CN105978711A - Best switching edge searching method based on minimum spanning tree - Google Patents

Best switching edge searching method based on minimum spanning tree Download PDF

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CN105978711A
CN105978711A CN201610286714.2A CN201610286714A CN105978711A CN 105978711 A CN105978711 A CN 105978711A CN 201610286714 A CN201610286714 A CN 201610286714A CN 105978711 A CN105978711 A CN 105978711A
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node
limit
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spanning tree
exchange
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CN105978711B (en
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王宇虹
陈志�
岳文静
陈雨诗
卜杰
田思明
龚凯
杨天明
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Nanjing Post and Telecommunication University
Nanjing University of Posts and Telecommunications
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/14—Network analysis or design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F16/00—Information retrieval; Database structures therefor; File system structures therefor
    • G06F16/20—Information retrieval; Database structures therefor; File system structures therefor of structured data, e.g. relational data
    • G06F16/22—Indexing; Data structures therefor; Storage structures
    • G06F16/2228—Indexing structures
    • G06F16/2246—Trees, e.g. B+trees
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/14—Network analysis or design
    • H04L41/145—Network analysis or design involving simulating, designing, planning or modelling of a network
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
    • H04L41/50—Network service management, e.g. ensuring proper service fulfilment according to agreements
    • H04L41/508—Network service management, e.g. ensuring proper service fulfilment according to agreements based on type of value added network service under agreement
    • H04L41/5096—Network service management, e.g. ensuring proper service fulfilment according to agreements based on type of value added network service under agreement wherein the managed service relates to distributed or central networked applications

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  • General Engineering & Computer Science (AREA)
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  • Information Retrieval, Db Structures And Fs Structures Therefor (AREA)

Abstract

The invention discloses a best switching edge searching method based on a minimum spanning tree. According to the method of the invention, the best switching edge searching problem is defined as a graph model, the best switching edge corresponding to a failure edge is solved from a global perspective, and feasible solution space is acquired through a strategy such as a distributed algorithm. The scheme of solving searching of the best switching edge corresponding to the failure edge in the graph model in the global condition can be formed, the problem of solving the best switching edge in the graph model is optimized during the solving process in aspects of time and space complexity, and premature convergence can be avoided. According to the to-be-solved best switching edge searching problem of the invention, a communication network is given, a certain edge in the minimum spanning tree in the network fails to result in temporary communication failure, the distributed algorithm is used to search the best switching edge in the network to replace the failure edge, the communication is kept as smooth as possible, and purposes such as minimum communication network recovery loss and minimum spanning tree diameter as small as possible can be achieved.

Description

A kind of optimal exchange limit based on minimum spanning tree lookup method
Technical field
The present invention relates to the optimal method for solving exchanging limit in communication network minimum spanning tree, mainly utilize distributed algorithm From the overall situation angle solve inefficacy limit graph model corresponding most preferably exchange limit, belong to computer technology, information technology, figure excavate, Distributed treatment interleaving techniques field.
Background technology
Distributed system refers to all computer utilitys on some computers of cooperating in some way or processor.Point Cloth processes different location, or has difference in functionality, or the multiple stage computer having different pieces of information passes through communication network Network couples together, and under the unified management of control system controls, completes the department of computer science of extensive information handling task in phase System.Distributed treatment utilizes network technology many minicomputers or microcomputer can be connected into and have high performance computer system, makes It has the ability solving challenge.
Distributed algorithm, it is simply that refer to when completing multiply-add function by the computing that each for each input data corresponding position is produced Result carries out being added the corresponding partial product of formation in advance, and each several part carries out cumulative formation final result the most again.Distributed calculation The research of method, derives from the basic research in Development of Distributed System activity, its Composition of contents core skill of Distributed Calculation Art.
Summary of the invention
Technical problem: the invention solves the problems that the optimal exchange limit in minimum spanning tree searches problem, this problem refers to give One communication network, i.e. graph model, and it is scheduled on the minimum spanning tree in this graph model and an inefficacy limit, from this graph model Select an optimal exchange limit, be used for replacing inefficacy limit so that the communication failure that inefficacy limit is caused recovers as early as possible, and make new The minimum spanning tree diameter produced is minimum.
