Introduction to Scheduling Theory


 Claude Anthony
 2 years ago
 Views:
Transcription
1 Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France November 8, / 26
2 Outline 1 Task graphs from outer space 2 Scheduling definitions and notions 3 Platform models and scheduling problems 2/ 26
3 Outline 1 Task graphs from outer space 2 Scheduling definitions and notions 3 Platform models and scheduling problems 3/ 26
4 Analyzing a simple code Solving A.x = B where A is lower triangular matrix 1: for i = 1 to n do 2: Task T i,i : x(i) b(i)/a(i, i) 3: for j = i + 1 to n do 4: Task T i,j : b(j) b(j) a(j, i) x(i) For a given value 1 i n, all tasks T i, are computations done during the i th iteration of the outer loop. < seq is the sequential order : T 1,1 < seq T 1,2 < seq T 1,3 < seq...< seq T 1,n < seq T 2,2 < seq T 2,3 < seq...< seq T n,n. 4/ 26
5 Analyzing a simple code Solving A.x = B where A is lower triangular matrix 1: for i = 1 to n do 2: Task T i,i : x(i) b(i)/a(i, i) 3: for j = i + 1 to n do 4: Task T i,j : b(j) b(j) a(j, i) x(i) For a given value 1 i n, all tasks T i, are computations done during the i th iteration of the outer loop. < seq is the sequential order : T 1,1 < seq T 1,2 < seq T 1,3 < seq...< seq T 1,n < seq T 2,2 < seq T 2,3 < seq...< seq T n,n. 4/ 26
6 Analyzing a simple code Solving A.x = B where A is lower triangular matrix 1: for i = 1 to n do 2: Task T i,i : x(i) b(i)/a(i, i) 3: for j = i + 1 to n do 4: Task T i,j : b(j) b(j) a(j, i) x(i) For a given value 1 i n, all tasks T i, are computations done during the i th iteration of the outer loop. < seq is the sequential order : T 1,1 < seq T 1,2 < seq T 1,3 < seq...< seq T 1,n < seq T 2,2 < seq T 2,3 < seq...< seq T n,n. 4/ 26
7 Independence However, some independent tasks could be executed in parallel. Independent tasks are the ones whose execution order can be changed without modifying the result of the program. Two independent tasks may read the value but never write to the same memory location. For a given task T, In(T ) denotes the set of input variables and Out(T ) the set of output variables. { In the previous example, we have : In(T i,i ) = {b(i), a(i, i)} for i = 1 to N do Out(T i,i ) = {x(i)} and Task T i,i : x(i) b(i)/a(i, i) { for j = i + 1 to N do In(T i,j ) = {b(j), a(j, i), x(i)} Task T i,j : b(j) b(j) a(j, i) x(i) Out(T i,j ) = {b(j)} for j > i. 5/ 26
8 Independence However, some independent tasks could be executed in parallel. Independent tasks are the ones whose execution order can be changed without modifying the result of the program. Two independent tasks may read the value but never write to the same memory location. For a given task T, In(T ) denotes the set of input variables and Out(T ) the set of output variables. { In the previous example, we have : In(T i,i ) = {b(i), a(i, i)} for i = 1 to N do Out(T i,i ) = {x(i)} and Task T i,i : x(i) b(i)/a(i, i) { for j = i + 1 to N do In(T i,j ) = {b(j), a(j, i), x(i)} Task T i,j : b(j) b(j) a(j, i) x(i) Out(T i,j ) = {b(j)} for j > i. 5/ 26
9 Independence However, some independent tasks could be executed in parallel. Independent tasks are the ones whose execution order can be changed without modifying the result of the program. Two independent tasks may read the value but never write to the same memory location. For a given task T, In(T ) denotes the set of input variables and Out(T ) the set of output variables. { In the previous example, we have : In(T i,i ) = {b(i), a(i, i)} for i = 1 to N do Out(T i,i ) = {x(i)} and Task T i,i : x(i) b(i)/a(i, i) { for j = i + 1 to N do In(T i,j ) = {b(j), a(j, i), x(i)} Task T i,j : b(j) b(j) a(j, i) x(i) Out(T i,j ) = {b(j)} for j > i. 5/ 26
10 Bernstein conditions Definition. Two tasks T and T are not independent ( T T ) whenever they share a written variable: In(T ) Out(T ) T T or Out(T ) In(T ). or Out(T ) Out(T ) Those conditions are known as Bernstein s conditions [Ber66]. We can check that: Out(T 1,1 ) In(T 1,2 ) = {x(1)} T 1,1 T 1,2. Out(T 1,3 ) Out(T 2,3 ) = {b(3)} T 1,3 T 2,3. for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) 6/ 26
11 Bernstein conditions Definition. Two tasks T and T are not independent ( T T ) whenever they share a written variable: In(T ) Out(T ) T T or Out(T ) In(T ). or Out(T ) Out(T ) Those conditions are known as Bernstein s conditions [Ber66]. We can check that: Out(T 1,1 ) In(T 1,2 ) = {x(1)} T 1,1 T 1,2. Out(T 1,3 ) Out(T 2,3 ) = {b(3)} T 1,3 T 2,3. for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) 6/ 26
12 Bernstein conditions Definition. Two tasks T and T are not independent ( T T ) whenever they share a written variable: In(T ) Out(T ) T T or Out(T ) In(T ). or Out(T ) Out(T ) Those conditions are known as Bernstein s conditions [Ber66]. We can check that: Out(T 1,1 ) In(T 1,2 ) = {x(1)} T 1,1 T 1,2. Out(T 1,3 ) Out(T 2,3 ) = {b(3)} T 1,3 T 2,3. for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) 6/ 26
13 Bernstein conditions Definition. Two tasks T and T are not independent ( T T ) whenever they share a written variable: In(T ) Out(T ) T T or Out(T ) In(T ). or Out(T ) Out(T ) Those conditions are known as Bernstein s conditions [Ber66]. We can check that: Out(T 1,1 ) In(T 1,2 ) = {x(1)} T 1,1 T 1,2. Out(T 1,3 ) Out(T 2,3 ) = {b(3)} T 1,3 T 2,3. for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) 6/ 26
