An efficient algorithm for scheduling instructions with deadline constraints on ilp machines
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An Efficient Algorithm for Scheduling Instructions with Deadline Constraints on ILP Machines. Wu Hui Joxan Jaffar School of Computing National University of Singapore. What is an ILP machine?. Multiple functional units of different types.

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An efficient algorithm for scheduling instructions with deadline constraints on ilp machines

An Efficient Algorithm for Scheduling Instructions with Deadline Constraints on ILP Machines

Wu Hui Joxan Jaffar

School of Computing

National University of Singapore


What is an ilp machine
What is an ILP machine? Deadline Constraints on ILP Machines

  • Multiple functional units of different types.

  • Issue an instruction every machine cycle on each functional unit.

  • Multiple instructions executed in parallel.

  • Latencies exist between instructions.

  • Two categories: Superscalar and VLIW (Very Long Instruction Word).

  • Typical Example: Intel Itanium processor (http://developer.intel.com/design/ia64/microarch_ovw/index.htm)


  • What is the problem? Deadline Constraints on ILP Machines

  • Given a problem instance P: a set of n UET instructions in a basic block with the following constraints:

    • precedence-latency constraints: DAG G = (V, E, W), where each latency lij -1,

    • deadline constraints: individual pre-assigned deadlines, and

    • m functional units with p different types,

  • compute a feasible schedule which satisfies all constraints whenever one exists, or a valid schedule with minimum lateness if no feasible schedule exists.


v Deadline Constraints on ILP Machines1 [4]

v2 [4]

v3 [4]

FU1

1

0

1

FU2

1

0

0

0

v4[5]

v5 [5]

v6 [5]

v7 [5]

0

0

0

0

0

v8 [6]

v9 [6]

v10 [6]

v11 [6]

v12 [6]

Example 1. A problem instance P with two functional units of different types.

Table 1. A feasible schedule for P.


What does our algorithm achieve? Deadline Constraints on ILP Machines

Our scheduling algorithm computes a feasible schedule whenever one exists for any problem instance of the following special cases.

1) Arbitrary DAG, latencies of 0 and two functional units of different types.

2) Monotone interval graph, latencies  -1 and multiple functional units of different types.

3) In-forest, equal latencies and multiple functional units of different types.


  • In the case that there is no feasible schedule, our algorithm computes a schedule with minimum lateness for all the above special cases.

  • Furthermore, by setting all deadlines to a constant, our algorithm will compute a schedule with minimum completion time for

    • any instance of the above special cases and

    • any instance of the special case of out-forest, equal latencies and multiple functional units of different types.


v algorithm computes a schedule with minimum lateness for all the above special cases. 2

v3

v1

v1

v4

v5

v2

v3

v6

v4

v5

v6

An in-tree. An out-tree

v1

v2

2

3

2

3

v3

v4

v5

-1

1

2

1

4

v6

v7

A monotone interval graph.


  • What is the Time Complexity ? algorithm computes a schedule with minimum lateness for all the above special cases.

  • Given the transitive closure of the precedence graph,

    • O(ne+nd) for the general model, where d is the maximum latency.

    • O(min{ne, de}+nd) if no latency of -1 exists.

    • O(n2) if for each instruction the latencies between it and all its immediate successors are equal.

  • Transitive closure can be computed in O(min(ne, n2.367)) time.


  • What has been done in the past? algorithm computes a schedule with minimum lateness for all the above special cases.

    • Palem and Simon’s algorithm on identical processors [ACM TOPLAS, 1993].

    • Wu, Joxan and Yap’s algorithm on identical processors [PACT 2000].

    • Berstein, Rodeh and Gertner’s work on two processors of different types [IEEE TOC, 1989].


  • What are the contributions of our work? algorithm computes a schedule with minimum lateness for all the above special cases.

    • Propose an efficient polynomial algorithm which solves several special cases for each of which no polynomial algorithm was known before.

    • Present the first approximation ratio, i.e. for any greedy algorithm, the length of any schedule computed never exceeds p+1, where p is the number of types of functional units.


  • What are the main ideas of our algorithm? algorithm computes a schedule with minimum lateness for all the above special cases.

    • Compute the lmax(vi)-successor-tree-consistent deadline for each instruction vi, where lmax(vi)is the maximum latency between vi and all its immediate successors.

    • Compute a schedule by using list scheduling, where the priority of each instruction is its successor-tree-consistent deadline and a smaller number implies higher priority.


  • What is the algorithm computes a schedule with minimum lateness for all the above special cases. lmax(vi)-successor-tree-consistentdeadline?

    • For each sink instruction, its lmax(vi)-successor-tree-consistentdeadline d´i is equal to its pre-assigned deadline.

    • For a non-sink instruction vi, d´i is the upper bound on its latest completion time in any feasible schedule for the relaxed problem instance P(i).


  • What is P(i)? algorithm computes a schedule with minimum lateness for all the above special cases.

  • P(i) consists of a set V(i)={vi}  Succ(vi) of instructions with following new constraints.

