Maximal independent set

In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. For example, we have {b}, {a,c} two MISs, then {a} is not a MIS since it was a subset of {a,c}. Figure

Figure 4-1. Transform the relationship between flows to conflict graph

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4-2 shows all the MISs of the five nodes, {2,4},{1,5},{3}.

4.2 Combination of schedulable link flows set

Each multicast has 2ⁿ-1 link patterns. n is the number of destinations of the multicast flow. The number of schedulable link flows of each flow is derived as

(14) The total number of schedulable link flows set is derived as

(15) N is the number of the flows in the system. Since the combination of schedulable link flows set is exponential, we would propose an efficient link pattern selection scheme for each multicast flow and select a good schedulable link flows set compare with the original link flows set and only unicast link flows set. The link pattern selection problem is highly correlated with scheduling problem and it would affect the final scheduling result.

4.3 Proof of minimal time frame length scheduling problem is NP-hard

We could transform the schedulable link flow set into conflict graph and find the maximal independent set to schedule in each slot. First we proof the minimal time frame length scheduling problem is a NP-hard problem and it also can be formulate as ILP problem which is also a NP-complete problem.

Theorem 1.0. In a general conflict graph, the minimal time frame length scheduling problem is a NP-hard problem.

Figure 4-2. Maximal independent set

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Proof. To minimize the time frame length, it should schedule maximum independent set of the conflict graph in each time slot. For instance, given a conflict graph G with two nodes and find a maximum independent set in graph G. We could reduce the problem to minimal time frame length problem. We create the G’ which is equal to G and set the time requirement of each node in G’ to 1. To find a maximum independent set in G if and only if solving the minimal time frame length scheduling problem in G’. G reduces to G’ is a polynomial time reduction.

We formulate the minimal time frame scheduling problem as integer linear programming (ILP) problem.

(16) is the set of maximal independent set after we select a schedulable link flow set. is the number of time slot allocated to MIS . The ILP problem is a NP-complete problem and the complexity is too high for practical system to adopt.

We define each entry of the concurrent link set of , CLS( ), as a set of link flows coexist with . We prove each link flow L belong to entry of set of maximal independent set (SMIS) of , SMIS( ), implies L belong to CLS( ).

Lemma 1.0.

Proof.

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4.4 Design of link pattern selection

We defined the relational symbols in Table I. We would split each multicast flow into multiple unicast link flows, transform it to the conflict graph and merge selected two link flows into one link flow called merged link flow Li,J depends on the defined benefit B. To solve the general case in conflict graph, we class the graph into one of following three classes with corresponding merge rule. The merge rule 1 is benefit is larger than zero, the link pattern selection operator would merge the selected

Table I. Description of notations

Notations Description

{ML} Merge link flows set.

V represents link flows and E represents conflict relationship.

A set of link flows coexist with . (CLS)

The defined benefit of merge rule i. It was defined as predicted difference of scheduling results between before merge and after merge.

18 rate of multicast flow is less than or equal to minimal transmission rate of unicast link flow of the multicast flow. First we delete the from the graph of before merge and schedule the {ML} of before merge and {ML} of after merge. The difference of scheduling result is equal to , after that attach back to graph of before merge and schedule it. The benefit of merge operation is the additional cost in

before merge and the benefit would be , for

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we can delete the co-conflict nodes without affecting the difference of scheduling result. Since the co-conflict node fulfill the condition one, would be included in maximal independent set (MIS) of , the difference of scheduling result would not change after we delete the nodes in type . The second type is the co-concurrent node which coexist with and its value is less than or equal to , for example likes in Fig. 4-3(e). Since

and is included in SMIS of , the scheduler allocate time slot to in before merge or allocate time slot to in after merge is equal to simultaneously allocate time slot to , so we can delete the without affecting the difference of scheduling result. The final step is scheduling the flows in remaining graph of before merge and after merge, take the difference as the benefit of merge rule 3. The following is the steps we calculate the benefit of merge rule 3.

When conflict with and .

(1) With condition that coexist with other merge link flows except and

1 ,j+

Li _and . Delete from

graph of before merge and after merge.

