Algorithms for Precomputing Constrained Widest Paths and Multicast Trees Paper by Stavroula...

Algorithms for Precomputing Constrained Widest Paths and Multicast Trees

Paper by Stavroula Siachalou and Leonidas Georgiadis

Presented by Jeremy Witmer

CS 622

Fall 2007

10.22.2007 Precomputing Multicast Trees - CS622

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Multicast Trees


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Large Multicast Trees

In large networks, adding nodes becomes inefficient

Adding on a widest-bandwidth path

Paths with QoS constraints


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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


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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


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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)

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)


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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*) > 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) > W(p2) and D(p1) < D(p2)

OR

W(p1) > W(p2) and D(p1) < 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:

1. Initialize queue D’(u)

2. Create all possible successor discontinuities to u

3. Store the discontinuities (d, W, u) for each u in P

Note: (d, W, u) is generally stored as (d, W, u, prev_node)


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Algorithm 1

4. Take the discontinuity in the minimum lexicographic order off of the queue.

5. If the current discontinuity pair isn’t dominated by any pair currently on D’(u), add the current pair to D’(u), otherwise, discard the pair.

6. Do this for all discontinuities in P


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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 discontinuity links

Note: P is implemented as a heap in this algorithm


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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 A[u,k] where 1 < k < K, K < M

Instead of storing possible discontinuities by node u, on queues D’(u), store on K heaps H(k)


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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 H(k), instead of being added

to the queue


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Algorithm 3

Given the same graph G = (V, E)

1. Find the widest-shortest path from s to all nodes in G

2. Let W* be the minimum among the widths of the paths pu

3. For all nodes u in V if W(pu) = W* then add (D(u), W(pu)) to the appropriate queue D’(u)

4. Remove from G all links with width at most W*

5. 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 a version of Dijkstra’s algorithm

Static-Heap Dijkstra’s algorithm has been shown to be the most efficient implementation.


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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 tree-calculation protocols do not solely optimize resources

Problem aggravated by the need to satisfy QoS restraints


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Computation of Constrained Trees

Obtain a multicast tree from the discontinuities previously calculated, with the following QoS constraints

1. Path width W(p) will be > Wmin

2. Path delay D(p) will be < d


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Assume that we need to create a multicast tree T

T is a subset U of the nodes V in G

Where D(T) < QoS constraint d

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


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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


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Algorithm 4

Assuming that D’(u) is an array

1. For each node u in U, determine W(p*u)

2. Determine Wmin of p*u

3. For each (d, W, u) in U determine the discontinuity having property 1

4. Construct G’ using the predecessor node information stored in D’(u)


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Algorithm 4 Performance

Running Time: O(max{|U|logN, N})


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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=άN, ά > 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.

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 = β(101 – d), where β 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 ά = 4, 8, 16

Real networks selected with M = 9360, 16568, 27792 and N = 2107, 4120, 6474


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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 A[u, 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.


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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.

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