Deterministic Network Coding by Matrix Completion Nick Harvey David Karger Kazuo Murota.
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Transcript of Deterministic Network Coding by Matrix Completion Nick Harvey David Karger Kazuo Murota.
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DeterministicNetwork Coding
by Matrix Completion
Nick HarveyDavid KargerKazuo Murota
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s tb1 b2
Nodes s and t want to swap bits b1 and b2
No disjoint paths from st and ts
bottleneck edge
Network Coding Example
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b1⊕b2
b1⊕b2b1⊕b2
Network Coding Example
s tb1 b2
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 [ACLY’00]: 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 linear
combinations of inputs
Thm [LYC’03]: Linear codes sufficient for multicast in DAGs
Thm [HKMK’03]: 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| * |#edges|3 )
Related Work:Jaggi et al: deterministic algOur 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 0I-F Bt
Nt :=
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 CompletionDef: 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 > 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 Fk+1 in time
O( |N| (n3 log n + |X| n2) )For a single matrix, the time is O( n3 log n )
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Let A = Q + T be a mixed matrix whereQ contains only numbersT contains only variables (or zeros)
A is equivalent to
Numbers and variables in disjoint rows a layered mixed matrix (LM-matrix)
Mixed Matrices
I QÃ :=
(i.e. rank A = rank à - n)
I T
diag(z1,…,zn)
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Computing rank of LM-matrices
Thm [M93]: rank à =
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 qd 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 doesn’t
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
z1
z2
z3
1 1
1
1
x1 x2
x3
à =
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Computing rank by matroid union
1
1
1
z1
z2
z3
1 1
1
1
x1 x2
x3
MT
MB
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Computing rank by matroid union
1
1
1
z1
z2
z3
1 1
1
1
x1 x2
x3
MT
MB
JT
JB
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Independent Matching Approach
z1
z2
z3
x1 x2
x3
MT
MB
CT
CB
RT
1
1
1
1 1
1
1
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Independent Matching Approach
z1
z2
z3
x1 x2
x3
MT
MB
RT
1
1
1
1 1
1
1
CT
CB
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Independent Matching Approach
z1
z2
z3
x1 x2
x3
MT
MB
CT
CB
RT
1
1
1
1 1
1
1
JT
JB
Cols must be indep.
Non-zeroswhere edgesbend
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Computing Rank of à Compute max indep matching, JT and JB
rank à = |JT| + |JB|
Time to find max indep matchingO(n3 log n): Cunningham 1986O(n2.62): Gabow-Xu 1996
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Better Completion Algorithm
Idea: Use structure of matching to helpfind completion
Let XM X be variables used by JB
BetterCompletion Given A, compute LM-matrix à Compute max indep matching M for Ã
…
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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
Better Completion Algorithm
z1
z2
z3
x2x1
x3
x1
x3
x1
x3
XM
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Better Completion Algorithm
BetterCompletion Given A, compute LM-matrix à Compute max indep matching M for à Foreach x X \ XM, set x := 0
…
Variables in XM are more tricky
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Handling XM
z1
z2
z3
x1
x3
1
1
1
1 1
1
1
Example• Assign x1 the value v
x1
x3
x1
x3
XM
CB
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Handling XM
z1
z2
z3 x3
1
1
1
1 1
1
1
x3x3
XM
v
Example• Assign x1 the value v• x1 and its edge disappear• v appears in top matrix
Because A = Q + T andCB
I Qdiag(z1,…,zn) T
à =
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Handling XM
z1
z2
z3 x3
1
1
1
1 1
1
1
x3x3
XM
v
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
CB
JT
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Handling XM
Swapping Lemmav s.t.JT + c2 – c4 andJB – c1 + c3 are indep
z2
1
1
1
1 1
1
1
x1
c1
c2
c3
c4
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Handling XM
Swapping Lemmav s.t.JT + c2 – c4 andJB – c1 + c3 are indep
z2
1
1
1
1 1
1
1
v
c1
c2
c3
c4
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Handling XM
Thm: The value v can be found in O(n2) time
Proof: Maintain Gauss-Jordan pivoted copies of Q. Finding v now trivial.
z2
1
1
1
1 1
1
1
v
c1
c2
c3
c4
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Better Completion Algorithm
BetterCompletion
1) Given A, compute LM-matrix Ã
2) Compute max indep matching M for Ã
3) Foreach x X \ XM, set x := 0
4) Foreach x XM,
5) Use swapping lemma and pivoted forms to find a good value v
6) 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 Fk+1
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