Deterministic Network Coding by Matrix Completion

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DeterministicNetwork Codingby Matrix Completion

• Nick HarveyDavid KargerKazuo Murota

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Network Coding Example
s
t
b1
b2

• Nodes s and t want to swap bits b1 and b2
• No disjoint paths from s?t and t?s

3
Network Coding Example
b1
b2
s
t

• Key idea allow nodes to encode data
• Send xor b1?b2 on bottleneck edge
• Nodes s and t can decode desired bit

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Multicast in DAGs

• 1 source, k sinks
• Source has r messages in alphabet ?
• Each sink wants all msgs
• Network code function node inputs ? outputs
• Thm ACLY00 Network coding solution exists iff
  connectivity r to each sink

m1
m2
mr

Source
Sinks
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Linear Network Codes

• Treat alphabet ? as finite field
• Node outputs linearcombinations of inputs
• Thm LYC03 Linear codes sufficient for
  multicast in DAGs
• Thm HKMK03 Random linear codes work

A
B
?A?B
?A?B
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Our Contribution (Network Coding)

• Deterministic algorithm for multicast problems
• Derandomization of HKMK algorithm
• Runtime Õ( sinks edges3 )
• Related Work
• Jaggi et al deterministic alg
• Our algebraic technique more general

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Characterizing Multicast Solutions

• Sink receives linear comb of source msgs
• Linear combs have full rank ? can decode!
• Def KM Transfer matrix Mt for sink t
• Thm KM Network coding solution exists iffdet
  Mt ? 0, for all sinks t.

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Expanded Transfer Matrices

• Def Expanded transfer matrix (for sink t)

A 0
Nt
I-F Bt

• Each entry is a number or a variable.
• Lemma det Mt ? det Nt
• Goal substitute values into expanded transfer
  matrices s.t. they have full rank

A matrix completion problem
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Matrix Completion

• Def A completion is an assignment of values to
  the variables that maximizes the rank.
• Thm L79 Choosing random values in Fn gives a
  completion with prob gt 1/2.
• Used in RNC bipartite matching
• Thm G99 Completion in Fn can be
  founddeterministically in O(n9) time.

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Simultaneous Completion

• Does Geelen solve our problem? No.
• We have set of mixed matrices N N1, , Nk
• Each variable appears at most once per matrix
• A variable can appear in several matrices
• Def A simultaneous completion for N is an
  assignment of values to the variables that
  preserves the rank of all matrices
• Hard part matrices overlap, not identical
• Randomized algorithm still works

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Our Contribution (Matrix Completion)

• Let N N1, , Nk be a set of mixed matrices
  of size n x n
• Let X be the variables in N
• Thm A simultaneous completion for N can be found
  deterministically over Fk1 in time
  O( N (n3 log n X n2) )For a single matrix,
  the time is O( n3 log n )

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Mixed Matrices

• Let A Q T be a mixed matrix where
• Q contains only numbers
• T contains only variables (or zeros)
• A is equivalent to
• Numbers and variables in disjoint rows? a
  layered mixed matrix (LM-matrix)

(i.e. rank A rank à - n)
I Q
Ã
diag(z1,,zn)
I T
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Computing rank of LM-matrices

• Thm M93 rank Ã
• maxcols JT,JB JT ? JB
• s.t. in top rows, columns JT are indep
• in bottom rows, columns JB are indep
• Find JT and JB by matroid union algorithm?
  efficient alg to compute rank Ã

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Matrix Completion

• Use black-box rank alg to find completion
• Lemma Let P(x1,,xt) be a multivariatepolynomial
  with all exponents ? d.If q?d then P(x1,,xt) ?
  0 has a solution in Fq.
• Proof Similar to Schwartz-Zippel.