Technical scheme: the optimal exchange limit in described minimum spanning tree searches problem and is described as follows: set a given communication Network, then give an inefficacy limit, it is optimal that optimal exchange limit based on minimum spanning tree lookup method finds in this graph model Exchange limit, in order to replace this inefficacy limit so that communication network resume operation as early as possible and newly generated minimum spanning tree diameter Little.It is assumed that this communication network is organized in graph model G=(V, E), this communication network is a weighted-graph.Each joint Point may be connected with multiple or 0 other node, and node is connected with node and constitutes a limit, and each edge is regarded as to be checked right As.Our target is to find an optimal exchange limit from graph model G=(V, E), is used for substituting inefficacy limit so that newly generated Minimum spanning tree diameter minimum.
Optimal exchange limit based on minimum spanning tree of the present invention lookup method is by the optimal exchange in communication network Limit inquiry problem definition becomes graph model, and uses distributed algorithm to obtain solution space.
Optimal exchange limit based on minimum spanning tree of the present invention querying method comprises the following steps:
Step 1) according to the information of user's input, build optimal exchange limit inquiry problem in network graph model G=(V, E), described V is set of node, and E is limit collection, and n=| V |, m=| E |, n refer to the nodes in this graph model, and m refers to this graph model G Limit number in=(V, E).Specifically comprise the following steps that
Step 11) user input comprise a group node collection, limit collection, minimum spanning tree and an inefficacy limit.Described user The set of node of input is designated as V={d1,d2,d3,…,dn, limit collection is designated as E={e1,e2,e3,…,em, n=| V |, m=| E |, often Corresponding non-negative length l (x) of bar limit e, path Ρ=< d1,d2,d3,…,dr> represent by node d1,d2,d3,…,drConstitute One paths, | Ρ | represents the length of this paths, and the length sum on the most all limits, minimum spanning tree is designated as T.Described n refers to Nodes in set of node；Described m refers to the limit number concentrated on limit.
Step 12) by set of node V={d1,d2,d3,…,dnJoint during all nodes regard graph model G=(V, E) as in } Point.
Step 13) regard the path between node u and node v as arc between graph model G=(V, E) two node, two save (u, v) as the weights of arc between node u and node v, (u v) is joint in graph model G=(V, E) to described d to distance d between point Point u and the weights of node v shortest path.
Step 14) given minimum spanning tree T=(V, E (T)), define D=< d1,d2,d3,…,dk>, | D | generates for minimum The diameter of tree T=(V, E (T)), i.e. longest path length in T.The Centroid d of definition Dc, define DL=< d1,d2,d3…dc >, define DR=< dc,dc+1,dc+2…dk>, above-mentioned node meets: | DL|≥|DR|.Described D refers in minimum spanning tree by saving Point d1,d2,d3,…,dkThe paths constituted, described DLRefer in minimum spanning tree by node d1,d2,d3,…,dcStructure one Paths, described DRRefer in minimum spanning tree by node dc,dc+1,dc+2,…,dkStructure one paths, described | DL| it is to show the way Footpath DLDiameter, described | DR| refer to path DRDiameter.
Step 15) make dcAs the center of minimum spanning tree T=(V, E (T)), for each node x ∈ V and x ≠ dc, fixed Father node p (x) of justice node x, child nodes collection C (x) of definition node x.Definition Tx=(V (Tx),E(Tx)) represent with node x A stalk tree for the minimum spanning tree T of root node.Definition VLRepresent with node dc-1For the subtree of root node, define VRRepresent with Node dc+1For the subtree of root node, define VcRepresent except VL、VROther all nodes outward.
Step 16) after user inputs inefficacy limit e '=(x ', p (x ')), minimum spanning tree T=(V, E (T)) divide into two Stalk tree Tx′With T Tx′, described T Tx′Do not include node x '.Described T Tx′Refer to by set of node V (T)/V (Tx′) and limit collection E (T)/E(Tx′)/{ (x ', p (x ')) } figure that constitutes.Our task is to find a limit for inefficacy limit e '=(x ', p (x ')) F, in order to substitute this inefficacy limit.Described f be E in E (T) any one can make two stalk tree Tx′With T Tx′It is reconnected in one The limit risen, referred to as exchanges limit.Definition S (e ') represents the set that exchange limit corresponding for limit e ' is constituted.For inefficacy limit e '= For (x ', p (x ')), optimal exchange limit f ' ∈ S (e '), newly generated minimum generation can be made after referring to add this exchange limit Tree | D (Te′/f′) | minimum.Described e '=(x ', f (x ')) represents given inefficacy limit, and f '=(z, z ') represents an exchange Limit.Described z refers at TxIn a node；Described z ' refer to T TxIn a node.
