Comments on Cut-Set Bounds on Network Function Computation

Abstract

A function computation problem in directed acyclic networks has been considered in the literature, where a sink node wants to compute a target function with the inputs generated at multiple source nodes. The network links are error-free but capacity-limited, and the intermediate network nodes perform network coding. The target function is required to be computed with zero error. The computing rate of a network code is measured by the average number of times that the target function can be computed for one use of the network, i.e., each link in the network is used at most once. In the papers [1], [2], two cut-set bounds were proposed on the computing rate. However, we show in this paper that these bounds are not valid for general network function computation problems. We analyze the arguments that lead to the invalidity of these bounds and fix the issue with a new cut-set bound, where a new equivalence relation associated with the inputs of the target function is used. Our bound is qualified for general target functions and network topologies. We also show that our bound is tight for some special cases where the computing capacity is known. Moreover, some results in [11], [12] were proved using the invalid upper bound in [1] and hence their correctness needs further justification. We also justify their validity in the paper.

Improved Upper Bound on the Network Function Computing Capacity

“…Appuswamy et al [34] investigated the fundamental computing capacity, i.e., the maximum average number of times that the function can be computed with zero error for one use of the network, and gave a cut-set based upper bound that is valid under certain constraints on either the network topology or the target function. Huang et al [37] obtained an enhancement of Appuswamy et al's upper bound that can be applied for arbitrary functions and arbitrary network topologies. Specifically, for the case of computing an arbitrary function of the source messages over a multi-edge tree network and the case of computing the identity function or the algebraic sum function of the source messages over an arbitrary network topology, the above two upper bounds coincide and are tight (see [34] and [37]).…”

“…Huang et al [37] obtained an enhancement of Appuswamy et al's upper bound that can be applied for arbitrary functions and arbitrary network topologies. Specifically, for the case of computing an arbitrary function of the source messages over a multi-edge tree network and the case of computing the identity function or the algebraic sum function of the source messages over an arbitrary network topology, the above two upper bounds coincide and are tight (see [34] and [37]). However, both of these bounds are in general quite loose.…”

“…Our improved upper bound not only is an enhancement of the previous upper bounds (cf. [34], [37]), but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain "non-tree" network (cf. Example 1 in Section II of the current paper).…”

“…Without loss of generality, we assume throughout the paper that A = B, so that k n log |B| |A| in the above is simplified to k n . Although the definition of the computing capacity C(N , f ) here is a little different from the one used in [34] and [37], i.e., sup k/n : k/n is achievable , it is easy to see that they are equivalent. Our definition has the advantage that it is more consistent with the usual concept of rate region in information theory problems.…”

“…In [34], Appuswamy et al gave a proof of an enhanced upper bound by considering an equivalence relation defined on the input vectors of the target function f with respect to the cut sets. However, it was subsequently pointed out by Huang et al [37] that the proof in [34] is incorrect and in fact the claimed upper bound is valid only for either arbitrary target functions but special network topologies or arbitrary network topologies but special target functions. Instead, they fixed the upper bound in [34] by modifying the equivalence relation considered in [34].…”