fast self-stabilizing minimum spanning tree construction

8/6/2019 Fast Self-stabilizing Minimum Spanning Tree Construction

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Presented by - Abiola Famurewa & Samuel Idowu {[email protected]}

Lelia Blin, Shlomi Dolev, Maria Gradinariu, Potop-Butucaru and

Stephane Rovedakis


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This paper is based on the extension ofGHS algorithm in self-

stabilizing setting

there already exists self-stabilizing solutions for the MSTconstruction, but none of them considered the extension of

Gallager, Humblet and Spira algorithm (GHS) to self-

Stabilizing settings.

Advantages Of Extending GHS Algorithm

It unifies the best properties of designing large scale MSTs

it is fast and totally decentralized

it does not rely on any global parameter of the system


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Gupta and Srimani presented in a journal titled Self-stabilizingmulticast protocols for ad hoc networks the first stabilizing

algorithm for MST problem.

Executing the algorithm requires that every node should store the

cost of all paths from it to all the other nodes.

Assumptions:

Every node knows the number n of nodes in the network

Every node u stores the weight of the edge eu,vplaced in the

MST for every node v u

Algorithm requires (vu log (eu,v)) bits of memory at node u

Memory requirement at each node is (n log n) bits


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Drawback

Lack of scalability Not scalable since each node has to know

and maintain information for all the nodes in the system

Time Complexity

The time complexity in synchronous setting is , O(n) round as

announced by the authors

In asynchronous setting the complexity is (n2) rounds


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Drawback:

Its computation is expensive in large scale systems and even

harder in dynamic settings.

Time Complexity:

Time complexity of this approach is O(mD) rounds.

where m and D are the number of edges and the diameter of the

network respectively.

O(n3) rounds in the worst case.


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The main challenge addresses in this paper is to design fast and

scalable self-stabilizing MSTwith little memory.

The paper extends the GHS algorithm to self-stabilizing settings

and keep compact its memory space by using a self-stabilizing

extension of the nearest common ancestor labeling scheme.

Assumption:

The proposed algorithm does not make any assumption on the

network size

No assumption on the existence of an priori known root.

Complexity:

The convergence time is O(n2) asynchronous rounds

Memory space complexity per node is O(log2

n) bits.


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Connected network G = (V,E,)

y : E R + is a positive cost function

y Nodesprocessors and edges bidirectional

communication links

y G = (V,E, ) is a network in which the weight of the

communication links may change value.

Processors asynchronously execute their programs consisting of

a set ofvariables and a nite set of rules

Each guarded rule contains a Guard (Boolean) and anAction(node variable update)

A local state of a node is the value of the local variables of the

node and the state of its program counter


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A conguration of the system G = (V,E) is the cross product of

the local states of all nodes in the system.

Execution of an action at a node leads to transition from a

configuration to the next one

Computation of the system:

y Weakly fair: if any action in G is continuously enabled along the

sequence, it is eventually chosen for execution

y maximal sequence of configurations: sequence is either innite,

or it is nite and no action of G is enabled in the nal global state.

* The system can start in any configuration. In a worst case, all

the nodes may start in a corrupted configuration, but these faults

can be tackled by using self-stabilization techniques.


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paper propose to extend the Gallager, Humblet and Spira (GHS)

algorithm to self-stabilizing setting via a compact informative

labeling scheme.

Advantages of result:

y Appealing for large scale systemsy compact since it uses only logarithmic memory in the size of the network

y Scales well since it does not rely on any global parameter of the system

y It is fast its time complexity is the better known in self-stabilizing settings

y Additionally, it self-recovers from any transient fault.

FRAGMENT is the central notion in the GHS approach. It is a

partial spanning tree of the graph, i.e., a fragment is a tree

which spans a subset of nodes.


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The GHS minimum spanning tree construction

y Initially each node is a fragment.

y fragment F, the GHS algorithm identies the MEF and merges the

two fragments endpoints of MEF

MEF - The minimum-weight outgoing edge of a fragment F

y

more than two fragments may be merged concurrently

y The merging process is recursively repeated until a single fragment

remains

y The result is a MST { blue rule for MST construction }

**The direct application of the blue rule allows the system to

reconstruct a MST and to recover from faults whichhave divided

the existing MST

With more severe faults, the application of blue-rule only is not

sufficient to reconstruct a MST


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A combination ofblue-rule with another method, referred to as red-

rule. The red-rule removes the heaviest edge from every cycle

The resulting configuration contains a MST.