14 Precedence If T T, then they should be ordered with the sequential execution order. T T if T T and T < seq T. More precisely is defined as the transitive closure of (< seq ). T 1,1 for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) A dependence graph G is used. (e : T T ) G means that T can start only if T has already been finished. T is a predecessor of T. Transitivity arc are generally omitted. T 1,2 T 1,3 T 1,4 T 1,5 T 1,6 T 2,2 T 2,3 T 2,4 T 2,5 T 2,6 T 3,3 T 3,4 T 3,5 T 3,6 T 4,4 T 4,5 T 5,5 T 4,6 T 5,6 T 6,6 7/ 26
15 Precedence If T T, then they should be ordered with the sequential execution order. T T if T T and T < seq T. More precisely is defined as the transitive closure of (< seq ). T 1,1 for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) A dependence graph G is used. (e : T T ) G means that T can start only if T has already been finished. T is a predecessor of T. Transitivity arc are generally omitted. T 1,2 T 1,3 T 1,4 T 1,5 T 1,6 T 2,2 T 2,3 T 2,4 T 2,5 T 2,6 T 3,3 T 3,4 T 3,5 T 3,6 T 4,4 T 4,5 T 5,5 T 4,6 T 5,6 T 6,6 7/ 26
16 Precedence If T T, then they should be ordered with the sequential execution order. T T if T T and T < seq T. More precisely is defined as the transitive closure of (< seq ). T 1,1 for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) A dependence graph G is used. (e : T T ) G means that T can start only if T has already been finished. T is a predecessor of T. Transitivity arc are generally omitted. T 1,2 T 1,3 T 1,4 T 1,5 T 1,6 T 2,2 T 2,3 T 2,4 T 2,5 T 2,6 T 3,3 T 3,4 T 3,5 T 3,6 T 4,4 T 4,5 T 5,5 T 4,6 T 5,6 T 6,6 7/ 26
17 Precedence If T T, then they should be ordered with the sequential execution order. T T if T T and T < seq T. More precisely is defined as the transitive closure of (< seq ). T 1,1 for i = 1 to N do Task T i,i: x(i) b(i)/a(i, i) for j = i + 1 to N do Task T i,j: b(j) b(j) a(j, i) x(i) A dependence graph G is used. (e : T T ) G means that T can start only if T has already been finished. T is a predecessor of T. Transitivity arc are generally omitted. T 1,2 T 1,3 T 1,4 T 1,5 T 1,6 T 2,2 T 2,3 T 2,4 T 2,5 T 2,6 T 3,3 T 3,4 T 3,5 T 3,6 T 4,4 T 4,5 T 5,5 T 4,6 T 5,6 T 6,6 7/ 26
18 Coarsegrain task graph Taskgraph do not necessarily come from instructionlevel analysis. scan(proteins p) (termgo: ) scan(proteinterms t) (termgo: ) select p.proteinid, blast(p.sequence) from proteins p, proteinterms t where p.proteinid = t.proteinid and t.term = GO: project (p.proteinid,p.sequence) exchange project (t.proteinid) exchange join (p.proteinid=t.proteinid) exchange operation call (blast(p.sequence)) project (p.proteinid,blast) Each task may be parallel, preemptable, divisible,... Each edge depicts a dependency i.e. most of the times some data to transfer. In the following, we will focus on simple models. 8/ 26
19 Coarsegrain task graph Taskgraph do not necessarily come from instructionlevel analysis. scan(proteins p) (termgo: ) scan(proteinterms t) (termgo: ) select p.proteinid, blast(p.sequence) from proteins p, proteinterms t where p.proteinid = t.proteinid and t.term = GO: project (p.proteinid,p.sequence) exchange project (t.proteinid) exchange join (p.proteinid=t.proteinid) exchange operation call (blast(p.sequence)) project (p.proteinid,blast) Each task may be parallel, preemptable, divisible,... Each edge depicts a dependency i.e. most of the times some data to transfer. In the following, we will focus on simple models. 8/ 26
20 Coarsegrain task graph Taskgraph do not necessarily come from instructionlevel analysis. scan(proteins p) (termgo: ) scan(proteinterms t) (termgo: ) select p.proteinid, blast(p.sequence) from proteins p, proteinterms t where p.proteinid = t.proteinid and t.term = GO: project (p.proteinid,p.sequence) exchange project (t.proteinid) exchange join (p.proteinid=t.proteinid) exchange operation call (blast(p.sequence)) project (p.proteinid,blast) Each task may be parallel, preemptable, divisible,... Each edge depicts a dependency i.e. most of the times some data to transfer. In the following, we will focus on simple models. 8/ 26
21 Coarsegrain task graph Taskgraph do not necessarily come from instructionlevel analysis. scan(proteins p) (termgo: ) scan(proteinterms t) (termgo: ) select p.proteinid, blast(p.sequence) from proteins p, proteinterms t where p.proteinid = t.proteinid and t.term = GO: project (p.proteinid,p.sequence) exchange project (t.proteinid) exchange join (p.proteinid=t.proteinid) exchange operation call (blast(p.sequence)) project (p.proteinid,blast) Each task may be parallel, preemptable, divisible,... Each edge depicts a dependency i.e. most of the times some data to transfer. In the following, we will focus on simple models. 8/ 26
22 Outline 1 Task graphs from outer space 2 Scheduling definitions and notions 3 Platform models and scheduling problems 9/ 26
23 Task system Definition (Task system). A task system is an directed graph G = (V, E, w) where : V is the set of tasks (V is finite) E represent the dependence constraints: e = (u, v) E iff u v w : V N is a time function that give the weight (or duration) of each task. T1,1 T1,2 T1,3 T1,4 T1,5 T1,6 We could set w(t i,j ) = 1 but also decide that performing a division is more expensive than a multiplication followed by an addition. T2,2 T2,3 T2,4 T2,5 T2,6 T3,3 T3,4 T3,5 T3,6 T4,4 T4,5 T5,5 T4,6 T5,6 T6,6 10/ 26
24 Task system Definition (Task system). A task system is an directed graph G = (V, E, w) where : V is the set of tasks (V is finite) E represent the dependence constraints: e = (u, v) E iff u v w : V N is a time function that give the weight (or duration) of each task. T1,1 T1,2 T1,3 T1,4 T1,5 T1,6 We could set w(t i,j ) = 1 but also decide that performing a division is more expensive than a multiplication followed by an addition. T2,2 T2,3 T2,4 T2,5 T2,6 T3,3 T3,4 T3,5 T3,6 T4,4 T4,5 T5,5 T4,6 T5,6 T6,6 10/ 26
