List Ranking Moon Jung Chung

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List RankingMoon Jung Chung
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Pointer Jumping Review
How to compute a prefix on a linked list?
If NEXTi ! NILL then Xi lt- Xi
XNEXTi NEXTi lt- NEXTNEXTi
work O(nlogn) is not optimal
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Pointer Jumping Review
How to compute a prefix on a linked list?
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A node does not not jump if it is jumped
over. How to know to jump or not?
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Random Pointer Jumping
pointer jump only if female-male
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4
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A-gtB-gtC is jumped to A-gtC. B is jumped over. No
jump crossing. i.e., A-gt B-gtC and B-gtC-gtD
simultaneously Once a node is jumped over, the
node becomes inactive. In reconstruction phase,
process are activated in reverse order.
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Random list ranking algorithm

• ranki 1, activei true, t1
• in parallel while nexthead ! nil
• if (activei and nexti! nil)
• sexi random M,F
• if sexi F and sexnexti M
• timenexti t
• activenexti false
• ranki ranki
  ranknexti
• nexti nextnexti
• t
• for kt down to 1
• if (timei t and nexti ! nil)
• ranki ranki ranknexti

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complexity of easy random list ranking

• Prob that a cell is not jumped over 3/4
• Probability that a cell is not jumped over after
  k-iteration (3/4)k
• Pi cell is not not jumped over after k rounds.
• At least one cell is not jumped over after k
  rounds
• P1 ? P2 ... ? Pn ? P1 P2 ... Pn 1/n(c-1)
  if we choose (3/4)k lt 1/nc
• So, for (3/4)k lt 1/nc ltgt k clog 4/3n,
• the error that it goes more than clog 4/3n rounds
  becomes 0.

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Random ranking algorithm with O(n/logn) PEs,
O(logn) time

• Shared memory model
• each processor handles logn cell as a queue
• processor only look at the top of the queue
• jump over top i only if jprevi is top.
• Then nexj nexti and remove top from the
  queue
• Still has memory imbalance
• Each PE can splice out
• Slice_out(i)
• prevnexti previ and nextptevi
  nexti
• rankprevi rankprevi ranki
• if nexti ! nil then prevnexti previ

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Algorithm

• Initially
• ranki,k 1, sexi,k F
• topi Qi.1 // Queue of PE is 1st element.
• sexnil M
• t1
• while nexthead ! nil
• if position of topi ? logn
• sextopi randomM,F
• if (sexprevtopi F and sextopi M
• spliceout(topi)
• position of topi points next in the queue
  of PE i
• splicetimetopi t
• t
• Observation Prob. that a particular queue is not
  empty after 16logn rounds is lt 1/n.

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Optimal Deterministic List Ranking Algorithm

• How to pick 1/4 PEs with guarantee?
• k-Ruling set
• G(V,E) be the directed graph representation of a
  linked list L with vertices v1,...,vn.
• A subset S of V is a ruling set if
• 1. No two vertices in S are adjacent.
• 2. For each vertex vi in V, there is a directed
  path from some vertex in S to vi with length at
  most k.
• S head of each segment of linked list, k is the
  length from the ruler.
• If we have a k-ruling set, every time we can
  select rulers (n/k elements) without conflict.
• Observation k-coloring of a linked list can be
  converted into k-ruling set.
• How to find k-ruling set?

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Deterministic Coin Tossing and k-Coloring
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List Ranking and Ruling Set

• N/logN PEs
• Each PE has logN data
• Each PE has N/logN data
• 1. Find a 6 ruling set
• 2. Apply splice operation for ruling set
• Each iteration
• c(N/logN) top element be eliminated
• C(N/logN) k N , k O(logN)
• Complexity O(logN (logN)logN)

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New Divide and Conquer Approach

• List-Ranking (A, m)
• 1. If m lt n/logn, apply list ranking with each PE
  one node
• 2. Find Ruling Set
• 3. Reduce list to A
• 4. List-Ranking(A, m)
• Depth of recursion O(loglogN)
• Complexity logn loglogN