simple bit
>>>>>>>>>>>> shift and a bit
>>>>>>>>>>>> >> >> > Well this is more than a bit more - we'll be doing this
>>>>>>>>>>>> each time an
>>>>>>>>>>>> >> >> > index
>>>>>>>>>>>> >> >> > is requested.  I'll spend some time trying to twiddle
>>>>>>>>>>>> the bits to see
>>>>>>>>>>>> >> >> > if
>>>>>>>>>>>> >> >> > I
>>>>>>>>>>>> >> >> > can eliminate/combine some of the operations.
>>>>>>>>>>>> >> >> >
>>>>>>>>>>>> >> >> >         for (int r = 1; r < 33; r++) {
>>>>>>>>>>>> >> >> >             int k = lg(r);
>>>>>>>>>>>> >> >> >             int floorKO2 = k >> 1;
>>>>>>>>>>>> >> >> >             int powFloorKO2 = (1 << floorKO2);
>>>>>>>>>>>> >> >> >             int p = ((1 << floorKO2) - 1) << 1;
>>>>>>>>>>>> >> >> >             int ceilKO2;
>>>>>>>>>>>> >> >> >             if ((k & 1) == 1) {
>>>>>>>>>>>> >> >> >                 ceilKO2 = floorKO2 + 1;
>>>>>>>>>>>> >> >> >                 p += powFloorKO2;
>>>>>>>>>>>> >> >> >             } else {
>>>>>>>>>>>> >> >> >                 ceilKO2 = floorKO2;
>>>>>>>>>>>> >> >> >             }
>>>>>>>>>>>> >> >> >             int e = r & ((1 << ceilKO2) - 1);
>>>>>>>>>>>> >> >> >             int b = (r >> ceilKO2) & (powFloorKO2 - 1);
>>>>>>>>>>>> >> >> >
>>>>>>>>>>>> >> >> >             System.out.println((r - 1) + " " + (p + b) +
>>>>>>>>>>>> " " + e);
>>>>>>>>>>>> >> >> >         }
>>>>>>>>>>>> >> >> >
>>>>>>>>>>>> >> >> > Kevin
>>>>>>>>>>>> >> >> >
>>>>>>>>>>>> >> >> > On Wed, Mar 31, 2010 at 7:08 PM, Kevin L. Stern
>>>>>>>>>>>> >> >> > <kevin.l.stern at gmail.com>
>>>>>>>>>>>> >> >> > wrote:
>>>>>>>>>>>> >> >> >>
>>>>>>>>>>>> >> >> >> I realize that 2 * (2^(k/2) - 1) only works for even
>>>>>>>>>>>> numbered
>>>>>>>>>>>> >> >> >> superblocks,
>>>>>>>>>>>> >> >> >> the odd numbered superblocks need an additional term
>>>>>>>>>>>> >> >> >> number
>>>>>>>>>>>> >> >> >> of
>>>>>>>>>>>> >> >> >> data blocks in SB_[k-1]) to jive with my
>>>>>>>>>>>> interpretation; anyhow, I
>>>>>>>>>>>> >> >> >> also
>>>>>>>>>>>> >> >> >> came
>>>>>>>>>>>> >> >> >> across an alternative characterization of superblock in
>>>>>>>>>>>> the paper
>>>>>>>>>>>> >> >> >> which
>>>>>>>>>>>> >> >> >> states that data blocks are grouped within a superblock
>>>>>>>>>>>> when they
>>>>>>>>>>>> >> >> >> are
>>>>>>>>>>>> >> >> >> the
>>>>>>>>>>>> >> >> >> same size - to me, though, that implies that my example
>>>>>>>>>>>> structure
>>>>>>>>>>>> >> >> >> below
>>>>>>>>>>>> >> >> >> would be
>>>>>>>>>>>> >> >> >>
>>>>>>>>>>>> >> >> >> SB_0: [1]
>>>>>>>>>>>> >> >> >> SB_1: [2][2][2]
>>>>>>>>>>>> >> >> >> SB_2: [4][4][4][4][4][4]
>>>>>>>>>>>> >> >> >>
>>>>>>>>>>>> >> >> >> which seems to contradict my understanding of (1)
>>>>>>>>>>>> below.  I must be
>>>>>>>>>>>> >> >> >> reading this upside down.
