METHOD FOR DISTRIBUTED MULTI-CHOICE VOTING/RANKING

Abstract

A method for distributed multi-choice voting/ranking in a network with a plurality of nodes associated to a set of choices is disclosed. The method includes setting a plurality of value sets for a plurality of nodes, setting a plurality of collections of memory sets for the plurality of nodes, and updating the plurality of value sets. In addition, the method includes updating the plurality of collections of memory sets, calculating a majority vote for the set of choices, and calculating a rank set for the set of choices.

Claims

1. A method for distributed multi-choice voting or ranking in a network with a plurality of nodes associated to a set of choices C {c.sub.1, . . . , c.sub.k, . . . , c.sub.n}, the method comprising: setting a plurality of value sets for the plurality of nodes at a moment t, including a first value set v.sub.i(t) for a first node i having a first size |v.sub.i(t)| and a second value set v.sub.j(t) for a second node j having a second size |v.sub.j(t)|; setting a plurality of collections of memory sets for the plurality of nodes at the moment t, including a first collection of memory sets m.sub.i,k(t) for the first node i and a second collection of memory sets m.sub.j,k(t) for the second node j; generating a first time t.sup.i for the first node i and a second time t.sup.j for the second node j; updating the first value set v.sub.i(t) at the first time t.sup.i to obtain an updated first value set v.sub.i(t.sup.i) with an updated first size |v.sub.i(t.sup.i)|, and updating the second value set v.sub.j(t) at the second time t.sup.j to obtain an updated second value set v.sub.j(t.sup.j) with an updated second size |v.sub.j(t.sup.j)|; updating the first collection of memory sets m.sub.i,k(t) at the time t.sup.i to obtain an updated first memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i), and updating the second collection of memory sets m.sub.i,k(t) at the time t.sup.j to obtain an updated second memory set m.sub.j,|Vj(t.sub.j.sub.)|(t.sup.j); calculating a majority vote π.sub.1 for the set of choices C; and calculating a rank set {π.sub.k} for the set of choices C.

2. The method of claim 1, wherein the moment t includes an initial moment t.sub.0 or a moment after the initial moment t.sub.0, and the first time t.sup.i and the second time t.sup.j are generated after the moment t.

3. The method of claim 2, wherein setting the plurality of value sets includes initializing the first value set v.sub.i(t) at the initial moment t.sub.0 with a first initial vote of the first node i, and initializing the second value set v.sub.j(t) at the initial moment t.sub.0 with a second initial vote of the second node j.

4. The method of claim 2, wherein setting the plurality of collections of memory sets includes initializing the first collection of memory sets m.sub.i,k(t) at the initial moment t.sub.0 with empty sets, and initializing the second collection of memory sets m.sub.j,k(t) at the initial moment t.sub.0 with empty sets.

5. The method of claim 1, wherein the second node j includes a random neighbor of the first node i.

6. The method of claim 1, wherein generating the first time t.sup.i and the second time t.sup.j, updating the first value set v.sub.i(t) and the second value set v.sub.j(t), updating the first collection of memory sets m.sub.i,k(t) and the second collection of memory sets m.sub.j,k(t), calculating the majority vote π.sub.1, and calculating the rank set {π.sub.k} are iterated until a convergence is achieved.

7. The method of claim 1, wherein the majority vote π.sub.1 includes the updated first memory set m.sub.i,|vi(t.sub.i.sub.)|(t.sup.i) when the updated first size |v.sub.i(t)| equals one.

8. The method of claim 1, wherein calculating the rank set {π.sub.k} includes operations defined by the following: $π_{k} = {\begin{matrix} m_{i, k} (t^{i}) .Math. \ .Math. m_{i, k - 1} (t^{i}), & if .Math. .Math. k > 1 \\ m_{i, k} (t^{i}), & if .Math. .Math. k = 1 \end{matrix}$ wherein k is an integer number that is associated to a choice c.sub.k from the set of choices C, m.sub.i,k(t.sup.i) is an updated first collection of memory sets, \ is the set-theoretic difference operator, and π.sub.k is a member of the rank set {π.sub.k} that shows the rank of the choice c.sub.k.

9. The method of claim 1, wherein updating the first value set v.sub.i(t), updating the second value set v.sub.j(t), updating the first collection of memory sets m.sub.i,k(t), and updating the second collection of memory sets m.sub.j,k(t) includes a procedure defined by the following: TABLE-US-00003 If |v.sub.i(t)| ≦ |v.sub.j(t)| then v.sub.i(t.sup.i) := v.sub.i(t) ∪ v.sub.j(t), v.sub.j(t.sup.j) := v.sub.i(t) ∩ v.sub.j(t) Else v.sub.i(t.sup.i) := v.sub.i(t) ∩ v.sub.j(t), v.sub.j(t.sup.j) := v.sub.i(t) ∪ v.sub.j(t) End if If v.sub.i(t.sup.i) ≠ ø then m.sub.i,|vi(t.sup.i.sub.)|(t.sup.i) := v.sub.i(t.sup.i), m.sub.j,|vj(t.sup.j.sub.)|(t.sup.j):= v.sub.j(t.sup.j) End if wherein v.sub.i(t)∪v.sub.j(t) is a union set that includes the union of the first value set v.sub.i(t) and the second value set v.sub.j(t), v.sub.i(t)∩v.sub.j(t) is an intersection set that includes the intersection of the first value set v.sub.i(t) and the second value set v.sub.j(t), the updated first memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i) is associated to the updated first size |v.sub.i(t.sup.i)|, and the updated second memory set m.sub.j,|Vj(t.sub.j.sub.)|(t.sup.j) is associated to the updated second size |v.sub.j(t.sup.j)|.

