Algorithms for Precomputing Constrained Widest Paths and Multicast Trees

1
Algorithms for Precomputing Constrained Widest
Paths and Multicast Trees

• Paper by Stavroula Siachalou and Leonidas
  Georgiadis
• Presented by Jeremy Witmer
• CS 622
• Fall 2007

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Overview

1. Large Multicast Trees
2. Proposed Solution for adding nodes
3. Network Model
4. Precomputation Algorithms
5. Time and Space Requirements
6. Multicast Tree Algorithm
7. Simulated Results
8. Conclusion

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Multicast Trees
4
Large Multicast Trees

• In large networks, adding nodes becomes
  inefficient
• Wish to add nodes on a widest-bandwidth path
• Wish to add nodes with QoS constraints

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Definitions

• Constrained Widest Path the path that provides
  the most leftover bandwidth, after bandwidth
  requirements are satisfied and delay constraints
  are guaranteed to be met

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Proposed Solution

• Precompute as many of the links in the tree as
  possible
• When adding a node, calculate delay and bandwidth
  for the links it has to other nodes in the
  network
• When a node is added to the multicast tree,
  choose the path with the highest available
  bandwidth while obeying QoS delay constraints

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Proposed Solution

• Solution defined as solutions to two separate
  problems
• First, compute the delay and width data for the
  links in the network, and when nodes are added to
  the network
• Second, selection of a new path from the
  precomputed data when a new node subscribes to
  the multicast tree

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Network Model

• Given a directed graph G (V, E)
• V is the set of nodes in the graph
• E is the set of edges in the graph
• N V
• M E

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Network Model

• Each edge in E has a corresponding delay and
  width, (d,W), where width is the available
  bandwidth in a link
• A path from source node s to another node u in
  the network with delay no greater than d is
  represented as Pu(d)
• The optimal path is represented as Pu(d)
• A path is defined in terms of the overall delay
  D(p) and minimum width W(p)

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Network Model
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Network Model
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Problem 1 Definition

• Find the path Pu(d) that has the greatest width
  of all the paths from s to u, meeting the
  bandwidth requirement W(pu) gt W(p) for all paths
  Pu(d)

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Dominated Pairs

• Pair (D(p1), W(p1)) dominates pair (D(p2), W(p2))
  or path p1 dominates path p2 iff
• W(p1) gt W(p2) and D(p1) lt D(p2)
• OR
• W(p1) gt W(p2) and D(p1) lt D(p2)

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Algorithm 1

• Create a heap P to store all possible
  discontinuities
• For each node u in G, except for the source node
  s
• Initialize queue D(u)
• Create all possible successor discontinuities to
  u
• Store the discontinuities (d, W, u) in P for each
  u in V
• Note (d, W, u) is actually stored as (d, W, u,
  prev_link)

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Algorithm 1

• Take the discontinuity in the minimum
  lexicographic order off of P, in the form (d, W,
  u).
• If the current discontinuity pair isnt dominated
  by any pair currently on D(u), add the current
  pair to D(u), otherwise, discard the pair.
• Do this for all discontinuities in P

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Algorithm 1
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Algorithm 1

• This will result in a set of queues D(u), one
  for each node u in G.
• Each queue is then sorted in lexicographical
  order, so the optimal discontinuity for each node
  u is at the head of the queue
• Because each discontinuity except for the source
  s has a predecessor discontinuity (d, W, v), the
  path can be found by keeping track of these
  predecessor discontinuities

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Algorithm 2

• P is an array Au,k where 1 lt k lt K, K lt M where
  k is a function of the width w of a link
• Initialize Hk heaps, one for each k
• Initialize D(u) queues, one for each node u

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Algorithm 2

1. For each node u, generate all discontinuities (d,
   k, u)
2. Add or replace the discontinuity at Au,k only
   if d is lt the discontinuity at Au,k, and add
   the discontinuity to Hk
3. Starting at the last column in A (largest width
   discontinuities), find the first non-empty heap
   Hk
4. Find the lexicographically smallest discontinuity
   (d, k, u) in Hk.
5. If d is lt the smallest discontinuity (dmin, kmin,
   u) currently on D(u), add (d, k, u) to D(u)
6. Set Au,k to null
7. Repeat until all entries in Au,k are null or
   all Hk are empty

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Algorithm 2
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Algorithm 2

• Since the algorithm works from largest to
  smallest widths k in A, and only adds links that
  have delay lt the current delay on D(u), D(u) is
  guaranteed to have the best possible links

