Jiwon Seo

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Parallel Max-Flow Algorithm

• Jiwon Seo
• May 31st, 2007

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Max-Flow Algorithm

• Finding maximum flow from source to sink from a
  directed graph (edges have capacities).
• Two possible approaches
• Finding augmenting path from src to sink
  (augmenting path)
• Push as much flow as possible from src and for
  nodes with excess, try to push excess towards
  sink

Augmenting path
Push from node to node
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Push Relabel Algorithm

• Make a height variable for each node analogous
  to water pressure
• Make a excess variable for each node to hold
  excess flow until pushing it to another node
• After first push of flow from the source
  (saturating all of its edge capacities) do one of
  two operations on each node(active node)
• Push push the excess to the node with lower
  height (water pressure)
• Relabel increment the height of the current node
  so that it can do push operation
• Constraints for algorithm correctness
• height(u) lt height(v) 1 for residual edge u -gt
  v

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Push Relabel Algorithm contd

• Continue push and relabel operation on the active
  nodes while there is any node with excess except
  source and sink
• Push operation look at the neighbor nodes and
  pick one node that satisfies the constraint
  height(v) height(u)1
• Relabel operation if push operation fails for a
  node, increment the height of that node to
  minimum height among neighbors 1

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Parallel Push Relabel Algorithm

• Sequential algorithm consists of many of
  operations on active nodes
• Parallel algorithm distribute active nodes to
  threads
• Have a shared queue of active nodes
• Have a local queue of active nodes for each
  thread
• Each thread should operate on nodes in its local
  queue, sometimes fetching from, queueing to
  shared queue to balance loads among threads.
• Synchronization
• Push operation acquire locks for two nodes (to
  avoid deadlock, acquire both locks only when both
  are free)
• Relabel operation acquire lock for the node
• Shared queue acquire lock when communicating to
  shared queue

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Algorithm Analysis

• Communication
• Explicit communication with shared queue
• Implicit communication with neighbor nodes
• Operations
• All arithmetic operations in push/relabel are
  very simple there are at most a few additions
  and one assignment for each operation
• Communication / computation is very high

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Performance Analysis

• Assumptions
• Its not good to leave an active node for long
  not being pushed experience
• of Active nodes can decrease or increase need
  to rebalance loads
• Communicating with shared queue too often will be
  an overhead
• Parameters
• Pr probability of inteference among active nodes
  (function of thread and local queue size)
• p1 penalty for not processing a node for long
  time
• p2 penalty for load inbalance
• f communication frequency
• Pc penalty for communication
• Communication overhead Pc f (f is
  communication frequency)
• Penalty for less communication Pr(p1p2) 1/f
• gt Performance Pcf Pr(p1p2)1/f

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Experiment

• Implemented parallel push/relabel max flow
  algorithm without any heuristics that speeds up
  sequential algorithm
• Run the algorithm with random grid graph (size
  10000)
• on Ultra Sparc 2 SMP 16 cpu
• and on Niagara CMP 8 cpu, 4 threads/core
• Lock overhead was around10

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Experimental Result
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Application Evaluation

• Programming Language Mapping
• PL should provide a way to optimize data layout
  in memory
• Various (concurrent) data structure will be handy
• Java gtgt C/C gt Matlab, Python, Perl, Ruby, etc
• Architecture Implication
• Transaction instead of lock TCC
• Multiple thread per core NIAGARA

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Reference

• Anderson and Setubal, On the parallel
  implementation of Goldberg's maximum flow
  algorithm
• Bader and Sachdeva, A Cache-Aware Parallel
  Implementation of the Push-Relabel Network Flow
  Algorithm and Experimental Evaluation of the Gap
  Relabeling Heuristic
• Quinn and Deo, Parallel Graph Algorithm
• Goldberg, Recent Developments in Maximum Flow
  Algorithms