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Reliability Analysis of Tree Networks Applied to Balanced Content Replication

Reliability Analysis of Tree Networks Applied to Balanced Content Replication. Mugurel Ionut Andreica , Nicolae Tapus Polytechnic University of Bucharest Computer Science Department. Summary. Motivation Unrestricted Vertex Multicut Problem on Trees problem definition

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Reliability Analysis of Tree Networks Applied to Balanced Content Replication

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  1. Reliability Analysis of Tree Networks Applied to Balanced Content Replication Mugurel Ionut Andreica, Nicolae Tapus Polytechnic University of Bucharest Computer Science Department

  2. Summary • Motivation • Unrestricted Vertex Multicut Problem on Trees • problem definition • suboptimal solution (known) • optimal solution (new) • Reliability Metric Definition • Balanced Content Replication on Trees • problem definition • k-equitable coloring of trees (new algorithm) • Evaluation Results • Conclusions Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  3. Motivation • Trees • multiple uses: • multicast content distribution • packet routing • content replication • distributed data indexing and storage • very fragile • a single node failure disconnects the network • need a special kind of reliability analysis (maybe node failures and network “splits” are acceptable under certain conditions) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  4. Unrestricted Vertex Multicut Problem on Trees (1/3) • problem definition • tree – V vertices • H pairs of critical vertices (si,ti) • remove the minimum number of vertices such that all the critical pairs are disconnected • suboptimal algorithm – O(V·H) [Guo et al., 2006] • Step 1.root the tree at some vertex r and compute the parent-son relationships for all the vertices. • Step 2.for each critical pair (si,ti) do: compute its LCA and the level of the LCA (the distance from the LCA to the root) • Step 3.sort all the critical pairs in non-increasing order of the level of their LCA • Step 4.for each critical pair (si, ti), in the sorted order, do • Step 4.1.if si and ti are not already disconnected then • Step 4.1.1.remove the LCA of si and ti from the tree Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  5. Unrestricted Vertex Multicut Problem on Trees (2/3) • optimal algorithm – O(V+H) • Step 1: choose a root r and compute the parent-son relationships for all the vertices of the tree – O(V) • Step 2: compute the arrays pLCA and level: pLCA[i] is the lowest common ancestor of the ith pair and level[i] is the level of their LCA in the tree – O(V+H) [Bender, Farach-Colton, 2000] • Step 3 – O(V+H) • forl=0toV-1doLL[l]=empty • fori=1toHdoLL[level[i]].add(i) • sorted_pairs=empty • forlev=V-1downto0do • if (LL[lev] is not empty) then • foriinLL[lev]dosorted_pairs.add(i) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  6. Unrestricted Vertex Multicut Problem on Trees (3/3) • optimal algorithm – O(V+H) (continued) • Step 4 – O(V+H), amortized analysis • forv=1toVdomarked[v]=False • num_removed=0 • fori=1toHdo • p=sorted_pair[i] • if ((notmarked[sp]) and (notmarked[tp])) then • num_removed = num_removed + 1 • TraverseAndMark(pLCA[p]) • returnnum_removed • TraverseAndMark(v): • marked[v]=True • for eachw in sons(v) do • if (not marked[w]) then • TraverseAndMark(w) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  7. Reliability Metric Definition • tree with V vertices • H critical pairs (problem dependent) • Q=total number of distinct vertices belonging to at least one pair • U=UVMC(V, H) • reliability=U/Q Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  8. Balanced Content Replication (1/5) • tree with V vertices (servers) • k pieces of content • distribute the pieces evenly among the servers • each server: 1 piece of content • nvi=the number of servers storing piece i • |nvi-nvj|≤1 • neighboring servers should store different pieces of content • find a valid assignment (piece of content – set of vertices) => k-equitable coloring • supplementary assumption: k≥D (D=the maximum degree of the tree) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  9. Balanced Content Replication (2/5) • new k-equitable coloring algorithm – O(V·k) • root the tree at a vertex r having <D sons => each vertex has at most D-1 sons • bottom-up traversal of the tree (from leaves to the root) • for each vertex i, compute: • nvtotali= the number of vertices in i’s subtree (including i) • ncplusi= (nvtotali mod k) – the number of surplus colors in an equitable coloring of vertex i’s subtree • colori = the color of vertex i in an equitable coloring of its subtree. • color_permi = a permutation which describes how the colors in vertex i’s subtree should be relabeled. • r=v1, v2, ..., vq=i -> the pathfrom r ro i • p=color_permv1 • color_permv2 • … • color_permvq • real_colori=p(colori) • if i is a leaf: • nvtotali=1; ncplusi=1 mod k; colori=1; color_permi=(1,2,...,k) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  10. Balanced Content Replication (3/5) • new k-equitable coloring algorithm – O(V·k) (continued) • if i has ns(i)>0 sons • combine the colorings of i’s sons (in some order s(i,1), s(i,2), ..., s(i,j), ..., s(i,ns(i))) • define cplus(j-1)=nvtotal(s(i,1))+...+nvtotal(s(i,j-1)) • if ncpluss(i,j)=0 => si,j is inactive; otherwise si,jisactive. • if (ncpluss(i,j)>0 and cplusj-1=0) or (cplusj-1 + ncpluss(i,j)>k), then all the sons si,1,…, si,j-1 are made inactive and si,j=the only active son • if s(i,j)=inactive then • swap 2 colors: colors(i,j) and k-1 in color_perms(i,j) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  11. Balanced Content Replication (4/5) • new k-equitable coloring algorithm – O(V·k) (continued) • if (vertex i has ns(i)>0 sons) and (s(i,j)=active) then • cyclic recoloring: color c -> color ((cplusj-1+c-1) mod k)+1 (in color_perms(i,j)) • maintain counters: • cactive=number of currently active sons • totalactive=total number of sons which were active at some point • if (color_perms(i,j)(colors(i,j))>totalactive) then • relabeling permutation with a cycle of length 3: color_perms(i,j)(colors(i,j)) -> cactive -> totalactive • else: relabeling permutation with two-color swap: color_perms(i,j)(colors(i,j)) , cactive • maintain a stack S of relabeling permutation (inactive sons push the identity permutation on the stack) • after the last son, compose the permutations in S top-down • plev=the composition from the top until level lev • color_perms(i,lev)=ComposePermutations(plev,color_perms(i,lev)) • assign to vertex i the color k (no son is colored with k) • swap the colors in color_permi, such that vertex i is colored with 1 Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  12. Balanced Content Replication (5/5) • new k-equitable coloring algorithm – O(V·k) (continued) • perform a second, top-down traversal of the tree • compose the permutations on the path from the root r to each vertex i (O(V·k) overall) • compute the “real color” of vertex i • overall complexity: O(V·k) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  13. Evaluation Results (1/5) • comparison between UVMC algorithms (Python implementation) • O(V·H) vs O((V+H)·log(V)) (a simpler, more practical, LCA finding method) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  14. Evaluation Results (2/5) • two types of test scenarios • type 1 test scenario • different values of V, D, k, leaf percentage • C – number of vertices serving client requests • we k-equitably colored the tree • critical pairs = (i,j) • j has color color(j) • dist(i,j)<dist(i,p), with color(p)=color(j) • reliability decreases with k • reliability depends on leaf percentage (tree structure) more than anything else Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  15. Evaluation Results (3/5) • V=10000 Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  16. Evaluation Results (4/5) • different values of V and different leaf percentages Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  17. Evaluation Results (5/5) • type 2 test scenario • critical pairs=(i,j) • color(i)=color(j) • motivated by the need of periodical synchronization between replicas • obtained similar results Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  18. Conclusions • first optimal algorithm for UVMC on trees • new k-equitable coloring algorithm • reliability metric depends on tree structure more than anything else • has potential to be a useful metric • future work • tests on real-life tree networks • content replication (might not be perfectly balanced) • multicast • peer-to-peer overlays (many possibilities) or physical network infrastructure (tree-like in certain situations) Reliability Analysis of Tree Networks Applied to Balanced Content Replication

  19. Thank You ! Reliability Analysis of Tree Networks Applied to Balanced Content Replication

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