36 research outputs found
Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures
The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon
capacity of a graph. This parameter of a graph has recently gained much more
attention from the research community after the work of Bar-Yossef et al.
(2006). In their paper, it was shown that the min-rank of a graph G
characterizes the optimal scalar linear solution of an instance of the Index
Coding with Side Information (ICSI) problem described by the graph G. It was
shown by Peeters (1996) that computing the min-rank of a general graph is an
NP-hard problem. There are very few known families of graphs whose min-ranks
can be found in polynomial time. In this work, we introduce a new family of
graphs with efficiently computed min-ranks. Specifically, we establish a
polynomial time dynamic programming algorithm to compute the min-ranks of
graphs having simple tree structures. Intuitively, such graphs are obtained by
gluing together, in a tree-like structure, any set of graphs for which the
min-ranks can be determined in polynomial time. A polynomial time algorithm to
recognize such graphs is also proposed.Comment: Accepted by Algorithmica, 30 page
Weakly Secure MDS Codes for Simple Multiple Access Networks
We consider a simple multiple access network (SMAN), where sources of
unit rates transmit their data to a common sink via relays. Each relay is
connected to the sink and to certain sources. A coding scheme (for the relays)
is weakly secure if a passive adversary who eavesdrops on less than
relay-sink links cannot reconstruct the data from each source. We show that
there exists a weakly secure maximum distance separable (MDS) coding scheme for
the relays if and only if every subset of relays must be collectively
connected to at least sources, for all . Moreover, we
prove that this condition can be verified in polynomial time in and .
Finally, given a SMAN satisfying the aforementioned condition, we provide
another polynomial time algorithm to trim the network until it has a sparsest
set of source-relay links that still supports a weakly secure MDS coding
scheme.Comment: Accepted at ISIT'1
On the Existence of MDS Codes Over Small Fields With Constrained Generator Matrices
We study the existence over small fields of Maximum Distance Separable (MDS)
codes with generator matrices having specified supports (i.e. having specified
locations of zero entries). This problem unifies and simplifies the problems
posed in recent works of Yan and Sprintson (NetCod'13) on weakly secure
cooperative data exchange, of Halbawi et al. (arxiv'13) on distributed
Reed-Solomon codes for simple multiple access networks, and of Dau et al.
(ISIT'13) on MDS codes with balanced and sparse generator matrices. We
conjecture that there exist such MDS codes as long as , if the specified supports of the generator matrices satisfy the so-called
MDS condition, which can be verified in polynomial time. We propose a
combinatorial approach to tackle the conjecture, and prove that the conjecture
holds for a special case when the sets of zero coordinates of rows of the
generator matrix share with each other (pairwise) at most one common element.
Based on our numerical result, the conjecture is also verified for all . Our approach is based on a novel generalization of the well-known Hall's
marriage theorem, which allows (overlapping) multiple representatives instead
of a single representative for each subset.Comment: 8 page
Balanced Sparsest Generator Matrices for MDS Codes
We show that given and , for sufficiently large, there always
exists an MDS code that has a generator matrix satisfying the
following two conditions: (C1) Sparsest: each row of has Hamming weight ; (C2) Balanced: Hamming weights of the columns of differ from each
other by at most one.Comment: 5 page
On the Security of Index Coding with Side Information
Security aspects of the Index Coding with Side Information (ICSI) problem are
investigated. Building on the results of Bar-Yossef et al. (2006), the
properties of linear index codes are further explored. The notion of weak
security, considered by Bhattad and Narayanan (2005) in the context of network
coding, is generalized to block security. It is shown that the linear index
code based on a matrix , whose column space code has length ,
minimum distance and dual distance , is -block secure
(and hence also weakly secure) if the adversary knows in advance
messages, and is completely insecure if the adversary knows in advance more
than messages. Strong security is examined under the conditions that
the adversary: (i) possesses messages in advance; (ii) eavesdrops at most
transmissions; (iii) corrupts at most transmissions. We prove
that for sufficiently large , an optimal linear index code which is strongly
secure against such an adversary has length . Here
is a generalization of the min-rank over of the side
information graph for the ICSI problem in its original formulation in the work
of Bar- Yossef et al.Comment: 14 page