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Simple Completion Algorithm

• Let A be a mixed matrix
• Let X be the variables in A
• det(A) linear in each variable? non-root exists
  over F2

• SimpleCompletion foreach x ? X
• set x 0
• compute rank of A
• if rank has decreased, set x 1

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Sharing work in computing rank

• When fill in one value, matrix doesnt change
  much
• So, reuse info from previous round to quickly
  update rank computation for next matrix entry

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Computing rank by matroid union
1
1
1
1
1
1
1
Ã
z1
z2
x1
x2
z3
x3
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Computing rank by matroid union
1
1
1
MT
1
1
1
1
z1
MB
z2
x1
x2
z3
x3
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Computing rank by matroid union
JT
1
1
1
MT
1
1
1
1
z1
MB
z2
x1
x2
z3
x3
JB
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Independent Matching Approach
CT
1
1
1
MT
1
1
1
1
RT
CB
z1
MB
z2
x1
x2
z3
x3
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Independent Matching Approach
CT
1
1
1
MT
1
1
1
1
RT
CB
z1
MB
z2
x1
x2
z3
x3
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Independent Matching Approach
CT
Cols must be indep.
JT
1
1
1
MT
1
1
1
1
RT
CB
Non-zeroswhere edgesbend
JB
z1
MB
z2
x1
x2
z3
x3
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Computing Rank of Ã

• Compute max indep matching, JT and JB
• rank à JT JB
• Time to find max indep matching
• O(n3 log n) Cunningham 1986
• O(n2.62) Gabow-Xu 1996

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Better Completion Algorithm

• BetterCompletion
• Given A, compute LM-matrix Ã
• Compute max indep matching M for Ã

• Idea Use structure of matching to helpfind
  completion
• Let XM ? X be variables used by JB

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Better Completion Algorithm
z1
z2
x2
x1
z3
x3

• Idea Use structure of matching to helpfind
  completion
• Let XM ? X be variables used by JB
• Lemma Setting x ? X \ XM to 0does not affect
  rank

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Better Completion Algorithm

• BetterCompletion
• Given A, compute LM-matrix Ã
• Compute max indep matching M for Ã
• Foreach x ? X \ XM, set x 0

• Variables in XM are more tricky

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Handling XM

• Example
• Assign x1 the value v

1
1
1
1
1
1
1
CB
z1
x1
z2
x1
x1
x3
z3
x3
x3
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Handling XM

• Example
• Assign x1 the value v
• x1 and its edge disappear
• v appears in top matrix Because A Q T and

1
1
1
v
1
1
1
1
CB
I Q
z1
Ã
diag(z1,,zn) T
z2
x3
z3
x3
x3
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Handling XM

• Example
• Must update M
• All vertices in CB must have a matching edge
• Must add this col to JT
• Must swap a col from JT and add it to JB

1
1
1
v
1
1
1
1
CB
z1
z2
x3
z3
x3
x3
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Handling XM
c2
c4
Swapping Lemma ?v s.t.JT c2 c4 andJB c1
c3 are indep
1
1
1
1
1
1
1
z2
c1
c3
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Handling XM
c2
c4
Swapping Lemma ?v s.t.JT c2 c4 andJB c1
c3 are indep
1
1
1
1
1
1
1
z2
c1
c3
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Handling XM
c2
c4
Thm The value v can be found in O(n2)
time Proof Maintain Gauss-Jordan pivoted copies
of Q. Finding v now trivial.
1
1
1
1
1
1
1
z2
c1
c3
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Better Completion Algorithm

• BetterCompletion
• Given A, compute LM-matrix Ã
• Compute max indep matching M for Ã
• Foreach x ? X \ XM, set x 0
• Foreach x ? XM,
• Use swapping lemma and pivoted forms to
  find a good value v
• Set x v

Time required O(n3 log n)
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Simultaneous Matrix Completion

• Let N N1, , Nk be a set of mixed matrices
• A simultaneous completion for N is an assignment
  of values to the variables that preserves the
  rank of all matrices
• Thm BetterCompletion alg can be extended to find
  simultaneous completion for N over Fk1
• Pf Sketch Run simultaneously on all Ni. Choose
  values v that are good for all Ni.

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Summary

• Network coding model for sending data
• Deterministic alg for multicast
• New deterministic alg for matrix completion