Step 2) use distributed algorithm, it is thus achieved that the optimal exchange limit in communication network search problem graph model G=(V, E) solution space on.Specifically comprise the following steps that
Step 21) for given inefficacy limit e '=(x ', f (x ')), traversal subtree T with x ' as rootx′In each joint Point z.
Step 22) wait the available information from father node, calculate | L (Tx′, z) |, it is that all child nodes calculate simultaneously Available information also sends this information.Described | L (Tx′, z) |, that refer to start with node z and at figure Tx′In the length of longest path Degree.
Step 23) to each local exchange limit f=(z, z '), right | L (T Tx′, z ') | carry out Distributed Calculation.Described Exchange limit, ground, refers in limit collection S (e ') and a limit of adjacent node z；Described | L (T Tx′, z ') |, refer to open with node z ' Begin and figure T Tx′In the length of longest path.Wherein,
|L(T\Tx′, z ') |=max{d (z ', dk),d(z′,dc)+γ,d(z′,dc)+λ(p(x′)),
d(z′,u′)+d(u′,ρR)+μR-d(dk,ρR)}
Described d (z ', dk) refer to node z ' and node dkA limit for end points；Described d (z ', dc) refer to node Z ' and node dcA limit for end points；Described d (z ', u ') refers to node z ' and a node u ' limit as end points；Described d (u′,ρR) refer to node u ' and node ρRA limit for end points；Described d (dk,ρR) refer to node dkWith node ρRFor end One limit of point；Described λ (p (x ')) refers to node dcStart, enter into vertex set VLIn but without going past under node p (x ') The length of the longest path of side；Described u ' refers on D (T) with the joint nearest for distance z ' on the longest path of node z ' beginning Point；Described ρRRefer to node dkStart and at set of node VRIn longest path on, enter into VRRoot joint in subtree Point；Described γ refers to node dcStart and only comprise set of node VCThe length of the longest path of interior joint；Described VCRefer to except joint Point dcThe node set that outer all nodes are constituted；Described μRRefer to node dkStart and at set of node VRIn longest path Length.
Step 24) to each local exchange limit f=(z, z '), right | D (Te′/f) | carry out Distributed Calculation: | D (Te′/f) |= max{|D(T)|,|L(Tx′,z)|+l(f)+|L(T\Tx′,z′)|}.Described L (Tx′, that z) refer to start with node z and at figure Tx In the length of longest path；Described L (T Tx', z ') refer to node z ' start and figure T TxIn the length of longest path Degree；Described l (f) refers to the length on local exchange limit f=(z, z ').
Step 25) from these local exchange limits above-mentioned, select one most preferably to exchange limit temporarilyAnd store addition should The new di that will produce behind exchange limitExist without local exchange limit, then create an illusory candidate Limit, and its new di is set to ∞.
Step 26) for each child nodes q ∈ C (z), the range information tried to achieve according to above-mentioned steps, each child is saved Point selection one optimal exchange limit candidateAnd the new di that will produce after storing this exchange limit of additionAgain from this The optimal exchange limit that a little child nodes are corresponding selects second and most preferably exchanges limit temporarilyDescribed
Step 27) definition target most preferably exchange limit fbest.Relatively first most preferably exchanges limit temporarilyOptimal with second Exchange limitSelect to add that exchange limit that corresponding new di is less.IfThen willAs Target most preferably exchanges limit fbest；IfThen willLimit f is most preferably exchanged as targetbest；IfThen willOrLimit f is most preferably exchanged as targetbest?.
Step 28) if subtree T with x ' as rootx′In each node z all do not traveled through, repeat step 22)～step 27)。
Step 29) determine last solution space Solution, this solution space comprises inefficacy limit e '=(x ', f (x ')) correspondence Most preferably exchange limit.
Beneficial effect: the present invention utilizes distributed algorithm to form efficient optimal exchange limit lookup method.Concrete embodiment is such as Lower beneficial effect:
1) present invention provides a kind of optimal exchange limit based on minimum spanning tree lookup method, its complete procedure bag Include and the optimal exchange limit inquiry problem definition in communication network is become graph model, and use distributed algorithm to obtain solution space.
2) in heretofore described modeling process, it is provided that one or the most abstract a set of graph model, it is possible to ask actual Relevant method for solving in topic is converted into the model form of mathematicization.
