LowerBound Study for Function Computation in Distributed Networks via VertexEccentricity
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Abstract
Distributed computing networksystems are modeled as graphs with vertices representing compute elements and adjacencyedges capturing their uni or bidirectional communication. Distributed function computation has a wide spectrum of major applications in distributed systems. Distributed computation over a networksystem proceeds in a sequence of timesteps in which vertices update and/or exchange their values based on the underlying algorithm constrained by the time(in)variant networktopology. For finite convergence of distributed information dissemination and function computation in the model, we present a lower bound on the number of timesteps for vertices to receive (initial) vertexvalues of all vertices regardless of underlying protocol or algorithmics in timeinvariant networks via the notion of vertexeccentricity in a graphtheoretic framework. We also address lower bounds on vertexeccentricity and its maximum version in terms of common graphparameters such as maximum degree, and order and size.
Keywords
Distributed function computation Linear iterative schemes Information dissemination Finite convergence VertexeccentricityPreliminaries
 1.
Quantized consensus [6]: Consider an ordern network with an initial networkstate in which each vertex assumes an initial (integer) value \(x_{i}[0]\) for \(i = 1, 2, \ldots , n\). The network achieves a quantized consensus when, at some later time, all the n vertices simultaneously arrive with almost equal values \(y_{i}\) for \(i = 1, 2, \ldots , n\) (that is, \(y_{i}  y_{j} \le 1\) for all \(i, j \in \{1, 2, \ldots , n\}\)) while preserving the sum of all initial values (that is, \(\sum _{i=1}^{n} x_{i}[0] = \sum _{i=1}^{n} y_{i}\)).
 2.
Collaborative distributed hypothesis testing [7]: Consider a networksystem of n vertices (sensors/agents) that collaboratively determine the probability measure of a random variable based on a number of available observations/measurements. For the binary setting in deciding two hypotheses, each vertex collects measurement(s) and makes a preliminary (local) decision \(d_{i} \in \{0, 1\}\) in favor of the two hypotheses for \(i = 1, 2, \ldots , n\). The n vertices are allowed to communicate, and the networksystem resolves with a final decision by, for example, the majority rule (that is, computes the indicator function of the event \(\sum _{i=1}^{n} d_{i} > \frac{n}{2}\)) in distributed fashion.
 3.
Solitude verification [4]: Consider an unlabeled network of n vertices (processes) in which each vertex is in one of a finite number of states: \(s_{i}\) for \(i = 1, 2, \ldots , n\). Solitude verification on the network checks if a unique vertex with a given state s exists in the network, that is, computes the Boolean function for the equality \(\{ i \in \{1, 2, \ldots , n\} \mid s_{i} = s \} = 1\).
 4.
Fundamental iterative limits of distributed function computation [11, 18]: Consider a generic distributed information processing system to attain collective goals via iterative or noniterative intervertex communication. Lower and upper bounds on numbers of iterations for achieving finite convergence of distributed information dissemination and function computation are studied via: structuralcontrollability and observability theories [11] and informationtheoretic techniques [18] in deterministic and probabilistic settings that capture initialvalue/input distribution, networktopological and communication constraints, and/or estimation/output performance.
Model of Distributed Computing Systems
Most graphtheoretic definitions in this article are given in [2]. We will abbreviate “directed graph” and “directed path” to digraph and dipath, respectively.
 1.
Networktopology: A distributed computing system is modeled as a digraph G with V(G) and E(G) denoting its sets of vertices and directed edges, respectively. Unidirectional communication on V(G) is captured by the adjacency relation represented by E(G): for all distinct vertices, \(u, v \in V(G)\), \((u, v) \in E(G)\) if and only if vertex u can send information to vertex v (and v can receive information from u). Note that bidirectional communication between u and v is viewed as the coexistence of the two directed edge (u, v) and (v, u) in E(G).
Distributed computation over the network proceeds in a sequence of timesteps. At each timestep, all vertices update and/or exchange their values based on the underlying algorithm constrained by the networktopology, which is assumed to be timeinvariant.
 2.
Resource capabilities in vertices: The digraph G of the networktopology is vertexlabeled such that messages are identified with senders and receivers. The vertices of V(G) are assumed to have sufficient computational capabilities and local storage. Generally, we assume that: (1) all communications/transmissions between vertices are reliable and in correct sequence, and (2) each vertex may, in the current timestep, receive the priorstep transmission(s) from its inneighbor(s), update, and send transmission(s) to its outneighbor(s) in accordance to the underlying algorithm.
