1. Dynamic Programming and Some VLSI CAD Applications
Shmuel Wimer
Bar Ilan Univ. Eng. Faculty
Technion, EE Faculty
May 2012
Dynamic Programming

2. Outline NP Completeness paradox
Efficient matrix multiplication by dynamic programming
Dynamic programming in a tree model
Optimal tree covering in technology mapping
Optimal floor planning
Optimal buffer insertion
Dynamic programming as sequential decision problem
Resource allocation
The knapsack problem
Automatic cell layout generation
Optimal wire sizing
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Dynamic Programming

3. NP Completeness Paradox
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Dynamic Programming

4. j i 1 2 3 4 5 6 7 8 9 10 11 12 13 T F
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Dynamic Programming

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7. Optimal Matrix-Chain Multiplication
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Dynamic Programming

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14. i=1 2 3 4 5 6
i=1 2 3 4 5 6
15125
10500
5375
3500
5000
11875
7125
2500
1000
9375
4375
750
7875
2625
15750 3 5 4 1 2
J=6 5 4 3 2 1
J=6 5 4 3 2 1
m[1..6,1..6]
s[1..6,1..6]
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Dynamic Programming

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Dynamic Programming

17. Elements of Dynamic Programming
A problem exhibits optimal substructure if an optimal solution to the problem contains within it optimal solutions to sub problems.
In a sequence of decisions the remaining ones must constitute optimal solutions regardless of past decisions. (principle of optimality).
The space of sub problems must be small, namely, a recursive solution must solve same problem many times. Optimization problem has overlapping sub-problems.
May 2012
Dynamic Programming

18. Optimal solution is constructed by backtracking.
Overlapping sub-problems called by recursive solution are memorized (encoded in a table), hence addressing their solution only once.
Optimal solution is constructed by backtracking.
May 2012
Dynamic Programming

19. Optimal Tree Covering
A problem occurring in mapping logic circuit into new cell library. Given:
Rooted binary tree T(V,E) called subject tree (cone of logic circuit), whose leaves are inputs, root is an output and internal nodes are logic gates with their I/O pins.
A family of rooted pattern trees (logic cells of library), each associated with a non-negative cost (area, power, delay). Root is cell’s output and leaves are its inputs.
May 2012
Dynamic Programming

20. Find a cover of subject tree whose total sum of costs is minimal.
A cover of the subject tree is a partitioning where every part is matching an element of library and every edge of the subject tree is covered exactly once.
Find a cover of subject tree whose total sum of costs is minimal.
May 2012
Dynamic Programming

21. r s t u
t1 (2)
t2 (3)
t5 (5)
t4 (4)
t3 (3) t4 t1 t3
4+2+3=9 t2 t1 t3
=10 t2 t5
3+5=8
May 2012
Dynamic Programming

22. AOI21 (3)
NAND3 (3)
NAND2 (2)
INV (1)
INV (1)
NAND2 (2) a b c d g e f j h i
NAND2 (5)
NAND2 (8)
INV (9)
AOI21(6)
NAND2 (11)
NAND3 (12)
INV (3)
NAND2 (5)
NAND3 (3)
Observation: pattern p rooted at the root of T(V,E) yields minimal cost only if the cost at any of p’s leaves is minimal, suggesting bottom-up matching algorithm.
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Dynamic Programming

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24. Optimal Buffer Insertionv
u , q ? ? ? ? ? ? ?
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Dynamic Programming

25. Delay Model May 2012 Dynamic Programming R4 4 C4 R2 2 R5 5 C3 R1 C5 11 3 2 4 5 6 7
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Dynamic Programming

26. Bottom-Up Solution RM CM M sub-tree (TM , LM) K (T’K , L’K) (TK , LK)RK CK RN CN N
sub-tree
(TN , LN)
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Dynamic Programming

27. Outline of Algorithm + =
With b nodes, 2b buffer insertions exist. There’s a polynomial solution!
Merging sub-tree solutions at a parent node takes linear time! LM TM LN TN
L’K
T’K + =
May 2012
Dynamic Programming

28. Interconnect Signal Model
driver’s resistance
receiver’s load
line resistance
signal's activity, 0<= AF <=1
line-to-line coupling
Using Elmore delay model, simple, inaccurate but with high fidelity
May 2012
Dynamic Programming

29. Interconnect Bus Modelσ1 A σi σn
σn-1 Wi Si
Si+1 Ri Ci L
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Dynamic Programming

30. Delay and Dynamic Power Minimization
signal’s delay:
signal’s dynamic power:
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Dynamic Programming

