27,102 research outputs found

### Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes

For the majority of the applications of Reed-Solomon (RS) codes, hard
decision decoding is based on syndromes. Recently, there has been renewed
interest in decoding RS codes without using syndromes. In this paper, we
investigate the complexity of syndromeless decoding for RS codes, and compare
it to that of syndrome-based decoding. Aiming to provide guidelines to
practical applications, our complexity analysis differs in several aspects from
existing asymptotic complexity analysis, which is typically based on
multiplicative fast Fourier transform (FFT) techniques and is usually in big O
notation. First, we focus on RS codes over characteristic-2 fields, over which
some multiplicative FFT techniques are not applicable. Secondly, due to
moderate block lengths of RS codes in practice, our analysis is complete since
all terms in the complexities are accounted for. Finally, in addition to fast
implementation using additive FFT techniques, we also consider direct
implementation, which is still relevant for RS codes with moderate lengths.
Comparing the complexities of both syndromeless and syndrome-based decoding
algorithms based on direct and fast implementations, we show that syndromeless
decoding algorithms have higher complexities than syndrome-based ones for high
rate RS codes regardless of the implementation. Both errors-only and
errors-and-erasures decoding are considered in this paper. We also derive
tighter bounds on the complexities of fast polynomial multiplications based on
Cantor's approach and the fast extended Euclidean algorithm.Comment: 11 pages, submitted to EURASIP Journal on Wireless Communications and
Networkin

### Hamilton cycles in almost distance-hereditary graphs

Let $G$ be a graph on $n\geq 3$ vertices. A graph $G$ is almost
distance-hereditary if each connected induced subgraph $H$ of $G$ has the
property $d_{H}(x,y)\leq d_{G}(x,y)+1$ for any pair of vertices $x,y\in V(H)$.
A graph $G$ is called 1-heavy (2-heavy) if at least one (two) of the end
vertices of each induced subgraph of $G$ isomorphic to $K_{1,3}$ (a claw) has
(have) degree at least $n/2$, and called claw-heavy if each claw of $G$ has a
pair of end vertices with degree sum at least $n$. Thus every 2-heavy graph is
claw-heavy. In this paper we prove the following two results: (1) Every
2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian.
(2) Every 3-connected, 1-heavy and almost distance-hereditary graph is
Hamiltonian. In particular, the first result improves a previous theorem of
Feng and Guo. Both results are sharp in some sense.Comment: 14 pages; 1 figure; a new theorem is adde

### A note on nowhere-zero 3-flow and Z_3-connectivity

There are many major open problems in integer flow theory, such as Tutte's
3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero
3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is
$Z_3$-connected and Kochol's conjecture that every bridgeless graph with at
most three 3-edge-cuts admits a nowhere-zero 3-flow (an equivalent version of
3-flow conjecture). Thomassen proved that every 8-edge-connected graph is
$Z_3$-connected and therefore admits a nowhere-zero 3-flow. Furthermore,
Lov$\acute{a}$sz, Thomassen, Wu and Zhang improved Thomassen's result to
6-edge-connected graphs. In this paper, we prove that: (1) Every
4-edge-connected graph with at most seven 5-edge-cuts admits a nowhere-zero
3-flow. (2) Every bridgeless graph containing no 5-edge-cuts but at most three
3-edge-cuts admits a nowhere-zero 3-flow. (3) Every 5-edge-connected graph with
at most five 5-edge-cuts is $Z_3$-connected. Our main theorems are partial
results to Tutte's 3-flow conjecture, Kochol's conjecture and Jaeger et al.'s
conjecture, respectively.Comment: 10 pages. Typos correcte

### On Nash Dynamics of Matching Market Equilibria

In this paper, we study the Nash dynamics of strategic interplays of n buyers
in a matching market setup by a seller, the market maker. Taking the standard
market equilibrium approach, upon receiving submitted bid vectors from the
buyers, the market maker will decide on a price vector to clear the market in
such a way that each buyer is allocated an item for which he desires the most
(a.k.a., a market equilibrium solution). While such equilibrium outcomes are
not unique, the market maker chooses one (maxeq) that optimizes its own
objective --- revenue maximization. The buyers in turn change bids to their
best interests in order to obtain higher utilities in the next round's market
equilibrium solution.
This is an (n+1)-person game where buyers place strategic bids to gain the
most from the market maker's equilibrium mechanism. The incentives of buyers in
deciding their bids and the market maker's choice of using the maxeq mechanism
create a wave of Nash dynamics involved in the market. We characterize Nash
equilibria in the dynamics in terms of the relationship between maxeq and mineq
(i.e., minimum revenue equilibrium), and develop convergence results for Nash
dynamics from the maxeq policy to a mineq solution, resulting an outcome
equivalent to the truthful VCG mechanism.
Our results imply revenue equivalence between maxeq and mineq, and address
the question that why short-term revenue maximization is a poor long run
strategy, in a deterministic and dynamic setting

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