The domain of all initial/input and observed/output values of the vertices of G is assumed to be an algebraic field \(\mathbb {F}\).
 3.Linear iterative scheme (for algorithmic lower and upperbound results): For a vertex \(v \in V(G)\), denote by \(x_{v}[k] \in \mathbb {F}\) the vertexvalue of v at timestep \(k = 0, 1, \ldots\). A function with domain \(\mathbb {F}^{V(G)}\) and codomain \(\mathbb {F}\) is computed in accordance to a linear iterative scheme. Given initial vertexvalues \(x_{v}[0] \in \mathbb {F}\) for all vertices \(v \in V(G)\) as arguments to the function, at each timestep \(k = 0, 1, \ldots\), each vertex \(v \in V(G)\) updates (and transmits) its vertexvalue via a weighted linear combination of the priorstep vertexvalues constrained by neighborstructures: for all \(v \in V(G)\) and \(k = 0, 1, \ldots\),where the prescribed weights \(w_{v u} \in \mathbb {F}\) for all \(v, u \in V(G)\) that are subject to the adjacencyconstraints \(w_{v u} = 0\) (the zeroelement of \(\mathbb {F}\)) if u is not adjacent to v (that is, \((u, v) \not \in E(G)\)); equivalently,$$\begin{aligned} x_{v}[k + 1] = \sum _{u \in V(G)} w_{v u} x_{u}[k], \end{aligned}$$where the two vectors of vertexvalues and W are indexed by a common discrete ordering of V(G) with \(W = [ w_{v u} ]_{(v, u) \in V(G) \times V(G)}\).$$\begin{aligned}&\text{transpose } \text{of } (x_{v}[k + 1] \mid v \in V(G))\\&\quad = W \cdot \text{ transpose } \text{of } (x_{v}[k] \mid v \in V(G)) \end{aligned}$$
Motivation of Our Study

Time(in)variance of networktopology.

Granularity of timestep: discrete versus continuous.

Choice of base field: special (real or complexes) versus arbitrary (finite or infinite).

Characterization of calculable functions.

Convergence criteria and rates (finite, asymptotic, and/or probabilistic).

Adoption and algebraic properties of weightmatrix for linear interactive schemes: random weightmatrix, spectrum of eigenvalues, base field, etc.

Resilience and robustness of computation algorithmics for networktopology in the presence/absence of malicious vertices.

Lower and upper bounds on (linear) iteration required for the convergence of calculable functions.
Sundaram and Hadjicostis [11, 12] present their research findings in the finite convergence of distributed information dissemination and function computation in the model with linear iterative algorithmics stated above, among other contributions in distributed function computation and datastream transmission in the presences of noise and malicious vertices. More specifically, (1) they employ structural theories in observability and invertibility of linear systems over arbitrary finite fields to obtain lower and upper bounds on the number of linear iterations for achieving networkconsensus for finite convergence of arbitrary functions, and (2) the bounds are valid for all initial vertexvalues of arbitrary finite fields as arguments to the functions in connected timeinvariant topologies with almost all random weightmatrices.
Sundaram conjectures in [11] that \(\alpha _{G, u}\) may also serve as a lower bound on the number of timesteps for a vertex \(u \in V(G)\) to receive the initial vertexvalues of all \(v \in \varGamma ^{*}_{G, \text {in}}(u)\) regardless of underlying protocol or algorithmics. Hence, linear iterative schemes are timeoptimal in disseminating information over arbitrary timeinvariant connected networks.
 1.
We view the annotated graph G in Fig. 1 in which each edge represents the coexistence of its two directed versions. For the vertex \(u \in V(G)\), the number of timesteps to receive the initial vertexvalues of all \(v \in \varGamma ^{*}_{G, \text {in}}(u)\) (\(= V(G)\)), regardless of underlying protocol or algorithmics, is \(\text {order}(S) + 2\)—which is realized by the dipath composed of the serial component S and vertices \(v_{5}\), \(v_{4}\), and u.
 2.
We show below that \(\alpha _{G, u} = \text {order}(S) + 3\), hence can not serve as a lower bound on the number of timesteps mentioned in item 1 above—as suggested in the conjecture [11].
Case when \(n= 3\): Consider the three subcases of saturation of vertices \(v_{6}\), \(v_{7}\), and \(v_{5}\) (hence the serial component S).
