Do-all computing in distributed systems by C Georgiou; Alex Allister Shvartsman

By C Georgiou; Alex Allister Shvartsman

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0. σ times Proof. To show the result it suffices to show that, after dropping one floor and strengthening the inequality: (⌊. . ⌊⌊n · κ⌋ · κ⌋ . . · κ⌋ · κ) − 1 > 0, or that σ−1 times ⌊. . ⌊⌊n · κ⌋ · κ⌋ . . · κ⌋ > κ1 . σ−1 times Applying this transformation for σ − 1 more steps, we see that it suf1 fices to show that n > κ1σ + κσ−1 + . . + κ1 , or, using geometric progression summation, that n > (κ−1 )σ+1 −(κ−1 ) . (κ−1 )−1 (κ−1 )σ+1 − (κ−1 ) for κ ≤ 21 , thus it (κ−1 ) − 1 is enough to show that n > (κ−1 )σ+1 .

I Compute: In the compute substage w sets Dw ← Dw and Pw ← Pwi for an arbitrary i ∈ {1, . . , kw } and updates its local view Lw (the update rule is described in detail in the text below). Send: The send substage is not used. Fig. 3. 4 gives a pictorial description of a single phase of algorithm AN with its three stages, each consisting of receive, compute, and send substages. Stage 1 Stage 2 receive report Coordinator Worker Perform one task ✒ report knows L,P,U,D update D,P Stage 3 summary ❘ receive update summary D,P,L ◆ Perform one task Fig.

By induction on the number of phases.

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