Maze Routing Algorithms with Exact
Matching Constraints for Analog and
Mixed Signal Designs
M. M. Ozdal and R. F. Hentschke
Intel Corporation
ICCAD 2012
Outline
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Introduction
Problem Formulation
Preliminaries
Proposed Maze Routing Algorithm
Scalable Route Matching Framework
Experimental Results
Conclusions
Introduction
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One of the most common classes of constraints in
analog and mixed-signal designs is the matching of
specific devices and interconnects between them.
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For matching the interconnect properties exactly,
the following route matching formulation was
proposed:
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The routes of all nets must have identical number of wire
segments.
The i-th segment of each net must have the same
wirelength and the same layer assignment.
Introduction
Introduction
Problem Formulation
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Given a set of 2-pin nets with driver and receiver
terminals.
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For exact route matching constraint to hold, the
following conditions must be satisfied:
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The segment counts are identical for all nets.
The lengths and layer assignments of the i-th segments of
all nets are identical.
Minimize the routing cost:
Preliminaries
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The exact route matching problem can be
formulated as a system of linear equations for
horizontal and vertical segments:
CH × LH = DH (for horizontal segments)
CV × LV = DV (for vertical segments)
LH[j] is the length of the j-th horizontal segment.
DH[n] is the horizontal distance between the driver
and receiver terminals of net n.
CH is the configuration matrix to determine the
direction of each horizontal segment for each net.
Preliminaries
Net 1
Net 2
Seg 1
Preliminaries
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A segment type t for a set of nets is defined as a
specific routing direction for each net.
A segment type can be either a horizontal type or a
vertical type.
(n1;n2;n3)
(→;→;→)
(→;→;←)
Proposed Maze Routing
Algorithm
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Us: a set of unit segments.
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ut : the number of unit segments in Us with type t.
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The set Us is feasible for N iff for each net in N, a path can be
constructed from driver to receiver by using all unit segments
in Us.
Proposed Maze Routing
Algorithm
Proposed Maze Routing
Algorithm
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Assume that there are m segment types in
Us.
A partial solution state is defined as {c1,
c2,…, cm,last_layer}.
Each ct (1 ≤ t ≤ m) corresponds to the
number of unit segments of type t in the
partial solution.
last_layer is the layer of the last segment in
the partial solution.
Proposed Maze Routing
Algorithm
Scalable Route Matching
Framework
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Start with a topology obtained by the algorithms in
[8].
Scalable Route Matching
Framework
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CR1 corresponds to the empty partial solution state. {c1=0;
c2=0; c3=0; c4=0}
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CR2 {c1=10; c2=0; c3=0; c4=0}
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CR5 {c1=10; c2=8; c3=6; c4=2}
Scalable Route Matching
Framework
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Define a graph, each node corresponds to a corner of the
topology, each edge corresponds to the best routing solution
between the corresponding corners.
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CR1→CR2 correspond to routing between the respective
corners with a single segment type.
Scalable Route Matching
Framework
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CR1→CR3 correspond to the best exactly matched maze routing
result between the respective corners using 2 segment types.
(segment types 1 and 2)
After computing the maze routing solution corresponding to each
graph edge, set the edge costs to be equal to the routing cost of the
route.
Computing the shortest path on this graph will give the best
combination of different maze routing solutions.
Experimental Results
Conclusions
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This paper proposed a maze routing algorithm that
satisfied exact matching constraints for analog nets.
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Experiments show significant congestion reductions
compared to the previous work.