On D-ary Fano Codes

Cicalese, Ferdinando; Rossi, Eros

doi:10.1109/isit44484.2020.9174023

Cited by 1 publication

(4 citation statements)

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“…The implication of the above theorem on the construction of low redundancy D-ary Fano codes, is given by the following result which had been conjectured in [1] and for which we are now giving a complete proof.…”

Section: Fact 1 Using the Above Notation It Holds Thatmentioning

confidence: 78%

“…In this paper we continue the investigation on efficient construction of low redundancy D-ary Fano codes. We significantly improve the results of [1,14] both in terms of the redundancy guarantee and the time complexity of the construction algorithm. We show that a somehow even simpler splitting criterion can be employed that can be implemented in time O(D log n).…”

Section: Alphabetic Codes and Fano Codesmentioning

confidence: 93%

“…However, it is not clear how to efficiently perform the optimization employed in [14], as a direct approach would result in time complexity exponential in D. On the basis of this observation, in [1], we proposed an alternative even splitting optimization criterion, which can be optimally solved via dynamic programming in time O(n D). This approach is shown to guarantee the redundancy bound L − H D (X) ≤ 1 − p min for D = 2, 3, 4 and in the special case where the resulting code tree is complete, so providing a partial extension of the result of [14].…”

Section: Alphabetic Codes and Fano Codesmentioning

confidence: 99%

“…We believe that an algorithm that considers both options would always succeed also for D = 3. Moreover, it is possible to show that the splitting criterion used in [1] can also be adapted to guarantee nice aggregation in time O(Dn) for all D ≥ 2. This, however, leads to the overall more expensive construction of a code with redundancy 1 − p n in overall O(n 2 D) time.…”

Section: Final Remarksmentioning

confidence: 99%

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On the Redundancy of D-Ary Fano Codes

Cicalese

Rossi

2021

SOFSEM 2021: Theory and Practice of Computer Science

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We study the redundancy of D-ary Fano source codes. We show that a novel splitting criterion allows to prove a bound on the redundancy of the resulting code which sharpens the guarantee provided by Shannon's classical result for the case of an optimal code.In particular we show that, for any D ≥ 2 and for every source distribution p = p1, . . . , pn, there is a D-ary Fano code that satisfies the redundancy boundwhere, L denotes the average codeword length, pmin = mini pi, andis the D-ary entropy of the source. The existence of D-ary Fano codes achieving such a bound had been conjectured in [ISIT2015], where, however, the construction proposed achieves the bound only for D = 2, 3, 4. In [ISIT2020], a novel construction was proposed leading to the proof that the redundancy bound in (1) above also holds for D = 5 (and some other special cases). This result was attained by a dynamic programming based algorithm with time complexity O(Dn) (per node of the codetree).Here, besides proving that the redundancy bound in (1) can be achieved, unconditionally, for every D > 3, we also significantly improve the time complexity of the algorithm building a D-ary Fano code tree achieving such a bound: We show that, for every D ≥ 4, a D-ary Fano code tree satisfying (1) can be constructed by an efficient greedy procedure that has complexity O(D log 2 n) per node of the codetree (i.e., improving from linear time to logarithmic time in n).

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Section: Fact 1 Using the Above Notation It Holds Thatmentioning

confidence: 78%

Section: Alphabetic Codes and Fano Codesmentioning

confidence: 93%