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Algorithm 3

• Given the same graph G (V, E)
• Find the widest-shortest path from s to each u
  node in G
• Let W be the minimum among the widths of the
  paths from step 1
• For all nodes, if W(pu) gt W then add (d, w, u)
  to the appropriate queue D(u)
• Remove from G all links with width lt W
• If s has no more outgoing links, then stop, else
  repeat

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Algorithm 3

• The widest-shortest paths in step 1 are found by
  Dijkstras Algorithm

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Time and Space Requirements
Worst Case Requirements Running Time Space Requirements
Algorithm 1 O(MNlogN M2logN) Space O(MN)
Algorithm 2 O(KNlogN K2) Space O(KN)
Algorithm 3 O(MNlogN M2) Space O(MN)
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Current Multicast Tree Design

• The optimization problem to conserve resources is
  known to be NP complete.
• Existing multicast tree-calculation protocols do
  not rely on trees that solely optimize resources
• Problem aggravated by the need to satisfy QoS
  restraints

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Problem 2

• Given a source node s, a destination node set U,
  and delay requirement dmax, find a multicast tree
  T that satisfies a width requirement Wmin
• Path width W(p) will be gt Wmin
• Path delay D(p) will be lt dmax

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Computation of Constrained Trees

• Assume that we need to create a multicast tree T
• T is a subset U of the nodes V in G
• Where D(T) lt QoS constraint dmax
• And W(T) is the width of the narrowest link in T

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Computation of Constrained Trees

• Any calculated tree T must satisfy Property 1
• The delay d of discontinuity (d, W, u) is the
  smallest one among the delays of the
  discontinuities in D(u) whose width is larger
  than or equal to Wmin

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Algorithm 4

• Assuming that D(u) is an array
• For all nodes in U, determine the minimum width
  Wmin that has a delay still less than dmin
• For each (d, W, u) in each of the arrays D,
  determine the discontinuity having property 1,
  and mark that discontinuity as belonging to the
  tree
• Construct T using the predecessor node
  information stored in D(u)

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Algorithm 4 Performance

• Running Time O(maxUlogN, N)
• Space Requirements O(U)

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Simulation Results

• Simulations were run on two different networks
• Power Law Networks a network with N nodes and M
  links, where M?N, ? gt 1
• Real Internet Networks observed internet
  topologies from 9/20/1998, 1/1/2000, and 2/1/2000

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Simulation Results

• The delays of the links in both network types
  were picked randomly in the interval 1,100.
• Width 1 networks width of each link chosen at
  random from the interval 1,100
• Width 2 networks link width is a function of
  link delay, based on w ß(101 d), where ß is
  random from the interval 1,10

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Simulation Results

• Power Law networks generated with 400, 800, and
  1200 nodes and ratios ? 4, 8, 16
• Real networks selected with M 9360, 16568,
  27792 and N 2107, 4120, 6474

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Simulation Results
35
Simulation Results
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Simulation Results
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Simulation Results
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Simulation Results

• Running times are increased using Width 2 method,
  as there are more available discontinuities
• Algorithm 2 has the best running time, Algorithm
  3 the worst
• Algorithm 1 takes up to 1.6 times as long as
  Algorithm 2
• Algorithm 3 takes up to 14 times as long as
  Algorithm 2
• Algorithm 2 performs the best, especially on
  larger networks

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Simulation Results

• Algorithm 3 has the smallest memory requirements,
  followed closely by Algorithm 1.
• Algorithm 2 requires significantly more space
  than either of Algorithms 1 and 3, due to the
  memory requirements of the two-dimensional array
  Au, k

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Conclusions

• The performance of all algorithms decreases
  rapidly as u increases
• Algorithm 1 presents the best trade-off between
  time and space requirements for precomputing tree
  paths.
• Algorithm 4 is simple by comparison, picking the
  multicast tree T is fairly quick, no matter the
  number of subscriber nodes

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References

• 1 S. Siachalou and L. Georgiadis. Algorithms
  for Precomputing Constrained Widest Paths and
  Multicast Trees. IEEE/ACM Transactions on
  Networking. Vol. 13, No. 5. pp 1174-1187.
  October 2005.
• 2 S. Siachalou and L. Georgiadis. Efficient
  QoS Routing. INFOCOM 2003. 22nd Annual Joint
  Conference of the IEEE Computer and
  Communications Societies. Vol. 2. pp 938-947.
  30 March-3 April 2003.