10. The method of claim 9, wherein the intersection set v.sub.i(t)∩v.sub.j(t) is replaced with the set of choices C if the intersection set v.sub.i(t)∩v.sub.j(t) is an empty set.

11. The method of claim 9, wherein the updated first memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i) is replaced with a set having one random element from the updated first value set v.sub.i(t.sup.i) if the updated first memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i) is not a subset of the updated first value set v.sub.i(t.sup.i).

12. The method of claim 9, wherein updating the first collection of memory sets m.sub.i,k(t) and updating the second collection of memory sets m.sub.j,k(t) further include a procedure defined by the following: TABLE-US-00004 If |v.sub.i(t.sup.i)| > 1 and |v.sub.j(t.sup.j)| > 1 then generate a random variable u from Bernoulli distribution with success probability 0.5 If u = 1 then m.sub.i,k(t.sup.i) := m.sub.j,k(t) Else m.sub.j,k(t.sup.j) := m.sub.i,k(t) End if End if wherein m.sub.i,k(t.sup.i) is an updated first collection of memory sets and m.sub.j,k(t.sup.j) is an updated second collection of memory sets.

13. A method for state-optimal ranking in a network with a plurality of nodes associated to a set of choices C {c.sub.1, . . . , c.sub.k, . . . , c.sub.n}, the method comprising: setting a plurality of ordered sets for the plurality of nodes at a moment t, including a first ordered set a.sub.i(t) for a first node i and a second ordered set a.sub.j(t) for a second node j; generating a first time t.sup.i for the first node i and a second time t.sup.j for the second node j; updating the first ordered set a.sub.i(t) at the first time t.sup.i and updating the second ordered set a.sub.j(t) at the second time t.sup.j according to a plurality of permutation sets, to obtain an updated first ordered set a.sub.i(t.sup.i) and an updated second ordered set a.sub.j(t.sup.j); and calculating a rank set {π.sub.k} for the set of choices C according to the updated first ordered set a.sub.i(t.sup.i) and the updated second ordered set a.sub.j(t.sup.j).

14. The method of claim 13, wherein the second node j includes a random neighbor of the first node i.

15. The method of claim 13, wherein the moment t includes an initial moment t.sub.0 or a moment after the initial moment t.sub.0, and the first time t.sup.i and the second time t.sup.j are generated after the moment t.

16. The method of claim 13, wherein generating the first time t.sup.i and the second time t.sup.j, updating the first ordered set a.sub.i(t), updating the second ordered set a.sub.j(t) and calculating the rank set {π.sub.k} are iterated until a convergence is achieved.

17. The method of claim 13, wherein the rank set {π.sub.k} includes the updated first ordered set a.sub.i(t.sup.i), or the updated second ordered set a.sub.j(t.sup.j).

18. The method of claim 13, wherein setting the plurality of ordered sets comprises: initializing the first ordered set a.sub.i(t)'s first entry at the initial moment t.sub.0 with the first node i's preferred element from the set of choices C, and initializing the first ordered set a.sub.i(t)'s other entries at the initial moment t.sub.0 with an arbitrary permutation of other elements that excludes the first node i's preferred element from the set of choices C; initializing the second ordered set a.sub.j(t)'s first entry with the second node j's preferred element from the set of choices C, and initializing the second ordered set a.sub.j(t)'s other entries at the initial moment t.sub.0 with an arbitrary permutation of other elements that excludes the second node j's preferred element from the set of choices C; initializing a first pointer p.sub.i(t) at the initial moment t.sub.0 to one, wherein the first pointer p.sub.i(t) includes an integer number; and initializing a second pointer p.sub.j(t) at the initial moment t.sub.0 to one, wherein the second pointer p.sub.j(t) includes an integer number that is equal or larger than the first pointer p.sub.i(t).

19. The method of claim 13, wherein updating the first ordered set a.sub.i(t) and updating the second ordered set a.sub.j(t) include operations defined by the following:
a.sub.i(t.sup.i):={Π.sub.1(t),Π.sub.2(t),Π.sub.3(t)},a.sub.j(t.sup.j):={Π.sub.4(t),Π.sub.5(t),Π.sub.6(t)}; wherein Π.sub.1(t) is a first permutation set that includes a permutation of an intersection set A.sub.1 that preserves the order of entries in accordance with the first ordered set a.sub.i(t), Π.sub.2(t) is a second permutation set that includes a permutation of the set-theoretic difference of a union set A.sub.2 and the intersection set A.sub.1 that preserves the order of entries in accordance with the first ordered set a.sub.i(t), Π.sub.3(t) is a third permutation set that includes a permutation of the set-theoretic difference of the set of choices C and the union set A.sub.2 that preserves the order of entries in accordance with the first ordered set a.sub.i(t), Π.sub.4(t) is a fourth permutation set that includes a permutation of the intersection set A.sub.1 that preserves the order of entries in accordance with the second ordered set a.sub.j(t), Π.sub.5(t) is a fifth permutation set that includes a permutation of the set-theoretic difference of the union set A.sub.2 and the intersection set A.sub.1 that preserves the order of entries in accordance with the second ordered set a.sub.j(t), and Π.sub.6(t) is a sixth permutation set that includes a permutation of the set-theoretic difference of the set of choices C and the union set A.sub.2 that preserves the order of entries in accordance with the second ordered set a.sub.j(t).

