E. N. Gilbert and E. F. Moore. Variable-length binary encoding. Bell Systems Technical Journal, 38:933--968, 1959.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is statistically synchronizable, since the probability of not observing the string tends to zero as the number of observations increases without bound. Gilbert [10] studied certain fixed length synchronizable codes where all codewords had common prefixes, and variable length codes were studied in [11]. Wei and Sholtz [28] showed that fixed length codes are statistically synchronizable if and only if they have a self synchronizing string, as well as some other characterizations. Capocelli, Gargano, and Vaccaro [2] proved that a variable length code is statistically synchronizable if and only if ....

....sequence for certain cases when the minimum codeword length is less than five. Montgomery and Abrahams [19] showed how to construct variable length codes with a self synchronizing string, whose average length is close to that of a Huffman code. Other work in these areas can be found in [11, 12, 14, 15, 16, 18, 20, 21, 22, 25, 26, 27, 29]. 2 Complete Prefix Codes A prefix code is a set 99 9 with the property that no element of is a prefix of any other element of . The elements of are called codewords. A prefix code is complete if for every , the string is a prefix of some codeword if and only if ....

E. N. Gilbert and E. F. Moore. Variable-length binary encodings. Bell System Technical Journal, 38:933--967, July 1959.

....implemented the following simple heuristics that still guarantee short labels. Method 1 (Choose arbitrary total order) For the special case where the partial order on the set of weights w 1 , w k is in fact a total order, i.e. w k there are simple polynomial time algorithms [11, 16, 13] for finding an optimal binary tree with w 1 , w k at the leaves from left to right that minimizes either # i=1. n (w i l i ) or max i=1. n (w i l i ) where l i is the number of edges from the root to the leaf containing w i ) A binary tree containing w 1 . w k at the ....

E. N. Gilbert and E. F. Moore. Variable length binary encoding. Bell systems technical journal, 38:933--968, 1959.

....relative frequencies were pre sorted. Schwartz [Sch64] presented a modification to Hu#man s algorithm that computes the code tree with minimum height that could result from the original algorithm. Bookstein and Klein [BK93] present an overview of such developments up to 1993. Gilbert and Moore [GM59] gave a standard O(n ) time dynamic programming algorithm for constructing an optimal binary search tree. It is interesting to note that this was first presented as an algorithm for constructing an optimal static, length unrestricted, alphabetic code tree. Knuth [Knu71] improved Gilbert and ....

E. N. Gilbert and E. Moore. Variable-length binary encodings. Bell System Technical Journal, 38:933--968, 1959.

....nodes (successful searches) only to external tree nodes (unsuccessful searches) or to both. Finally, we can consider a more general model, where both access costs and weights are included. An algorithm for constructing an optimal binary search tree has been rst described by Gilbert and Moore [1], for the case in which to each key is assigned a weight. The complexity of this algorithm is O(n ) Knuth [3] considered the model of access probabilities including successful and unsuccessful searches. He proved an elegant monotonicity principle, which decreased the complexity by a factor of ....

....optimal binary search trees in which the access cost to a key x q depends on the k preceding keys which were reached in the path to x q . We permit arbitrary access probabilities (independent on the preceding keys) as well. The classical optimal binary search tree construction by Gilbert and Moore [1] and Knuth [3] corresponds thus to the fundamental case k = 0. In this work we are concerned with the values k 1. Two kinds of optimal trees are considered, namely optimal worst case trees and weighted average case trees. The inputs of these prob2 lems are a number n of keys, the value k, 1 k ....

[Article contains additional citation context not shown here]

E. N. Gilbert and E. F. Moore, Variable-length binary encoding, Bell System Tech. J. 38 (1959), pp. 933-968.

.... f i g of the codewords, rather than the codewords themselves; the latter are easily obtained as follows: the i th codeword consists of the first i bits immediately to the right of the binary point in the infinite binary expansion of P i Gamma1 j=1 2 Gamma j , for i = 1; n [12]. Many properties of canonical codes are mentioned in [15, 3] The following will be used as a running example in this paper. Consider the probability distribution implied by Zipf s law, defined by the weights p i = 1= i H n ) for 1 i n, where H n = P n j=1 (1=j) is the n th harmonic number. ....

Gilbert E.N., Moore E.F., Variable-length binary encodings, The Bell System Technical Journal 38 (1959) 933--968.

....for an LS code C with maximum length L we have 0 L Gamma1 (d Gamma 1) 2 C, and if 0 L 2 C then 0 L 1 is in the augmented code. Lemma 2 Any augmented code is feasible and has at least a s word. Example 3 Consider the English alphabet. The frequencies of letters and space (as given in [9]) and the binary augmented code C are given in Table 2. C has 14 synchronizing codewords (marked by in the table) The average codeword length of the Huffman code is 4:1195. The average codeword length of C is 4:2965. The sum of the probabilities of letters encoded with a synchronizing word is ....

