reliability analysis of tree networks applied to balanced content replication mugurel ionut...

Reliability Analysis of Tree Reliability Analysis of Tree Networks Applied to Balanced Networks Applied to Balanced

Content ReplicationContent Replication

Mugurel Ionut AndreicaMugurel Ionut Andreica, Nicolae Tapus, Nicolae TapusPolytechnic University of BucharestPolytechnic University of Bucharest

Computer Science DepartmentComputer Science Department

Reliability Analysis of Tree Networks Applied to Balanced Content ReplicationReliability Analysis of Tree Networks Applied to Balanced Content Replication 22

SummarySummary

MotivationMotivation Unrestricted Vertex Multicut Problem on TreesUnrestricted Vertex Multicut Problem on Trees

problem definitionproblem definition suboptimal solution (known)suboptimal solution (known) optimal solution (new)optimal solution (new)

Reliability Metric DefinitionReliability Metric Definition Balanced Content Replication on TreesBalanced Content Replication on Trees

problem definitionproblem definition k-equitable coloring of trees (new algorithm)k-equitable coloring of trees (new algorithm)

Evaluation ResultsEvaluation Results ConclusionsConclusions


MotivationMotivation

TreesTrees multiple uses: multiple uses:

• multicast content distributionmulticast content distribution• packet routingpacket routing• content replicationcontent replication• distributed data indexing and storagedistributed data indexing and storage

very fragilevery fragile• a single node failure disconnects the networka single node failure disconnects the network

need a special kind of reliability analysis need a special kind of reliability analysis (maybe node failures and network “splits” are (maybe node failures and network “splits” are acceptable under certain conditions)acceptable under certain conditions)


Unrestricted Vertex Multicut Unrestricted Vertex Multicut Problem on Trees (1/3)Problem on Trees (1/3)

problem definitionproblem definition tree – V verticestree – V vertices H pairs of critical vertices (sH pairs of critical vertices (sii,t,tii)) remove the minimum number of vertices such that all the remove the minimum number of vertices such that all the

critical pairs are disconnectedcritical pairs are disconnected suboptimal algorithm – O(Vsuboptimal algorithm – O(V··H) [Guo et al., 2006]H) [Guo et al., 2006]

Step 1.Step 1. rootroot the tree at some vertex r and compute the the tree at some vertex r and compute the parent-son relationships for all the vertices.parent-son relationships for all the vertices.

Step 2.Step 2. forfor each critical pair (s each critical pair (sii,t,tii) ) dodo: compute its LCA and : compute its LCA and the level of the LCA (the distance from the LCA to the root)the level of the LCA (the distance from the LCA to the root)

Step 3.Step 3. sortsort all the critical pairs in non-increasing order of the all the critical pairs in non-increasing order of the level of their LCAlevel of their LCA

Step 4.Step 4. forfor each critical pair (s each critical pair (sii, t, tii), in the sorted order, ), in the sorted order, dodo Step 4.1.Step 4.1. ifif s sii and t and tii are not already disconnected are not already disconnected thenthen Step 4.1.1.Step 4.1.1. removeremove the LCA of s the LCA of sii and t and tii from the tree from the tree



optimal algorithm – O(V+H)optimal algorithm – O(V+H) Step 1Step 1: : cchoose a root r and compute the parent-son hoose a root r and compute the parent-son

relationships for all the vertices of the treerelationships for all the vertices of the tree – O(V) – O(V) Step 2Step 2: : ccompute the arrays pLCA and level: pLCA[i] is ompute the arrays pLCA and level: pLCA[i] is

the lowest common ancestor of the ithe lowest common ancestor of the ithth pair and level[i] is pair and level[i] is the level of their LCA in the treethe level of their LCA in the tree – O(V+H) [Bender, – O(V+H) [Bender, Farach-Colton, 2000]Farach-Colton, 2000]