THE RED RULE

y Given a spanning tree TofG, edge e ofG T, e is added to T, thus

creating a (unique) cycle

y This cycle is called a fundamental cycle, denoted by Ce

y If e is not the edge of maximum weight in Ce

y The edge of maximum weight can be removed since it is not part of any

MST.

Proposed algorithm uses both blue-rule and red-rule

It uses the blue-rule to construct a spanning tree the red rule to

recover from invalid congurations.


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Both red & blue rules use self-stabilizing labeling scheme algorithm

called NCA-L to identify both

fragments and fundamental cycles.

NCA-L : Nearest Common AncestorLabeling

For any node v V(G), we denote by N(v) the set of all neighbors ofvin

G.

Pv: the parent ofvin the current spanning tree

Lv : the label of v composed of a list of pairs of integers where each pair is

an identier and a distance

sizev : a pair of variables, the 1st one is an integer the number of nodes in

the sub-tree rooted at v and the 2nd is the identier of the child u of v with

the maximum number of nodes in the sub-tree rooted at u.

mwev : the minimum weighted edge composed by a pair of variables


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heavy edge - is an edge between a node u and one of its children v of

maximum number of nodes in its sub-tree

light edges - The other edges between u and its other children

y heavy node: if the edge between v and its parent is a heavy edge,

otherwise v is called light node.

The idea of the scheme is as follows:

T is recursively divided into disjoint paths: heavy and light paths

each node v maintains a variable named sizev which is a pair of integers

based on the value of variable sizev at each node v T, each node ofT

can compute its label.

The label of a node v stored in lv is a list of pair of integers

Each pair of the list contains the identier of a node and a distance to the

root of the heavy path (i.e. path including only heavy edges)


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NCA-L algorithm is used to find the minimum weighted edges

it is also used to detect and destroy cycles since the initial configuration

may not be a spanning tree.


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NCA-L algorithm is composed of two rules

y Rule R creates a root or breaks cycles [Root creation]

y Rule Rl - produce a proper labeling [label correction]

A node vwith an incoherent parent or present in a cycle executes

R. Following the execution of this rule node vbecomes a root

node

Rule Rl helps a node vto compute the number of nodes in its

sub-tree (stored in variable sizev ) and provides to va coherent

label.


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The algorithm executes two phases:

y The MST correctiony The MST fragments merging

**Assumptions: merging operations have a higher priority than the

recovering operations. i.e the system recovers from an invalidconfiguration if and only if no merging operations is possible.

The minimum weighted edge and MSTcorrection:

For two nodes u and v with e = (u,v), a non tree edge, if the nearestcommon ancestor does not exist then u and v are in two distinct

fragments. Otherwise u and v are in the same fragment F and the

addition of e to F generates a cycle

.


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The minimum weighted edge and MSTcorrection:

Two rules used:

y Rule Rmin [Minimum computation]

y Rule R [MSTCorrection ]

Rule Rmin computes from the leaves to the root the minimum

outgoing edge e = (u,v)

**In general: Rule Rmin computes the outgoing-edges and the

fundamental cycles, while Rule R is used to recover from a false

tree.


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FRAGMENTS MERGING

In this phase, 2 rules are executed:

y Rule R [merging]

y Rule R [End merging]

Wh

en a root r of Fu (fragment)h

as stabilized its variable mwe i.e.the weight of the edge and a common ancestor

** The information about e are stored in the variable mwe

it starts a merging phase (Rule R ), here the nodes in the path

between r and u are reoriented from r to u

The merging process is repeated until a single fragment is obtained

Rule R is executed


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The extension of GHS algorithm gave a result with

advantages appealing to large scale systems. Time

complexity is O(n2) rounds and space complexity is

o(log2n).

fast self-stabilizing minimum spanning tree construction

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