Dynamic Programming on SIMD

SIMD (SSE)

A register has say 4 parts, each

part has 8 bits or 16 bits

Each part executes the

same operation but in parallel

Examplex:=x+y

x,y are memory locations of appropriate alignment and each refers to 4 different

variables

Edit Distance between Strings

D

Q

𝐻 (𝑖 , 𝑗 )=𝑚𝑎𝑥 ¿Edit distance between Q[1..j]

and D[1..i]

𝑄 [ 𝑖 ]𝐷 [ 𝑗 ]

𝐷 [ 𝑗 ]

𝑄 [ 𝑖 ]

E(i,j)

E

𝐷 [ 𝑗−1 ]𝐷 [ 𝑗 ]

𝐷 [ 𝑗−1 ]𝐷 [ 𝑗 ]Q

F(i,j)

F

Q Q

𝐷 [ 𝑗 ]QQ

Summarizing

E

F

𝐻 (𝑖 , 𝑗 )=𝑚𝑎𝑥 ¿ A G C T

A -1 2 2 2

G 2 -1 2 2

C 2 2 -1 2

T 2 2 2 -1

Parallel Computing

Mismatch

Insertion in Q

Insertion in D

Find the max value path from each location to

the top left corner

Parallel Computing

Anti-diagonal can be done in parallel

Δ (𝑄 [ 𝑖 ] ,𝐷 [ 𝑗 ])

SIMD?

Δ (𝑄 [ 𝑖 ] ,𝐷 [ 𝑗 ])

The 4 parts go to different memory

locations to pick up

One SIMD register can handle all

values in this chunkPartition into chunks, each chunk should have proper

alignment

Fix: Go Vertical

Δ (𝑄 [ 𝑖 ] , 𝐴 )

Partition into vertical chunks, each chunk should have proper

alignment

𝐴

, , , can be pre-computed and

chunked

Δ (𝑄 [0 ] ,𝐴 )Δ (𝑄 [1 ] , 𝐴)

Δ (𝑄 [𝑚 ] , 𝐴 )

Problem 1: Boundaries𝐴

Each chunk needs a shift

One value comes from the previous chunk y

Δ (𝑄 [0 ] ,𝐴 )Δ (𝑄 [1 ] , 𝐴)

Δ (𝑄 [𝑚 ] , 𝐴 )

x z

• Shift

x ′

Solution 1: Striping𝐴

x’s belong to one chunk, y’s to another

and so on

• ”

xyz

xyz

xyz

x ′y ′z ′

x ′y ′z ′

x ′y ′z ′

xyz

xyz

xyz

Δ (𝑄 [∗] , 𝐴 )

Problem 2: Computing F𝐴

Sequential Dependency

x ′y ′z ′

x ′y ′z ′

Repeat At most number of blocks

Problem 2: Computing F𝐴

Sequential Dependency

x ′y ′z ′

x ′y ′z ′

Repeat Only if F-z’<<1 -

has an entry > H- x’ - -init

Thank You