25 Schedule and Allocation Definition (Schedule). A schedule of a task system G = (V, E, w) is a time function σ : V N such that: (u, v) E, σ(u) + w(u) σ(v) Let us denote by P = {P 1,..., P p } the set of processors. Definition (Allocation). An allocation of a task system G = (V, E, w) is a function π : V P such that: { π(t ) = π(t σ(t ) + w(t ) σ(t ) or ) σ(t ) + w(t ) σ(t ) Depending on the application and platform model, much more complex definitions can be proposed. 11/ 26
26 Schedule and Allocation Definition (Schedule). A schedule of a task system G = (V, E, w) is a time function σ : V N such that: (u, v) E, σ(u) + w(u) σ(v) Let us denote by P = {P 1,..., P p } the set of processors. Definition (Allocation). An allocation of a task system G = (V, E, w) is a function π : V P such that: { π(t ) = π(t σ(t ) + w(t ) σ(t ) or ) σ(t ) + w(t ) σ(t ) Depending on the application and platform model, much more complex definitions can be proposed. 11/ 26
27 Schedule and Allocation Definition (Schedule). A schedule of a task system G = (V, E, w) is a time function σ : V N such that: (u, v) E, σ(u) + w(u) σ(v) Let us denote by P = {P 1,..., P p } the set of processors. Definition (Allocation). An allocation of a task system G = (V, E, w) is a function π : V P such that: { π(t ) = π(t σ(t ) + w(t ) σ(t ) or ) σ(t ) + w(t ) σ(t ) Depending on the application and platform model, much more complex definitions can be proposed. 11/ 26
28 Ganttchart Manipulating functions is generally not very convenient. That is why Ganttchart are used to depict schedules and allocations. T 1 Processors T 2 T 3 T 4 P 3 P 2 T 5 T 6 T 7 P 1 Time T 8 12/ 26
29 Basic feasibility condition Theorem. Let G = (V, E, w) be a task system. There exists a valid schedule of G iff G has no cycle. Sketch of the proof. Assume that G has a cycle v 1 v 2... v k v 1. Then v 1 v 1 and a valid schedule σ should hold σ(v 1 ) + w(v 1 ) σ(v 1 ) true, which is impossible because w(v 1 ) > 0. If G is acyclic, then some tasks have no predecessor. They can be scheduled first. More precisely, we sort topologically the vertexes and schedule them one after the other on the same processor. Dependences are then fulfilled. Therefore all task systems we will be considering in the following are Directed Acyclic Graphs. 13/ 26
30 Basic feasibility condition Theorem. Let G = (V, E, w) be a task system. There exists a valid schedule of G iff G has no cycle. Sketch of the proof. Assume that G has a cycle v 1 v 2... v k v 1. Then v 1 v 1 and a valid schedule σ should hold σ(v 1 ) + w(v 1 ) σ(v 1 ) true, which is impossible because w(v 1 ) > 0. If G is acyclic, then some tasks have no predecessor. They can be scheduled first. More precisely, we sort topologically the vertexes and schedule them one after the other on the same processor. Dependences are then fulfilled. Therefore all task systems we will be considering in the following are Directed Acyclic Graphs. 13/ 26
31 Basic feasibility condition Theorem. Let G = (V, E, w) be a task system. There exists a valid schedule of G iff G has no cycle. Sketch of the proof. Assume that G has a cycle v 1 v 2... v k v 1. Then v 1 v 1 and a valid schedule σ should hold σ(v 1 ) + w(v 1 ) σ(v 1 ) true, which is impossible because w(v 1 ) > 0. If G is acyclic, then some tasks have no predecessor. They can be scheduled first. More precisely, we sort topologically the vertexes and schedule them one after the other on the same processor. Dependences are then fulfilled. Therefore all task systems we will be considering in the following are Directed Acyclic Graphs. 13/ 26
32 Makespan Definition (Makespan). The makespan of a schedule is the total execution time : Processors P3 MS(σ) = max{σ(v) + w(v)} min {σ(v)}. v V v V The makespan is also often referred as C max in the literature. P2 P1 Time C max = max v V C v P b(p): find a schedule with the smallest possible makespan, using at most p processors. MS opt (p) denotes the optimal makespan using only p processors. P b( ): find a schedule with the smallest makespan when the number of processors that can be used is not bounded. We note MS opt ( ) the corresponding makespan. 14/ 26
33 Makespan Definition (Makespan). The makespan of a schedule is the total execution time : Processors P3 MS(σ) = max{σ(v) + w(v)} min {σ(v)}. v V v V The makespan is also often referred as C max in the literature. P2 P1 Time C max = max v V C v P b(p): find a schedule with the smallest possible makespan, using at most p processors. MS opt (p) denotes the optimal makespan using only p processors. P b( ): find a schedule with the smallest makespan when the number of processors that can be used is not bounded. We note MS opt ( ) the corresponding makespan. 14/ 26
34 Makespan Definition (Makespan). The makespan of a schedule is the total execution time : Processors P3 MS(σ) = max{σ(v) + w(v)} min {σ(v)}. v V v V The makespan is also often referred as C max in the literature. P2 P1 Time C max = max v V C v P b(p): find a schedule with the smallest possible makespan, using at most p processors. MS opt (p) denotes the optimal makespan using only p processors. P b( ): find a schedule with the smallest makespan when the number of processors that can be used is not bounded. We note MS opt ( ) the corresponding makespan. 14/ 26