20. The method of claim 19, wherein calculating the intersection set A.sub.1 and the union set A.sub.2 includes operations defined by the following:
A.sub.1=a.sub.i.sup.1:p.sup.i.sup.(t)∩a.sub.j.sup.1:p.sup.j.sup.(t),A.sub.2=a.sub.i.sup.1:p.sup.i.sup.(t)∪a.sub.j.sup.1:p.sup.j.sup.(t) wherein a.sub.i.sup.1:p.sup.i.sup.(t) includes a permutation of the first ordered set a.sub.i(t) that comprises the first p.sub.i(t) entries of the first ordered set a.sub.i(t), and a.sub.j.sup.1:p.sup.j.sup.(t) includes a permutation of the second ordered set a.sup.j(t) that comprises the first p.sub.j(t) entries of the second ordered set a.sub.j(t).

21. The method of claim 19, wherein updating the first ordered set a.sub.i(t) and updating the second ordered set a.sub.j(t) further includes updating the first pointer p.sub.i(t) and the second pointer p.sub.j(t) according to operations defined by the following:
p.sub.i(t.sup.i)=|A.sub.2|,p.sub.j(t)=|A.sub.1|; wherein |A.sub.1| is the intersection set A.sub.1's size, |A.sub.2| is the union set A.sub.2's size, p.sub.i(t.sup.i) is an updated first pointer and p.sub.j(t.sup.j) is an updated second pointer.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The drawing figures depict one or more implementations in accord with the present teachings, by way of example only, not by way of limitation. In the figures, like reference numerals refer to the same or similar elements.

[0014] FIG. 1 is a flowchart illustrating an implementation of a method for distributed multi-choice voting/ranking in a network with a plurality of nodes;

[0015] FIG. 2 is a flowchart illustrating an implementation of a method for state-optimal ranking in a network with a plurality of nodes;

[0016] FIG. 3 is a graph illustrating average running times for implementations of three algorithms for binary voting in a complete graph with n=100 nodes;

[0017] FIG. 4A is a graph illustrating average running times for implementations of two algorithms for binary voting in a ring network;

[0018] FIG. 4B is a graph illustrating average running time for implementations of two algorithms for binary voting in a torus network;

[0019] FIG. 5 is a graph illustrating variations of average running times of implementations of two algorithms for ternary voting in a network with n=198 nodes; and

[0020] FIG. 6 is a flow chart depicting an implementation of the method for distributed multi-choice voting/ranking in a network with a plurality of nodes.

DETAILED DESCRIPTION

[0021] In the following detailed description, numerous specific details are set forth by way of examples in order to provide a thorough understanding of the relevant teachings. However, it should be apparent that the present teachings may be practiced without such details. In other instances, well known methods, procedures, components, and/or circuitry have been described at a relatively high-level, without detail, in order to avoid unnecessarily obscuring aspects of the present teachings.

[0022] A method for distributed multi-choice voting/ranking is disclosed. The disclosed distributed multi-choice voting/ranking method, also referred to herein as the DMVR method, calculates a majority vote and a rank set for a set of choices associated to a network with a plurality of nodes. The present disclosure further includes a design and simulation of iterative procedures for distributed multi-choice voting/ranking, which may include initializing and updating a plurality of value sets, collections of memory sets, and/or ordered sets. In some implementations, the majority vote and the rank set may be obtained after the iterative procedures reach a convergence.

[0023] For purposes of clarity, FIG. 1 provides a flowchart of an implementation of a method 100 for distributed multi-choice voting/ranking (DMVR) in a network with n nodes. In FIG. 1, the topology of the network is represented by a connected undirected graph where any two nodes of the network, for example, node i and node j, can communicate directly if there is an edge between them. In this example, the nodes are associated to a set of choices C with K elements (choices) that can be identified by values c.sub.1, . . . , c.sub.k, . . . , c.sub.K.

[0024] For purposes of this description, #c.sub.k can be understood to refer to the number of nodes that select the choice c.sub.k. The goal of the voting problem can include finding a choice c.sub.m, 1≦m≦K, that is in majority. In other words, ascertaining the choice c.sub.m satisfying #c.sub.m≧#c.sub.n, nε{1, . . . , K}. In a ranking problem, the desired output may include a permutation {π.sub.k} of C such that #π.sub.k≧#π.sub.k+1, ∀kε{1, . . . , K}.