3) heretofore described model searches optimal exchange limit method from the angle of the overall situation, makes most preferably to exchange limit and searches problem Finally can obtain optimum accurately solution.
4) present invention uses nodal information to prestore and distributed algorithm, thus it is complicated and empty effectively to reduce the algorithm time Between complexity.
Accompanying drawing explanation
Fig. 1 is the flow chart that optimal exchange limit based on minimum spanning tree lookup method is corresponding.
Fig. 2 is the exemplary plot that minimum spanning tree model is corresponding.
Fig. 3 is that example is searched on most preferably exchange limit.
Detailed description of the invention
Below some embodiment of accompanying drawing of the present invention is described in more detail.
Shown in optimal exchange limit based on minimum spanning tree lookup method shown in 1 is corresponding with reference to the accompanying drawings flow chart, Fig. 2 The exemplary plot that little generation tree-model is corresponding, the specific embodiment of the invention is:
1). the optimal exchange limit in communication network is searched problem and is defined as graph model.
11). user's input comprises a group node collection, limit collection, minimum spanning tree and an inefficacy limit.As it is shown on figure 3, joint Point set V={1,2,3,4,5,6}, node 1 is connected with node 2, its distance d (1,2)=1；Node 1 is connected with node 6, its distance D (1,6)=3；Node 2 is connected with node 3, its distance d (2,3)=1；Node 3 is connected with node 4, its distance d (3,4)=3； Node 4 is connected with node 5, its distance d (4,5)=2；Node 4 is connected with node 6, its distance d (4,6)=3；Node 5 and node 6 are connected, and its distance d (5,6)=2, one has 7 limits.Limit between all above node and node is connected, pie graph model G =(V, E).Wherein, minimum spanning tree T is that graph model G=(V, E) removes the limit between node 4 and node 6, node 5 and node 6 Between limit after the new figure that produces.Limit between its interior joint 2 and node 3 is inefficacy limit, and node 3 represents the lower end segment on inefficacy limit Point, node 2 represents the father node of node 3.
Inefficacy limit is denoted as e=(3,2).
12). regard nodes all in set of node V={1,2,3,4,5,6} as node in graph model G=(V, E).
Described attributed graph model G is after foundation, and the shortest path between any two summit has corresponding weights, represents two The adjacency on summit.
2). use distributed algorithm, it is thus achieved that the optimal exchange limit in communication network searches problem at graph model G=(V, E) On solution space.Specifically comprise the following steps that
21). for given inefficacy limit e=(3,2), subtree T with node 3 as root3By node 4, node 5 and node Limit between 4 and node 5 is constituted, subtree T3Have two nodes, be node 4 and node 5 respectively, node 4 is designated as z, will joint Point 5 is designated as q.To this subtree T3In node travel through.
22). first pass travels through, and operation object is node z.Now local exchange limit is f (4,6), right | L (T3, z) | carry out Distributed Calculation, obtains | L (T3, z) |=| L (T3, 4) |=3.Right | L (T T3, z ') | carry out Distributed Calculation, obtain | L (T T3, z ') |=| L (T T3, 6) |=max{0,3+1,3+7,3}=10.To local exchange limit f (4,6), right | D (Te/f) | carry out point Cloth calculates, and obtains | D (Te/f) |=max{10,3+3+10}=16.Record most preferably exchanges limit temporarilyAnd the new di that will produce after storing this exchange limit of addition
Next child nodes q ∈ C (z) of z, the now child nodes only one of which of node 4, i.e. node 5 are traveled through.Now Local exchange limit is f (5,6), right | L (T3, q) | carry out Distributed Calculation, obtain | L (T3, q) |=L (T3, 5) |=5.Right | L (T\T3, z ') | carry out Distributed Calculation, obtain | L (T T3, z ') |=| L (T T3, 6) |=max{0,3+1,3+7,3}=10. To local exchange limit f (5,6), right | D (Te/f) | carry out Distributed Calculation, obtain | D (Te/f) |=max{10,5+2+10}=17. Record second and most preferably exchange limit temporarilyAnd store and add will produce behind this exchange limit new Diameter
23). definition target most preferably exchanges limit fbest.Relatively first most preferably exchanges limit temporarilyWith second optimal friendship Change sidesSelect to add that limit that corresponding new di is less.CauseSo will Limit f is most preferably exchanged as targetbest, i.e. fbest=f (z, z ')=f (4,6).
