Tree Contraction

1
Tree Contraction

• Label leaf nodes 1...n
• Rake odd indexed leaf nodes
• Left Compress
• Right Compress
• Left Compress
• Right Compress
• Key avoid memory conflicts

2
Tree Contraction for arithmetic expression

• each node has an expression two variables x,y,
• and four coefficients a,b,c,d,e,f,g
• Rake replace the variable with the corresponding
  value
• Compress
• note that node v has two children x and y, and x
  is compressed to v, then the expression
  associated with x has only one variable.
• v has expression is f(x,y), and x has expression
  g(x),
• the expression of v gt f(g(x),y)
• general formula

Thus, compressing corresponds to computing the
new coefficients.
3
Work Complexity of Compression

• Efficient? each compress involves 20 operations
• Note that the leaf node contains only variable or
  constant
• The coefficients are mostly zero if the
  operations do not involve division
• Worth while to compare this technique with other
  know optimization techniques (Cook etc.,)

4
Tree Compression Applications

• In general, let S be a set.
• F?f f S?S ?S
• If S is finite, any tree contraction can be done
  efficiently
• Use finite table
• Number of children
• rake x to v add value of x to y
• compress x to v add value of x to y
• Height?

5
Vertex Cover of a tree and Tree Contraction

• Vertex cover of a tree
• F not selected
• R selected
• Mark all nodes with F
• Let x be the label of parent, y be the label of
  child
• f(x,y) new label after raking or compressing
  the child

Sequential algorithm computing f
6
VC of a tree

• How to do in parallel?
• Apply tree contraction by applying f whenever
  rake or compress is done.
• How to handle node degree more than 2?
• Convert it into a binary tree
• For VC, many dummy nodes.
• Labeling function should be adjusted? gt Not
  necessary

a
a
?
a2
d
c
d
b
c
a1
b
a0
7
Other applications of Tree Contraction

• Minimum Independent set for a tree
• Dominating set for a tree
• For Eulerian tour technique
• Biconnected components
• Least Common Ancestor

8
Least Common Ancestor

• To be prepared

9
Evaluating Straight Line Program in
parallel(Dynamically)

• Straight Line Program