[0025] Referring to the implementation shown in FIG. 1, the method 100 may include setting or establishing a plurality of value sets for the plurality of nodes at a moment t (a first step 102) such as for example, a first value set v.sub.i(t) for a first node i having a first size |v.sub.i(t)| and a second value set v.sub.j(t) for a second node j having a second size |v.sub.j(t)|. A second step 104 includes setting or establishing a plurality of collections of memory sets for the plurality of nodes at the moment t, such as for example, a first collection of memory sets m.sub.i,k(t) for the first node i and a second collection of memory sets m.sub.j,k(t) for the second node j. The method 100 also includes generating updated time instants for at least two of the plurality of nodes, such as for example, a time t.sup.i for node i and a time t.sup.j for node j (a third step 105). A fourth step 106 comprises updating at least two of the plurality of value sets at respective updated time instants, such as for example, updating the first value set v.sub.i(t) at time t.sup.i to obtain an updated first value set v.sub.i(t.sup.i) with an updated first size |v.sub.i(t.sup.i)|, and updating the second value set v.sub.j(t) at time t.sup.j to obtain an updated second value set v.sub.j(t) with an updated second size |v.sub.j(t.sup.j)|. The method further includes updating at least two of the plurality of collections of memory sets at respective updated time instants, such as for example, updating the first collection of memory sets m.sub.i,k(t) at time t.sup.i to obtain an updated first memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i), and updating the second collection of memory sets m.sub.i,k(t) at time t.sup.j to obtain an updated second memory set m.sub.j,|Vj(t.sub.j.sub.)|(t.sup.j) (a fifth step 108). In addition, a sixth step 110 involves calculating a majority vote π.sub.1 for the set of choices C, and calculating a rank set {π.sub.k} for the set of choices C.

[0026] In some implementations of first step 102, n value sets v.sub.i(t), . . . , v.sub.n(t) for the n nodes may be set at a moment t. For example, value set v.sub.i(t) for node i, and value set v.sub.j(t) for node j, may be set. Corresponding sizes for the n value sets, for example size |v.sub.i(t)| for node i and size |v.sub.j(t)| for node j may also be calculated. In different implementations of second step 104, n collections of memory sets, m.sub.1,k(t), . . . , m.sub.n,k(t), each associated to a choice k from the set of choices C, where 1≦k≦K, may be set for the n nodes at the moment t. For example, collection m.sub.i,k(t) may be set for node i, and collection m.sub.j,k(t) may be set for node j. Each memory set may be a subset of the set of choices C.

[0027] Furthermore, in some implementations, in third step 105 updated time instants may be generated after the moment t, and each value set may be updated at the respective updated time instant at fourth step 106, to obtain updated value sets. For example, value set v.sub.i(t) may be updated at time t.sup.i to obtain the updated value set v.sub.i(t.sup.i) with the updated size |v.sub.i(t.sup.i)|, and value set v.sub.j(t) may be updated at time t.sup.j to obtain the updated value set v.sub.j(t.sup.j).

[0028] In some implementations, during fifth step 108 each memory set from the collections of memory sets that is associated to a choice k equal to size of a corresponding value set may be updated. For example, at time t.sup.i, memory set m.sub.i,|vi(t.sub.i.sub.)|(t) from the collection m.sub.i,k(t), which is associated to the updated size |v.sub.i(t.sup.i)|, may be updated to obtain the updated memory set m.sub.i,|vi(t.sub.i.sub.)|(t.sup.i). Similarly, memory set m.sub.j,|vj(t.sub.j.sub.)|(t.sup.j) from collection m.sub.j,k(t), which is associated to the updated size |v.sub.j(t.sup.j)|, may be updated to obtain the updated memory set m.sub.j,|vj(t.sub.j.sub.)|(t.sup.j).

[0029] In sixth step 110, the method 100 may calculate a majority vote π.sub.1 and a rank set {π.sub.κ} for the set of choices C. The members of the set of choices C may be sorted in the rank set {π.sub.k} according to their votes in a descending order. In one implementation, the majority vote π.sub.1 may be the first member of the rank set {π.sub.k}.

[0030] In different implementations, the procedure of updating the value sets and the collections of memory sets may be expressed as follows:

TABLE-US-00001 If |v.sub.i(t)| ≦ |v.sub.j(t)| then v.sub.i(t.sup.i) := v.sub.i(t) ∪ v.sub.j(t), v.sub.j(t.sup.j) := v.sub.i(t) ∩ v.sub.j(t) Else v.sub.i(t.sup.i) := v.sub.i(t) ∩ v.sub.j(t), v.sub.j(t.sup.i) := v.sub.i(t) ∪ v.sub.j(t) End if If v.sub.i(t.sup.i) ≠ ø Then m.sub.i,|vi(t.sup.i.sub.)|(t.sup.i) := v.sub.i(t.sup.i), m.sub.j,|vj(t.sup.j.sub.)|(t.sup.j):= v.sub.j(t.sup.j) End if
where (1) the moment t may be an initial moment, for example t.sub.0=0, or a moment after the initial moment t.sub.0; (2) the time t.sup.i and the time t.sup.j are generated after the moment t; (3) the node j may be a random neighbor of the node i, that is nodes i and j may communicate directly; (4) v.sub.i(t)∪v.sub.j(t) is a union set, that is, the union of the value set v.sub.i(t) and the value set v.sub.j(t); (5) v.sub.i(t)∩v.sub.j(t) is an intersection set, that is, the intersection of the value set v.sub.i(t) and the value set v.sub.j(t); (6) the updated memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i) is associated to the updated size |v.sub.i(t.sup.i)|; and (7) the updated memory set m.sub.j,|Vj(t.sub.j.sub.)|(t.sup.j) is associated to the updated size |v.sub.j(t.sup.j)|. It should be understood that updating the memory sets may be ignored if the value set v.sub.i(t) is an empty set. The union and the intersection operations may be considered as a consolidating function that consolidates node choices across the network. Updating the memory sets may be understood to provide a disseminating function that disseminates the consolidated results throughout the network. The consolidating and disseminating functions may be executed in parallel.

