Algorithms for computing optimal multipath connections

1. Algorithms for computing  optimal multipath connections By Cai, Yu and Wu, Jing optimal multipath connections

2. Introduction • The existing routing algorithm in Internet typically finds a routing path between two end points by randomly selecting between all the available intermediate nodes.  • We are trying to find out, by given a set of intermediate nodes with known paths and costs, which is optimum path between the two end points. optimal multipath connections

3. S E {N1, N2, N3… Nm} Abstracted Problem • Given a start node S, an end node E, and a set of intermediate nodes {N1, N2, N3… Nm}, find the optimum path starting from node S to node E through node set {N1, N2, N3… Nm}. optimal multipath connections

4. 1 2 3 8 S A Simplified Example 4 E 5 6 9 7 optimal multipath connections

5. A Simplified Example • The path cost matrix is as below: (C [i, j] is the cost of path from node i to node j.) C [S, 1] =2; C [S, 2] =4; C [S, 3] =5; C [1, 6] =4; C [1, 8] =17; C [1, E] =20; C [2, 4] =3; C [2, 5] =1; C [2, 7] =15; C [3, 6] =8; C [3, 9] =10; C [4, 8] =3; C [5, 8] =8; C [5, 9] =6; C [6, 9] =13; C [7, E] =10; C [8, E] =6; C [9, E] =5; optimal multipath connections

6. Dynamic Algorithm Starting from the end nodes E, we trace back. Cost [E]=0, For node 8, the only path is from E. Same for node 9, so Cost [8]=6, P [8]=E, Cost [9]=5, P [9]=E, For node 7, the only path is from E, so Cost [7]=C[7,E]+Cost [E]=10+0=10, P [7]=E, Cost [6]=C[6,9]+Cost [9]=13+5=18, P [6]=9, For node 5, there are two paths from 8 or 9, so Cost [5]=minimum(C[5, 8]+Cost[8], C[5, 9]+Cost[9])=minimum(8+6, 6+5)=minimum(14,11)=11, In the above calculation, the minimum comes from node 9, so P [5]=9, Same, we can calculate Cost [4], Cost [3], Cost [2], Cost [1], Cost [S]. The final optimum cost is 16, and the path is S—2—4—8—E. optimal multipath connections

7. Breath First Search (BFS) • It is important that the cost of all the previous nodes are available before we can get cost of a certain node. For example, if we want to get cost of node 3, then we must get the cost of all previous node of 3, which are node 6, 9. Here is the algorithm to number the nodes: breath-first search. (BFS). • Starting from the end node E, find all the available nodes {K1,K2,…Km}, which has a connection to node E. Number the nodes K1, K2… Km as n, n-1…, n-m+1. Then from nodes K1, do the same numbering action decreasing from number (n-m), until all the nodes are numbered. If a node is already numbered before, or it is the start node S, then ignore it. optimal multipath connections

8. Final Algorithm So the general process of finding the optimum path from node S to node E through node set {N1, N2, N3… Nm} is as following: • Use BFS algorithm to number all the nodes {N1, N2, N3… Nm}, starting from E, end with S. • Use Dynamic algorithm to find the optimum path and the optimum cost. optimal multipath connections

9. Running Result 5 nodes 400s = 6 minutes 10nodes 920s =15 minutes 15nodes 7,280s = 2 hours 20nodes 128,040s = 35 hours O(n2n) Intractable problem!! optimal multipath connections

10. Genetic Algorithm • A GA is based on the principle of evolution – survival of the fittest in a Darwinian sense. • Populations of potential solutions undergo a sequence of crossover type transformations as well as unary mutation transformations. A selection scheme biased towards fitter solutions determines the next generation of potential solutions. After many generations, the goal is to produce a convergence towards an optimum solution • Not discussed in detail. optimal multipath connections

11. References • ? optimal multipath connections