Combining the three cases of \(n \in \{1, 2, 3\}\), we conclude that \(\alpha _{G, u} \ge \text {order}(S) + 3\). Hence, the two inequalities on \(\alpha _{G, u}\) yield that \(\alpha _{G, u} = \text {order}(S) + 3\), which confirms the counterexample graph G in Fig. 1.
To complement the explicitly constructed counterexample to the lowerbound conjecture on the number of timesteps for distributed function computation and information dissemination with respect to a given vertex, we present in this article a lower bound on the number of timesteps for a vertex \(u \in V(G)\) to receive the initial vertexvalues of all \(v \in \varGamma ^{*}_{G, \text {in}}(u)\) regardless of underlying protocol or algorithmics in a timeinvariant network via the notion of vertexeccentricity.
Revised Lower Bound for Distributed Function Computation and Information Dissemination
 1.
For every (linear or nonlinear) iteration scheme, in which a vertex’s value or information is transmitted to its outneighbors via their incidence directed edges in unit timestep, requires at least \(e_{G, \text {in}}(u)\) timesteps for vertex u to access values/information of all the vertices in \(\varGamma ^{*}_{G, \text {in}}(u)\). Thus, \(e_{G, \text {in}}(u)\) serves as a lower bound on the number of timesteps required for function computation by vertex u via such iteration scheme.
 2.In accordance with the distributed framework for our function computation, we show below that:We illustrate an example organization of \(\varGamma ^{*}_{G, \text {in}}(u)  \{u\}\) in a family of vertexdisjoint directed trees in Fig. 3.$$\begin{aligned} \begin{array}{l} e_{G, \text {in}}(u) = 1 + \min \{\max \{ \underbrace{e_{T_{i}, \text {in}}(\text {root}(T_{i}))}_{= \, \text {depth}(T_{i})} \mid 1 \le i \le n \} \mid \\ \quad \{T_{i}\}_{i=1}^{n}\hbox { is a family of directed trees that are:}\\ \quad \hbox {(1)~a vertexdecomposition of } \varGamma ^{*}_{G, \text {in}}(u)  \{u\}\hbox {, and}\\ \quad \hbox {(2)~rooted in (as subset of) }\varGamma _{G, \text {in}}(u) \}. \\ \end{array} \end{aligned}$$
To show the above equality for \(e_{G, \text {in}}(u)\), we prove the two embedded inequalities in the following sections.
Upper Bound for VertexEccentricity
Lower Bound for VertexEccentricity
 1.
The family \(\{ T_{1}, T_{2}, \ldots , T_{i} \}\) consists of mutually vertexdisjoint directed trees with their roots in \(\varGamma _{G, \text {in}}(u)\),
 2.Each directed tree in the family provides a shortest dipath (in G) for each of its vertices to u, that is, for every vertex \(v \in V(T_{j})\) where \(j \in \{1, 2, \ldots , i\}\), the (unique) dipath from v to \(\text {root}(T_{j})\) in \(T_{j}\) yields \(\overrightarrow{d}_{G}(v, u)\):and$$\begin{aligned} \begin{array}{l} {\text {length}}{\text {((unique) dipath from }}v{\text{ to root}}(T_{j}){\text{ concatenated }}\\ {\text{with directed edge }}({\text {root}}(T_{j}), u)) = \overrightarrow{d}_{G}(v, u), \end{array} \end{aligned}$$
 3.The ineccentricity of u in G is bounded below as:$$\begin{aligned} e_{G, \text {in}}(u) \ge 1 + \max \{ \text {depth}(T_{1}), \text {depth}(T_{2}), \ldots , \text {depth}(T_{i}) \}. \end{aligned}$$
Basis step: For \(P_{1}\), we may employ a dipath from a vertex, say v, in \(\varGamma ^{*}_{G, \text {in}}(u)  \{u\}\) to u in G that realizes \(\overrightarrow{d}_{G}(v, u)\). Then, designate \(P_{1}\) as such a path, and \(T_{1} = \{ P_{1}\{u\} \}\).
For the family \(\{ T_{1} \}\), we can verify the abovestated three items 1, 2 (via “shortest dipath in G” enjoys “optimal substructure property in G” by typical cutandpaste argument), and 3.
