Transcript Fast computation of maximum - uni

Lecture 4 : Accelerated Cascading
and Parallel List Ranking
• We will first discuss a technique called
accelerated cascading for designing very
fast parallel algorithms.
• We will then study a very important
technique for ranking the elements of a list
in parallel.
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Advanced Topics in Algorithms and Data Structures
Fast computation of maximum
Input: An array A holding p elements from a linearly
ordered universe S. We assume that all the elements in
A are distinct.
Output: The maximum element from the array A.
We use a boolean array M such that M(k)=1 if and only
if A(k) is the maximum element in A.
Initialization: We allocate p processors to set each entry
in M to 1.
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Advanced Topics in Algorithms and Data Structures
Fast computation of maximum
Step 1: Assign p processors for each element in A,
p2 processors overall.
•Consider the p processors allocated to A(j). We
name these processors as P1, P2,..., Pi,..., Pp.
•Pi compares A(j) with A(i) :
If A(i) > A(j) then M(j) := 0
else do nothing.
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Advanced Topics in Algorithms and Data Structures
Fast computation of maximum
Step 2: At the end of Step 1, M(k) , 1  k  p
will be 1 if and only if A(k) is the maximum
element.
•We allocate p processors, one for each
entry in M.
•If the entry is 0, the processor does nothing.
•If the entry is 1, it outputs the index k of the
maximum element.
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Advanced Topics in Algorithms and Data Structures
Fast computation of maximum
Complexity: The processor requirement is p2
and the time complexity is O(1).
• We need concurrent write facility and
hence the Common CRCW PRAM model.
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Advanced Topics in Algorithms and Data Structures
Optimal computation of
maximum
• This is the same algorithm which we used
for adding n numbers.
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Advanced Topics in Algorithms and Data Structures
Optimal computation of
maximum
• This algorithm takes O(n) processors and
O(log n) time.
• We can reduce the processor complexity
to O(n / log n). Hence the algorithm does
optimal O(n) work.
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Advanced Topics in Algorithms and Data Structures
An O(log log n) time algorithm
• Instead of a binary tree, we use a more complex
2k
tree. Assume that n  2 .
2k 1
• The root of the tree has 2  n children.
2k i1
• Each node at the i-th level has 2
children
for 0  i  k  1.
• Each node at level k has two children.
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Advanced Topics in Algorithms and Data Structures
An O(log log n) time algorithm
Some Properties
• The depth of the tree is k. Since
n  2 , k    loglog n 
2k
• The number of nodes at the i-th level is
2k 2k i
2
,for 0  i  k.
Prove this by induction.
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Advanced Topics in Algorithms and Data Structures
An O(log log n) time algorithm
The Algorithm
• The algorithm proceeds level by level,
starting from the leaves.
• At every level, we compute the maximum
of all the children of an internal node by
the O(1) time algorithm.
• The time complexity is O(log log n) since
the depth of the tree is O(log log n).
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Advanced Topics in Algorithms and Data Structures
An O(log log n) time algorithm
Total Work:
• Recall that the O(1) time algorithm needs
O(p2) work for p elements.
2k i1
• Each node at the i-th level has 2 children.
• So the total work for each node at the i-th
2k i1 2
level is O(2 ) .
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Advanced Topics in Algorithms and Data Structures
An O(log log n) time algorithm
Total Work:
2k 2k i
• There are 2
nodes at the i-th level.
Hence the total work for the i-th level is:
2k i1 2
O(2
2k 2k i
) 2
 O(2 )  O(n)
2k
• For O(log log n) levels, the total work is
O(n log log n) . This is suboptimal.
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Advanced Topics in Algorithms and Data Structures
Accelerated cascading
• The first algorithm which is based on a
binary tree, is optimal but slow.
• The second algorithm is suboptimal, but
very fast.
• We combine these two algorithms through
the accelerated cascading strategy.
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Advanced Topics in Algorithms and Data Structures
Accelerated cascading
• We start with the optimal algorithm until
the size of the problem is reduced to a
certain value.
• Then we use the suboptimal but very fast
algorithm.
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Advanced Topics in Algorithms and Data Structures
Accelerated cascading
Phase 1.
• We apply the binary tree algorithm, starting
from the leaves and upto log log log n
levels.
• The number of candidates reduces to
n
2log log log n
n

.
log log n
• The total work done so far is O(n) and the
total time is O(log log log n) .
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Advanced Topics in Algorithms and Data Structures
Accelerated cascading
Phase 2.
• In this phase, we use the fast algorithm on
n

) remaining candidates.
the n  O(
log log n
• The total work is O(n log log n)  O(n) .
• The total time is O(log log n)  O(log log n) .
• Theorem: Maximum of n elements can be
computed in O(log log n) time and O(n)
work on the Common CRCW PRAM.
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Advanced Topics in Algorithms and Data Structures