Optimal Placement of Bank SelectionInstructions in Polynomial Time

Philipp Klaus Krause

2013-06-19

Table of Contents

1 Bank Selection

2 Tree Decompositions

3 Algorithm

4 Example

5 Summary

Table of Contents

1 Bank Selection

2 Tree Decompositions

3 Algorithm

4 Example

5 Summary

Partitioned Memory Architectures

Allow using a physical address space larger than the logicalone

Common in embedded systems

Bank selection instructions change which part of the physicaladdress space is mapped into a window in logical addressspace

Example Architecture: Z180

Z8018xFamily MPU User Manual

56

UM005003-0703

Figure 24. Physical Address Transition

MMU Block Diagram

The MMU block diagram is depicted in Figure 25. The MMU translates internal 16-bit logical addresses to external 20-bit physical addresses.

Figure 25. MMU Block Diagram

Logical Address Space

Physical Address Space

Common Area 1

Bank Area

Common Area 0

Common Base

Bank Base

FFFFH

0000H

FFFFFH

00000H

0

+

+

+x y z

z

y

x

MMU Common/Bank AreaRegister; CBAR (8) Memory

Management

8PA12—PA19

LA12—LA154

MMU Common Base Register; CBR (8)MMU Bank Base Register; BBR (8)

LA: Logical AddressPA: Physical Address

Internal Address/Data Bus

Unit

0Zilog Z180 manual

Placement of Bank Selection Instructions

Inserting bank selection instructions optimally is hard

Previous approaches are not optimal or do not havepolynomial runtime

Graph structure theory allows an efficient, optimal approach

Table of Contents

1 Bank Selection

2 Tree Decompositions

3 Algorithm

4 Example

5 Summary

C Code

int binomial_coefficient(int n, int k){ int i, delta, max, c; if(n < k) return(0); if(n == k) return(1); if(k < n - k) { delta = n - k; max = k; } else { delta = k; max = n - k; } c = delta + 1; for(i = 2; i <= max; i++) c = (c * (delta + i)) / i; return(c);}

Control Flow Graph

int binomial_coefficient(int n, int k){ int i, delta, max, c; if(n < k) return(0); if(n == k) return(1); if(k < n - k) { delta = n - k; max = k; } else { delta = k; max = n - k; } c = delta + 1; for(i = 2; i <= max; i++) c = (c * (delta + i)) / i; return(c);}

Tree Decomposition

Given a graph G with nodes Π a tree decomposition of G is a pair(T ,X ) of a tree T and a family X = {Xi | i node of T} ofsubsets of Π:⋃

i node of T Xi = Π,

∀{x , y} edge in Π: ∃i node of T : x , y ∈ Xi ,

∀x ∈ Π: {i node of T | x ∈ Xi} is connected in T .

a

b

c

d

e

f

g

h

i

j k

a, f, b

b, f, g

b, g, c

c, h, d

b, f, j

j, i, k

a, e, f

f, i, j

g, c, h

Tree Width

The width of a tree decomposition (T ,X ) ismax{|Xi | | i node of T} − 1.

The tree-width tw(G ) of a graph G is the minimum widthover all tree decompositions of G .

The tree-width of control-flow graphs is bounded1.

1M. Thorup, 1998

Nice Tree Decomposition

A tree decomposition (T ,X ) of a graph G is called nice, iff

T is oriented, with root t, Xt = ∅.Each node i of T is of one of the following types:

Leaf node, no childrenIntroduce node, one child j ,Xj ( Xi

Forget node, one child j ,Xj ) Xi

Join node, two children j1, j2,Xi = Xj1 = Xj2

a

b

c

d

e

f

g

h

i

j k

a, f, b

b, f, g

b, g, c

c, h, d

b, f, j

j, i, k

a, e, f

f, i, j

g, c, h

a, f, b

b, f, g

b, g, c

c, h, d

b, f, j

j, i, k

a, e, f

f, i, j

a, f

f, b

aa, e

b, f, gb, f, g

b, g

g, c, h

g, c

c, h

j, f

i, j

b, f

Table of Contents

1 Bank Selection

2 Tree Decompositions

3 Algorithm

4 Example

5 Summary

Notation

Let B be the set of memory banks in the architecture.

Let G = (Π,E ) be the control-flow graph of the program.

Let c : E ×B×B→ R be the cost function.

Let (T ,X ) be a nice tree decomposition of minimum width ofG with root t.

Placement of Bank Selection Instructions

For a fixed set of memory banks B, the problemof Optimal Placement of Bank SelectionInstructions with input program consisting of apartially colored control-flow graph G = (Π,E )and cost function c is the following: Find acompatible coloring f : Π→ B of G , such that∑

e∈Ec(e, f (e0), f (e1))

is minimal over all compatible colorings.

Accumulated cost function

s(i , f ) gives the minimum possible cost for the edges covered inthe subtree rooted at i , excluding the edges between nodes in inXi , when using f as the coloring for Xi .

Leaf: s(i , f ) := 0.

Introduce: s(i , f ) :=

{∞ if coloring incompatible

s(j , f |Xj) otherwise.

Forget: s(i , f ) :=

min

s(j , g) +∑

e∈(E∩(X 2j rX 2

i ))

c(e, g(e0), g(e1))

∣∣∣∣∣∣∣ g |Xi= f

.

Join: s(i , f ) := s(j1, f ) + s(j2, f ).

The optimal coloring in polynomial time

The nice tree-decomposition of minimum width can becalculated in linear time2.

s can be calculated in time O(|T |tw(G )2|B|tw(G)+1).

Extend the single remaining f at the root t to all nodes toobtain the optimal coloring.

2H. Bodlaender 1996

Table of Contents

1 Bank Selection

2 Tree Decompositions

3 Algorithm

4 Example

5 Summary

Example code

i f ( . . . )a c c e s s b ank a ;

e l s edo not acce s s banked memory ;

sw i t c h ( . . . ){ca se 0 :

do not acce s s banked memory ;b reak ;

ca s e 1 :a c c e s s b ank b ; b reak ;

ca s e 2 :do not acce s s banked memory ;b reak ;

.

.

.c a s e n−1:

a c c e s s b ank b ;b reak ;

d e f a u l t :do not acce s s banked memory ;b reak ;

}

a c c e s s b ank a ;

Table of Contents

1 Bank Selection

2 Tree Decompositions

3 Algorithm

4 Example

5 Summary

Summary

Optimal placement of bank selection instructions inpolynomial time

Linear time for fixed number of banks

Previous approaches were not optimal or had no polynomialruntime bounds

Uses concepts from graph-structure theory(tree-decompositions)

Implemented in sdcc