35 Critical path Let Φ = (T 1, T 2,..., T n ) be a path in G. w can be extended to paths in the following way : Lemma. w(φ) = n i=1 w(t i) Let G = (V, E, w) be a DAG and σ p a schedule of G using p processors. For any path Φ in G, we have MS(σ p ) w(φ). Proof. Let Φ = (T 1, T 2,..., T n ) be a path in G: (T i, T i+1 ) E for 1 i < n. Therefore we have σ p (T i ) + w(t i ) σ p (T i+1 ) for 1 i < n, hence n MS(σ p ) w(t n ) + σ p (T n ) σ p (T 1 ) w(t i ) = w(φ). i=1 15/ 26
36 Critical path Let Φ = (T 1, T 2,..., T n ) be a path in G. w can be extended to paths in the following way : Lemma. w(φ) = n i=1 w(t i) Let G = (V, E, w) be a DAG and σ p a schedule of G using p processors. For any path Φ in G, we have MS(σ p ) w(φ). Proof. Let Φ = (T 1, T 2,..., T n ) be a path in G: (T i, T i+1 ) E for 1 i < n. Therefore we have σ p (T i ) + w(t i ) σ p (T i+1 ) for 1 i < n, hence n MS(σ p ) w(t n ) + σ p (T n ) σ p (T 1 ) w(t i ) = w(φ). i=1 15/ 26
37 Critical path Let Φ = (T 1, T 2,..., T n ) be a path in G. w can be extended to paths in the following way : Lemma. w(φ) = n i=1 w(t i) Let G = (V, E, w) be a DAG and σ p a schedule of G using p processors. For any path Φ in G, we have MS(σ p ) w(φ). Proof. Let Φ = (T 1, T 2,..., T n ) be a path in G: (T i, T i+1 ) E for 1 i < n. Therefore we have σ p (T i ) + w(t i ) σ p (T i+1 ) for 1 i < n, hence n MS(σ p ) w(t n ) + σ p (T n ) σ p (T 1 ) w(t i ) = w(φ). i=1 15/ 26
38 Speedup and Efficiency Definition. Let G = (V, E, w) be a DAG and σ p a schedule of G using only p processors: Speedup: s(σ p ) = Efficiency: e(σ p ) = s(σ p) p Seq MS(σ p ), where Seq = MS opt(1) = v V = Seq p MS(σ p ). w(v). Theorem. Let G = (V, E, w) be a DAG. For any schedule σ p using p processors: 0 e(σ p ) 1. 16/ 26
39 Speedup and Efficiency Definition. Let G = (V, E, w) be a DAG and σ p a schedule of G using only p processors: Speedup: s(σ p ) = Efficiency: e(σ p ) = s(σ p) p Seq MS(σ p ), where Seq = MS opt(1) = v V = Seq p MS(σ p ). w(v). Theorem. Let G = (V, E, w) be a DAG. For any schedule σ p using p processors: 0 e(σ p ) 1. 16/ 26
40 Speedup and Efficiency Definition. Let G = (V, E, w) be a DAG and σ p a schedule of G using only p processors: Speedup: s(σ p ) = Efficiency: e(σ p ) = s(σ p) p Seq MS(σ p ), where Seq = MS opt(1) = v V = Seq p MS(σ p ). w(v). Theorem. Let G = (V, E, w) be a DAG. For any schedule σ p using p processors: 0 e(σ p ) 1. 16/ 26
41 Speedup and Efficiency (Cont d) Theorem. Let G = (V, E, w) be a DAG. For any schedule σ p using p processors: 0 e(σ p ) 1. Proof. Processors P4 P3 P2 P1 Time active idle Let Idle denote the total idle time. Seq + Idle is then equal to the total surface of the rectangle, i.e. p MS(σ p ). Seq Therefore e(σ p ) = p MS(σ p ) 1. The speedup is thus bound the number of processors. No supralinear speedup in our model! 17/ 26
42 A trivial result Theorem. Let G = (V, E, w) be a DAG. We have Seq = MS opt (1)... MS opt (p) MS opt (p + 1)... MS opt ( ). Allowing to use more processors cannot hurt. However, using more processors may hurt, especially in a model where communications are taken into account. If we define MS (p) as the smallest makespan of schedules using exactly p processors, we may have MS (p) > MS (p ) with p < p. 18/ 26
43 A trivial result Theorem. Let G = (V, E, w) be a DAG. We have Seq = MS opt (1)... MS opt (p) MS opt (p + 1)... MS opt ( ). Allowing to use more processors cannot hurt. However, using more processors may hurt, especially in a model where communications are taken into account. If we define MS (p) as the smallest makespan of schedules using exactly p processors, we may have MS (p) > MS (p ) with p < p. 18/ 26
44 A trivial result Theorem. Let G = (V, E, w) be a DAG. We have Seq = MS opt (1)... MS opt (p) MS opt (p + 1)... MS opt ( ). Allowing to use more processors cannot hurt. However, using more processors may hurt, especially in a model where communications are taken into account. If we define MS (p) as the smallest makespan of schedules using exactly p processors, we may have MS (p) > MS (p ) with p < p. 18/ 26
45 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
46 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
47 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
48 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
49 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
50 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
51 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
52 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
53 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
54 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
55 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
56 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
57 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
58 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
59 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
60 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
61 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
62 Graham Notation Many parameter can change in a scheduling problem. Graham has then proposed the following classification : α β γ [Bru98] α is the processor environment (a few examples): : single processor; P : identical processors; Q: uniform processors; R: unrelated processors; β describe task and resource characteristics (a few examples): pmtn: preemption; prec, tree or chains: general precedence constraints, tree constraints, set of chain constraints and independent tasks otherwise; rj : tasks have release dates; pj = p or p p j p: all task have processing time equal to p, or comprised between p and p, or have arbitrary processing times otherwise; d: deadlines; γ denotes the optimization criterion (a few examples): Cmax : makespan; Ci : average completion time; w i C i : weighted A.C.T; Lmax : maximum lateness (max C i d i );... 19/ 26
