1. Maximum Reliability K-Hop Multicast Strategy in Tree Networks Mugurel Ionut Andreica, Nicolae Tapus Polytechnic University of Bucharest Computer Science Department

2. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 2 Summary  Maximum Reliability K-Hop Multicast in Directed Tree Networks  2 Dynamic Programming Solutions O(k·n 2 ) and O(k·n 3 ) O(k·n 2 ) and O(k·n 3 )  Extensions to General Directed Graphs  A Max-Min Reliability Metric 2 situations 2 situations  Conclusions

3. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 3 Maximum Reliability K-Hop Multicast in Directed Trees (1/2)  directed tree – n vertices, n-1 edges  edge u->v: r u,v => cost(u,v)=-log(r u,v )  root of the tree -> distributes content to all the leaves  minimum total cost  any number of simultaneous connections to nodes in the subtree  at most k time units  intermediate nodes S 0 ={root} S 0 ={root} during time unit i: S i-1 -> S i during time unit i: S i-1 -> S i S k ={ x | x is a leaf } S k ={ x | x is a leaf }

4. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 4  n=14, k=2, 11 messages sent  cost=10+12+3+11+17+1+3+5+3+12+1=75 Maximum Reliability K-Hop Multicast in Directed Trees (2/2)

5. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 5 Existing Solutions  Galdi, Kaklamanis, Montangero, Persiano (2001): O(k·n·M(n))  M(n)=the time complexity of a min- cut algorithm (in graphs with n vertices and O(n) edges)  recently (Borradaile, Klein, 2006): M(n)=O(n·log(n))  O(k·n 2 ·log(n))

6. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 6 O(k·n 3 ) Dynamic Programming Algorithm (1/3)  turn tree into binary tree (standard)  C(i,j,p) = the minimum cost of distributing the content to all the leaves in node i’s subtree (starting from i) at most j (0≤j≤k) time units at most j (0≤j≤k) time units if 1<p≤n: i is on the paths between an intermediate node x above i and p (in 1,..., in p ) if 1<p≤n: i is on the paths between an intermediate node x above i and p (in 1,..., in p ) x sends messages to in 1,..., in px sends messages to in 1,..., in p if p=1 if p=1 i is an intermediate nodei is an intermediate node i is not an intermediate node (set p=1 in the case above)i is not an intermediate node (set p=1 in the case above)

7. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 7 O(k·n 3 ) Dynamic Programming Algorithm (2/3)  C(i,-1,p)=+∞  i is a leaf C(i,j,1)=0 C(i,j,1)=0 C(i,j,p)=+∞ C(i,j,p)=+∞  i has one son s

8. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 8 O(k·n 3 ) Dynamic Programming Algorithm (3/3)  i has two sons: s1, s2  root: special case for p=1 and only one son (must be an intermediate node)

9. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 9 O(k·n 2 ) Dynamic Programming Algorithm (1/2)  DFS traversal => assign a label to each leaf (starting from 1)  [lmin(i),lmax(i)]=the interval of labels of the leaves in node i’s subtree (ST i )  C(i,j,p)=the minimum cost of distributing the content to the first p leaves in ST i at most j time units at most j time units i is an intermediate node i is an intermediate node

10. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 10 O(k·n 2 ) Dynamic Programming Algorithm (2/2)   L(i,p)=a list with all the nodes x in ST i with lmax(x)=lmin(i)+p-1  for a pair(i,j): O(n) for all the values of p  can get to O(1) amortized time for each tuple (i,j,p)  O(k·n 2 ) overall

11. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 11 Extensions to General Directed Graphs (1/2)  directed graph: n vertices, n·(n-1) directed edges  src=source of the content  D=set of destinations  C(i,j,S)=the minimum cost of distributing the content to all the nodes in S, starting from node i at most j time units at most j time units S is a subset of nodes from D. S is a subset of nodes from D.

12. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 12 Extensions to General Directed Graphs (2/2)    O(k·n 2 ·3 |D| )  can also extend the O(k·n 3 ) algorithm => O(k·n 3 ·3 |D| )

13. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 13 A Max-Min Reliability Metric (1/3)  directed graph – n vertices  edge u->v: reliability: r u,v reliability: r u,v duration: du u,v duration: du u,v transmission processing time (lag): tp u,v transmission processing time (lag): tp u,v  src=source of the content  D=set of destinations – {d 1,d 2,...,d C }  maximize the minimum reliability of the traversed edges

14. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 14 A Max-Min Reliability Metric (2/3)  binary search R (the minimum reliability) => feasibility test minimum time multicast strategy minimum time multicast strategy  the general case: Tmin(i,S)=the minimum duration of distributing the content to all the destinations in S, starting from the node i Tmin(i,S)=the minimum duration of distributing the content to all the destinations in S, starting from the node i S is the set of nodes in node i’s subtree S is the set of nodes in node i’s subtree O(n 2 ·3 n ) – similar to the directed graph optimal k-hop multicast strategy O(n 2 ·3 n ) – similar to the directed graph optimal k-hop multicast strategy

15. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 15 A Max-Min Reliability Metric (3/3)  A particular case: d u,v =tp u,v =1 O s,u, O r,u, O s+r,u operations per time unit for node u (send, receive, send+receive) => O s,u =O r,u =O s+r,u =1 in this case O s,u, O r,u, O s+r,u operations per time unit for node u (send, receive, send+receive) => O s,u =O r,u =O s+r,u =1 in this case C=|D| - bounded by a constant C=|D| - bounded by a constant Tmin[(v 1,v 2,…,v C )]=the minimum time after which the message destined to each node d i reaches the node v i. Tmin[(v 1,v 2,…,v C )]=the minimum time after which the message destined to each node d i reaches the node v i. transitions from a state S 1 to another state S 2 (1 time unit) transitions from a state S 1 to another state S 2 (1 time unit) every node u i in S 1 sends the message to the receiving node v i in S 2 (if v i ≠u i ) or does not send anything (v i =u i )every node u i in S 1 sends the message to the receiving node v i in S 2 (if v i ≠u i ) or does not send anything (v i =u i ) for every node u: count the number of send and receive operations => invalidate transition if the numbers are too largefor every node u: count the number of send and receive operations => invalidate transition if the numbers are too large a queue of states Q a queue of states Q O(C·n 2·C ) algorithm O(C·n 2·C ) algorithm

16. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 16 Conclusions  O(k·n 2 ) and O(k·n 3 ) DP solutions for the maximum reliability k-hop multicast strategy in directed trees  O(k·n 2 ·3 |D| ) extension to general directed graphs  max-min reliability metric => minimum time multicast general and particular case general and particular case  applications to: design of multicast strategies in distributed systems (e.g. peer-to-peer systems) design of multicast strategies in distributed systems (e.g. peer-to-peer systems) performance evaluation of other multicast strategies (by comparison) performance evaluation of other multicast strategies (by comparison)

17. Maximum Reliability K-Hop Multicast Strategy in Tree Networks 17 Thank You !