Acyclic orientations do not lead

1
Acyclic orientations do not lead to optimal
deadlock-free packet routing algorithms.
Daniel Štefankovic
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Packet routing
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1
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4
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7
8
3
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Routing one packet
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1
to 6
2
4
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7
8
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Where should I forward the packet?
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Routing one packet
6
1
to 6
2
4
5
7
8
3
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Routing one packet
to 6
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1
2
4
5
7
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3
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Routing more packets
to 6
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1
2
4
5
7
to 1
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3
to 8
to 2
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Routing more packets
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1
to 6
2
4
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to 1
to 2
to 8
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3
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Routing more packets
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1
to 6
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4
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to 1
to 2
to 8
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3
We need to store the packets
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Buffers
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Routing with buffers
to 6
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to 1
to 2
to 8
8
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Storeforward deadlock
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to 6
to 1
to 8
to 2
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The model (buffer reservation)

• the packets must go along some shortest route
• the devil controls the race conditions
• we control to which buffer the packet goes

Can we avoid storeforward deadlock?
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Yes, we can
we control to which buffer the packet goes
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4
5
7
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How many buffers per node suffice?
diam(G)1 buffers are enough (always move from
buffer i to buffer i1)
Can we do better?
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Can we do better?
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Acyclic orientations
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Acyclic orientations
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Acyclic orientations
If all packets are travelling along acyclic
orientation then there cannot be deadlock.
Split rounting into phases, in each phase packets
would travel along an acyclic orientation.
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Orientation cover of a set of paths P
Set of acyclic orientations A1 A2 ....A k such
that each shortest path p?P can be written as
p1p2...pk where pi is the longest prefix of
pi...pk which can be traversed in Ai.
P for any two vertices u,v at least one
shortest path between u,v
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Orientation cover of a grid
1.
2.
3.
4.
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The question
Is it always possible to design (assymptotically)
optimal solution using acyclic orientations?
There is a graph G with N vertices which has
buffer reservation scheme with 8 buffers per
vertex but any orientation cover of G has size
O(log N/log log N).
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The graph G
0
0
1
1
1
1
Machine M consists of a tape with n cells and a
head which can be positioned above any cell. In
one step head can either change the content of
the occupied cell or move to the left or to the
right.
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Shortest paths in G
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Solution with 8 buffers
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2
3
2
3
4
shuffle a?a
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The lower bound
1. Concentrate only on the orientation of the
shuffle edges, the other edges are free. 2.
Consider only paths from vertices having head on
the leftmost cell to vertices having head on the
rightmost cell.
unique shortest paths
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The lower bound
A path in hypercube Qn is monotone if the bits
are changed from left to right.
What is the minimal size of a cover of the set of
all monotone paths in the hypercube Qn?
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The lower bound
Everybody sends packet to everybody. We are
going to observe the system after each
orientation.
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d-active vertex
x
y
v
d
If for each vertex u of the form uxz there is a
packet in the system destined to u which has to
pass through v. (i.e. it was sent by somebody of
the form wy.) If u has not yet received a packet
from somebody of the form wy gt it is active.
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The lower bound
Let klog2 n. We will construct a sequence of
hypercubes Q0,Q1... Qn/k such that after moving
according to the i-th orientation the ratio of
ki-passive vertices in Qi is at most i/2k.
Qi will have dimension n-ki and will consist of
vertices starting with some string of length ki.
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Qi ? Qi1
x
w
k
ki
Aw ki-active vertices after phase i Nw
k(i1)-passive vertices after phase i1
2l gt Nw gt 2l-1
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Qi ? Qi1
x
w
k
ki
2lgtNwgt2l-1
a
0
a
1
Cannot both be in Aw
Both are in Nw
z1
e
0
e
1
Awlt2k-l
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Qi ? Qi1
Nw-(2k-Aw)lt 2l2k-l-2k lt1
( k(i1)-passive vertices in Qi after
(i1)-st phase ) - ( ki-passive vertices in in
Qi after i-th phase) lt 2n-k(i1)
M
M/2n-ki ? ((M 2n-k(i1))/2k)/2n-k(i1)
(1/2k)
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Questions
Can the gap be improved? Better lower bound for
plain-buffer technique?