IOComplexity of Graph Algorithms

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I/O-Complexity of Graph Algorithms
Kameshwar MunaglaAbhiram Ranade Presenter Li
Rui
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Outline

• Parallel disk model
• Lower bound of graph problem (edge list version)
• Connected components, etc.
• Computational Model for lower bound
• P-way Indexed I/O tree
• Decision tree
• Duplicate elimination (DE)
• Lower bound
• Upper bound
• Connected component labeling (CC)
• Lower bound reduce DE to CC
• Upper bounds
• BFS, Preprocessing step for sparse graph

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Parallel Disk Model
D disks
M
P
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Parallel Disk Model

• Global disk address space is striped across the
  disks
• Record i in track j of disk k is
  iB(k-1)BD(j-1) record in the sequence
• Parallel access of different track in different
  disks

disk 1
disk 2
disk D
Track 1

Track 2
Track 3
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Basic algorithms

• M gt BD
• Scan(N) O(N/BD)
• Number of disk accesses needed to read N records
• N/BD tracks, or parallel steps
• Sort(N)
• Optimum to sort N records
• phases using
  I/Os each

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Connected Component Labeling

• Input An undirected graph G(V, E)
• As edge list (1, 3), (1, 4), (2, 1), (2, 5),
• In range 1,V
• Output A array L1V, such that L(i)L(j) iff u
  and v are in the same component
• CC(E,V)

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Duplicate elimination

• Remove duplicates in a multiset with elements in
  a finite domain
• Used for lower bounds for many graph problems
• Input a sequence of N integers records each in
  range 1..P
• Output C1P in P records on disk, Ci 1 if i
  occurs in the input, and 0 otherwise.
• DE(N, P)

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Main Results

• Assumption
• V, P are larger than M, logV and log P are
  smaller than Blog(M/B)
• Duplicate elimination
• DE(N,P)
• A matching upper bound
• Connected Components Labeling
• CC(E,V)
• Upper bound
• Breadth First Search
• Optimum for E/V gt BD
• Preprocessing for sparse graph
• Transforms V vertices to E/BD vertices

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Decision tree recall

• Comparison based
• Binary tree
• Tree height
• Lower bound for sorting B elements, B! possible
  leafs,
• Merge two sorted lists with B, and M-B elements
  respectively,

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P-way indexed I/O tree

• Extend I/O Tree
• Allows P-way indexed accesses to disk
• Restricted model D1
• Three types of nodes
• Comparison node (of main memory records i j)
• Indexed Input node (I/O node)
• Which track is to read lti,m1,m2,,mB,t1,t2,,tPgt
• Indexed Output node (I/O node)
• Which track is to write lti,m1,m2,,mB,t1,t2,,tPgt
  
• Given such an I/O tree T
• Let I/OT(x) be I/O nodes for an input instance
  x,
• Let I/OT be the maximum of I/OT(x). To Lower
  Bound!
• Can be transformed to a binary decision tree??

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P-Way indexed I/O tree and decision tree

• Lemma 2.1 (D1)
• N the number of records in the problem of I/O
  tree T, theres a decision tree Tc which can be
  constructed from T and
• Theorem 2.1 (general D)
• The transformation satisfies total ordering of
  records in main memory!
• It means the decision tree has some constraint!

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Duplicate elimination lower bounder

• Lemma 3.1
• The depth of any decision tree solving DE(N,P) is
  at least Nlog(P/2)
• there are (P/2)N odd instances. Take logarithm!
• having
• So
• With some rearranging

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Duplicate elimination Upper bound

• Upper bound Algorithm
• Divide N input into N/P groups of P records each
• Scan(N)
• Sort each group construct the solution of each
  group, get N/P bit vectors for each group,
  C1..P
• In each group sorting Sort(P), construction
  Scan(P)
• Total N/P Sort(P) Scan(N)
• Compute the OR of all bit vectors
• Scan(N)
• Total time O(N/PSort(P) Scan(N)) O(N/P
  Sort(P))
• Match the lower bound

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Connected Components Lower Bound

• Idea
• Reduce a similar DE(N,P) problem to a CC(E,V)
  problem.
• Lower bound CC(E,V) to lower bound of DE(N,P)
• Another Duplicate elimination problem DE(N,P)
• A sequence of N records in P1,2P, P lt N lt P2
• b can be divided into P contiguous sequences
  each of length N/P, each sequence has distinct
  elements.
• The depth of any decision tree solving DE(N,P)
  is
• Similarly we have

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DE(N,P) to CC(E,V)

• Reduce DE(E-V/2,V/2) to CC(E,V)
• Let bj to be the jth subsequence with length
  (E-V/2)/(V/2)
• If V/2 i appears in bj, edge (j, V/2i)
• edge(i, i 1) for every i 1..V/2-1
• Exactly E-1 edges and V nodes
• If we solve CC(E,V) and get L1..V
• V/2i appears in b if and only if L1
  LV/2i
• So from lower bound of DE(E-V/2,V/2) we have
  lower bound of CC(E,V)

j

V/22
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3
V/2
V/21
V
V/2i
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Connected Components Upper Bound

• Algorithm (BFS)
• First convert edge list to adjacent list form
• In sorting time
• Using same idea taught in the lecture (L7)
• L(0) r
• d 0
• While L(d) not empty do
• Find L(d)s neighbors, remove duplicates
• Remove all vertices in (L(d) U L(d 1))
• The resulting set is L(d1)
• d d1
• Plug in the expression for parallel disk model,
  similar result

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Connected Components Upper Bound

• Upper bound does not match lower bound
• Optimization for sparse graph? By preprocessing
• PRAM algorithm, Cole and Vishkin
• Chi et al.
• a)identify a group of vertices in same group
• b)identify a leader in the group
• c)move edges incident on every vertex to its
  leaer
• Construct the labeling from the labeling of
  induced graph
• Transform V vertices to E/BD vertices
• Iterative steps Halve of leaders each time
• Nearly optimal algorithm for sparse graphs

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Other graph problems

• Lower bound also applies for
• Connected components in bipartite graphs
• Biconnected components labeling
• Minimum spanning trees
• Ear decomposition
• Their decisions versions
• All related to DE(N,P) problem

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Open problems

• Lower bound assumes input of edge-list format
• Other format
• Gap between CC(E,V)s Upper Lower bounds.

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Summary

• Lower bound of graph problem
• edge list parallel disks model
• P-way Indexed I/O tree, Decision Tree
• Duplicate elimination
• Connected Component