[0031] In some implementations, at each iteration of the distributed multi-choice voting/ranking, the updated time instants such as time t.sup.i and time t.sup.j may be generated according to a random process, such as a Poisson process. In an example implementation, local clocks may be assigned to each of the plurality of nodes, and each clock may tick according to a random process at a respective updated time instant of the respective node.

[0032] In different implementations, generating time t.sup.i and the time t.sup.j (third step 105), updating the first value set v.sub.i(t) and the second value set v.sub.j(t) (fourth step 106), updating the first collection of memory sets m.sub.i,k(t) and the second collection of memory sets m.sub.i,k(t) (fifth step 108), and calculating the majority vote π.sub.1 and the rank set {π.sub.k} (sixth step 110) may be iterated until a convergence is achieved. Accordingly, the updated value sets and updated memory sets of the previous iteration replace the value sets and memory sets at the moment t for the current iteration to obtain new updated value sets and updated memory sets. This process may be iterated until the convergence is achieved.

[0033] In some implementations, the majority vote π.sub.1 and the rank set {π.sub.k} may be calculated by the following equation:

[00002] $\begin{matrix} π_{k} = {\begin{matrix} m_{i, k} (t^{i}) .Math. \ .Math. m_{i, k - 1} (t^{i}), & if .Math. .Math. k > 1 \\ m_{i, k} (t^{i}), & if .Math. .Math. k = 1 \end{matrix} & Equation .Math. .Math. (1) \end{matrix}$

where (1) k is an integer number that is associated to an element c.sub.k from the set of choices C, (2) m.sub.i,k(t.sup.i) is an updated collection at the time t.sup.i from the collection m.sub.i,k(t) at the moment t, (3) \ is the set-theoretic difference operator, and (4) a.sub.k is a member of the rank set {π.sub.k} that shows the rank of the choice c.sub.k. The majority vote π.sub.1 may be the first member of the rank {π.sub.k}, that is m.sub.i,1(t.sup.i), which may have the highest number of votes among the choices from the set of choices C. In some cases, updating memory sets in the distributed multi-choice voting/ranking may be limited to memory sets that are associated to value set sizes, for example the memory set m.sub.i,|vi(t.sub.i.sub.)|(t.sup.i) that is associated to the updated size |v.sub.i(t)|. In these types of cases, any updated memory set that is associated to a value set size equal to one, that is the corresponding value set having only one element, may be selected as the majority vote π.sub.1. In other words, the operations of Equation (1) may be performed on any node in the network.

[0034] In some implementations, prior to the first step 102 of the distributed multi-choice voting/ranking, the value sets and memory sets may be initialized at the initial moment t.sub.0. Each value set may be initialized with an initial vote, that is, a preferred choice from the set of choices C that is associated to the corresponding node of each value set. For example, the set v.sub.i(t) may be initialized with a first initial vote of node i at the moment t=t.sub.0, and the set v.sub.j(t) may be initialized with a second initial vote of the node j at the moment t=t.sub.0. Following initialization in the distributed multi-choice voting/ranking, the value sets may remain subsets of the set of choices C.

[0035] In one implementation, for initialization of the memory sets, each collection of memory sets may be initialized with empty sets at the initial moment. For example, the collection m.sub.i,k(t) and collection m.sub.j,k(t) may be initialized with empty sets at the moment t=t.sub.0.

[0036] For each node of the network in the distributed multi-choice voting/ranking, a state can be defined as consisting of a pair of a collection of memory sets and a value set. For example, for the case of the majority vote in, the pair of the collection m.sub.i,1(t) and the value set v.sub.i(t), that is (m.sub.i,1(t), v.sub.i(t)) may be the state of the node i, leading to the total number of K×2.sup.K sates.

[0037] Furthermore, the number of states may be reduced by adding the following rules to the distributed multi-choice voting/ranking at each iteration: [0038] If the intersection set v.sub.i(t)∩v.sub.j(t) is an empty set, it can be replaced with the set of choices C; and [0039] If the updated memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i) is not a subset of the updated value set v.sub.i(t.sup.i), the updated memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i) can be replaced with a set having one random element from the updated first value set v.sub.i(t.sup.i).

[0040] Updating the collections of memory sets may be further modified to increase the execution speed of the distributed multi-choice voting/ranking. For example, the following procedure may be used:

TABLE-US-00002 If |v.sub.i(t.sup.i)| > 1 and |v.sub.j(t.sup.j)| > 1 then Generate a random variable u from Bernoulli distribution with success probability 0.5; If u = 1 then m.sub.i,k(t.sup.i) := m.sub.j,k(t); Else m.sub.j,k(t.sup.j) := m.sub.i,k(t); End if; End if
where m.sub.i,k(t.sup.i) is an update for the collection m.sub.i,k(t) at the time t.sup.i, and m.sub.j,k(t.sup.j) is an update for the collection m.sub.j,k(t) at the time t.sup.j.