Induction step: Assume that we have constructed a sequence \((P_{1}, P_{2}, \ldots , P_{j})\) of dipaths with common terminal vertices u such that the sequence \((P_{1}\{u\}, P_{2}\{u\}, \ldots , P_{j}\{u\})\) is organized as a family \(\{ T_{1}, T_{2}, \ldots , T_{i} \}\), where \(j \ge i \ge 1\), of directed trees that satisfies the abovestated items 1, 2, and 3.
If the family \(\{ T_{1}, T_{2}, \ldots , T_{i} \}\) yields a vertexdecomposition of \(\varGamma ^{*}_{G, \text {in}}(u) \{u\}\), then the inductive construction is complete. Thus, we may assume that there exists a vertex \(v \in (\varGamma ^{*}_{G, \text {in}}(u) \{u\})  \cup _{\eta =1}^{i} V(T_{\eta })\). We construct a desired dipath \(P_{j+1}\) from v to u in G as follows.
Case 1: \(V(P) \cap \cup _{\eta =1}^{i} V(T_{\eta }) = \emptyset\). From the above observation that \(\text {length}(P) \le e_{G, \text {in}}(u)\), hence for \(P_{j+1}\), we may employ P by designating \(P_{j+1} = P\) and \(T_{i+1} = \{ P_{j+1}\{u\} \}\) as in the basis step. We can verify the abovestated items 1, 2, and 3 for the augmented family \(\{ T_{1}, T_{2}, \ldots , T_{i+1} \}\).
 1.
Suppose that \(\text {length}(Q) < \text {length}(P_{2})\): The dipath from v, via w, to u formed by the concatenation of \(P_{1}\) (v to w) and Q (w, via \(\text {root}(T_{k})\), to u) is a shorter dipath than P—which contradicts to the assumption that \(\text {length}(P) = \overrightarrow{d}_{G}(v, u)\).
 2.
Suppose that \(\text {length}(Q) > \text {length}(P_{2})\): The existence of such dipath \(P_{2}\) from v to u in G contradicts to the above item 2 that \(\text {length}(Q) = \overrightarrow{d}_{G}(v, u)\).
 1.
The statement is obvious,
 2.
The condition follows from that “shortest dipath in G” enjoys “optimal substructure property in G”, and
 3.By noting that:$$\begin{aligned} e_{G, \text {in}}(u) \ge \overrightarrow{d}_{G}(v, u) = \text {length}(P). \end{aligned}$$
Lower Bounds for VertexEccentricity via GraphParameters
The \(\min\)\(\max\) formulation of \(e_{G, \text {in}}(u)\), which was developed above for lowerbounding the number of timesteps for function computation by vertex u in \(\varGamma ^{*}_{G, \text {in}}(u)\), motivates us to study lower bounds for (maximum) vertexeccentricity in terms of common graphparameters of the underlying graph G: (1) maximum indegree, and (2) order and size.
VertexEccentricity and Maximum InDegree
We derive a lower bound on \(e_{G, \text {in}}(u)\) from the knowledge of the maximum indegree of G (vertexspanned by \(\varGamma ^{*}_{G, \text {in}}(u)\)), which yields a (possibly weaker) lower bound on the number of timesteps for vertex u to access values/information of all the vertices in \(\varGamma ^{*}_{G, \text {in}}(u)\).
Denote by \(\varDelta _{G, \text {in}}(u)\) (\(\ge 1\)) the maximum indegree of the subdigraph of G vertexspanned by \(\varGamma ^{*}_{G, \text {in}}(u)\).
Theorem 1
Proof
Consider the two cases for the \(\varDelta _{G, \text {in}}(u)\)value.
We can obtain desired lower bounds in analogous fashion with similar graphparameters such a regularity indegree, and maximum and regularity degrees.
Maximum VertexEccentricity and GraphOrder and Size
For a strongly connected digraph, we study a lower bound on its diameter in terms of its order and size, and show the optimality of the diameterbound with a family of explicitly constructed strongly connected digraphs.
Theorem 2
Proof
We study the general case, henceforth, \(\text {dia}(G) \le V(G)  2\). Our approach in deriving a desired lower bound on E(G), hence on \(\text {dia}(G)\), relies on: (1) embedding in G a vertexspanning rooted directed tree with depth of \(\text {dia}(G)\), and (2) then relating the order V(G), size E(G), and diameter \(\text {dia}(G)\) of G via a classification/enumeration of all leaftoroot dipaths into families of dipaths with shared versus nonshared suffixes (towards the root).
