A Fusion Method Implementation

In the CoordSatCap algorithm, the saturation capacity in dimensionφ_i is determined

by solving min{f_α#(X)−r(X): φi ∈ X⊆V}, where each element inVis the index of

a user in the system. In this section, we show that this problem can be solved over a merged user set where each non-singleton element denotes a super user, i.e., the CoordSatCap algorithm can be implemented by a fusion method. The validity of this fusion method is based on Lemma 5.5.10 and Lemma 5.5.11 below with the proofs in Appendix 5.D and Appendix 5.E, respectively.

Lemma 5.5.10. Let the CoordSatCap algorithm starts with a rate vectorrV ∈ P(fα#,≤)such thatrV≤ 0, where0= (0, . . . , 0)∈R|V|. We have

min{f_α#(X)−r(X): φi ∈ X⊆V}=min{fα#(X)−r(X): φi ∈X ⊆Vi}, (5.13) where V_i ={φ1, . . . ,φ_i}and the minimal minimizerXˆφi ⊆ Vi for all i∈ {1, . . . ,|V|}.

The equality (5.13) in Lemma 5.5.10 was originally derived in [51], based on which the authors in [67, 68] apply min{f_α#(X)−r(X): φi ∈ X ⊆ Vi} in the CoordSatCap

algorithm for solving the non-asymptotic minimum sum-rate problem in the finite linear source model and CCDE.23 However, the studies in [51] did not show that

ˆ

Xφ_i ⊆Vi for alli, neither did [67, 68].

Lemma 5.5.11. In the CoordSatCap algorithm,

min{f_α#(X)−r(X): φi ∈X⊆V} (5.14)

=min{f_α#(U˜)−r(U˜):U ⊆ P∗,φi ∈U˜} (5.15)

for all i ∈ {1, . . . ,|V|}. LetXφˆ _i and U_φ∗_i be the minimal minimizer of the(5.14)and(5.15), respectively. X˜ =U˜_φ∗_i, whereX ={C∈ P∗_{: C∩Xˆ}

φi 6=∅}.

Both minimization problems, (5.14) and (5.15), in Lemma 5.5.11 are SFM problems due to the intersecting submodularity of f#

α. But,P

∗_{is a fused user set since some of} the users have been merged into a super user set, which is treated as one dimension in the SFM problem (5.15). So,|P∗_{| ≤ |V|. We will show in Section 5.5.5 that mini-} mizing over the fused user set contributes to a reduction in computation complexity 23_{The algorithms proposed in [67,68] for solving the non-asymptotic minimum sum-rate problem are} based on the CoordSatCap algorithm. See Section 5.5.4.

of the SFM algorithms. In addition, according to Lemma 5.5.11, if we do the min- imization problem (5.15) to determine the saturation capacity in the CoordSatCap algorithm, the update of P∗ is easier: Since (P∗\U_φ∗_i)t {U˜_φ∗_i} = (P∗\ X)t {X }˜

whereX ={C∈ P∗_{: C∩Xˆ} φi 6=∅}, simply doP ∗ _←(P∗_\U∗ φ)t {U˜ ∗ φ}, which does not require the determination ofX.

Remark 5.5.12. Lemma 5.5.11 is based on the fact: IfXˆφi is a non-singleton minimal min-

imizer of min{f_α#(X)−r(X): φi ∈ X ⊆ V} for some φi, then, for any partition P that

crosses Xˆφi, there exists a P

0 _{that does not cross Xˆ}

φi such that fα#[P

0_{] < f}#

α[P].24 It means that, to determine the minimal/finest minimizer of the Dilworth truncation problem minP ∈Π0_(V) f_α#[P], we just need to consider all the partitions in Π(V) that do not cross

ˆ

Xφ_i.25 See Appendix 5.F for the explanation.

Example 5.5.13. In Exmaple 5.5.8, we have Xˆ5 = {4, 5} when α = 23₄. For P =

{{1, 4},{2, 3},{5}} that crosses {4, 5}, we have P0 = {{1, 4, 5},{2, 3}} that does not cross {4, 5} such that f_α#[P0_{] =} 15

2 < fα#[P] =

37

4; For P = {{1, 4},{2, 3, 5}} that

crosses {4, 5}, we have P0 = {{4, 5},{1},{2, 3}} that does not cross {4, 5} such that f_α#[P0_{] =} 29

4 < fα#[P] =

17

2. One can show that for each partitionP that crosses{4, 5}there

exists a partitionP0 _{that does not cross{4, 5}such that f}#

α[P 0_{]< f}#

α[P]. Therefore, we just need to consider all partitions that do not cross{4, 5}to determine the minimal minimizer of min_{P ∈Π}0_(V) f_α#[P]. It is equivalent to considering all partitions ofP∗, whereP∗ is updated to{{4, 5},{1},{2},{3}}when φ2 =5in Exmaple 5.5.8.

