ThroughputCompetitive OnLine Routing

1
Throughput-Competitive On-Line Routing

• Presentation by Gur Hildesheim
• Based on a paper by Awerbuch, Azar and Plotkin

2
Background and motivation

• High speed networks (ATM)
• Algorithm for admission control (accept the call
  or not) and routing of the accepted calls
• Online (no future knowledge, no rerouting, no
  interrupts)

3
Background and motivation (cont.)

• This algorithm tries to maximize the throughput
  (in the competitive attitude)
• How to measure throughput?
• Each call request has a profit that the algorithm
  would gain IFF it will admit it
• We assume the profit (throughput) is proportional
  to the product rate-time (bits)

4
Model details and notations

• Given
• The network as graph G(V,E)
• n nodes
• u(e) is edge capacity (bandwidth)
• t - time unit (integer)
• T - Maximal call duration

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Model details and notations (cont.)

• The input
• Sequence of call requests b1, b2,
• Each request bi contain
• s(i) , d(i) - source and destination
• Ts(i), Tf(i) - starting and completion times
• r(i) - the traffic rate of the call (bandwidth)
• w(i) - the profit of the algorithm if the request
  is admitted

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Model details and notations (cont.)

• The output
• p(i) - the set of edges in the path from s(i) to
  d(i)
• (empty set in case of blocking)
• We denote A the indices set of admitted requests.
• We denote Q the indices set of requests admitted
  by (best) offline algorithm and not by the online
  algorithm.

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Algorithm goal

• The algorithm tries to maximize the profit,
• i.e. maximize

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Example
n5 T2
u3
u4
v
d
b
a
u
u3
e
c
u3
u2
b1 su, dv, Ts0, Tf2, r1.5, w1.5
p a,c,e
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Model details and notations (cont.)

• Relative load
• For each edge e we define its relative load just
  before the kth request by

We demand for every e,t,k
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Back to the example
u3 L0
u4 L0
r(1)1.5
v
d
b
a
u
u3 L0.5
e
c
u3 L0.5
u2 L0.75
b2 sv, du, Ts1, Tf2, r2, w2
p
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Assumptions

• We will use use two assumptions

This means the profit is proportional to the
product rate-time, normalized by 1/n
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Assumptions (cont.)

• We denote M 2nTF 1
• And assume

(2)
This means that the transmission rate of the call
must be relatively small compares to the minimal
bandwidth.
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The algorithm

• We will show O(lognT) TC algorithm
• We are looking at edge e at time t, after a
  sequence of requests b1-bj-1.
• Let us define the cost of e as

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The algorithm (cont.)

• Upon connection request bj(s, d, Ts, Tf, r(j),
  w(j))

Then route the connection on p, and set
else Block the connection
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Analysis

• For convenience, we will prove correctness and
  performance, assuming
• u(e) 1 (all edges have the same capacitance)
• T1 (a call is for endless time)
• w(j) n r(j)
• Hence,

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Analysis (cont.)

• The load is
• The cost is
• The algorithm checks if

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Correctness

• Lemma the algorithm never admit a request that
  overload the capacity of any edge, namely

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Correctness (cont.)

• Proof
• Assume the contrary, and let bj be the first
  connection causing overload on edge e on time t.

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Correctness (cont.)

• Contradiction to the admission of the request.
• QED

20
Performance

• We will now show that the algorithm is O(lognT)
  throughput-competitive.
• This will be done in 2 stages
• Proving a lower bound on the profit accrued by
  the online algorithm.
• Proving an upper bound on the profit accrued by
  the (best) offline algorithm.

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Performance (cont.)

• Lemma Denote k maxA Then
• Proof
• By induction on k.
• Base k0. No request was admitted yet so both
  the costs sum and the profit are 0.

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Performance (cont.)

• Step to keep the inequality true it is enough to
  prove

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Performance (cont.)
Indeed,
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Performance (cont.)
QED.
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Performance (cont.)

• Now the upper bound
• Reminder Q is the indices set of requests
  admitted by the off-line algorithm and not by the
  online algorithm.
• Lemma Let h maxQ. Then

Proof exercise ?
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Performance (cont.)

• But,

So the offline algorithm cannot obtain a total
profit of more than
And we get that the on-line algorithm is O(lognT)
throughput-competitive.
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Remarks

• The original proof is for every u(e), T, rj(t),
  w(j)
• This algorithm is optimal with respect to TC.
• The assumptions (1) and (2) are essential.
  Without them there is no polylogaritmic TC online
  algorithm.

28
References

• B. Awerbuch, Y. Azzar, S. Plotkin
• Throughput-Competitive On-Line Routing, 1993