To have a refined derivation, the rooted directed tree is seeded with a maximumlength stemming dipath to ensure the consideration of nearleaf proximity of the first/lowest common descendants of shared suffixes of the leaftoroot dipaths.
Denote by R a longest dipath of vertices with indegree of 1 in G, that is, (1) for every vertex \(u \in V(R)\), \(\text {deg}_{G, \text {in}}(u) = 1\), and (2) \(\text {length}(R)\) is the maximum among those of dipaths satisfying item (1). Note that \(0 \le \text {length}(R) \le \text {dia}(G)  1\). Denote by r the terminal vertex of R, and by \(\hat{R}\) the dipath that concatenates the unique vertex s (and its directed edge) adjacent to the initial vertex of R together with the dipath R.
Employing \(\hat{R}\) as a seedstructure, we grow a vertexspanning rooted (at the terminal vertex r of R) directed intree \(T_{r, \text {in}}\) of G [9] in which for every vertex \(u \in T_{r, \text {in}}\) (\(= V(G)\)), there exists a unique dipath from u to r in \(T_{r, \text {in}}\).
Furthermore, we limit the depth of \(T_{r, \text {in}}\) with \(\text {depth}(T_{r, \text {in}}) \le \text {dia}(G)\). For instance, we apply Dijkstra’s singlesource shortestdipaths algorithm (see, for example, [3]) with the seedstructure of \(\hat{R}\), which results in a vertexspanning directed intree \(T_{r, \text {in}}\) of G such that: (1) \(\text {root}(T_{r, \text {in}}) = r\), and (2) for every vertex \(u \in V(T_{r, \text {in}})\) (\(= V(G)\)), the unique dipath P from u to the root r of \(T_{r, \text {in}}\) satisfies that: (2.1) P is a shortest dipath from u to r in G with \(\text {length}(P) \le \text {dia}(G)\), and (2.2) if \(u \in V(\hat{R})\) then P is the suffix of \(\hat{R}\) (with initial vertex u) else P contains \(\hat{R}\) as its suffix. We depict in Fig. 9b the topological structure of a vertexspanning rooted directed intree \(T_{r, \text {in}}\) of a strongly connected digraph G.
Note that, due to the maximality of \(\text {length}(R)\) imposed above, every leaftoroot dipath P must contain a vertex \(u \in V(P)\) that satisfies the following sharingcondition: (1) \(\text {deg}_{G, \text {in}}(u) \ge 2\), and (2) the (first) appearance of such u is within a directed distance of \(\text {length}(\hat{R})\) from the initial leafvertex of P, that is, \(\overrightarrow{d}_{T_{r, \text {in}}}(\text {initial}(P), u) \le \text {length}(\hat{R})\).
 1.Upperbounding V(G):
 1.1.For every leaftoroot dipath \(P \in \mathcal {S}\), there exists a vertex \(u \in V(P)\) such that \(\overrightarrow{d}_{T_{r, \text {in}}}(\text {initial}(P), u) \le \text {length}(\hat{R})\) and u is a first/lowest common descendant shared with at least one other leaftoroot dipath \(P' \in \mathcal {S}\). For the dipathpair P and \(P'\), we include: (1) all the vertices of the prefix of P preceding to s (\(= \text {initial}(\hat{R})\)), that is, \(V(P  \hat{R})\), in the enumeration of V(G) with:and (2) all the vertices of the prefix of \(P'\) preceding to u, that is, \(\overrightarrow{d}_{T_{r, \text {in}}}(\text {initial}(P), u)\) such vertices, in the enumeration of V(G) with:$$\begin{aligned} V(P  \hat{R})&= (\text {length}(P) + 1)  (\text {length}(R) + 2) \\&= \text {length}(P)  \text {length}(R)  1 \\&\le \text {depth}(T_{r, \text {in}})  \text {length}(R)  1 \\&= \text {dia}(G)  \text {length}(R)  1, \end{aligned}$$Therefore, an upper bound on the number of vertices of the prefixes \(P  \hat{R}\) for all \(P \in \mathcal {S}\) that are included in the enumeration of V(G) is:$$\begin{aligned} \overrightarrow{d}_{T_{r, \text {in}}}(\text {initial}(P), u) \le \text {length}(\hat{R}) = \text {length}(R) + 1. \end{aligned}$$$$\begin{aligned}&\frac{\mathcal {S}}{2} (\text {dia}(G)  \text {length}(R)  1)\\&\quad + \frac{\mathcal {S}}{2} (\text {length}(R) + 1) = \frac{\mathcal {S}}{2} \text {dia}(G). \end{aligned}$$
 1.2.For every leaftoroot dipath \(P \in \mathcal {L}  \mathcal {S}\), the vertex s (\(= \text {initial}(\hat{R})\)) may be the first/lowest common descendant that is shared with other leaftoroot dipaths of \(\mathcal {S}\). Therefore, an upper bound on the number of vertices of the prefixes \(P  \hat{R}\) for all \(P \in \mathcal {L}  \mathcal {S}\) that are included in the enumeration of V(G) is:$$\begin{aligned} \mathcal {L}  \mathcal {S} (\text {dia}(G)  \text {length}(R)  1). \end{aligned}$$
 1.3.