    • Precedence-latency constraints: The lmax(vi)-successor-tree of vi.

    • Deadline constraints: The deadline of each instruction vj in Succ(vi) is its lmax(vj)-successor-tree-consistentdeadline and the deadline of vi is its pre-assigned deadline.


  • What is algorithm computes a schedule with minimum lateness for all the above special cases. the k-successor-tree of vi ?

  • Given a weighted graph G=(V, E, W), an integer k and vi V, the k-successor-tree of vi is a subgraph G= (V, E, W), where

    • V ={vi}  {vj: vj Succ(vi)},

    • E={(vi, vj): vj Succ(vi)} and

    • each edge weight l´ij in W is defined as follows.

    • 1) In the case that k= -1, if l+ij = -1, then l´ij = -1; otherwise l´ij = 0.

    • 2) In the case that k  -1, if l+ij < k, then l´ij = l+ij; otherwise, l´ij = k.


v algorithm computes a schedule with minimum lateness for all the above special cases. 1

v2

4

2

1

-1

1

v3

v4

v5

1

0

1

v7

v8

v6

Figure 1: The precedence-latency constraints.

v2

4

4

1

2

-1

1

v3

v6

v4

v7

v5

v8

Figure 2: The 4-successor tree of v2.


  • How to compute algorithm computes a schedule with minimum lateness for all the above special cases. lmax(vi)-successor-tree-consistent deadline for vi ?

  • Key idea: Backward Scheduling

  • At any time t, among all ready instructions, an instruction vk with the largest latency in P(i) is chosen and scheduled as late as possible on a functional unit of the same type. In case of ties, among all instructions with the same latency, an instruction with the latest deadline is chosen.

  • A schedule computed by backward scheduling is called a backward schedule.


FU1 algorithm computes a schedule with minimum lateness for all the above special cases.

FU2

v1 [2]

3

3

1

2

-1

1

v2[5]

v3[6]

v4[5]

v5 [3]

v6[4]

v7[3]

Example 2: A relaxed problem instance P(1).

Table 2. A backward schedule for P(1).


  • Scheduling Algorithm algorithm computes a schedule with minimum lateness for all the above special cases.

  • repeat

  • choose an instruction vi satisfying that 1) its lmax(vi)-successor-tree-consistent deadline d´i has not been computed; and 2) either vi is a sink or the successor-tree-consistent deadlines of all its successors have been computed;

  • if vi is a sink then d´i = di;

  • else

  • { if vi has only one immediate successor vj and lij  -1

  • then d´i = min{di, dj - lij - 1};

    • else

    • { compute a backward schedule b for P(i);

    • d´i = min{di, min{b(vj) - lij : vj Succ(vi) }};

    • } }

    • until the successor-tree-consistent deadlines of all instructions have been computed;

    • use list scheduling to compute a schedule for P, where the priority of each instruction vi is d´i and a smaller number implies higher priority;


v algorithm computes a schedule with minimum lateness for all the above special cases. 1 [4, ?]

v2 [4]

v3 [4]

FU1

1

0

1

FU2

1

0

0

0

v4[5, 4]

v5 [5, 5]

v6 [5, 5]

v7 [5, 5]

0

0

0

0

0

v8 [6, 6]

v9 [6, 6]

v10 [6, 6]

v11 [6, 6]

v12 [6, 6]

Example 1. A problem instance P with two functional units of different types.

V1[4]

0

0

1

1

1

1

1

V4 [4]

V5 [5]

V 6[5]

V8 [6]

V9 [6]

V10 [6]

V11 [6]

Figure 4: The relaxed problem P(1).


Table 3: A backward schedule algorithm computes a schedule with minimum lateness for all the above special cases. b for Succ(v1).

Since min{b(vj) - l1j : vj  Succ(v1)}= 2, the lmax(v1)-successor-tree-consistent deadline of v1is min{d1, 2}= min{4, 2}= 2.


v algorithm computes a schedule with minimum lateness for all the above special cases. 1 [4, 2]

v2 [4, 3]

v3 [4, 3]

FU1

1

0

1

FU2

1

0

0

0

v4[5, 4]

v5 [5, 5]

v6 [5, 5]

v7 [5, 5]

0

0

0

0

0

v8 [6, 6]

v9 [6, 6]

v10 [6, 6]

v11 [6, 6]

v12 [6, 6]

Example 1. A problem instance P with two functional units of different types.

Table 3. A feasible schedule computed by list scheduling.


  • Conclusion algorithm computes a schedule with minimum lateness for all the above special cases.

    • K-successor-tree-consistency:

    • A general technique for instruction scheduling problem.

    • Approximating precedence-latency constraints by using priorities which are k-successor-tree consistent.

    • Successfully used to solve several open instruction scheduling problems such as two processor scheduling with equal execution times and release time-deadline constraints.

    • Open Problem:

    • What is the tight worst-case approximation ratio of our algorithm (Conjecture: Lours / Lopt = 4/3)?


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