Step 3Step 3 – O(V+H) – O(V+H)• forfor l=0l=0 toto V-1V-1 dodo LL[l]=emptyLL[l]=empty• forfor i=1i=1 toto HH dodo LL[level[i]].add(i)LL[level[i]].add(i)• sorted_pairs=emptysorted_pairs=empty• forfor lev=V-1lev=V-1 downtodownto 00 dodo

ifif ( (LL[lev] is not emptyLL[lev] is not empty) ) thenthen• forfor ii inin LL[lev]LL[lev] dodo sorted_pairs.add(i)sorted_pairs.add(i)



optimal algorithm – O(V+H) (continued)optimal algorithm – O(V+H) (continued) Step 4 – O(V+H), amortized analysisStep 4 – O(V+H), amortized analysis

• forfor v=1v=1 toto VV dodo marked[v]=Falsemarked[v]=False• num_removed=0num_removed=0• forfor i=1i=1 toto HH dodo

p=sorted_pair[i]p=sorted_pair[i] ifif (( ((notnot marked[smarked[spp]]) ) andand ( (notnot marked[tmarked[tpp]])) )) thenthen

• num_removed = num_removed + 1num_removed = num_removed + 1• TraverseAndMarkTraverseAndMark((pLCA[p]pLCA[p]))

• returnreturn num_removednum_removed TraverseAndMark(v):TraverseAndMark(v):

• marked[v]=Truemarked[v]=True• for eachfor each w w inin sons(v) sons(v) dodo

ifif ( (notnot marked[w]) marked[w]) thenthen• TraverseAndMarkTraverseAndMark(w)(w)


Reliability Metric DefinitionReliability Metric Definition

tree with V verticestree with V vertices H critical pairs (problem dependent)H critical pairs (problem dependent) Q=total number of distinct vertices Q=total number of distinct vertices

belonging to at least one pairbelonging to at least one pair U=UVMC(V, H)U=UVMC(V, H) reliability=U/Qreliability=U/Q


Balanced Content Replication Balanced Content Replication (1/5)(1/5)

tree with V vertices (servers)tree with V vertices (servers) k pieces of contentk pieces of content distribute the pieces evenly among the serversdistribute the pieces evenly among the servers

each server: 1 piece of contenteach server: 1 piece of content nvnvii=the number of servers storing piece i=the number of servers storing piece i |nv|nvii-nv-nvjj|≤1|≤1 neighboring servers should store different pieces of neighboring servers should store different pieces of

contentcontent find a valid assignment (piece of content – set of find a valid assignment (piece of content – set of

vertices) => k-equitable coloringvertices) => k-equitable coloring supplementary assumption: k≥D (D=the maximum supplementary assumption: k≥D (D=the maximum

degree of the tree)degree of the tree)



new k-equitable coloring algorithm – O(Vnew k-equitable coloring algorithm – O(V··k)k) root the tree at a vertex r having <D sons => each vertex has root the tree at a vertex r having <D sons => each vertex has

at most D-1 sonsat most D-1 sons bottom-up traversal of the tree (from leaves to the root)bottom-up traversal of the tree (from leaves to the root) for each vertex i, compute:for each vertex i, compute:

• nvtotalnvtotalii = the number of vertices in i’s subtree (including i)= the number of vertices in i’s subtree (including i)• ncplusncplusii = (= (nvtotalnvtotalii mod k mod k) – the number of surplus colors in an ) – the number of surplus colors in an

equitable coloring of vertex i’s subtreeequitable coloring of vertex i’s subtree• colorcolorii = the color of vertex i in an equitable coloring of its subtree. = the color of vertex i in an equitable coloring of its subtree.• color_permcolor_permii = a permutation which describes how the colors in = a permutation which describes how the colors in

vertex i’s subtree should be relabeled.vertex i’s subtree should be relabeled. r=vr=v11, v, v22, ..., v, ..., vqq==i -> the pathi -> the path from r ro ifrom r ro i