All the leaftoroot dipaths of \(\mathcal {L}\) share the common suffix \(\hat{R}\), which contributes \(\text {length}(\hat{R}) + 1 = \text {length}(R) + 2\) vertices in the enumeration of V(G).
Hence, the three abovederived upper bounds yield that:$$\begin{aligned} V(G)\le & \, \frac{\mathcal {S}}{2} \text {dia}(G) + \mathcal {L}  \mathcal {S} (\text {dia}(G)  \text {length}(R)  1) \\ & \, + \text {length}(R) + 2. \end{aligned}$$  1.1.
 2.Lowerbounding E(G):
 2.1.
The vertexspanning rooted directed intree \(T_{r, \text {in}}\) of G contributes \(E(T_{r, \text {in}}) = V(T_{r, \text {in}})  1 = V(G)  1\) treeedges in the enumeration of E(G).
 2.2.
Every leaftoroot dipath P of \(\mathcal {L}\) satisfies the abovestated sharingcondition in the context of G: there exists a vertex u of P, within a directed distance of \(\text {length}(\hat{R})\) from \(\text {initial}(P)\), has \(\text {deg}_{G, \text {in}}(u) \ge 2\). The membership of a leaftoroot dipath P of \(\mathcal {S}\) demands the degreerequirement in the context of \(T_{r, \text {in}}\): \(\text {deg}_{T_{r, \text {in}}, \text {in}}(u) \ge 2\). Hence, every leaftoroot dipath \(P \in \mathcal {L}  \mathcal {S}\) must contain a vertex u (within a directed distance of \(\text {length}(\hat{R})\) from \(\text {initial}(P)\)) to which at least one directed nontreeedge of \(E(G)  E(T_{r, \text {in}})\) is incident/convergent. Therefore, a lower bound on the number of directed nontreeedges of \(E(G)  E(T_{r, \text {in}})\) incident to P (within a directed distance of \(\text {length}(\hat{R})\) from \(\text {initial}(P)\)) for all \(P \in \mathcal {L}  \mathcal {S}\) that are included in the enumeration of E(G) is \(\mathcal {L}  \mathcal {S}\).
 2.3.
The strong connectedness of G implies that for every leafvertex of \(T_{r, \text {in}}\) has its indegree of at least 1. Therefore, a lower bound on the number of directed nontreeedges incident to all leafvertices of \(T_{r, \text {in}}\) that are included in the enumeration of E(G) is \(\mathcal {L}\).
 2.4.
The strong connectedness of G also entails that for the rootvertex r of \(T_{r, \text {in}}\), there exists a directed nontreeedge incident from r to a vertex \(u \in V(G)\) (\(= V(T_{r, \text {in}})\)). Consider the two cases of the containment of \(\mathcal {S}\) in \(\mathcal {L}\) that decides if the directed nontreeedge (r, u) may contribute additionally to the enumeration of E(G).
Case when \(\mathcal {S} \subsetneq \mathcal {L}\): Since \(\mathcal {L}  \mathcal {S} \not = \emptyset\), the vertex u may appear in a leaftoroot dipath \(P \in \mathcal {L}  \mathcal {S}\) (within a directed distance of \(\text {length}(\hat{R})\) from \(\text {initial}(P)\)), and the directed nontreeedge (r, u) may have been included in the enumeration of E(G) above in item 2.2. In addition, if u is a leafvertex in \(T_{r, \text {in}}\), the directed nontreeedge (r, u) may have been accounted for in item 2.3.
