3,294 research outputs found
Dynamic Integer Sets with Optimal Rank, Select, and Predecessor Search
We present a data structure representing a dynamic set S of w-bit integers on
a w-bit word RAM. With |S|=n and w > log n and space O(n), we support the
following standard operations in O(log n / log w) time:
- insert(x) sets S = S + {x}. - delete(x) sets S = S - {x}. - predecessor(x)
returns max{y in S | y= x}. -
rank(x) returns #{y in S | y< x}. - select(i) returns y in S with rank(y)=i, if
any.
Our O(log n/log w) bound is optimal for dynamic rank and select, matching a
lower bound of Fredman and Saks [STOC'89]. When the word length is large, our
time bound is also optimal for dynamic predecessor, matching a static lower
bound of Beame and Fich [STOC'99] whenever log n/log w=O(log w/loglog w).
Technically, the most interesting aspect of our data structure is that it
supports all the above operations in constant time for sets of size n=w^{O(1)}.
This resolves a main open problem of Ajtai, Komlos, and Fredman [FOCS'83].
Ajtai et al. presented such a data structure in Yao's abstract cell-probe model
with w-bit cells/words, but pointed out that the functions used could not be
implemented. As a partial solution to the problem, Fredman and Willard
[STOC'90] introduced a fusion node that could handle queries in constant time,
but used polynomial time on the updates. We call our small set data structure a
dynamic fusion node as it does both queries and updates in constant time.Comment: Presented with different formatting in Proceedings of the 55nd IEEE
Symposium on Foundations of Computer Science (FOCS), 2014, pp. 166--175. The
new version fixes a bug in one of the bounds stated for predecessor search,
pointed out to me by Djamal Belazzougu
The Cost of Address Translation
Modern computers are not random access machines (RAMs). They have a memory
hierarchy, multiple cores, and virtual memory. In this paper, we address the
computational cost of address translation in virtual memory. Starting point for
our work is the observation that the analysis of some simple algorithms (random
scan of an array, binary search, heapsort) in either the RAM model or the EM
model (external memory model) does not correctly predict growth rates of actual
running times. We propose the VAT model (virtual address translation) to
account for the cost of address translations and analyze the algorithms
mentioned above and others in the model. The predictions agree with the
measurements. We also analyze the VAT-cost of cache-oblivious algorithms.Comment: A extended abstract of this paper was published in the proceedings of
ALENEX13, New Orleans, US
Dynamic Ordered Sets with Exponential Search Trees
We introduce exponential search trees as a novel technique for converting
static polynomial space search structures for ordered sets into fully-dynamic
linear space data structures.
This leads to an optimal bound of O(sqrt(log n/loglog n)) for searching and
updating a dynamic set of n integer keys in linear space. Here searching an
integer y means finding the maximum key in the set which is smaller than or
equal to y. This problem is equivalent to the standard text book problem of
maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein:
Introduction to Algorithms, 2nd ed., MIT Press, 2001).
The best previous deterministic linear space bound was O(log n/loglog n) due
Fredman and Willard from STOC 1990. No better deterministic search bound was
known using polynomial space.
We also get the following worst-case linear space trade-offs between the
number n, the word length w, and the maximal key U < 2^w: O(min{loglog n+log
n/log w, (loglog n)(loglog U)/(logloglog U)}). These trade-offs are, however,
not likely to be optimal.
Our results are generalized to finger searching and string searching,
providing optimal results for both in terms of n.Comment: Revision corrects some typoes and state things better for
applications in subsequent paper
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