• p=color_permp=color_permv1v1 • color_perm • color_permv2v2 • … • color_perm • … • color_permvqvq

• real_colorreal_colorii=p(color=p(colorii)) if i is a leaf:if i is a leaf:

• nvtotalnvtotalii=1; ncplus=1; ncplusii=1 mod k; color=1 mod k; colorii=1; color_perm=1; color_permii=(1,2,...,k)=(1,2,...,k)



new k-equitable coloring algorithm – new k-equitable coloring algorithm – O(VO(V··k) (continued)k) (continued) if i has ns(i)>0 sonsif i has ns(i)>0 sons

• combine the colorings of i’s sons (in some order combine the colorings of i’s sons (in some order s(i,1), s(i,2), ..., s(i,j), ..., s(i,ns(i)))s(i,1), s(i,2), ..., s(i,j), ..., s(i,ns(i)))

• define define cplus(j-1)=nvtotal(s(i,1))+...+nvtotal(s(i,j-1))cplus(j-1)=nvtotal(s(i,1))+...+nvtotal(s(i,j-1))• iif ncplusf ncpluss(i,j)s(i,j)=0=0 => => ssi,ji,j is is inactiveinactive;; otherwise s otherwise si,ji,j isis

activeactive..• iif (ncplusf (ncpluss(i,j)s(i,j)>0 and cplus>0 and cplusj-1j-1=0) or (cplus=0) or (cplusj-1j-1 + +

ncplusncpluss(i,j)s(i,j)>k), then all the sons s>k), then all the sons s i,1i,1,…, s,…, si,j-1i,j-1 are made are made inactiveinactive and s and si,ji,j==the only the only activeactive son son

• if s(i,j)=if s(i,j)=inactiveinactive then then swap 2 colors: colorswap 2 colors: colors(i,j)s(i,j) and k-1 in color_perm and k-1 in color_perms(i,j)s(i,j)



new k-equitable coloring algorithm – O(Vnew k-equitable coloring algorithm – O(V··k) (continued)k) (continued) if (vertex i has ns(i)>0 sons) and (s(i,j)=if (vertex i has ns(i)>0 sons) and (s(i,j)=active)active) then then

• cyclic recoloring: color c -> color cyclic recoloring: color c -> color ((cplus((cplusj-1j-1+c-1) mod k)+1+c-1) mod k)+1 (in (in color_permcolor_perms(i,j)s(i,j)))

• maintain counters:maintain counters: ccactiveactive=number of currently active sons=number of currently active sons totaltotalactiveactive=total number of sons which were active at some point=total number of sons which were active at some point

• if (if (color_permcolor_perms(i,j)s(i,j)(color(colors(i,j)s(i,j)))>total>totalactiveactive) then) then relabeling permutation with a cycle of length 3: relabeling permutation with a cycle of length 3: color_permcolor_perms(i,j)s(i,j)

(color(colors(i,j)s(i,j))) -> -> ccactiveactive -> -> total totalactiveactive else: relabeling permutation with two-color swap: else: relabeling permutation with two-color swap: color_permcolor_perms(i,j)s(i,j)

(color(colors(i,j)s(i,j))) , , ccactiveactive • maintain a stack S of relabeling permutation (inactive sons push maintain a stack S of relabeling permutation (inactive sons push

the identity permutation on the stack)the identity permutation on the stack)• after the last son, compose the permutations in S top-downafter the last son, compose the permutations in S top-down

pplevlev=the composition from the top until level =the composition from the top until level levlev color_permcolor_perms(i,lev)s(i,lev)==ComposePermutations(pComposePermutations(plevlev,color_perm,color_perms(i,lev)s(i,lev)))

