H. S. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Trans. Inform. Theory, 22(5):592--593, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....work We can broadly characterize the different techniques based on their model of the differences between two reconciling hosts. One model involves synchronizing two discrete random variables with some known joint prob ability distribution using a minimum communication complexity. Witsenhausen [10] followed by Alon and Orlitsky [11] show a connection between such random variable reconciliation and graph coloring, giving results analogous to those of Section 3.1 and 4.1. In addition, Orlitsky [12] showed how to use linear error correcting codes for a specific instance of data ....

....is to determine whether M = MB, subject to the sole a priori assumption that MB is in the G vicinity of M. The data verification problem is thus to determine the minimum amount of information A should send to B so that B can decide whether or not M = MB. Consider a natural graph structure [10] corresponding to a given error set . Definition 8. The characteristic graph of an error set is the undirected graph Gs = V, E) whose vertices are characteristic vectors of multi sets M C Z. Any two vertices v, v2 6 V are connected by an edge in this graph iff there exists a non identity error ....

H.S. Witsenhausen, "The zero-error side information problem and chromatic numbers," IEEE Trans. on Info. Theory, vol. 22, no. 5, September 1976.

....message of at most 690 bits from server to client and that works on all and 429 with 723 and 70 29280 . The result is obtained by considering the characteristic hypergraph for the problem, first defined by Witsenhausen [51] and obtained by adding a vertex for each file , and for each h 5 a hyperedge 610 358 and 10 23380 . Since each vertex is adjacent to at most edges and each edge contains at most 420 vertices, the chromatic number ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592-593, September 1976.

....on the authenticated channel to Bob. Finally, using this information and his continuous value XB , Bob is able to determine K = K) XB ) with high probability. In the source coding terminology, is a quantizer [11] and the pair ( is a lossless code with side information at the receiver [27, 31]. We have therefore split our problem in two main parts: 1) design a good quantizer , 2) design a good lossless code ( We set XA ; XB 2 R ; K 2 K N. The functions involved are : R K (1) K f0; 1g (2) f0; 1g R K: 3) A summary of the scheme is provided in ....

....discrete side information is usually assumed, unlike XB in our case, but this aspect will be covered in Sec. D, allowing the reader to assume a discrete XB in the current section. We divide explicit constructions of codes in three main families. First there are constructions for zero error codes [31, 33, 14, 15, 16, 34], some of which are based on graph coloring, and their near zero error variants [34] Then, some are based on sending a syndrome of an error correcting code [22, 23, 24, 30] Finally, in the context of QKD, a mention of interactive codes [6, 7] will be made. B.1 Zero error codes Zero error codes ....

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H. S. WITSENHAUSEN, The zero-error side information problem and chromatic numbers, IEEE Trans. Inform. Theory, 22 (1976), pp. 592-- 593.

....that the number of bits that P:c must transmit in the worst case to convey n instances of without error is rr( def log X(6 ) If G cn be colored with X colors. G 2 cn be colored with X colors. Therefore, X(G ) X(G) for every graph G, nd a( 2a (1) for every dual source. Witsenhusen [21] showed that br some dual sources, fewer bits suce. Example 2(b) For a completely correlated source we saw that is the complete graph over . Hence is the complete graph on X( I1 nd ) 1og I1 ir ll . For a completely correlated dual source we saw that 6 is the empty graph over . Hence ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592 593, September 1976.

....We can broadly characterize the di erent techniques based on their model of the di erences between two reconciling hosts. One model involves synchronizing two discrete random variables with some known joint probability distribution using a minimum communication complexity. Witsenhausen [8] followed by Alon and Orlitsky [9] show a connection between such random variable reconciliation and graph coloring, giving results analogous to those of Section 3.1 and 4.1. Another model involves two hosts reconciling les (or strings) that di er by a bounded number of insertions, deletions, or ....

H.S. Witsenhausen, \The zero-error side information problem and chromatic numbers," IEEE Trans. on Information Theory, vol. 22, no. 5, September 1976.

