Communication-aware Heterogeneous Multiprocessor Mapping for Real-time Streaming Systems

Abstract

Real-time streaming signal processing systems typically desire high throughput and low latency. Many such systems can be modeled as synchronous data flow graphs. In this paper, we address the problem of multi-objective mapping of SDF graphs onto heterogeneous multiprocessor platforms, where we account for the overhead of bus-based inter-processor communication. The primary contributions include (1) an integer linear programming (ILP) model that globally optimizes throughput, latency and cost; (2) low-complexity two-stage heuristics based on a combination of an evolutionary algorithm with an ILP to generate either a single sub-optimal mapping solution or a Pareto front for design space optimization. In our simulations, the proposed heuristic shows up to 12x run-time efficiency compared to the global ILP while maintaining a 10^− 6 optimality gap in throughput.

Appendix: Scheduling ILP

When given a partition, the execution time of conventional SDF actors as well as the communication delays of the send and receive actors become fixed parameters. For conciseness, we slightly abuse the notation by defining parameters

$$\begin{array}{rll} D_i&=\sum\limits_j{a^*_{ij}D_{ij}}, \\ DS_k&=ipc^*_k\cdot\sum\limits_l{bs^*_{kl}DC_{kl}}, \\ DR_k&=ipc^*_k\cdot ch^*_k\cdot \sum\limits_l{br^*_{kl}DC_{kl}}, \end{array} $$

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where v ^* denotes a specific value of a variable v. Let us define the index set $\mathcal{T}_p=[T_{\rm max}-Period^*,T_{\rm max}]$, in which Period ^* is the minimum period determined by the partition. Also denote $\mathcal{K}_s\stackrel{\triangle}{=}\{k|ipc^*_k=1\}$ and $\mathcal{K}_r\stackrel{\triangle}{=}\{k|ipc^*_k\cdot ch^*_k=1\}$, i.e. the set of edges with an active send or an active receive actor, respectively.

The scheduling sub-problem can then be formalized as an ILP that minimizes

$$\begin{array}{rll} Latency=&\sum\limits_{t\in \mathcal{T}_p}t[s_I(t)-s_I(t-1)]+D_I \\ &-\sum\limits_{t\in\mathcal{T}_p}t[s_1(t)-s_1(t-1)]\ \\ &+(s_1(T_{\rm max})-s_I(T_{\rm max}))Period^*, \end{array} $$

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subject to

Precedence

$$\begin{array}{rll} s_{src(k)}(t-D_i)P_k-ss_k(t) &\geq 0, \\[3pt] ss_k(t-DS_k)-sr_k(t) &\geq 0, \\[3pt] sr_k(t-DR_k)-s_{dst(k)}(t)C_k+O_k &\geq 0, \forall k. \end{array} $$

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Resource sharing

$$\begin{array}{rll} & \sum\limits_i{[s_i(t)-s_i(t-D_i)]} \\ &\sum\limits_{k\in\mathcal{K}_s}{[ss_k(t)-ss_k(t-DS_k)]a^*_{src(k)j}} \\ &\sum\limits_{k\in\mathcal{K}_r}{[sr_k(t)-sr_k(t-DR_k)]a^*_{dst(k)j}}\leq 1, \forall j;\\ \end{array} $$

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$$\begin{array}{rll} &\sum\limits_{k\in\mathcal{K}_s}{[ss_k(t)-ss_k(t-DS_k)]bs^*_{kl}} \\ &\sum\limits_{k\in\mathcal{K}_r}{[sr_k(t)-sr_k(t-DR_k)]br^*_{kl}} \leq 1, \forall l. \end{array} $$

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Periodicity

$$\begin{array}{rll} &\forall i, t\in[T_{\rm max}-D_i, T_{\rm max}] \\ &\quad s_i(t)-s_i(t-Period^*) = N_i; \\ \end{array} $$

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$$\begin{array}{rll} &\forall k, t\in[T_{\rm max}-DS_k, T_{\rm max}], \\ &\quad ss_k(t)-ss_k(t-Period^*) = N_{src(k)}P_k; \\ \end{array} $$

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$$\begin{array}{rll} &\forall k, t\in[T_{\rm max}-DR_k, T_{\rm max}], \\ &\quad sr_k(t)-sr_k(t-Period^*) = N_{dst(k)}C_k. \end{array} $$

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