Case when \(\mathcal {S} = \mathcal {L}\): The vertex u must appear in a leaftoroot dipath \(P \in \mathcal {S}\). We claim that there exists a directed nontreeedge that contributes additionally to the enumeration of E(G). Consider the two subcases of \(u \in V(P)\): (1) If u is not the initial leafvertex of P, then the directed nontreeedge (r, u) has not been included in the enumeration of E(G) above in item 2.3 nor item 2.2 (as \(\mathcal {L}  \mathcal {S} = \emptyset\)), or (2) If \(u = \text {initial}(P)\), then the maximality of \(\text {length}(R)\) imposed above implies that \(\text {deg}_{G, \text {in}}(u) \not = 1\) (\(> 1\), in fact, due to the directed nontreeedge (r, u) incident to u). Thus, there exists an additional directed nontreeedge incident to u that has not been included in the enumeration of E(G) above in item 2.3.
Hence, the four abovederived lower bounds yield that:$$\begin{aligned} E(G) \ge {\left\{ \begin{array}{ll} V(G) 1 + \mathcal {L}  \mathcal {S} + \mathcal {L} &{} \text {if }\mathcal {S} \subsetneq \mathcal {L}, \\ V(G) 1 + \mathcal {L}  \mathcal {S} + \mathcal {L} + 1 &{} \text {otherwise }(\mathcal {S} = \mathcal {L}). \end{array}\right. } \end{aligned}$$  2.1.
Otherwise, assume that \(1  \frac{2 (\text {length}(R) + 1)}{\text {dia}(G)} < 0\). We revise our approach above in upperbounding V(G) (items 1.2 and 1.3) and derive a lower bound on \(\mathcal {L}\) as follows.
Corollary 3
The family of strongly connected digraphs \(G_{k, \gamma }\) for all positive integers k and \(\gamma\) is optimal for the relationship of the graphparameters: order, size, and diameter established in Theorem 2.
Proof
Conclusion
Distributed function computation has a wide spectrum of major applications in distributed systems. There is a natural need to understand and approximate, if possible, lower and upper bounds on the number of timesteps for some or all vertices to receive (initial) vertexvalues of all vertices of the networkgraph, regardless of the underlying protocol or algorithmics.
The number of timesteps for a vertex u of a networkgraph G to collect values/information from all the vertices in \(\varGamma ^{*}_{G, \text {in}}(u)\) is lowerbounded by the ineccentricity of u, \(e_{G, \text {in}}(u)\), in G. In accordance with the abovestated \(\min\)\(\max\) distributed framework, we have proved that \(e_{G, \text {in}}(u)\) is the minimum of the maximum treedepth of a directed forest among all possible directed forests that are vertexdecomposition of \(\varGamma ^{*}_{G, \text {in}}(u)  \{u\}\) and rooted in (as subset of) \(\varGamma _{G, \text {in}}(u)\).
 1.
For a digraph G and a vertex \(u \in V(G)\), we have shown a lower bound on \(e_{G, \text {in}}(u)\) in terms of the graphorder and maximum indegree of G, and
 2.
For a strongly connected digraph G, we have proved a lower bound on \(\text {dia}(G)\) in terms of the graphorder and size of G, and have demonstrated the optimality of the diameterbound for a family of (explicitly constructed) strongly connected digraphs.
Toulouse and Minh [15] study the linear functional case with prescribed timeinvariant networktopology over random weightmatrices, and obtain various empirical upperbound results.
In accordance with an informationtheoretic framework, Xu and Raginsky [18, 19] study the fundamental timestep limits of distributed function computation in a constrained probabilistic setting. The lower and upperbound results are based on tradeoffs between: (1) the minimal amount of information necessarily extracted about the function value by any accuracy and confidenceconstrained algorithm, and (2) the maximal amount of information about the function value obtained by any algorithm within specified timestep and communication bounds. The lowerbound analysis indicates the dependence of computation timesteps on the diameter of the underlying networkgraph, while the upperbound one relies on cutsetcapacity arguments.
In addition, there have been several other recent theoretical developments in distributed computation and optimization. Olshevsky and Tsitsiklis [10] prove lower bounds on the worstcase convergence time for various classes of linear, timeinvariant (in networktopology), distributed consensus methods. Kuhn, Moscibroda, and Wattenhofer [8] study lower and upper bounds on local/distributed computability and approximability (amount of local information, approximation ratio, communication round) for a large class of optimization problems: minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching.
Notes
Funding
This study was not supported by any funding.
Compliance with ethical standards
Conflict of Interest
The authors declare that they have no conflict of interest.
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