24). second time traversal, operation object is node q, and now local exchange limit is f (5,6), right | L (T3, q) | carry out Distributed Calculation, obtains | L (T3, q) |=| L (T3, 5) |=5.Right | L (T T3, z ') | carry out Distributed Calculation, obtain | L (T T3, z ') |=| L (T T3, 6) |=max{0,3+1,3+7,3}=10.To local exchange limit f (5,6), right | D (Te/f) | carry out point Cloth calculates, and obtains | D (Te/f) |=max{10,5+2+10}=17.Record second and most preferably exchange limit temporarilyAnd the new di that will produce after storing this exchange limit of additionNode q does not has There is child nodes, possible most preferably exchanging limit so need not to search for from child nodes.Because local exchange limit is f (5,6) The diameter that corresponding diameter is corresponding more than f (4,6), so target most preferably exchanges limit fbestConstant, it is still f (4,6).
25). determine last solution space Solution, comprise inefficacy limit e=(3,2) correspondence in this solution space most preferably exchanges limit f(4,6)。

Claims (3)

1. optimal exchange limit based on a minimum spanning tree lookup method, it is characterised in that the method comprises the following steps:
Step 1) according to the information of user's input, build the graph model G=(V, E) of optimal exchange limit inquiry problem in network, Described V is set of node, and E is limit collection, and n=| V |, m=| E |, n refer to the nodes in this graph model, and m refers to this graph model G= Limit number in (V, E)；
Step 2) use distributed algorithm, it is thus achieved that and the optimal exchange limit in communication network searches problem on graph model G=(V, E) Solution space.
A kind of optimal exchange limit based on minimum spanning tree the most according to claim 1 lookup method, it is characterised in that institute State step 1) specifically comprise the following steps that
Step 11) user's input comprises a group node collection, limit collection, minimum spanning tree and an inefficacy limit, and described user inputs Set of node be designated as V={d1,d2,d3,…,dn, limit collection is designated as E={e1,e2,e3,…,em, n=| V |, m=| E |, each edge Corresponding non-negative length l (x) of e, path P=< d1,d2,d3,…,dr> represent by node d1,d2,d3,…,drOne constituted Path, | P | represents the length of this paths, the length sum on the most all limits, and minimum spanning tree is designated as T, described n and refers to set of node In nodes；Described m refers to the limit number concentrated on limit；
Step 12) by set of node V={d1,d2,d3,…,dnNode during all nodes regard graph model G=(V, E) as in }；
Step 13) path between node u and node v regarded as the arc between graph model G=(V, E) two node, two nodes it Between distance d (u, v) as the weights of arc between node u and node v, (u v) is graph model G=(V, E) interior joint u to described d Weights with node v shortest path；
Step 14) given minimum spanning tree T=(V, E (T)), define D=< d1,d2,d3,…,dk>, | D | is minimum spanning tree T= The diameter of (V, E (T)), i.e. longest path length in T, the Centroid d of definition Dc, define DL=< d1,d2,d3…dc>, fixed Justice DR=< dc,dc+1,dc+2…dk>, above-mentioned node meets: | DL|≥|DR|, described D refers in minimum spanning tree by node d1, d2,d3,…,dkThe paths constituted, described DLRefer in minimum spanning tree by node d1,d2,d3,…,dcGou Yitiao road Footpath, described DRRefer in minimum spanning tree by node dc,dc+1,dc+2,…,dkStructure one paths, described | DL| refer to path DL Diameter, described | DR| refer to path DRDiameter；
Step 15) make dcAs the center of minimum spanning tree T=(V, E (T)), for each node x ∈ V and x ≠ dc, definition joint Father node p (x) of some x, child nodes collection C (x) of definition node x, define Tx=(V (Tx),E(Tx)) represent with node x as root The one stalk tree of the minimum spanning tree T of node, defines VLRepresent with node dc-1For the subtree of root node, define VRRepresent with node dc+1For the subtree of root node, define VcRepresent except VL、VROther all nodes outward；
Step 16) after user inputs inefficacy limit e '=(x ', p (x ')), minimum spanning tree T=(V, E (T)) divide into two stalk Tree Tx′With T Tx′, described T Tx′Do not include node x ', described T Tx′Refer to by set of node V (T)/V (Tx′) and limit collection E (T)/E (Tx′)/{ (x ', p (x ')) } figure that constitutes, task is to find a limit f, in order to substitute for inefficacy limit e '=(x ', p (x ')) This inefficacy limit, described f be E in E (T) any one can make two stalk tree Tx′With T Tx′Reconnect limit together, claim For exchange limit, definition S (e ') represents the set that exchange limit corresponding for limit e ' is constituted, comes for inefficacy limit e '=(x ', p (x ')) Say that optimal exchange limit f ' ∈ S (e ') can make newly generated minimum spanning tree after referring to add this exchange limit | D (Te′/f′) | minimum, described e '=(x ', f (x ')) represents given inefficacy limit, and f '=(z, z ') represents an exchange limit, and described z refers to TxIn a node；Described z ' refer to T TxIn a node.
