Algorithms for computing optimal multipath connections

1
Algorithms for computing  optimal multipath
connections

• By Cai, Yu
• and Wu, Jing

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.

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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.

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A Simplified Example

4
E
5
6
9
7
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

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Dynamic Algorithm

• Starting from the end nodes E, we trace back.
  Cost E0,
• For node 8, the only path is from E. Same for
  node 9, so
• Cost 86, P 8E, Cost 95, P 9E,
• For node 7, the only path is from E, so
• Cost 7C7,ECost E10010, P 7E,
• Cost 6C6,9Cost 913518, P 69,
• For node 5, there are two paths from 8 or 9, so
• Cost 5minimum(C5, 8Cost8, C5,
  9Cost9)minimum(86, 65)minimum(14,11)11,
• In the above calculation, the minimum comes from
  node 9, so P 59,
• Same, we can calculate Cost 4, Cost 3, Cost
  2, Cost 1, Cost S.
• The final optimum cost is 16, and the path is
  S248E.

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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-m1. 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.

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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.

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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!!

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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.

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References

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