Based on Lemmas 5.5.10 and 5.5.11, the CoordSatCap algorithm in Algorithm 5.2 is equivalent to the CoordSatCapFus algorithm in Algorithm 5.3. Since the satura- tion capacity in the CoordSatCapFus algorithm is always obtained by a minimization problem over a fused user set P∗ ∈ Π(Vi) for all i ∈ {1, . . . ,|V|}, we call the Co-

ordSatCapFus algorithm a fusion method implementation of the CoordSatCap algo- rithm.

Example 5.5.14. Consider the Dilworth truncation problem minP ∈Π(V) fα#[P] when α =

23

4 in Example 5.5.6. We apply the CoordSatCapFus algorithm by initiating rV = (α−

H(V))χV = (−17₄, . . . ,−17₄).

24_{A multi-way cut or partitionP ∈Π}0_{(V)does not cross a setX⊂Vif there existC∈ Psuch that}

X⊆C. For example, forX={1, 3},{{1, 3, 4},{2}}and{{1, 3},{2},{4}}are the partitions that do not crossX, while{{1, 2},{3},{4}}and{{1},{2, 3, 4}}are the partitions that crossX.

25_{In addition, the fundamental partitionP}∗_{must be a multi-way cut ofVthat does not cross ˆX} φ. See Appendix 5.F.

Algorithm 5.3: Coordinate-wise Saturation Capacity Algorithm by Fusion Method (CoordSatCapFus)

input :the ground setV, an oracle that returns the value ofH(X)for a givenX⊆V,

αwhich is an estimation ofRACO(V) output: rV which is a rate vector inB(fˆα#,≤)andP

∗_{which is the minimal/finest} minimizer of minP ∈Π(V)fα#[P]

1 letr_V ←(α−H(V))χV so thatrV ∈P(fα#,≤)and choose a linear ordering Φ= (φ1, . . . ,φ|V|);

2 initiater_φ1 ← f_α#({φ1})andP∗← {{φ1}};

3 fori=2to|V|do 4 P∗← P∗t {{φi}};

5 determine the saturation capacity

ˆ

ξ←min{f_α#(U˜)−r(U˜):{φi} ∈U⊆ P∗}

and the minimal/smallest minimizerU_φ∗_i;

6 r_V ←r_V+ξχˆ φi;

7 /* merge/fuse all subsets in U_φ∗

i into one subset U˜ ∗ φi =tX∈U∗φ_iX */ 8 P∗←(P∗\U_φ∗ i)t {U˜ ∗ φi}; 9 endfor 10 returnrV andP∗;

The linear ordering is set toΦ= (4, 5, 3, 2, 1)and we have the following results.

• Forφ1 = 4, we assign r4 = f_23/4# ({4}) = 15₄ so thatrV = (−17₄,−17₄,−17₄,15₄,−17₄)

and letP∗ _={{4}};

• For φ2 = 5, we update P∗ to {{4},{5}} and consider the problem min{fα#(U˜)− r(U˜): {5} ∈U⊆ P∗}. Since{U: {5} ∈U⊆ P∗}={{{5}},{{4},{5}}}and

f_23/4# ({5})−r({5}) =6, f_23/4# ({4, 5})−r({4, 5}) = 17

4 ,

we haveξˆ= 17₄ and U₅∗ ={{4},{5}}. rVandP∗are updated to(−17₄,−17₄,−17₄,15₄, 0)

and{{4, 5}}, respectively;

• Forφ3 = 2, we updateP∗ to{{2},{4, 5}} and consider the problem min{fα#(U˜)− r(U˜): {2} ∈U⊆ P∗}. Since{U: {2} ∈U⊆ P∗}={{{2}},{{2},{4, 5}}}and

f_23/4# ({2})−r({2}) =4, f_23/4# ({2, 4, 5})−r({2, 4, 5}) = 25

4 ,

we haveξˆ=4and U₂∗ = {{2}}. rVis updated to(−17₄,−1₄,−17₄,15₄, 0). By executing

• In the same way, we can show that:

– Forφ4=3,rV = (−1₄,−1₄,−17₄,15₄, 0)andP∗ ={{2},{3},{4, 5}};

– Forφ5=1,rV = (₄3,−1₄,−17₄,15₄, 0)andP∗ ={{1},{2},{3},{4, 5}}.

At the end, we have r = (3₄,−1₄,−17₄,15₄, 0)andP∗ = {{1},{2},{3},{4, 5}}, which are the same as in Example 5.5.8. It can be seen that the CoordSatCapFus outputs the same results as in Example 5.5.9 for the Dilworth truncation problemminP ∈Π(V) f_13/2# [P].