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## Algorithms for Precomputing Constrained Widest Paths and Multicast Trees

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Title: 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
• Presented by Jeremy Witmer
• CS 622
• Fall 2007

2
Multicast Trees
3
Large Multicast Trees
• In large networks, adding nodes becomes
inefficient
• Adding on a widest-bandwidth path
• Paths with QoS constraints

4
Proposed Solution
• Precompute as much of the tree as possible
• When a node is added, choose the path with the
highest available bandwidth while obeying QoS
delay constraints

5
Proposed Solution
• Solution defined as solutions to two separate
problems
• First, the precomputation of the links in the
tree
• Second, selection of a new path when a new node
subscribes to the multicast tree
• The paper proposes three algorithms to accomplish
the first goal

6
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

7
Network Model
• Each edge in E has a corresponding delay and
width, (d,W)
• A path from source node s to another node in the
network u is with delay no greater than d
represented as Pu(d)
• The optimal path is represented as Pu(d)

8
Network Model
9
Network Model
10
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)

11
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)

12
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) for each u in
P
• Note (d, W, u) is generally stored as (d, W, u,
prev_node)

13
Algorithm 1
• Take the discontinuity in the minimum
lexicographic order off of the queue.
• 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

14
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
• Note P is implemented as a heap in this algorithm

15
Algorithm 2
• Operation is similar to Algorithm 1
• Instead of the heap/queue data structures,
discontinuities are stored in arrays indexed by a
function of the link width w
• P is an array Au,k where 1 lt k lt K, K lt M
• Instead of storing possible discontinuities by
node u, on queues D(u), store on K heaps H(k)

16
Algorithm 2
• Algorithm execution is identical to Algorithm 1
except that the heaps H(k) only need to contain
one possible discontinuity at a time
• When a new discontinuity (d, k, u) is found, it
can replace the current discontinuity on heap

17
Algorithm 3
• Given the same graph G (V, E)
• Find the widest-shortest path from s to all nodes
in G
• Let W be the minimum among the widths of the
paths pu
• For all nodes u in V if W(pu) W then add
(D(u), W(pu)) to the appropriate queue D(u)
• Remove from G all links with width at most W
• If s has no more outgoing links, then stop, else
repeat

18
Algorithm 3
• The widest-shortest paths in step 1 are found by
a version of Dijkstras algorithm
• Static-Heap Dijkstras algorithm has been shown
to be the most efficient implementation.

19
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)
20
Current Multicast Tree Design
• The optimization problem to conserve resources is
known to be NP complete.
• Existing tree-calculation protocols do not solely
optimize resources
• Problem aggravated by the need to satisfy QoS
restraints

21
Computation of Constrained Trees
• Obtain a multicast tree from the discontinuities
previously calculated, with the following QoS
constraints
• Path width W(p) will be gt Wmin
• Path delay D(p) will be lt d

22
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 d
• And W(T) is the width of the narrowest link in T

23
Computation of Constrained Trees
• Any calculated tree T must satisfy Property 1
• The delay du of discontinuity (du, Wu) is the
smallest one among the delays of the
discontinuities in D(u) whose width is larger
than or equal to Wmin

24
Algorithm 4
• Assuming that D(u) is an array
• For each node u in U, determine W(pu)
• Determine Wmin of pu
• For each (d, W, u) in U determine the
discontinuity having property 1
• Construct G using the predecessor node
information stored in D(u)

25
Algorithm 4 Performance
• Running Time O(maxUlogN, N)

26
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

27
Simulation Results
• The delays of the links in both network types
were picked randomly.
• 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

28
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

29
Simulation Results
30
Simulation Results
31
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

32
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

33
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.

34
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.