A kind of optimal exchange limit based on minimum spanning tree the most according to claim 1 lookup method, it is characterised in that institute State step 2) use distributed algorithm, it is thus achieved that and the optimal exchange limit in communication network searches problem on graph model G=(V, E) Solution space, specifically comprises the following steps that
Step 21) for given inefficacy limit e '=(x ', f (x ')), traversal subtree T with x ' as rootx′In each node z；
Step 22) wait the available information from father node, calculate | L (Tx′, z) |, it is that the calculating of all child nodes is available simultaneously Information also sends this information, described | L (Tx′, z) |, that refer to start with node z and at figure Tx′In the length of longest path；
Step 23) to each local exchange limit f=(z, z '), right | L (T Tx′, z ') | carry out Distributed Calculation, described local friendship Change sides, refer in limit collection S (e ') and a limit of adjacent node z；Described | L (T Tx′, z ') |, refer to node z ' beginning And figure T TxThe length of the longest path in ', wherein,
|L(T\Tx′, z ') |=max{d (z ', dk),d(z′,dc)+γ,d(z′,dc)+λ(p(x′)),
d(z′,u′)+d(u′,ρR)+μR-d(dk,ρR)}
Described d (z ', dk) refer to node z ' and node dkA limit for end points；Described d (z ', dc) refer to node z ' and Node dcA limit for end points；Described d (z ', u ') refers to node z ' and a node u ' limit as end points；Described d (u ', ρR) refer to node u ' and node ρRA limit for end points；Described d (dk,ρR) refer to node dkWith node ρRFor end points Article one, limit；Described λ (p (x ')) refers to node dcStart, enter into vertex set VLIn but without going past node p (x ') lower section The length of longest path；Described u ' refers on D (T) with the node nearest for distance z ' on the longest path of node z ' beginning； Described ρRRefer to node dkStart and at set of node VRIn longest path on, enter into VRRoot node in subtree；Institute State γ to refer to node dcStart and only comprise set of node VCThe length of the longest path of interior joint；Described VCRefer to except node dc The node set that outer all nodes are constituted；Described μRRefer to node dkStart and at set of node VRIn the length of longest path；
Step 24) to each local exchange limit f=(z, z '), right | D (Te′/f) | carry out Distributed Calculation: | D (Te′/f) |=max {|D(T)|,|L(Tx′,z)|+l(f)+|L(T\Tx′, z ') | }, described L (Tx′, that z) refer to start with node z and at figure TxIn The length of longest path；Described L (T Tx', z ') refer to node z ' start and figure T TxIn the length of longest path Degree；Described l (f) refers to the length on local exchange limit f=(z, z ')；
Step 25) from these local exchange limits above-mentioned, select one most preferably to exchange limit temporarilyAnd store this exchange of addition The new di that will produce behind limitExist without local exchange limit, then create an illusory candidate limit, And its new di is set to ∞；
Step 26) for each child nodes q ∈ C (z), the range information tried to achieve according to above-mentioned steps, to each child nodes Select an optimal exchange limit candidateAnd the new di that will produce after storing this exchange limit of additionAgain from these children The optimal exchange limit that child node is corresponding selects second and most preferably exchanges limit temporarilyDescribed
Step 27) definition target most preferably exchange limit fbest, compare first and most preferably exchange limit temporarilyWith second optimal exchange LimitSelect to add that exchange limit that corresponding new di is less, ifThen willAs target Optimal exchange limit fbest；IfThen willLimit f is most preferably exchanged as targetbest；IfThen willOrLimit f is most preferably exchanged as targetbest?；
Step 28) if subtree T with x ' as rootx′In each node z all do not traveled through, repeat step 22)～step 27)；
Step 29) determine last solution space Solution, this solution space comprises optimal friendship corresponding to inefficacy limit e '=(x ', f (x ')) Change sides.
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