>>>>>>>>>>>> >> >> >>
>>>>>>>>>>>> >> >> >> On Wed, Mar 31, 2010 at 6:36 PM, Kevin L. Stern
>>>>>>>>>>>> >> >> >> <kevin.l.stern at gmail.com>
>>>>>>>>>>>> >> >> >> wrote:
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> What am I missing here?  In "Resizable arrays in
>>>>>>>>>>>> optimal time and
>>>>>>>>>>>> >> >> >>> space"
>>>>>>>>>>>> >> >> >>> the authors define their data structure with the
>>>>>>>>>>>> following
>>>>>>>>>>>> >> >> >>> property:
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> (1)  "When superblock SB_k is fully allocated, it
>>>>>>>>>>>> consists of
>>>>>>>>>>>> >> >> >>> 2^(floor(k/2)) data blocks, each of size
>>>>>>>>>>>> 2^(ceil(k/2))."
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> Since the superblock is zero-based indexed this
>>>>>>>>>>>> implies the
>>>>>>>>>>>> >> >> >>> following
>>>>>>>>>>>> >> >> >>> structure:
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> SB_0: [1]
>>>>>>>>>>>> >> >> >>> SB_1: [2]
>>>>>>>>>>>> >> >> >>> SB_2: [2][2]
>>>>>>>>>>>> >> >> >>> SB_3: [4][4]
>>>>>>>>>>>> >> >> >>> SB_4: [4][4][4][4]
>>>>>>>>>>>> >> >> >>> [...]
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> Let's have a look at Algorithm 3, Locate(i), with i =
>>>>>>>>>>>> 3:
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> r = 100 (the binary expansion of i + 1)
>>>>>>>>>>>> >> >> >>> k = |r| - 1 = 2
>>>>>>>>>>>> >> >> >>> p = 2^k - 1 = 3
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> What concerns me is their statement that p represents
>>>>>>>>>>>> "the number
>>>>>>>>>>>> >> >> >>> of
>>>>>>>>>>>> >> >> >>> data
>>>>>>>>>>>> >> >> >>> blocks in superblocks prior to SB_k."  There are only
>>>>>>>>>>>> two data
>>>>>>>>>>>> >> >> >>> blocks
>>>>>>>>>>>> >> >> >>> in
>>>>>>>>>>>> >> >> >>> superblocks prior to SB_2, not three.  Given (1)
>>>>>>>>>>>> above, unless I'm
>>>>>>>>>>>> >> >> >>> misinterpreting it, the number of data blocks in
>>>>>>>>>>>> superblocks prior
>>>>>>>>>>>> >> >> >>> to
>>>>>>>>>>>> >> >> >>> SB_k
>>>>>>>>>>>> >> >> >>> should be:
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> 2 * Sum[i=0->k/2-1] 2^i = 2 * (2^(k/2) - 1)
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> This, of course, seems to work out much better in my
>>>>>>>>>>>> example above,
>>>>>>>>>>>> >> >> >>> giving the correct answer to my interpretation of
>>>>>>>>>>>> their data
>>>>>>>>>>>> >> >> >>> structure, but
>>>>>>>>>>>> >> >> >>> I have a hard time believing that this is their
>>>>>>>>>>>> mistake rather than
>>>>>>>>>>>> >> >> >>> my
>>>>>>>>>>>> >> >> >>> misinterpretation.
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> Thoughts?