(2) With condition that coexist with and , . Delete from graph of before merge and after merge.

(3) Create the scheduler FSA. .

Figure 4-3. Conflict graph of each merge rule

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In Fig. 4-4 (a), it contains a unicast flow L_2,1 with 3 time slot and a multicast flow L1,{1,2} with 5 time slot and it was split into two unicast link flows respectively L_1,1 with 5 time slot and L_1,2 with 4 time slot. The scheduling result of after merge is 5 time slot which is better than before merge. By the merge rule 1, the benefit B1= 5+4-5 = 4 which is larger than zero. In Fig. 4-4 (c), it contains a multicast flow L_1,{1,2}

with 9 time slot, a unicast flow L2,1 with 5 time slot and a unicast flow L3,1 with 4 time slot. By the merge rule 2, the benefit B₂=7-9+max(9-5,0)=2 which is larger than zero.

In Fig. 4-4 (e), it contains a multicast flow L1,{1,2} with 9 time slot, a unicast flow L2,1

with 7 time slot, a unicast flow L_3,1 with 6 time slot and L_4,1 with 5 time slot. By the merge rule 3, we could delete co-conflict node L3,1 and co-concurrent node L4,1

without affecting the difference of scheduling results between before merge and after merge.

Figure 4-4. Illustration of each merge rule

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4.5 Link pattern selection algorithm

First we get the unicast and multicast flow requests with time requirement T. The link pattern selection module would split each multicast flow into multiple unicast link flows with new time requirement fulfill the same data size of the original scheduling algorithm. The following is the proposed LPS algorithm.

• Input: Unicast link flows set {ML}

• Step 1: Sort {ML} by time requirement in descendent order.

• Step 2: Find S1 and S2 of each unicast link flows. Select of one multicast link flows from second entry to last entry in {ML}.

• Step 3: Search of the same multicast flow from first entry of {ML} to entry before . If exists, check the merge rule and invoke the merge function if pass one of the merge rule.

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1. If coexist with S1 (Li,j ) and S1 (Li,j+1 ) 2.

3. If , invoke merge function and goto step 5.

Merge rule 2:

1. If coexist with S¹ (Li,j+1 ) and conflict with S1 (Li,j ).

2.

3. If , invoke merge function and goto step 5.

Merge rule 3:

1. If conflict with S2 (Li,j ) and S2 (Li,j+1 ).

2. If coexist with other flows except and and . Delete from graph of before merge and after merge.

3. If coexist with and , . Delete from

graph of before merge and after merge.

4. Create the scheduler FSA. .

5. If , invoke merge function and goto step 5.

• Step 5: Repeat step 3 and step 4 until finish a round.

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4.6 Proposed fair slot assignment (FSA) algorithm

As mention in section 4.3, we prove the minimal time frame length scheduling problem is NP-hard. To reduce the time complexity and achieve high throughput and fairness, the proposed scheduling scheme gives high priority for flows with minimal allocated time and maximal time requirement to schedule first. In the conflict graph view, we always select the concurrent flows with large time requirement which maximize the CTAP utilization of allocated slots. The time complexity of FSA is . The complexity of sorting operation is and the number of operation in slot allocation is , F is the maximal time requirement which is a constant number and the time complexity of FSA would be . The proposed FSA algorithm is presented as algorithm 1.

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3: Sort {ML} by time requirement in descendent order 4: While size of {ML} 1

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Chapter 5 Performance Evaluation

5.1 Simulation environment and metric definition

To match the practical environment the WPAN deploy in, we set the topology as 10m x 10m indoor office room. In the WPAN system, the flows were generated with random source and destination, and we alternatively add the multicast and unicast Original with FSA and Original with conventional TDMA scheduler in comparison.

The SubOpt is a local optimal link pattern selection scheme and we derive the schedulable merge link flows set by recursively take two merge link flows of a multicast to merge if the scheduling result is better. LPS with FSA is our proposed link pattern selection and scheduling scheme. LPSREX is LPS cooperate with REX scheduler [5]. The metrics we concern about is throughput and fairness. We formulate the system throughput as

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(17) is transmission rate of flow i. is time requirement of flow i. is number of destination of flow i. N is total allocated time in the system. To measure the impact of the operation of deleting flows in merge rule 3, we define the efficiency of merge rule 3 as

(18) N is number of link flows before merge rule 3, is the number of link flows after merge rule 3, the power two is proportional to the complexity of scheduling scheme.

