Graph Theory

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Chapter 13

• Graph Theory

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Graph Theory Basics
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Spanning Tree
A spanning tree is a tree that contains all of
the vertices, but does not have cycles
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Breadth-first search
Produces shortest hop distance spanning
tree Dont add edges if they form a cycle
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Least Cost Routing Algorithms

• Dijkstra
• Find the least cost spanning tree by
• Selecting closest node not in spanning tree
• Updating the cost to all other nodes if they are
  made lower by passing through this new node.
• Bellman-Ford
• Find shortest paths from a source vertex with one
  link, then find the shortest paths with two links

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Dijkstra
V41,V22,V35
V22,V34,V52
V34,V52,
V33, V64,
V64,
Just determined cost, not path
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Bellman-Ford
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Comparison

• Dijkstra
• O(V2)
• Requires global knowledge from all routers
• Bellman Ford
• O(LV)
• Requires updates from neighbors at each step

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Multi-criteria Routing

• Bandwidth is concave delay and delay jitter are
  additive.
• Bandwidth m(P) minfm(s i)m(i j) m(l
  t)g.

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NP-complete

• There are two NP-complete problem classes,
  path-constrained path-optimization routing(PCPO)
  and multi-path-constrained routing (MPC)
• An example of PCPO is the delay-constrained
  least-cost routing.
• find the least-cost path with bounded delay.
• An example of MPC is the delay-delayjitter-constra
  ined routing.
• find a path with both bounded delay and bounded
  delay jitter.
• Shigang Chen dissertation, ROUTING SUPPORT FOR
  PROVIDING GUARANTEED END-TO-END
  QUALITY-OF-SERVICE, Computer Science in the
  Engineering College of the University of Illinois
  at Urbana-Champaign, 1999

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NP-complete

• NP-complete, we have two assumptions
• the QoS metrics are independent, and
• they are allowed to be real numbers or unbounded
  integer numbers.
• If all metrics except one take bounded integer
  values, then the problems are solvable in
  polynomial time by running an extended Dijkstra's
  (or Bellman-Ford) algorithm.
• If all metrics are dependent on a common metric,
  then the problems may also be solvable in
  polynomial time.
• For example, if the worst-case delay and delay
  jitter are functions of bandwidth in networks
  using the WFQ scheduling.

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Wang-Crowcroft algorithm

• Wang-Crowcroft algorithm finds a
  bandwidth-delay-constrained path by Dijkstra's
  shortest-path algorithm.
• First, all links with a bandwidth less than the
  requirement are eliminated so that any paths in
  the resulting graph will satisfy the bandwidth
  constraint.
• Then, the shortest path in terms of delay is
  found. The path is feasible if and only if it
  satisfies the delay constraint.

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Ma-Steenkiste algorithm

• Ma and Steenkiste showed that when a class of WFQ
  are used, the end-to-end delay, delay-jitter, and
  buffer space bounds are not independent.
• They are functions of the reserved bandwidth, the
  selected path and the traffic characteristics.
• Therefore, the problem of finding a path
  satisfying bandwidth, delay, delay-jitter and
  buffer space constraints, which is NP-complete in
  general can be simplified.

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Awerbuch et al algorithm

• Awerbuch et al. proposed a throughput-competitive
  routing algorithm for bandwidth-constrained
  connections.
• The algorithm tries to maximize the amortized
  (average) throughput of the network over time.
• Every link is associated with a cost function
  that is exponential to the bandwidth utilization.
  
• A new connection is admitted into the network
  only if there exists a path whose accumulated
  cost over the duration of the connection does not
  exceed the profit that is measured by the
  bandwidth-duration product of the connection.

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multi-path-constrained routing (MPC) problem

• An example of the MPC problem is the
  delay-cost-constrained routing, i.e., finding a
  route between two nodes in the network with both
  end-to-end delay and end-to-end cost bounds.
• The MPC problem is difficult because different
  path constraints can conflict with one another
  and create an exponentially large searching space.

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Multi-path-constrained routing problem (MPC)

• Given a directed graph G(V,E), a source node s, a
  destination node t, two weight functions w1 E
  ?R and w2 E ? R
• two constants c1? R and c2 ? R
• the problem, denoted as
• MPC(G s t w1 w2 c1 c2),
• is to find a path P from s to t such that
• w1(P)ltc1 and w2(P)ltc2
• if such a path exists.

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Heuristic
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Example
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Experimental Network
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Results
Provable lower bound