63 Outline 1 Task graphs from outer space 2 Scheduling definitions and notions 3 Platform models and scheduling problems 20/ 26
64 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
65 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
66 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
67 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
68 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
69 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
70 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
71 Complexity results If we have an infinite number of processors, the assoonaspossible schedule is optimal. MS opt ( ) = max w(φ). Φ path in G P, 2 C max is weakly NPcomplete (2Partition); Proof. By reduction to 2Partition: can A = {a 1,..., a n } be partitioned into two sets A 1, A 2 such a A 1 a = a A 2 a. p = 2, G = (V, E, w with V = {v 1,..., v n }, E = and w(v i ) = a i, 1 i n. Finding a schedule of makespan smaller or equal to 1 2 i a i is equivalent to solve the instance of 2Partition. P, 3 prec C max is strongly NPcomplete (3DM); P prec, p j = 1 C max is strongly NPcomplete (3DM); P, p 3 prec, p j = 1 C max is open; P, 2 prec, 1 p j 2 C max is strongly NPcomplete; 21/ 26
72 List scheduling When simple problems are hard, we should try to find good approximation heuristics. Natural idea: using greedy strategy like trying to allocate the most possible task at a given timestep. However at some point we may face a choice (when there is more ready tasks than available processors). Any strategy that does not let on purpose a processor idle is efficient. Such a schedule is called listschedule. Theorem (Coffman). Let G = (V, E, w) be a DAG, p the number of processors, and σ p a listschedule of G. ( ) MS(σ p ) 2 1 p MS opt (p). One can actually prove that this bound cannot be improved. Most of the time, listheuristics are based on the critical path. 22/ 26
73 List scheduling When simple problems are hard, we should try to find good approximation heuristics. Natural idea: using greedy strategy like trying to allocate the most possible task at a given timestep. However at some point we may face a choice (when there is more ready tasks than available processors). Any strategy that does not let on purpose a processor idle is efficient. Such a schedule is called listschedule. Theorem (Coffman). Let G = (V, E, w) be a DAG, p the number of processors, and σ p a listschedule of G. ( ) MS(σ p ) 2 1 p MS opt (p). One can actually prove that this bound cannot be improved. Most of the time, listheuristics are based on the critical path. 22/ 26
74 List scheduling When simple problems are hard, we should try to find good approximation heuristics. Natural idea: using greedy strategy like trying to allocate the most possible task at a given timestep. However at some point we may face a choice (when there is more ready tasks than available processors). Any strategy that does not let on purpose a processor idle is efficient. Such a schedule is called listschedule. Theorem (Coffman). Let G = (V, E, w) be a DAG, p the number of processors, and σ p a listschedule of G. ( ) MS(σ p ) 2 1 p MS opt (p). One can actually prove that this bound cannot be improved. Most of the time, listheuristics are based on the critical path. 22/ 26
75 Taking communications into account A very simple model (things are already complicated enough): the macrodata flow model. If there is some datadependence between T and T, the communication cost is { c(t, T 0 if alloc(t ) = alloc(t ) ) = c(t, T ) otherwise Definition. A DAG with communication cost (say cdag) is a directed acyclic graph G = (V, E, w, c) where vertexes represent tasks and edges represent dependence constraints. w : V N is the computation time function and c : E N is the communication time function. Any valid schedule has to respect the dependence constraints. e = (v, v ) E, { σ(v) + w(v) σ(v ) if alloc(v) = alloc(v ) σ(v) + w(v) + c(v; v ) σ(v ) otherwise. 23/ 26
76 Taking communications into account A very simple model (things are already complicated enough): the macrodata flow model. If there is some datadependence between T and T, the communication cost is { c(t, T 0 if alloc(t ) = alloc(t ) ) = c(t, T ) otherwise Definition. A DAG with communication cost (say cdag) is a directed acyclic graph G = (V, E, w, c) where vertexes represent tasks and edges represent dependence constraints. w : V N is the computation time function and c : E N is the communication time function. Any valid schedule has to respect the dependence constraints. e = (v, v ) E, { σ(v) + w(v) σ(v ) if alloc(v) = alloc(v ) σ(v) + w(v) + c(v; v ) σ(v ) otherwise. 23/ 26
77 Taking communications into account (cont d) Even Pb( ) is NPcomplete!!! You constantly have to figure out whether you should use more processor (but then pay more fore communications) or not. Finding the good tradeoff is a real challenge. 4/3approximation if all communication times are smaller than computation times. Finding guaranteed approximations for other settings is really hard, but really useful. That must be the reason why there is so much research about it! 24/ 26
