CS137: Dynamic Programming Electronic Design Automation Solution - - PDF document

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CS137: Dynamic Programming Electronic Design Automation Solution - - PDF document

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CS137b: Day2 CS137: Dynamic Programming Electronic Design Automation Solution Solution described is general instance of dynamic programming Require: optimal solution to subproblems is optimal solution Day 10: February 1, 2006

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CALTECH CS137 Winter2006 -- DeHon

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CS137: Electronic Design Automation

Day 10: February 1, 2006 Dynamic Programming

CALTECH CS137 Winter2006 -- DeHon

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Dynamic Programming Solution

  • Solution described is general instance of

dynamic programming

  • Require:

– optimal solution to subproblems is optimal solution to whole problem – (all optimal solutions equally good) – divide-and-conquer gets same (finite/small) number of subproblems

  • Same technique used for instruction selection

CS137b: Day2

CALTECH CS137 Winter2006 -- DeHon

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Dynamic Programming

  • Two Examples

– SPLASH sequence matching/edit distances

  • O(N2) operation in O(N) time with O(N)

hardware

– CMU parenthesis matching

  • O(N3) operation in O(N) time with O(N2)

hardware

CALTECH CS137 Winter2006 -- DeHon

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Sequence Matching

  • Find edit distance between two strings

– E.g.

  • Insert cost 1
  • Delete cost 1
  • Replace cost 2
  • Match 0
  • Primary Application:

– DNA Sequence comparison – Often compare new sequence against database

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Edit Example

  • SHMOO

– Add E (cost 1)

  • SHMOOE

– Remove M (cost 1 + 1=2)

  • SHOOE

– Replace O with R (cost 2+2=4)

  • SHORE

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Dynamic Programming

  • Build a table representing string prefixes

– Only m×n cases compute

  • Fill in costs
  • Cell (m,n) is

result

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Local Move Costs

  • D(0,0) = 0
  • D(i,0)=D(i-1,0)+Delete(Si)
  • D(0,j)=D(0,j-1)+Insert(Tj)
  • D(i,j)=min

D(i-1,j)+Delete(Si) D(i,j-1)+Insert(Tj) D(i-1,j-1)+Replace(Si,Tj)

[Constant work per cell to fillin]

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Edit Distance Table

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Systolic Array

  • Feed Strings from opposite ends
  • Compute along diagonals

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Systolic Array

  • Feed Strings from opposite ends
  • Compute along diagonals

O O M H S S H O R E

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Systolic Array

  • Feed Strings from opposite ends
  • Compute along diagonals

O O M H S S H O R E

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Systolic Array

  • Feed Strings from opposite ends
  • Compute along diagonals

O O M H S S H O R E

When Src[i] and Targ[j] line up, Compute cell (i,j)

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Systolic Array

  • PEDist = minimum of

TDin+Cost(Delete(SCin)) SDin+Cost(Insert(TCin)) PEDist + Cost(Substitute(TCin,SCin))

  • SDout=TDout=PEDist
  • ….plus details for edge cases

D(i,j)=min D(i-1,j)+Delete(Si) D(i,j-1)+Insert(Tj) D(i-1,j-1)+Replace(Si, On previous cycle: Cell computes (i-1,j-1) T-neighbor (i-1,j) S-neighbor (i,j-1)

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In Operation

(2,0) (1,0) (1,0) (0,0) (0,1) (0,1) (0,2) (0,2) (2,0) (3,0) 2 1 1 1 1 2 2 2 3 S H H S M

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(2,0) (1,0) (1,0) (0,0) (0,1) (0,1) (0,2) (0,2) (2,0) (3,0) 2 1 1 1 1 2 2 2 3 S H H S M (2,0) (2,0) (1,0) (1,1) (0,1) (0,2) (0,2) (0,3) (3,0) (3,0) S H O H S M

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(2,0) (2,0) (1,0) (1,1) (0,1) (0,2) (0,2) (0,3) (3,0) (3,0) 2 2 1 1 2 2 3 3 3 S H O H S M

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(3,0) (2,0) (2,1) (1,1) (1,2) (0,2) (0,3) (0,3) (3,0) (4,0) S H O M H S O (2,0) (2,0) (1,0) (1,1) (0,1) (0,2) (0,2) (0,3) (3,0) (3,0) 2 2 1 1 2 2 3 3 3 S H O H S M

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(3,0) (2,0) (2,1) (1,1) (1,2) (0,2) (0,3) (0,3) (3,0) (4,0) 3 2 1 1 2 3 3 3 4 S H O M H S O

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(3,0) (3,1) (2,1) (2,2) (1,2) (1,3) (0,3) (0,4) (4,0) (4,0) S H O R M H S O (3,0) (2,0) (2,1) (1,1) (1,2) (0,2) (0,3) (0,3) (3,0) (4,0) 3 2 1 1 2 3 3 3 4 S H O M H S O

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(3,0) (3,1) (2,1) (2,2) (1,2) (1,3) (0,3) (0,4) (4,0) (4,0) 3 2 1 1 2 3 4 4 S H O R M H S O

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Edit Distance Table

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Details/Variations

  • Can have one stationary

– Unidirectional: compute row at a time – Most useful when matching single target to large collection of sources

  • Only need constant state per cell

– Delta from neighbor bounded – Similarly, small, constant-width cost datapaths

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Implementation

  • On Splash2
  • Board with 16 XC4010s
  • 12MHz
  • 16PEs per XC4010

– XC4010 has 400CLBs=800 4-LUTs

  • 50 4-LUTs/PE

– Entire Splash 2: 12,800 4-LUTs

  • Less than XC2V6000 (40%)

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Performance (un-normalized)

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Normalized Computational Density

(CM5 used Sparc Processors)

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Non-Local

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Sub-dividing into Trees

  • Optimal parenthesization
  • Deciding where to split a tree

– Perhaps for covering

  • Search Tree

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Parenthesization

  • Have an associative sequence of operations
  • What is least cost way to parenthesize?