To measure the fairness between scheduling schemes, we define the Jain fairness index as

(19)

is the allocated time for flow i. The fairness index is calculated when total allocated time is larger than half the total time requirement.

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5.2 Simulation results

5.2.1 Number of flows and system throughput

In Figure 5-1, when the number of flows increases, it has the better spatial reusability and the throughput also increases. The link flows set selected by proposed LPS scheme is better than the only unicast link flows set and original link flows set.

Also, it is approximate to suboptimal link pattern selection scheme. FSA is better than REX scheduler in throughput. When the number of flows is small, the original link flow set would be better than only unicast link flows set since the conflict probability derived from mutual cover is low and the conflict factor is mainly dominated by the same source rule. The multicast flows avoid the same source rule and make more spatial reusability. Inversely, when the number of flows increases, the mutual cover rule would dominate the spatial reusability.

Figure 5-1. Throughput vs number of flows

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5.2.2 Minimal beamwidth and system throughput

In Figure 5-2, we can find out the throughput decrease when the minimal beamwidth increases. The larger the minimal beamwidth lead to lower transmission rate of each flow and lower spatial reusability. The LPS is better than the others and approximate to SubOpt. When the minimal beamwidth enlarge, the LPS prefer not merge since there is no spatial reusability if it merge to multicast flows. We also can find out the original link flow set with FSA almost become conventional TDMA scheduling state when the minimal beamwidth is about 60 degree.

5.2.3 Number of flows and efficiency of merge rule 3

In Figure 5-3, the flow deletion operation in merge rule 3 reduce almost half the time complexity compare with SubOpt link pattern selection scheme when the number of flows large enough. When the number of flow less than ten, because the high spatial reusability lead to small number of co-conflict flows. It’s not harmful since we should take more concern about the scenario with large number of flows.

Figure 5-2. Throughput vs minimal beamwidth

29 5.2.4 Number of flows and fairness index

We calculate the fairness index when the total allocated time larger than half the total time requirement of all flows. In Figure 5-4, we can take the conventional TDMA scheduling scheme as the upper bound in fairness index. The LPSREX is better than the others since it always schedule the flows with smallest allocated time first. The proposed FSA loss some fairness but keeps higher throughput and lower time complexity. The complexity of REX scheduler is , N is number of flows, since it iteratively sorting the flows for scheduling the flows with minimal allocated time. The fairness index decrease when number of flows increases, it’s because it has more flows with small time requirement scheduled before we take the fairness index value.

Figure 5-3. Efficiency of merge rule 3 vs number of flows

30 5.2.5 Minimal beamwidth and fairness index

In Figure 5.5, there is litter difference in fairness when the minimal beamwidth increases. The larger the beamwdith make larger difference in time requirement between multicast flows and unicast flows, and it also reduces the spatial reusability.

The difference would decrease the fairness since some small time requirement flows (unicast flows) has been scheduled before we take the value.

Figure 5-4. Fairness index vs number of flows

Figure 5-5. Fairness index vs minimal beamwidth

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Chapter 6 Conclusion and Future Work

In this work, with the motivation of the multicast data transmission paradigm is more suitable for multimedia multicast applications and the scheduling result is highly correlated with link pattern selection of multicast flows and the cooperated scheduler, we proposed a link pattern selection (LPS) scheme which transform the link pattern selection problem into conflict graph and find some rules to merge the link flows to the better state. In merge rule 3, we delete the flows to reduce the time complexity and approximate to suboptimal link pattern selection scheme in throughput. We also proposed the fair slot assignment (FSA) scheduling algorithm which maintains the fairness without throughput reduction. From the simulation results, the proposed LPS with FSA is better than the others and approximate to suboptimal scheme in throughput but with minor loss in fairness than LPS with REX scheduler. To get up to more practical scenario in WPAN system, we would take the NLOS problem into consideration in the future work.

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