The guessing number of a directed graph (digraph), equivalent to the entropy of that digraph, was introduced as a direct criterion on the solvability of a network coding instance. This paper makes two contributions on the guessing number. First, we introduce an undirected graph on all possible configurations of the digraph, referred to as the guessing graph, which encapsulates the essence of dependence amongst configurations. We prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph. Therefore, network coding solvability is no more a problem on the operations made by each node, but is simplified into a problem on the messages that can transit through the network. By studying the guessing graph of a given digraph, and how to combine digraphs or alphabets, we are thus able to derive bounds on the guessing number of digraphs. Second, we construct specific digraphs with high guessing numbers, yielding network coding instances where a large amount of information can transit. We first propose a construction of digraphs with finite parameters based on cyclic codes, with guessing number equal to the degree of the generator polynomial. We then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one, despite these digraphs being arbitrarily sparse. These constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear, although these instances are sparse and the sources are arbitrarily far from their corresponding sinks.
Network coding [1] is a protocol which outperforms routing for multicast networks by letting the intermediate nodes manipulate the packets they receive. In particular, linear network coding [2] is optimal in the case of one source; however, it is not the case for multiple sources [3], [4]. Although for large dynamic networks, good heuristics such as random linear network coding [5], [6] can be used, for a given static network maximizing the amount of information that can be transmitted is fundamental. Solving this problem by brute force, i.e. considering all possible operations at all nodes, is computationally prohibitive. In this paper, we reduce this problem to finding a maximum independent set in an undirected graph determined by the network coding instance.
Network coding also opens many new questions about network design (see [7], [8] for examples of networks with interesting properties). Clearly, dense graphs with a large number of edges between the nodes can transmit a large amount of information; similarly, a small diameter is a good property for information transfer; finally, a large number of intermediate nodes between the sources and the sinks is preferable. However, in this paper, we introduce classes of networks that are arbitrarily sparse, with arbitrarily high diameters, and with a relatively small number of intermediate nodes, yet on which all the requested information can be transmitted. Furthermore, for these graphs, the demands of the sinks can be satisfied over any alphabet, and linear combinations are sufficient. Therefore, our work provides different guidelines on the design of networks which take advantage of network coding. The results in this paper are based on the study of the guessing number of digraphs, reviewed below.
The guessing number of digraphs is a concept introduced in [9], which connects graph theory, network coding, and circuit complexity theory. In [9] it was proved that an instance of network coding with n sources and n sinks on an acyclic network (referred to as a multiple unicast network) is solvable over a given alphabet if and only if the guessing number of a related digraph is equal to n. Moreover, it is proved in [9], [10] that any network coding instance can be reduced into a multiple unicast network. Therefore, the guessing number is a direct criterion on the solvability of network coding. Similarly, the linear guessing number evaluates the solvability of a network coding instance by using linear combinations only. By determining these two quantities, the performance of linear network coding can then be compared to that of general network coding. In [11], the guessing number is also used to disprove a long-standing open conjecture on circuit complexity. In [12], the guessing number and linear guessing number of digraphs were studied, and bounds on the guessing number of some particular digraphs were derived.
The guessing number is equal to the entropy of the same digraph [11], thus tying this quantity with fundamental problems of information theory. For instance, by relying heavily on [13], [14] and [15], it was shown that the entropy of a digraph might not be determined by the use of Shannon inequalities alone [16]. Similarly, the information defect is related to the so-called public entropy [16]. We would like to emphasize that the graph entropy for digraphs considered in this paper is fundamentally different to the graph entropy for undirected graph introduced by Körner in [17] (see [18] for a review of that quantity).
Let us give a brief description of the guessing game with n players, viewed as vertices on a digraph D, and an alphabet of size s. All the players are assigned an element of the alphabet (collectively referred to as a configuration), and each player knows the values assigned to all the players in its in-neighborhood.
It does not, however, know its own value, and the goal of the game is to guess it correctly. Clearly, the values cannot all be guessed correctly every time. If the players do not collaborate, the probability that all guesses are correct is exactly s -n . However, the players may elaborate a collaborative strategy (referred to as a protocol) which increases the probability of success. For instance, suppose we play the game on the clique K n , where each player knows the values assigned to all the other vertices. A common strategy could be the following: each player guesses the opposite of the sum (modulo s) of all the values it sees. Any configuration whose sum modulo s is zero will be correctly guessed, hence raising the success probability to s -1 = s (n-1)-n (this is, in fact, optimal). The guessing number is then defined as the maximum over all protocols of the gain from the trivial guessing strategy. For instance, the guessing number of the clique on n vertices is n -1.
Suppose now the players have a helper, whose aim is to make all players guess correctly every time.
This helper is limited: he or she can only send the
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