....our proofs that may lead to such a result. 1.2 Dual source coding A dual source consists of a finite set X , a set Y , and a support set S X Theta Y . In each dual source instance, a sender PX is given an x 2 X and a receiver P Y is given a y 2 Y such that (x; y) 2 S. Following Witsenhausen [17] and Ferguson and Bailey [6] we study the number of bits that PX must transmit in the worst case in order for P Y to learn x without error. See Orlitsky [13] for the case where PX and P Y are allowed to interact. The fan out of x 2 X is the set S x def = fy : x; y) 2 Sg of y s that are ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

....set S Y of Y is similarly defined. Instrumental in determining C 1 (XjY ) is G(XjY ) the characteristic hypergraph of (X; Y ) Its vertex set is SX and for every y 2 S Y , it contains the edge: fx : x; y) 2 SX;Y g. The characteristic hypergraph is equivalent to a graph defined by Witsenhausen [5] who considered the onemessage version of this problem. For the league problem with t teams, G(X t jY t ) has t vertices, one corresponding to each value of X t . It has i t 2 j edges, one corresponding to each possible value of Y t . Each edge contains two vertices (the two possible winning ....

....are assigned the same color. The chromatic number of G is the 1 2 3 4 (1; 2) 1; 3) 1; 4) 2; 3) 2; 4) 3; 4) SX 4 (teams) S Y 4 (games) u u u u u u u u u u u u Figure 1: Characteristic table of the league problem with four teams. minimum number of colors required to color G. Witsenhausen [5] showed that in one way communication, the minimum number of different messages sent by PX is G. Although the messages transmitted do not have to be prefix free (by the implicit termination property they have to be prefix free only given the value of X) it can be shown that: Result 2 (Lemma 2 in ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

....x satisfies f(x; y) D then any z 2 D will do. If for some x, jf(x; y)j = 1 then we take z = f(x; y) For any other x 0 colored by c since there is no edge between x and x 0 it follows from the construction that z 2 f(x 0 ; y) On the other hand, every 2 Similar reductions appear in [16, 21]. In these works the two parties have an input (x; y) in some domain A and P1 has to transmit its input x to P2 . This problem corresponds in our setting to the problem of computing the specific function f which is defined as f(x; y) x if (x; y) 2 A and f(x;y) D otherwise. protocol induces ....

Witsenhausen, H. S., "The Zero-Error Side Information Problem and Chromatic Numbers", IEEE Transactions on Information Theory, 1976, pp. 592-593.

....instrumental in several subsequent proofs. As a first application we show that it determines the one way complexity of (X; Y ) Lemma 2 For all (X; Y ) pairs, C 1 (XjY ) dlog Ge where (G) is the chromatic number of a hypergraph G. 2 The proof slightly strengthens a result in Witsenhausen [1] which showed that the minimum number of different messages sent in one way communication is the chromatic number of a graph equivalent to G(XjY ) The characteristic hypergraph of the league problem with t teams is K t , the complete graph on t vertices. This proves, again, that C 1 (X t jY t ) ....

....k colorable. Let (X; Y ) be a random pair. For each y in S Y , the support set of Y , Equation (1) defined SXjY (y) to be the set of possible x values when Y = y. The characteristic hypergraph G(XjY ) of (X; Y ) has SX as its vertex set and the hyperedge SX jY (y) for each y 2 S Y . Witsenhausen [1] defined a graph which is closely related to G(XjY ) and linked its chromatic number to one way communication. Let OE be a one way protocol for a random pair (X; Y ) The codeword oe OE (x; y) of an input (x; y) consists of the message oe 1 (x; y) By the separate transmissions property, oe OE (x; ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

....lead to such a result. 1.2 Dual source coding A dual source consists of a finite set X , a (possibly infinite) set Y , and a support set S X Theta Y . In each dual source instance, a sender PX is given an x 2 X and a receiver P Y is given a y 2 Y such that (x; y) 2 S. Following Witsenhausen [21] and Ferguson and Bailey [8] we study the number of bits that PX must transmit in the worst case in order for P Y to learn x without error. See Orlitsky [17] for the case where PX and P Y are allowed to interact. The fan out of x 2 X is the set S x def = fy : x; y) 2 Sg of y s that are ....

....case to convey n instances of S without error is oe (n) def = log (G Delta n ) If G can be colored with colors, G Delta 2 can be colored with 2 colors. Therefore, G Delta 2 ) G) 2 for every graph G, and oe (2) 2oe (1) for every dual source. Witsenhausen [21] showed that for some dual sources, fewer bits suffice. Example 2(b) For a completely correlated source we saw that G is the complete graph over X . Hence G Delta n is the complete graph on X n , G Delta n ) jX j n , and oe (n) n log jX j for all n 2 N 1 . For a ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

....can be significantly smaller than either L or Lam . Since the bounds for unrestricted inputs are simpler to state than those for restricted inputs, from now on, we describe them first. 1. 4 Results Associated with the dual source (X; Y ) is a characteristic graph G, defined by Witsenhausen [17]. Its vertex set is X and two distinct vertices x and x 0 are connected if they are confusable. Example 1(c) For the distribution p ffl defined in part (a) of the example, the characteristic graph G consists of the vertex set f1; ng and X is distributed uniformly over its vertices. When ....

