TY - JOUR

T1 - Optimal monotone encodings

AU - Alon, Noga

AU - Hod, Rani

N1 - Funding Information:
Manuscript received April 29, 2008; revised October 23, 2008. Current version published February 25, 2009. This work was supported in part by the Israel Science Foundation and by a USA-Israeli BSF Grant. The material in this paper was presented in part at the 35th International Colloquium on Automata, Languages and Programming, Reykjavik, Iceland, July 2008.

PY - 2009

Y1 - 2009

N2 - Moran, Naor, and Segev have asked what is the minimal r = r(n,k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1,2,...,n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multiuser tracing ((k, α)-FUT (fraction user-tracing) families). We show that r(n, k) = Θ(k log(n/k) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant.

AB - Moran, Naor, and Segev have asked what is the minimal r = r(n,k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1,2,...,n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multiuser tracing ((k, α)-FUT (fraction user-tracing) families). We show that r(n, k) = Θ(k log(n/k) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant.

KW - Monotone encoding

KW - Multiuser tracing

KW - Superimposed codes

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U2 - 10.1109/TIT.2008.2011507

DO - 10.1109/TIT.2008.2011507

M3 - Article

AN - SCOPUS:62749144483

SN - 0018-9448

VL - 55

SP - 1343

EP - 1353

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 3

ER -