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r Add a new part vertex ui with zero weight for each part i, and x those vertices in respective parts, that is, f (ui ) = i for i = 1, 2, , k Hence, V j becomes j {u |i = 1, 2, , k} V i r For each vertex v V j , add a migration net between v and ui if v is assigned to part i at the beginning of epoch j Set the migration net s cost to the size of the data associated with v, since this migration net represents the cost of moving vertex v to a different part Figure 151 illustrates a sample hypergraph H j 1 for epoch j 1 (left) and a repartitioning hypergraph H j for epoch j (right).

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A nice feature of our model is that no distinction is required between communication and migration vertices as well as nets However, for clarity in this gure, we represent computation vertices with circles and part vertices with hexagons Similarly, nets representing communication during computation are represented with squares, and migration nets representing data to be migrated due to changing vertex assignments are represented with diamonds In this example, there are nine unit weight vertices partitioned into three parts with a perfect balance at epoch j 1 Three cut nets represent data that need to be communicated among the parts Two of these nets have connectivity of two and one has three Assuming unit cost for each net, total communication volume is four (equation 152) In other words, each iteration of epoch j 1 incurs a communication cost of four.

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The computational structure changes in epoch j, as shown in Figure 151 (right) In H j , new vertices a and b are added, while vertices 3 and 5 are deleted To construct the repartitioning hypergraph H j from H j , three part vertices u1 , u2 , and u3 are added and net weights in H j are scaled by j Then, each of the seven computation vertices inherited from H j 1 is connected to the part vertex associated with the part to which the computation vertex was assigned in epoch j 1, via a migration net Once the new repartitioning hypergraph H j that encodes both communication and migration costs is obtained, the repartitioning problem reduces to hypergraph partitioning with xed vertices Here, the constraint is that vertex ui must be assigned, or xed, to part i Let P = {V1 , V2 , .

Adding nonzero entries The operation S,, = S,, +value, where ual ue is an arbitrary real number, is performed via the command S->Add(i, j , v a l u e ) . The indices start

, Vk } be a valid partition for this problem Assume that a vertex v is assigned to part Vp in epoch j 1 and Vq in epoch j, where p = q Then, / the migration net between v and uq that represents the migration cost of vertex v s.

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