....are connected if for some i 2 f1; ng such that v i 6= v 0 i , v i is connected to v 0 i in G i . The n fold OR product of G with itself is denoted by G Delta n . 4.1 Asymptotic per instance number of bits Let (X; Y ) be a random pair with characteristic graph G. As in Witsenhausen [17], n instances of the restricted inputs scenario can be viewed as a single restricted inputs instance of a larger dual source with characteristic graph G Delta n . From Theorem 1, H (G Delta n ; X (n) Gamma log(H (G Delta n ; X (n) 1) Gamma log e L n H ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

....Y of Y is similarly defined. Instrumental in determining C 1 is G, the characteristic hypergraph of (X; Y ) Its vertex set is SX and for every y 2 S Y , it contains the hyperedge: E(y) def = fx : x; y) 2 Sg : 5) The characteristic hypergraph is equivalent to a graph defined by Witsenhausen [10] who considered the one message version of this problem. For the league problem with t teams, for example, G has t vertices, one corresponding to each (team) value of X. It has i t 2 j edges, one corresponding to each possible (game) value of Y . Each edge contains two vertices (the possible ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

....(k) m for large k. The m message worst case amortized complexity of (X; Y ) is Am def = lim k 1 C (k) m k ; where the limit exists by subadditivity. Intuitively, Am is the number of bits per repetition of (X; Y ) required in the worst case when m messages are allowed. Witsenhausen [11] considered one message amortized complexity. We are more concerned with the number of bits required when PX and P Y can interact. Using results of Section 2, we find upper bounds on C (k) 4 and apply them to show that for all random pairs A 4 = A1 = log where is P Y s ....

.... Instrumental in analyzing the worst case complexity measures is G, the characteristic hypergraph of (X; Y ) Its vertex set is SX and for every y 2 S Y , it contains the hyperedge: E(y) def = fx : x; y) 2 Sg : The characteristic hypergraph is equivalent to a graph defined by Witsenhausen [11] who considered the one message version of this problem. For the league problem with t teams, for example, G has t vertices, one corresponding to each (team) value of X. It has i t 2 j edges, one corresponding to each possible (game) value of Y . Each edge contains two vertices (the two ....

[Article contains additional citation context not shown here]

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

....of the Slepian Wolf Theorem [19] L f (XjY ) minfH(g(X)jY ) g(X) and Y determine f(X; Y )g. Both bounds are tight in special cases, but not in general. Drawing on rate distortion results, we show that for every X, Y , and f , L f (XjY ) HG (XjY ) 1) The graph G, defined by Witsenhausen [20] and used in [14, 6] is the characteristic graph of X, Y , and f . HG (XjY ) is the conditional G entropy of X given Y . It extends HG (X) the G entropy of X, or graph entropy of (G; X) defined by Korner [10] Graph entropy has recently been used to derive an alternative characterization of ....

....f1; 2g and f2; 3g. By convexity, I(W ; XjY ) is minimized when p(f1; 2gj2) p(f2; 3gj2) 1 2 . Therefore, HG (XjY ) H(W jY ) Gamma H(W jXY ) 1 3 2 3 Delta h( 1 4 ) Gamma 1 3 = 2 3 Delta h( 1 4 ) 2 2. 3 G The characteristic graph G of X, Y , and f was defined by Witsenhausen [20]. Its vertex set is the support set of X, and distinct vertices x; x 0 are connected if there is a y such that p(x; y) p(x 0 ; y) 0 and f(x; y) 6= f(x 0 ; y) 3 Motivation We motivate the characterization L f (XjY ) HG (XjY ) in (1) via three examples and two naive bounds that are ....

H. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592--593, September 1976.

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H. S. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Trans. Inform. Theory, 22(5):592--593, 1976.

No context found.

H.S. Witsenhausen, "The zero-error side information problem and chromatic numbers," IEEE Trans. on Info. Theory, vol. 22, no. 5, September 1976.

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H. S. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Trans. Inform. Theory, 22(5):592--593, 1976.

No context found.

H. S. WITSENHAUSEN, The zero-error side information problem and chromatic numbers, IEEE Trans. Inform. Theory, 22 (1976), pp. 592-- 593.

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