78 Conclusion Most of the time, the only thing we can done is to compare heuristics. There are three ways of doing that: Theory: being able to guarantee your heuristic; Experiment: Generating random graphs and/or typical application graphs along with platform graphs to compare your heuristics. Smart: proving that your heuristic is optimal for a particular class of graphs (fork, join, forkjoin, bounded degree,... ). However, remember that the first thing to do is to look whether your problem is NPcomplete or not. Who knows? You may be lucky... 25/ 26
79 Bibliography A.J. Bernstein. Analysis of programs for parallel processing. IEEE Transactions on Electronic Computers, 15: , October Peter Brucker. Scheduling Algorithms. Springer, Heidelberg, 2 edition, / 26
Scheduling Shop Scheduling. Tim Nieberg
Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations
More informationReal Time Scheduling Basic Concepts. Radek Pelánek
Real Time Scheduling Basic Concepts Radek Pelánek Basic Elements Model of RT System abstraction focus only on timing constraints idealization (e.g., zero switching time) Basic Elements Basic Notions task
More informationApproximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine
Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NPhard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Tradeoff between time
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More informationClassification  Examples
Lecture 2 Scheduling 1 Classification  Examples 1 r j C max given: n jobs with processing times p 1,...,p n and release dates r 1,...,r n jobs have to be scheduled without preemption on one machine taking
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationCost Model: Work, Span and Parallelism. 1 The RAM model for sequential computation:
CSE341T 08/31/2015 Lecture 3 Cost Model: Work, Span and Parallelism In this lecture, we will look at how one analyze a parallel program written using Cilk Plus. When we analyze the cost of an algorithm
More informationScheduling Single Machine Scheduling. Tim Nieberg
Scheduling Single Machine Scheduling Tim Nieberg Single machine models Observation: for nonpreemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe
More informationSingle machine parallel batch scheduling with unbounded capacity
Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University
More informationClassification  Examples 1 1 r j C max given: n jobs with processing times p 1,..., p n and release dates
Lecture 2 Scheduling 1 Classification  Examples 11 r j C max given: n jobs with processing times p 1,..., p n and release dates r 1,..., r n jobs have to be scheduled without preemption on one machine
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More information11. APPROXIMATION ALGORITHMS
11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005
More informationCommon Approaches to RealTime Scheduling
Common Approaches to RealTime Scheduling Clockdriven timedriven schedulers Prioritydriven schedulers Examples of priority driven schedulers Effective timing constraints The EarliestDeadlineFirst
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationComplexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar
Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationA STUDY OF TASK SCHEDULING IN MULTIPROCESSOR ENVIROMENT Ranjit Rajak 1, C.P.Katti 2, Nidhi Rajak 3
A STUDY OF TASK SCHEDULING IN MULTIPROCESSOR ENVIROMENT Ranjit Rajak 1, C.P.Katti, Nidhi Rajak 1 Department of Computer Science & Applications, Dr.H.S.Gour Central University, Sagar, India, ranjit.jnu@gmail.com
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationMapReduce and Distributed Data Analysis. Sergei Vassilvitskii Google Research
MapReduce and Distributed Data Analysis Google Research 1 Dealing With Massive Data 2 2 Dealing With Massive Data Polynomial Memory Sublinear RAM Sketches External Memory Property Testing 3 3 Dealing With
More informationIntroducción a Calendarización en Sistemas Paralelas, Grids y Nubes
CICESE Research Center Ensenada, Baja California Mexico Introducción a Calendarización en Sistemas Paralelas, Grids y Nubes Dr. Andrei Tchernykh CICESE Centro de Investigación Científica y de Educación
More informationMatching and Scheduling Algorithms on Large Scale Dynamic Graphs
Matching and Scheduling Algorithms on Large Scale Dynamic Graphs Author: Xander van den Eelaart Supervisors: Prof. dr. Rob H. Bisseling ir. Marcel F. van den Elst Second reader: dr. Tobias Müller Master
More informationDuplicating and its Applications in Batch Scheduling
Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China
More informationDiversity Coloring for Distributed Data Storage in Networks 1
Diversity Coloring for Distributed Data Storage in Networks 1 Anxiao (Andrew) Jiang and Jehoshua Bruck California Institute of Technology Pasadena, CA 9115, U.S.A. {jax, bruck}@paradise.caltech.edu Abstract
More informationDistributed Load Balancing for Machines Fully Heterogeneous
Internship Report 2 nd of June  22 th of August 2014 Distributed Load Balancing for Machines Fully Heterogeneous Nathanaël Cheriere nathanael.cheriere@ensrennes.fr ENS Rennes Academic Year 20132014
More informationPartitioned realtime scheduling on heterogeneous sharedmemory multiprocessors
Partitioned realtime scheduling on heterogeneous sharedmemory multiprocessors Martin Niemeier École Polytechnique Fédérale de Lausanne Discrete Optimization Group Lausanne, Switzerland martin.niemeier@epfl.ch
More informationSingle machine models: Maximum Lateness 12 Approximation ratio for EDD for problem 1 r j,d j < 0 L max. structure of a schedule Q...