[0041] In some implementations, the disclosed methods can include provisions for obtaining an optimal number of states. For example, the distributed multi-choice voting/ranking can be modified to obtain a state-optimal ranking, which also may be referred to as enhanced DMVR, in a network with n nodes {1, . . . , i, . . . , j, . . . , n} associated to a set of choices C {c.sub.1, . . . , c.sub.k, . . . , c.sub.K}. In the state-optimal ranking, the value sets and the collections of memory sets may be replaced with ordered sets. For example, an ordered set a.sub.i(t)={a.sub.i.sup.1(t), . . . , a.sub.i.sup.K(t)} may be set at a moment t for a node i, and an ordered set a.sub.j(t)={a.sub.j.sup.1(t), . . . , a.sub.j.sup.K(t)} may be set at the moment t for a node j of the network. Similar to the distributed multi-choice voting/ranking, updated time instants may be generated for respective nodes. For example, a time t.sup.i may be generated for node i and a time t.sup.j may be generated for node j after the moment t in the state-optimal ranking. The ordered sets may be updated at the updated time instants to obtain updated ordered sets. For example, the ordered set a.sub.i(t) may be updated at time t.sup.i to obtain an updated ordered set a.sub.i(t.sup.i) for the node i, and the ordered set a.sub.j(t) may be updated at time t.sup.j to obtain an updated ordered set a.sub.j(t.sup.j) for the node j. Next, the method may calculate a majority vote π.sub.1 and a rank set {π.sub.κ} for the set of choices C. The members of the set of choices C may be sorted in the rank set {π.sub.k} according to their votes, in a descending order. The majority vote in may be the first member of the rank set {π.sub.k}.

[0042] In order to better illustrate this process, FIG. 2 presents a schematic of an enhanced method 200 for state-optimal ranking (enhanced DMVR) in a network with a plurality of nodes {1, . . . , i, . . . , j, . . . , n} associated to a set of choices C {c.sub.1, . . . , c.sub.k, . . . , c.sub.K}. In contrast to the method 100 described above, in the enhanced method 200 the value sets and memory sets are replaced with a plurality of ordered sets to minimize the number of states, that is, possible values of the pair of memory set and value set for each node.

[0043] The enhanced method 200 may include setting or establishing a plurality of ordered sets for the plurality of nodes at a moment t (a first step 202), for example, a first ordered set a.sub.i(t)={a.sub.i.sup.1(t), . . . , a.sub.i.sup.K (t)} for a first node i and a second ordered set a.sub.j(t)={a.sub.j.sup.1(t), . . . , a.sub.j.sup.K (t)} for a second node j. In addition, the enhanced method 200 includes generating updated time instants for at least two of the plurality of nodes, such as for example, a time t.sup.i for node i and a time t.sup.j for node j (a second step 204), and updating at least two of the plurality of ordered sets at respective updated time instants according to a plurality of permutation sets (a third step 206), such as for example, updating the first ordered set a.sub.i(t) at time t.sup.i to obtain the updated ordered set a.sub.i(t.sup.i) for node i, and updating the second ordered set a.sub.j(t) at time t.sup.j to obtain the updated second ordered set a.sub.i(t.sup.j) for node j. A fourth step 208 involves calculating the rank set {π.sub.k} for the set of choices C according to the updated first ordered set a.sub.i(t.sup.i) and the updated second ordered set a.sub.i(t).

[0044] In some implementations, updating the ordered sets may be performed through the following operations:

a.sub.i(t.sup.i):={Π.sub.1(t),Π.sub.2(t),Π.sub.3(t)},a.sub.j(t.sup.j):={Π.sub.4(t),Π.sub.5(t),Π.sub.6(t)};

where (1) the moment t includes an initial moment t.sub.0 or a moment after the initial moment t.sub.0; (2) the time t.sup.i and the time t.sup.j are generated after the first moment t; (3) node j may be a random neighbor of node i, that is nodes i and j may communicate directly; (4) Π.sub.1(t) is a permutation of an intersection set A.sub.1 that preserves the order of entries in accordance with the ordered set a.sub.i(t); (5) Π.sub.2(t) is a permutation of the set-theoretic difference of A.sub.2 and A.sub.1, that is, A.sub.2\A.sub.1, which preserves the order of entries in accordance with a.sub.i(t); (6) Π.sub.3(t) is a permutation of the set-theoretic difference of the set of choices C and A.sub.2, that is, C\A.sub.2, which preserves the order of entries in accordance with a.sub.i(t); (7) Π.sub.4(t) is a permutation of A.sub.1 that preserves the order of entries in accordance with the ordered set a.sub.j(t); (8) Π.sub.5(t) is a permutation of A.sub.2\A.sub.1 that preserves the order of entries in accordance with a.sub.j(t); and (9) Π.sub.6(t) is a permutation of C\A.sub.2 that preserves the order of entries in accordance with a.sub.j(t).

[0045] In addition, in one implementation, the intersection set A.sub.1 and the union set A.sub.2 may be expressed by the following Equation:

A.sub.1=a.sub.i.sup.1:p.sup.i.sup.(t)∩a.sub.j.sup.1:p.sup.j.sup.(t),A.sub.2=a.sub.i.sup.1:p.sup.i.sup.(t)∪a.sub.j.sup.1:p.sup.j.sup.(t) Equation (2)

where (1) p.sub.i(t) is an integer number that can be perceived as a pointer to an entry of a.sub.i(t); (2) p.sub.j(t) is an integer number that can be perceived as a pointer to an entry of a.sub.j(t); (3) a.sub.i.sup.1:p.sup.i.sup.(t) is a permutation of a.sub.i(t) that may consist of the first p.sub.i(t) entries of a.sub.i(t), and (4) a.sub.j.sup.1:p.sup.j.sup.(t) is a permutation of a.sub.j(t) that may consist of the first p.sub.j(t) entries of a.sub.j(t). It may be assumed in equation (1) and equation (2), without a loss of generality, that p.sub.j(t)≧p.sub.i(t). Furthermore, in some implementations, the pointers may be updated as expressed below:

p.sub.i(t.sup.i):=|A.sub.2|,p.sub.j(t.sup.j):=|A.sub.1|

where (1) |A.sub.1| is the intersection set A.sub.1's size; (2) |A.sub.2| is the of the union set A.sub.2's size; (3) p.sub.i(t.sup.i) is the updated pointer of node i at time t.sup.i; and (4) p.sub.j(t.sup.j) is the updated pointer of node j at t.sup.j.