E.N. Gilbert and E.F. Moore, "Variable length binary encodings," Bell Syst. Tech. J., vol. 38, pp. 933--967, 1960.

....engineering, and the subjects have been studied extensively for more than forty years. In 1952, Huffman [7] discovered the algorithm for constructing an optimum variable length binary code or the binary tree with minimum weighted path length known as the Huffman s tree. In 1959, Gilbert and Moore [3] considered an alphabetic constraint which restricts the ordering of leaves in the binary tree and they found an algorithm for constructing an optimum alphabetic binary tree in O(n 3 ) time. Their technique is based on dynamic programming and is illustrated in many computer science textbooks. In ....

....node in the weight sequence during later combinations. Lemma9 Gilbert Moore. Let w a ; w 1 ; w 2 ; w k ; w d be a weight sequence where k X i=1 w i min(w a ; w d ) then the subsequence w 1 ; w 2 ; w k will form a binary tree whose root is a permanent node. Proof. See [3]. Lemma 10. Let w a ; w b ; w c ; w d be any consecutive four nodes in a sequence and 1. w a w c 2. w b w d 3. w b w c ) max(w a ; w d ) Then the parent of w b and w c is a permanent node. Proof. Conditions 1 and 2 imply that w b and w c are a local minimum compatible pair. Without ....

E. N. Gilbert and E. F. Moore. Variable length binary encodings. Bell System Technical Journal, 38:933--968, 1959.

....on the decoded text seems to be garbled. The lost tail can be restored at least partially by backward decoding of the compressed text, starting at the end of the message, so that the damage of the error can be restricted to a local perturbation. Affix codes are already mentioned in Gilbert Moore [6] (where they are called never self synchronizing) and in Schutzenberger [16] they are extensively studied in Berstel Perrin [1, Chapter III] under the name of biprefix codes. We further restrict attention to binary codes, but all the results are easily extendable to k ary codes for any k 2. ....

....has the affix property. For example, the left subtree of Figure 1a is not affix by Proposition 2. Hence a bottom up construction as the one used by Huffman will not work in our case. A first cut back in the number of potential affix codes is obtained from the following theorem which is cited in [6] and proved in [16] For a source S = hn 1 ; n i, let d(S) 1 n 1 2 Gamma1 2 n 2 2 Gamma2 Delta Delta Delta n 2 Gamma : The quantity d(S) is called the degree of the source S. Theorem (Gilbert, Moore, Schutzenberger) The degree of any complete affix code is ....

Gilbert E.N., Moore E.F., Variable-length binary encodings, The Bell System Technical Journal 38 (1959) 933--968. -- 23 --

....solves the problem in O Gamma (B Gamma log 2 n) n 2 Delta ; this bound applies for both time and space. A completely different dynamic programming solution is given by Garey [7] with O(Bn 2 ) time and space complexity. Garey s algorithm is based on a procedure proposed by Gilbert Moore [9] for alphabetical encodings, using time O(n 3 ) The latter pro 2 cedure was improved by Knuth [15] to O(n 2 ) in an application to optimum binary search trees, for which records can be stored also in internal nodes, but with no restriction on the depth of the tree. Garey shows how to ....

Gilbert E.N., Moore E.F., Variable-length binary encodings, The Bell System Technical Journal 38 (1959) 933--968.

....on the other hand, the frequencies need not be transmitted, nor the codewords themselves. In fact, it suffices for both encoder and decoder to know the lengths of the codewords, as both could construct, based on those lengths, the same optimal code. A natural choice would be a canonical code [39] [21], 16] An easy way to generate such a code, which is needed in the encoding phase, is as follows [21] given are the lengths 1 ; n of the Huffman codewords in non decreasing order (thus corresponding to the probabilities that have been sorted into non increasing order) the i th ....

....suffices for both encoder and decoder to know the lengths of the codewords, as both could construct, based on those lengths, the same optimal code. A natural choice would be a canonical code [39] 21] 16] An easy way to generate such a code, which is needed in the encoding phase, is as follows [21]: given are the lengths 1 ; n of the Huffman codewords in non decreasing order (thus corresponding to the probabilities that have been sorted into non increasing order) the i th codeword consists of the i first bits to the right of the binary point in the binary representation ....

[Article contains additional citation context not shown here]

Gilbert E.N., Moore E.F., Variable-length binary encodings, The Bell System Technical Journal 38 (1959) 933--968.