31. In 32nm node and beyond spaces and widths are very few discrete values
Minimize bus delay
Minimize bus power
Subject to:
In 32nm node and beyond spaces and widths are very few discrete values
Continuous optimization and its well-known results are invalid. The sizing problem is NP-complete. A pseudo polynomial resource allocation dynamic programming solution is suitable.
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Dynamic Programming

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34. Floorplan and Layout Floorplan Graph representationB1 B2 B8 B7 B2 B8 B9 B7 B1 B9
B12
B10 B5 B3
B10 B3 B5 B4
B12 B6
B11
B11 B6 B4
Floorplan is represented by a planar graph.
Vertices - vertical lines. Arcs - rectangular areas where blocks are embedded.
A dual graph is implied.
May 2012
Dynamic Programming

35. From Floorplan to Layout
Actual layout is obtained by embedding real blocks into floorplan cells.
Blocks’ adjacency relations are maintained
Blocks are not perfectly matched, thus white area (waste) results
Layout width and height are obtained by assigning blocks’ dimensions to corresponding arcs.
Width and height are derived from longest paths
Different block sizes yield different layout area, even if block sizes are area invariant.
May 2012
Dynamic Programming

36. Optimal Slicing Floorplan
Top block’s area is divided by vertical and horizontal cut-lines
Slicing tree. Leaf blocks are associated with areas. v B1 B2 B8 B7 h v B9 B1 B2 B7 h
B12
B12
B10
B11 B3 B4 B5 B6 B8 B9
B10 B3 B5
B11 B6 B4
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Dynamic Programming

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Dynamic Programming

38. Merge horizontally two width-height sets (vertical cut-line)+ = + = + =
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Dynamic Programming

39. h
Size of new width-height list equals sum of lengths of children lists, rather than their product.
May 2012
Dynamic Programming

40. Sketch of Proof
Problem is solved by a bottom-up dynamic programming algorithm working on corresponding slicing tree.
Each node maintains a set of width-height pairs, none of which can be ruled out until root of tree is reached. Size of sets is in the order of node’s leaf count. Sets in leaves are just Bi’s two orientations.
The sets of width-height pairs at each node is created by merging the sets of left-son and right-son sub-trees in time linear in their size.
Width-height pair sets are maintained as a sorted list in one dimension (hence sorted inversely in the other dimension).
Final implementation is obtained by backtracking from the root.
May 2012
Dynamic Programming

41. Automatic Cell Layout Generation
3 step process:
Transistor placement
Interconnect completion
Design rule adherence
Transistor placement comprises:
Transistor P-N pairing
Pair ordering
Pair flipping – optimize cell area, node cap, potential cell abutment, cell’s internal routing b a
Vcc
Vss
Cost=0 a b
Vcc
Vss
Cost=1 a
Vcc b
Vss
Cost=2 a
Vcc b
Vss
May 2012
Dynamic Programming

42. Most cells unfortunately contain more than 4 transistors.
A flip configuration of a pair depends on the flip of its left and right neighbors.
Seek the flip configuration yielding minimal sum of abutment cost.
With n pairs, there are 2n solutions to consider.
Observation: An optimal flip of j+1 pairs subject to given right end configuration of pair j necessitates that the first j pairs have been optimally flipped.
Principle of optimality, optimal sub problem solutions.
Observation: The optimal flip of rest n – j pairs is independent of the first j flips except the right end configuration of pair j.
This defines a state for which only the lowest cost flip of j pairs is of interest.
Dynamic Programming solution is in order. (Bar-Yehuda et. al.)
May 2012
Dynamic Programming

43. State Augmentation stage j stage j+1 a b c d a b c d abutment cost
May 2012
Dynamic Programming

44. Dynamic programming takes O(n) time.
Can be extended to multi-row cell (double height, etc.).
It can be combined in a DFS algorithm which considers simultaneously paring, pair ordering and optimal flip, without any complexity overhead (state augmentation takes O(1) time)
Dynamic programming is solving in fact a shortest path algorithm on the state transition graph.
New litho rules in 32nm and smaller feature size offer many optimization opportunities.
May 2012
Dynamic Programming

45. Resource Allocation
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Dynamic Programming

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Dynamic Programming

47. Elements of Dynamic Programming
Sequential decision making process.
Transition occurs from state to state.
A state is a summary of prior history of the process sufficiently detailed to enable evolution of current alternatives.
Sequential decision process evolves from state to state.
The pair (j,y) is a state in the resource allocation process.
The elements encoded in a state are called state variables.
Principle of optimality states that whatever the initial state is and decisions were, the remaining decisions must constitute an optimal policy.
May 2012
Dynamic Programming

48. Linear Case: Knapsack Problem
May 2012
Dynamic Programming

49. Linear Case: Knapsack Problem
May 2012
Dynamic Programming