Lecture 4 Scheduling 1 Single machine models: Maximum Lateness 12 Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule 0 Q 1100 11 00 11 000 111 0 0 1 1 00 11 00 11 00
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationMinimum Makespan Scheduling
Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,
More informationParallel machines scheduling with applications to Internet adslot placement
UNLV Theses/Dissertations/Professional Papers/Capstones 122011 Parallel machines scheduling with applications to Internet adslot placement Shaista Lubna University of Nevada, Las Vegas Follow this and
More informationScheduling Parallel Machine Scheduling. Tim Nieberg
Scheduling Parallel Machine Scheduling Tim Nieberg Problem P C max : m machines n jobs with processing times p 1,..., p n Problem P C max : m machines n jobs with processing times p 1,..., p { n 1 if job
More information19/04/2016. What does it mean? ResponseTime Analysis of Conditional DAG Tasks in Multiprocessor Systems. What does it mean? What does it mean?
9/04/20 What does it mean? «Responsetime analysis» ResponseTime Analysis of Conditional DAG Tasks in Multiprocessor Systems «conditional» «DAG tasks» «multiprocessor systems» Alessandra Melani 2 What
More information1 Approximating Set Cover
CS 05: Algorithms (Grad) Feb 224, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the setcovering problem consists of a finite set X and a family F of subset of X, such that every elemennt
More informationFIND ALL LONGER AND SHORTER BOUNDARY DURATION VECTORS UNDER PROJECT TIME AND BUDGET CONSTRAINTS
Journal of the Operations Research Society of Japan Vol. 45, No. 3, September 2002 2002 The Operations Research Society of Japan FIND ALL LONGER AND SHORTER BOUNDARY DURATION VECTORS UNDER PROJECT TIME
More informationEcient approximation algorithm for minimizing makespan. on uniformly related machines. Chandra Chekuri. November 25, 1997.
Ecient approximation algorithm for minimizing makespan on uniformly related machines Chandra Chekuri November 25, 1997 Abstract We obtain a new ecient approximation algorithm for scheduling precedence
More informationAnswers to some of the exercises.
Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the kcenter algorithm first to D and then for each center in D
More informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationImproving Experiments by Optimal Blocking: Minimizing the Maximum Withinblock Distance
Improving Experiments by Optimal Blocking: Minimizing the Maximum Withinblock Distance Michael J. Higgins Jasjeet Sekhon April 12, 2014 EGAP XI A New Blocking Method A new blocking method with nice theoretical
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationFairness in Routing and Load Balancing
Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationScheduling Parallel Jobs with Monotone Speedup 1
Scheduling Parallel Jobs with Monotone Speedup 1 Alexander Grigoriev, Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands, {a.grigoriev@ke.unimaas.nl,
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationCompletion Time Scheduling and the WSRPT Algorithm
Completion Time Scheduling and the WSRPT Algorithm Bo Xiong, Christine Chung Department of Computer Science, Connecticut College, New London, CT {bxiong,cchung}@conncoll.edu Abstract. We consider the online
More informationDimensioning an inbound call center using constraint programming
Dimensioning an inbound call center using constraint programming Cyril Canon 1,2, JeanCharles Billaut 2, and JeanLouis Bouquard 2 1 Vitalicom, 643 avenue du grain d or, 41350 Vineuil, France ccanon@fr.snt.com
More informationScheduling Parallel Jobs with Linear Speedup
Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev,m.uetz}@ke.unimaas.nl
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More informationScheduling Realtime Tasks: Algorithms and Complexity
Scheduling Realtime Tasks: Algorithms and Complexity Sanjoy Baruah The University of North Carolina at Chapel Hill Email: baruah@cs.unc.edu Joël Goossens Université Libre de Bruxelles Email: joel.goossens@ulb.ac.be
More informationPh.D. Thesis. Judit NagyGyörgy. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit NagyGyörgy Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
More informationLecture 6: Approximation via LP Rounding
Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other
More informationLecture Notes 12: Scheduling  Cont.
Online Algorithms 18.1.2012 Professor: Yossi Azar Lecture Notes 12: Scheduling  Cont. Scribe:Inna Kalp 1 Introduction In this Lecture we discuss 2 scheduling models. We review the scheduling over time
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationLecture Outline Overview of realtime scheduling algorithms Outline relative strengths, weaknesses
Overview of RealTime Scheduling Embedded RealTime Software Lecture 3 Lecture Outline Overview of realtime scheduling algorithms Clockdriven Weighted roundrobin Prioritydriven Dynamic vs. static Deadline
More informationMincost flow problems and network simplex algorithm
Mincost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.
More informationLecture 4 Online and streaming algorithms for clustering
CSE 291: Geometric algorithms Spring 2013 Lecture 4 Online and streaming algorithms for clustering 4.1 Online kclustering To the extent that clustering takes place in the brain, it happens in an online
More information1 Digraphs. Definition 1
1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together
More informationLoad Balancing. Load Balancing 1 / 24
Load Balancing Backtracking, branch & bound and alphabeta pruning: how to assign work to idle processes without much communication? Additionally for alphabeta pruning: implementing the youngbrotherswait
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the FordFulkerson
More informationScheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling
Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open
More informationExponential time algorithms for graph coloring
Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A klabeling of vertices of a graph G(V, E) is a function V [k].