....: l oe(N) r oe(N) is the path vector of a binary tree. Consider the tree T oe ( p; q) If oe is the identical permutation then the cost of the U R system corresponding to T oe ( p; q) is upper bounded by H( p) H( q) 5 where H is the entropy function 1 . Proof. It is well known [5] that it is possible to construct weighted binary trees T p whose cost is upper bounded by H( p) 2. So, we can assume that C(T p L ) H( p) 2 and C(T q R ) H( q) 2. Since oe is the identical permutation we have that the path vector of T oe ( p; q) is ( l 1 ; r 1 ) l 2 ; r ....

E.N. Gilbert and E.F. Moore. Variable-length binary encodings. Bell Syst Tech Journal 38, 4 (1959), pp.933--968.

....code used in the examples above is completely selfsynchronizing, and has universal synchronizing sequence 00000011000. Gilbert and Moore prove that the existence of a universal synchronizing sequence is a necessary as well as a sufficient condition for a code to be completely self synchronizing [Gilbert and Moore 1959]. They also state that any prefix code which is completely self synchronizing will synchronize itself with probability 1 if the source ensemble consists of successive messages independently chosen with any given set of probabilities. This is true since the probability of occurrence of the ....

Gilbert, E. N., and Moore, E. F. 1959. Variable-Length Binary Encodings. Bell System Tech. J. 38, 4 (July), 933--967.

....engineering, and the subject has been studied extensively for more than forty years. In 1952, Huffmann [7] discovered the algorithm for constructing an optimum variable length binary code or the binary tree with minimum weighted path length known as the Huffman s tree. In 1959, Gilbert and Moore [3] considered an alphabetic constraint which restricts the ordering of leaves in the binary tree and they found an algorithm for constructing an optimum alphabetic binary tree in O(n 3 ) time. Their technique is based on dynamic programming and is illustrated in many computer science textbooks. In ....

E. N. Gilbert and E. F. Moore. Variable length binary encodings. Bell System Technical Journal, 38:933-968, 1959.

....algorithm can be used to find an optimum extension when all extension roots of C have the same length, in particular when C has only one extension root. In this paper we present an algorithm that finds an optimum extension for an arbitrary extendible prefix code. Starting with Gilbert and Moore [4], dynamic programming has been successfully applied to several prefix coding problems (see for example [3,7,8,12] Particularly efficient algorithms are obtained with a speed up technique devised by Knuth [8] the speed up is based on the fact that a certain cost matrix satisfies the quadrangle ....

E.N. Gilbert and E.F. Moore, Variable-length binary encodings, Bell Systems Tech. J. 38 (1959), pp. 933--968.

....) l oe(N) r oe(N) is the path vector of a binary tree. Consider the tree T oe ( p; q) If oe is the identical permutation then the cost of the U R system corresponding to T oe ( p; q) is upper bounded by H( p) H( q) 5 where H is the entropy function 1 . Proof. It is well known [5] that it is possible to construct weighted binary trees T p whose cost is upper bounded by H( p) 2. So, we can assume that C(T p L ) H( p) 2 and C(T q R ) H( q) 2. Since oe is the identical permutation we have that the path vector of T oe ( p; q) is ( l 1 ; r 1 ) l 2 ; r ....

E.N. Gilbert and E.F. Moore. Variable-length binary encodings. Bell Syst Tech Journal 38, 4 (1959), pp.933--968.

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E. N. Gilbert and E. F. Moore. Variable-length binary encoding. Bell Systems Technical Journal, 38:933--968, 1959.

No context found.

E.N. Gilbert and E. Moore. Variable-length binary encodings. Bell System Technical Journal, 38:933--968, 1959.

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Gilbert, E. N. and Moore, E. F. (1959) Variable-length binary encodings. Bell Syst. Tech. J., 38, 933--968.

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E. N. Gilbert and E. F. Moore, "Variable-length binary encodings," Bell Syst. Tech. J., vol. 38, pp. 933--967, July 1959.

No context found.

E. N. Gilbert and E. F. Moore. Variable length binary encoding. Bell systems technical journal, 38:933-968, 1959.

No context found.

Gilbert E.N., & Moore E.F. (1959). Variable-length Binary Encodings. The Bell System Technical Journal , 38, 933--968.

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E.N. Gilbert and E.F. Moore, Variable-length binary encodings, Bell System Tech J. 38 (1959), pp. 933---968.

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E.N. Gilbert and E.F. Moore, Variable-length binary encodings, Bell System Tech J. 38 (1959), pp. 933---968.

No context found.

E. N. Gilbert and E. F. Moore, Variable-length binary encodings, Bell System Tech. J., 38 (1959) 933--968.

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E.N. Gilbert and E.F. Moore, Variable-length binary encodings, Bell System Tech J. 38 (1959), pp. 933---968.

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