[0046] In some implementations, at each iteration of the state-optimal ranking, the updated time instants such as time t.sup.i and time t.sup.j may be generated according to a random process, such as a Poisson process. In an example implementation, local clocks may be assigned to each of the plurality of nodes, and each clock may tick according to a random process at a respective updated time instant of the respective node.

[0047] In some implementations, generating time t.sup.i and time t.sup.j (second step 204), updating the first ordered set a.sub.i(t) and updating the second ordered set a.sub.j(t) (third step 206), and calculating the rank set {π.sub.k} (fourth step 208) may be iterated or repeated until a convergence is achieved. Accordingly, the updated ordered sets of the previous iteration replace the ordered sets at the moment t for the current iteration to obtain new updated ordered sets. This process may be iterated until the convergence is achieved.

[0048] In some implementations, any ordered set that is associated with an arbitrary node in the network, for example the updated ordered set a.sub.i(t.sup.i) associated to the node i, or the updated ordered set a.sub.j(t.sup.j) associated to the node j, may be selected as the rank set {π.sub.k}. In one implementation, the majority vote π.sub.1 may be the first member of the rank {.sub.k}, having the highest number of votes among the choices from the set of choices C.

[0049] In addition, in some cases prior to the first iteration of the state-optimal ranking, the ordered sets and the pointers may be initialized at the initial moment t.sub.0. The first entry of each ordered set may be initialized with a preferred choice from the set of choices C that is associated to the corresponding node of each ordered set. For example, the first entry of a.sub.i(t) may be initialized with a preferred choice of the node i at the moment t=t.sub.0, and the first entry of a.sub.j(t) may be initialized with a preferred choice of the node j at the moment t=t.sub.0. Other entries of each ordered set may be initialized with an arbitrary permutation of other elements of C, excluding the preferred choice of the corresponding node of each ordered set. As one example, a.sub.i(t)'s other entries may be initialized with an arbitrary permutation of other elements of C that excludes the node i's preferred element, and a.sub.j(t)'s other entries may be initialized with an arbitrary permutation of other elements of C that excludes the node j's preferred element. Also, each pointer, for example p.sub.i(t) or p.sub.j(t), may be initialized to 1 at the moment t=t.sub.0.

Example 1: Simulations of DMVR and Enhanced DMVR for Binary Voting/Ranking

[0050] In the following example, DMVR and enhanced DMVR are evaluated through simulations and compared with the Pairwise Asynchronous Graph Automata (PAGA) algorithm for binary voting/ranking in networks with n=100 nodes. Each point in simulations is averaged over 1000 runs.

[0051] Referring to FIG. 3, the average running time E{τ} of DMVR, enhanced DMVR, and PAGA versus the percentage ρ.sub.1 of a vote c.sub.1 in a complete graph with n nodes is shown, where ρ.sub.1=# c.sub.1/n. As seen in FIG. 3, the performance of DMVR and PAGA are very close to each other, in some cases overlapping. However, it is important to note that the enhanced version of DMVR algorithm clearly outperforms the two other algorithms as ρ.sub.1 gets close to 0.5.

[0052] FIGS. 4A and 4B further illustrate results for this example. FIG. 4A presents the average running time E{τ} of enhanced DMVR and PAGA versus ρ.sub.1 in a ring network, while FIG. 4B shows the average running time E{τ} of enhanced DMVR and PAGA versus ρ.sub.1 in a torus network. As noted above with respect to FIG. 3, the enhanced DMVR algorithm provides a better or faster performance than the PAGA algorithm. In both the ring network and the torus network, it can be seen that as ρ.sub.1 approaches 0.5, the E{τ} of enhanced DMVR is significantly less than the E{τ} of PAGA.

Example 2: Simulations of Enhanced DMVR for Ternary Voting/Ranking

[0053] In the following example, enhanced DMVR is evaluated through simulations and compared PAGA for ternary voting/ranking in a network with n=198 nodes. Each point in the simulations is averaged over 1000 runs. The percentage of initial votes is considered in the form of [ρ.sub.1, ρ.sub.2, ρ.sub.3]=[1/3+δ, 1/3, 1/3−δ] where 0<δ<1/3.

[0054] In the graph of FIG. 5, the average running time E{τ} of enhanced DMVR and PAGA for δε[0.005, 0.041] is presented. It can be seen that, once again, enhanced DMVR outperforms PAGA for small δ, such that the E{τ} of enhanced DMVR is significantly less than the E{τ} of PAGA as δ approaches 0.005.