• assign to vertex i the color k (no son is colored with k)assign to vertex i the color k (no son is colored with k)• swap the colors in color_permswap the colors in color_perm ii, such that vertex i is colored with 1, such that vertex i is colored with 1



new k-equitable coloring algorithm – new k-equitable coloring algorithm – O(VO(V··k) (continued)k) (continued) perform a second, top-down traversal of perform a second, top-down traversal of

the treethe tree compose the permutations on the path compose the permutations on the path

from the root r to each vertex i (O(Vfrom the root r to each vertex i (O(V··k) k) overall)overall)

compute the “real color” of vertex icompute the “real color” of vertex i overall complexity: O(Voverall complexity: O(V··k)k)


Evaluation Results (1/5)Evaluation Results (1/5)

comparison between UVMC algorithms comparison between UVMC algorithms (Python implementation)(Python implementation) O(VO(V··H) vs O((V+H)H) vs O((V+H)··log(V)) (a simpler, more log(V)) (a simpler, more

practical, LCA finding method)practical, LCA finding method)

VV HH O(V·H) O(V·H) algorithmalgorithm

O((V+H)·log(V)) O((V+H)·log(V)) algorithmalgorithm

1000010000 99989998 10,25 sec10,25 sec 0,37 sec0,37 sec

3000030000 2999929999 86 sec86 sec 1,39 sec1,39 sec

5000050000 5000150001 229,58 sec229,58 sec 2,18 sec2,18 sec

6666666666 6666666666 1149,22 sec1149,22 sec 3,70 sec3,70 sec

9999999999 9999899998 1896,78 sec1896,78 sec 5,71 sec5,71 sec



two types of test scenariostwo types of test scenarios type 1 test scenariotype 1 test scenario

different values of V, D, k, leaf percentagedifferent values of V, D, k, leaf percentage C – number of vertices serving client requestsC – number of vertices serving client requests we k-equitably colored the treewe k-equitably colored the tree critical pairs = (i,j)critical pairs = (i,j)

• j has color color(j)j has color color(j)• dist(i,j)<dist(i,p), with color(p)=color(j)dist(i,j)<dist(i,p), with color(p)=color(j)

reliability decreases with kreliability decreases with k reliability depends on leaf percentage (tree reliability depends on leaf percentage (tree

structure) more than anything elsestructure) more than anything else



V=10000V=10000

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

0 10 20 30 40 50 60

Number of colors

50%leaves

66%leaves

90%leaves

c



different values of V and different different values of V and different leaf percentagesleaf percentages

0,04

0,045

0,05

0,055

0,06

0,065

0,07

0,075

0,08

0,085

0 10 20 30 40 50 60

Number of colors

66% leaves, V=5000

66% leaves, V=10000

66% leaves, V=25000

90% leaves, V=5000

90% leaves, V=10000

90% leaves, V=25000



type 2 test scenariotype 2 test scenario critical pairs=(i,j)critical pairs=(i,j)

• color(i)=color(j)color(i)=color(j)• motivated by the need of periodical motivated by the need of periodical

synchronization between replicassynchronization between replicas obtained similar resultsobtained similar results


ConclusionsConclusions first optimal algorithm for UVMC on treesfirst optimal algorithm for UVMC on trees new k-equitable coloring algorithmnew k-equitable coloring algorithm reliability metric depends on tree structure reliability metric depends on tree structure

more than anything elsemore than anything else has potential to be a useful metrichas potential to be a useful metric

future workfuture work tests on real-life tree networkstests on real-life tree networks

• content replication (might not be perfectly balanced)content replication (might not be perfectly balanced)• multicastmulticast• peer-to-peer overlays (many possibilities) or peer-to-peer overlays (many possibilities) or

physical network infrastructure (tree-like in certain physical network infrastructure (tree-like in certain situations)situations)


Thank You !Thank You !

reliability analysis of tree networks applied to balanced content replication mugurel ionut...

Documents

vertex i

vertex r

lca of si

critical pair si

v vertices serversk

tree ov h bender

minimum number of vertices

tree ovstep