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> Kevin
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>> On Tue, Mar 30, 2010 at 5:20 PM, Martin Buchholz
>>>>>>>>>>>> >> >> >>> <martinrb at google.com>
>>>>>>>>>>>> >> >> >>> wrote:
>>>>>>>>>>>> >> >> >>>>
>>>>>>>>>>>> >> >> >>>> On Tue, Mar 30, 2010 at 04:25, Kevin L. Stern
>>>>>>>>>>>> >> >> >>>> <kevin.l.stern at gmail.com>
>>>>>>>>>>>> >> >> >>>> wrote:
>>>>>>>>>>>> >> >> >>>> > Hi Martin,
>>>>>>>>>>>> >> >> >>>> >
>>>>>>>>>>>> >> >> >>>> > Thanks much for your feedback.  The first approach
>>>>>>>>>>>> that comes to
>>>>>>>>>>>> >> >> >>>> > mind
>>>>>>>>>>>> >> >> >>>> > to
>>>>>>>>>>>> >> >> >>>> > implement O(1) time front as well as rear insertion
>>>>>>>>>>>> is to create
>>>>>>>>>>>> >> >> >>>> > a
>>>>>>>>>>>> >> >> >>>> > cyclic
>>>>>>>>>>>> >> >> >>>> > list structure with a front/rear pointer - to
>>>>>>>>>>>> insert at the
>>>>>>>>>>>> >> >> >>>> > front
>>>>>>>>>>>> >> >> >>>> > requires
>>>>>>>>>>>> >> >> >>>> > decrementing the front pointer (modulo the size)
>>>>>>>>>>>> and to insert
>>>>>>>>>>>> >> >> >>>> > at
>>>>>>>>>>>> >> >> >>>> > the
>>>>>>>>>>>> >> >> >>>> > rear
>>>>>>>>>>>> >> >> >>>> > requires incrementing the rear pointer (modulo the
>>>>>>>>>>>> size).  We
>>>>>>>>>>>> >> >> >>>> > need
>>>>>>>>>>>> >> >> >>>> > to
>>>>>>>>>>>> >> >> >>>> > resize
>>>>>>>>>>>> >> >> >>>> > when the two pointers bump into each other.  Could
>>>>>>>>>>>> you explain
>>>>>>>>>>>> >> >> >>>> > more
>>>>>>>>>>>> >> >> >>>> > about
>>>>>>>>>>>> >> >> >>>> > your suggestion of introducing an arraylet that is
>>>>>>>>>>>> shared by the
>>>>>>>>>>>> >> >> >>>> > front
>>>>>>>>>>>> >> >> >>>> > and
>>>>>>>>>>>> >> >> >>>> > the rear?
>>>>>>>>>>>> >> >> >>>>
>>>>>>>>>>>> >> >> >>>> It was a half-baked idea - I don't know if there's a
>>>>>>>>>>>> way to turn
>>>>>>>>>>>> >> >> >>>> it
>>>>>>>>>>>> >> >> >>>> into
>>>>>>>>>>>> >> >> >>>> something useful.  I was thinking of the ArrayDeque
>>>>>>>>>>>> >> >> >>>> implementation,
>>>>>>>>>>>> >> >> >>>> where all the elements live in a single array.
>>>>>>>>>>>> >> >> >>>>
>>>>>>>>>>>> >> >> >>>> >  It's not clear to me how that would help and/or be
>>>>>>>>>>>> a better
>>>>>>>>>>>> >> >> >>>> > approach than the cyclic list.  Anyhow, the paper
>>>>>>>>>>>> that you
>>>>>>>>>>>> >> >> >>>> > reference,
>>>>>>>>>>>> >> >> >>>> > "Resizable arrays in optimal time and space", gives
>>>>>>>>>>>> a deque so
>>>>>>>>>>>> >> >> >>>> > if
>>>>>>>>>>>> >> >> >>>> > we
>>>>>>>>>>>> >> >> >>>> > take
>>>>>>>>>>>> >> >> >>>> > that approach then the deque is specified.
>>>>>>>>>>>> >> >> >>>>
>>>>>>>>>>>> >> >> >>>> Technically, ArrayList also supports the Deque
>>>>>>>>>>>> operations -
>>>>>>>>>>>> >> >> >>>> just not efficiently.
>>>>>>>>>>>> >> >> >>>
>>>>>>>>>>>> >> >> >>
>>>>>>>>>>>> >> >> >
>>>>>>>>>>>> >> >> >
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >> >
>>>>>>>>>>>> >
>>>>>>>>>>>> >
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>
>>>>>
>>>>
>>>
>>
>
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