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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
    Georgiadis
  • 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
    discontinuity links
  • 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
    H(k), instead of being added to the queue

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.
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