[0055] For purposes of clarity, FIG. 6 presents a flow chart illustrating an implementation of the methods described herein. In FIG. 6, a method for distributed multi-choice voting/ranking in a network with a plurality of nodes {1, . . . , i, . . . , j, . . . , n} associated to a set of choices C {c.sub.1, . . . , c.sub.k, . . . , c.sub.K} is provided. For example, a first step 610 can comprise setting or establishing a plurality of value sets and memory sets at time t (including an initial moment t.sub.0 for initialization). In other words, first step 610 can involve setting or establishing a plurality of value sets for the plurality of nodes at a moment t, including a first value set v.sub.i(t) for a first node i having a first size |v.sub.i(t)| and a second value set v.sub.j(t) for a second node j having a second size |v.sub.j(t)|, as well as setting or establishing a plurality of collections of memory sets for the plurality of nodes at the moment t, including a first collection of memory sets m.sub.i,k(t) for the first node i and a second collection of memory sets m.sub.j,k(t) for the second node j.

[0056] After generation of updated times for each node, for example time t.sup.i for node i and time t.sup.j for node j, in a second step 620, the value sets and memory sets are updated at the updated times through operations that include union and intersection of the value sets of two neighboring nodes in the network. For example, third step 630 can include updating the first value set v.sub.i(t) at the time t.sup.i to obtain an updated first value set v.sub.i(t.sup.i) with an updated first size |v.sub.i(t.sup.i)|, updating the second value set v.sub.j(t) at the time t to obtain an updated second value set v.sub.j(t.sup.j) with an updated second size |v.sub.j(t.sup.j)|, updating the first collection of memory sets m.sub.i,k(t) at the time t.sup.i to obtain an updated first memory set m.sub.i,|Vi(t.sub.i.sub.)|(t.sup.i), and updating the second collection of memory sets m.sub.j,k(t) at the time t.sup.j to obtain an updated second memory set m.sub.j,|Vj(t.sub.j.sub.)|(t.sup.j). Finally, fourth step 640 can comprise calculating the majority vote and rank set from the memory sets. Thus, a majority vote π.sub.1 for the set of choices C can be calculated, and a rank set {π.sub.k} for the set of choices C can also be calculated.

[0057] The disclosed methods for distributed voting/ranking can provide a more effective approach in distributed function computation. The methods described herein provide for scalability of the voting/ranking method for any number of choices. In addition, the methods can be utilized to enhance the execution speed of distributed voting/ranking, as well as facilitate independence of the voting/ranking method from the network size, that is, the number of nodes in the network.

[0058] While the foregoing has described what are considered to be the best mode and/or other examples, it is understood that various modifications may be made therein and that the subject matter disclosed herein may be implemented in various forms and examples, and that the teachings may be applied in numerous applications, only some of which have been described herein. It is intended by the following claims to claim any and all applications, modifications and variations that fall within the true scope of the present teachings.

[0059] Unless otherwise stated, all measurements, values, ratings, positions, magnitudes, sizes, and other specifications that are set forth in this specification, including in the claims that follow, are approximate, not exact. They are intended to have a reasonable range that is consistent with the functions to which they relate and with what is customary in the art to which they pertain.

[0060] The scope of protection is limited solely by the claims that now follow. That scope is intended and should be interpreted to be as broad as is consistent with the ordinary meaning of the language that is used in the claims when interpreted in light of this specification and the prosecution history that follows and to encompass all structural and functional equivalents. Notwithstanding, none of the claims are intended to embrace subject matter that fails to satisfy the requirement of Sections 101, 102, or 103 of the Patent Act, nor should they be interpreted in such a way. Any unintended embracement of such subject matter is hereby disclaimed.

[0061] Except as stated immediately above, nothing that has been stated or illustrated is intended or should be interpreted to cause a dedication of any component, step, feature, object, benefit, advantage, or equivalent to the public, regardless of whether it is or is not recited in the claims.

[0062] It will be understood that the terms and expressions used herein have the ordinary meaning as is accorded to such terms and expressions with respect to their corresponding respective areas of inquiry and study except where specific meanings have otherwise been set forth herein. Relational terms such as first and second and the like may be used solely to distinguish one entity or action from another without necessarily requiring or implying any actual such relationship or order between such entities or actions. The terms “comprises,” “comprising,” or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. An element proceeded by “a” or “an” does not, without further constraints, preclude the existence of additional identical elements in the process, method, article, or apparatus that comprises the element.

[0063] The Abstract of the Disclosure is provided to allow the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in various implementations. This is for purposes of streamlining the disclosure, and is not to be interpreted as reflecting an intention that the claimed implementations require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed implementation. Thus, the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separately claimed subject matter.

[0064] While various implementations have been described, the description is intended to be exemplary, rather than limiting and it will be apparent to those of ordinary skill in the art that many more implementations and implementations are possible that are within the scope of the implementations. Although many possible combinations of features are shown in the accompanying figures and discussed in this detailed description, many other combinations of the disclosed features are possible. Any feature of any implementation may be used in combination with or substituted for any other feature or element in any other implementation unless specifically restricted. Therefore, it will be understood that any of the features shown and/or discussed in the present disclosure may be implemented together in any suitable combination. Accordingly, the implementations are not to be restricted except in light of the attached claims and their equivalents. Also, various modifications and changes may be made within the scope of the attached claims.

METHOD FOR DISTRIBUTED MULTI-CHOICE VOTING/RANKING

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G06F16/27

PHYSICS

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G06F16/951

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H04L45/46

ELECTRICITY

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H04L45/02

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International classification

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G06F17/30

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H04L12/751

ELECTRICITY

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H04L12/715

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Abstract

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Description