4,784 research outputs found
A Second Step Towards Complexity-Theoretic Analogs of Rice's Theorem
Rice's Theorem states that every nontrivial language property of the
recursively enumerable sets is undecidable. Borchert and Stephan initiated the
search for complexity-theoretic analogs of Rice's Theorem. In particular, they
proved that every nontrivial counting property of circuits is UP-hard, and that
a number of closely related problems are SPP-hard.
The present paper studies whether their UP-hardness result itself can be
improved to SPP-hardness. We show that their UP-hardness result cannot be
strengthened to SPP-hardness unless unlikely complexity class containments
hold. Nonetheless, we prove that every P-constructibly bi-infinite counting
property of circuits is SPP-hard. We also raise their general lower bound from
unambiguous nondeterminism to constant-ambiguity nondeterminism.Comment: 14 pages. To appear in Theoretical Computer Scienc
Recursive n-gram hashing is pairwise independent, at best
Many applications use sequences of n consecutive symbols (n-grams). Hashing
these n-grams can be a performance bottleneck. For more speed, recursive hash
families compute hash values by updating previous values. We prove that
recursive hash families cannot be more than pairwise independent. While hashing
by irreducible polynomials is pairwise independent, our implementations either
run in time O(n) or use an exponential amount of memory. As a more scalable
alternative, we make hashing by cyclic polynomials pairwise independent by
ignoring n-1 bits. Experimentally, we show that hashing by cyclic polynomials
is is twice as fast as hashing by irreducible polynomials. We also show that
randomized Karp-Rabin hash families are not pairwise independent.Comment: See software at https://github.com/lemire/rollinghashcp
Computational Complexity in Electronic Structure
In quantum chemistry, the price paid by all known efficient model chemistries
is either the truncation of the Hilbert space or uncontrolled approximations.
Theoretical computer science suggests that these restrictions are not mere
shortcomings of the algorithm designers and programmers but could stem from the
inherent difficulty of simulating quantum systems. Extensions of computer
science and information processing exploiting quantum mechanics has led to new
ways of understanding the ultimate limitations of computational power.
Interestingly, this perspective helps us understand widely used model
chemistries in a new light. In this article, the fundamentals of computational
complexity will be reviewed and motivated from the vantage point of chemistry.
Then recent results from the computational complexity literature regarding
common model chemistries including Hartree-Fock and density functional theory
are discussed.Comment: 14 pages, 2 figures, 1 table. Comments welcom
Dismantling sparse random graphs
We consider the number of vertices that must be removed from a graph G in
order that the remaining subgraph has no component with more than k vertices.
Our principal observation is that, if G is a sparse random graph or a random
regular graph on n vertices with n tending to infinity, then the number in
question is essentially the same for all values of k such that k tends to
infinity but k=o(n).Comment: 7 page
Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching
We present a deterministic distributed algorithm that computes a
-edge-coloring, or even list-edge-coloring, in any -node graph
with maximum degree , in rounds. This answers
one of the long-standing open questions of \emph{distributed graph algorithms}
from the late 1980s, which asked for a polylogarithmic-time algorithm. See,
e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and
Elkin. The previous best round complexities were by
Panconesi and Srinivasan [STOC'92] and
by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our
deterministic list-edge-coloring also improves the randomized complexity of
-edge-coloring to poly rounds.
The key technical ingredient is a deterministic distributed algorithm for
\emph{hypergraph maximal matching}, which we believe will be of interest beyond
this result. In any hypergraph of rank --- where each hyperedge has at most
vertices --- with nodes and maximum degree , this algorithm
computes a maximal matching in rounds.
This hypergraph matching algorithm and its extensions lead to a number of
other results. In particular, a polylogarithmic-time deterministic distributed
maximal independent set algorithm for graphs with bounded neighborhood
independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a
-round deterministic
algorithm for -approximation of maximum matching, and a
quasi-polylogarithmic-time deterministic distributed algorithm for orienting
-arboricity graphs with out-degree at most ,
for any constant , hence partially answering Open Problem 10 of
Barenboim and Elkin's book
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