– E.g. 1 2 3 4 – (((1 2) 3 4) – (((1 (2 3)) 4) – ((1 2) (3 4)) – (1 ((2 3) 4)) – (1 (2 (3 4)))

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Abstract Covering Problem

  • Given: Graph (V,E) with a single weight

(area) on each node and two weights (IO, cost) on the edges.

  • Cluster nodes into subsets Vi, such that

Σ (Cost(Vi)) minimized IO(Vi) < IO limit A(Vi) < Area limit Cost(Vi) = Σ(cost(e) | e ∈ E st. e1 ∈ Vi and e2 ∉Vi) PEs PE Communication

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Idea

  • If we had an ordering of nodes

– (wishful thinking)

  • Then easy to know how to include more

– Just pick the next node

  • Order: 1D list of nodes
  • Cluster: a contiguous sequence of

nodes in list

– Specify start, finish

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From Sequence to Clusters

  • Easy to know if a contiguous

subsequence

– Meets area constraints – Meets io constraints

  • Cover

– Set of (non-overlapping) subsequences – Include all nodes

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Feasible Clusters (mult16a)

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Covering

  • Not clear when to put more or less stuff

in a cluster…versus leave with next cluster

– Can’t build clusters greedily

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Dynamic Programming

  • For each subsequence start,end

– Either the area and io match – OR want to find a breakpoint between cluster sets

  • Cluster sets startmidpoint, midpointend may

each either be single or multiple clusters

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Minimization Problem

  • c(i,j)=w(i,j)+mink(c(i,k)+c(k,j))
  • Also filling in a table
  • Work per table entry O(N)

– Must look at all k’s

  • O(N3) total work

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Systolic Solution

  • Solve in O(N) time
  • Using O(N2) hardware
  • Each PE is one cell in table

– PE computes the local min

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Challenge

  • Getting right pair of data to show up on each

cycle

  • c(i,j)=w(i,j)+mink(c(i,k)+c(k+1,j))
  • E.g. (5,0) wants to see

– (5,1) (0,0) – (5,2) (1,0) – (5,3) (2,0) – (5,4) (3,0) – (5,5) (4,0)

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Computational Array

(6,6) (6,5) (5,5) (6,4) (5,4) (4,4) (6,3) (5,3) (4,3) (3,3) (6,2) (5,2) (4,2) (3,2) (2,2) (6,1) (5,1) (4,1) (3,1) (2,1) (1,1) (6,0) (5,0) (4,0) (3,0) (2,0) (1,0) (0,0)

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Computational Array

(6,6) (6,5) (5,5) (6,4) (5,4) (4,4) (6,3) (5,3) (4,3) (3,3) (6,2) (5,2) (4,2) (3,2) (2,2) (6,1) (5,1) (4,1) (3,1) (2,1) (1,1) (6,0) (5,0) (4,0) (3,0) (2,0) (1,0) (0,0)

(5,1) (0,0) (5,2) (1,0) (5,3) (2,0) (5,4) (3,0) (5,5) (4,0)

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Systolic Algorithm

  • For cell at distance t from edge

– Has completed data at time 2t – Send data along row and column

  • Data travels 1 cell / cycle for t units of time
  • Then travels 2 cells / cycle for rest of time

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Computational Array

(6,6) (6,5) (5,5) (6,4) (5,4) (4,4) (6,3) (5,3) (4,3) (3,3) (6,2) (5,2) (4,2) (3,2) (2,2) (6,1) (5,1) 8 (4,1) (3,1) (2,1) (1,1) (6,0) (5,0) 7 5 3 1 (0,0)

(5,1) (0,0) (5,2) (1,0) (5,3) (2,0) (5,4) (3,0) (5,5) (4,0)

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Computational Array

(6,6) (6,5) (5,5) (6,4) (5,4) (4,4) (6,3) (5,3) (4,3) (3,3) (6,2) (5,2) 6 (4,2) (3,2) (2,2) (6,1) 7 (4,1) (3,1) (2,1) (1,1) (6,0) (5,0) 7 5 3 (1,0) 2 (0,0)

(5,1) (0,0) (5,2) (1,0) (5,3) (2,0) (5,4) (3,0) (5,5) (4,0)

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Computational Array

(6,6) (6,5) (5,5) (6,4) (5,4) (4,4) (6,3) (5,3) 4 (4,3) (3,3) (6,2) 5 (4,2) (3,2) (2,2) (6,1) 6 (4,1) (3,1) (2,1) (1,1) (6,0) (5,0) 6 5 (2,0) 4 (1,0) (0,0)

(5,1) (0,0) (5,2) (1,0) (5,3) (2,0) (5,4) (3,0) (5,5) (4,0)

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Computational Array

(6,6) (6,5) (5,5) (6,4) (5,4) 2 (4,4) (6,3) 3 (4,3) (3,3) (6,2) 5 (4,2) (3,2) (2,2) (6,1) 7 (4,1) (3,1) (2,1) (1,1) (6,0) (5,0) 7 (3,0) 6 (2,0) (1,0) (0,0)

(5,1) (0,0) (5,2) (1,0) (5,3) (2,0) (5,4) (3,0) (5,5) (4,0)

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Message

  • With a little bit of coordination

– Can solve dynamic programming problems with parallelism – Using simple, nearest-neighbor connections

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Admin

  • Friday: GraphStep Programming
  • Baseline comparison due Friday
  • Monday: Sorting