More informationDecentralized Utilitybased Sensor Network Design
Decentralized Utilitybased Sensor Network Design Narayanan Sadagopan and Bhaskar Krishnamachari University of Southern California, Los Angeles, CA 900890781, USA narayans@cs.usc.edu, bkrishna@usc.edu
More informationApproximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai
Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More informationIntegrating job parallelism in realtime scheduling theory
Integrating job parallelism in realtime scheduling theory Sébastien Collette Liliana Cucu Joël Goossens Abstract We investigate the global scheduling of sporadic, implicit deadline, realtime task systems
More informationAnalysis of Algorithms I: Optimal Binary Search Trees
Analysis of Algorithms I: Optimal Binary Search Trees Xi Chen Columbia University Given a set of n keys K = {k 1,..., k n } in sorted order: k 1 < k 2 < < k n we wish to build an optimal binary search
More informationAnalysis of Binary Search algorithm and Selection Sort algorithm
Analysis of Binary Search algorithm and Selection Sort algorithm In this section we shall take up two representative problems in computer science, work out the algorithms based on the best strategy to
More informationNetwork File Storage with Graceful Performance Degradation
Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed
More informationModeling and Performance Evaluation of Computer Systems Security Operation 1
Modeling and Performance Evaluation of Computer Systems Security Operation 1 D. Guster 2 St.Cloud State University 3 N.K. Krivulin 4 St.Petersburg State University 5 Abstract A model of computer system
More informationResource Allocation with Time Intervals
Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes
More informationNPHardness Results Related to PPAD
NPHardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China EMail: mecdang@cityu.edu.hk Yinyu
More informationSolutions to Homework 6
Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example
More informationIntroduction to production scheduling. Industrial Management Group School of Engineering University of Seville
Introduction to production scheduling Industrial Management Group School of Engineering University of Seville 1 Introduction to production scheduling Scheduling Production scheduling Gantt Chart Scheduling
More informationOnline Bipartite Perfect Matching With Augmentations
Online Bipartite Perfect Matching With Augmentations Kamalika Chaudhuri, Constantinos Daskalakis, Robert D. Kleinberg, and Henry Lin Information Theory and Applications Center, U.C. San Diego Email: kamalika@soe.ucsd.edu
More informationOPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION
OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION Sérgio Pequito, Stephen Kruzick, Soummya Kar, José M. F. Moura, A. Pedro Aguiar Department of Electrical and Computer Engineering
More informationPrivate Approximation of Clustering and Vertex Cover
Private Approximation of Clustering and Vertex Cover Amos Beimel, Renen Hallak, and Kobbi Nissim Department of Computer Science, BenGurion University of the Negev Abstract. Private approximation of search
More informationeach college c i C has a capacity q i  the maximum number of students it will admit
n colleges in a set C, m applicants in a set A, where m is much larger than n. each college c i C has a capacity q i  the maximum number of students it will admit each college c i has a strict order i
More informationIE 680 Special Topics in Production Systems: Networks, Routing and Logistics*
IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti
More informationCyclic scheduling: an introduction
Cyclic scheduling: an introduction Claire Hanen Claire.Hanen@lip6.fr Université Paris X Nanterre / LIP6 UPMC Cyclic scheduling for EPIT 00 p. /? Summary Definition Uniform precedence constraints Feasibility
More informationCSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24
CSL851: Algorithmic Graph Theory Semester I 20132014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have
More informationR u t c o r Research R e p o r t. A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS.
R u t c o r Research R e p o r t A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS Navid Hashemian a Béla Vizvári b RRR 32011, February 21, 2011 RUTCOR Rutgers
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More informationA Practical Scheme for Wireless Network Operation
A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving
More informationEmbedded Systems 20 REVIEW. Multiprocessor Scheduling
Embedded Systems 0   Multiprocessor Scheduling REVIEW Given n equivalent processors, a finite set M of aperiodic/periodic tasks find a schedule such that each task always meets its deadline. Assumptions:
More information3. Markov property. Markov property for MRFs. HammersleyClifford theorem. Markov property for Bayesian networks. Imap, Pmap, and chordal graphs
Markov property 33. Markov property Markov property for MRFs HammersleyClifford theorem Markov property for ayesian networks Imap, Pmap, and chordal graphs Markov Chain X Y Z X = Z Y µ(x, Y, Z) = f(x,
More informationLecture 11: AllPairs Shortest Paths
Lecture 11: AllPairs Shortest Paths Introduction Different types of algorithms can be used to solve the allpairs shortest paths problem: Dynamic programming Matrix multiplication FloydWarshall algorithm
More informationMinimizing the Number of Machines in a UnitTime Scheduling Problem
Minimizing the Number of Machines in a UnitTime Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.basnet.by Frank
More informationDependability Driven Integration of Mixed Criticality SW Components
Dependability Driven Integration of Mixed Criticality SW Components Shariful Islam, Robert Lindström and Neeraj Suri Department of Computer Science TU Darmstadt, Germany {ripon,rl,suri}@informatik.tudarmstadt.de
More informationReliability Guarantees in Automata Based Scheduling for Embedded Control Software
1 Reliability Guarantees in Automata Based Scheduling for Embedded Control Software Santhosh Prabhu, Aritra Hazra, Pallab Dasgupta Department of CSE, IIT Kharagpur West Bengal, India  721302. Email: {santhosh.prabhu,
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationOptimized Asynchronous Passive MultiChannel Discovery of BeaconEnabled Networks
t t Technische Universität Berlin Telecommunication Networks Group arxiv:1506.05255v1 [cs.ni] 17 Jun 2015 Optimized Asynchronous Passive MultiChannel Discovery of BeaconEnabled Networks Niels Karowski,
More information10.1 Integer Programming and LP relaxation
CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely
More informationLoad balancing in a heterogeneous computer system by selforganizing Kohonen network
Bull. Nov. Comp. Center, Comp. Science, 25 (2006), 69 74 c 2006 NCC Publisher Load balancing in a heterogeneous computer system by selforganizing Kohonen network Mikhail S. Tarkov, Yakov S. Bezrukov Abstract.
More informationRUN: Optimal Multiprocessor RealTime Scheduling via Reduction to Uniprocessor
1 RUN: Optimal Multiprocessor RealTime Scheduling via Reduction to Uniprocessor Paul Regnier, George Lima, Ernesto Massa Federal University of Bahia, State University of Bahia Salvador, Bahia, Brasil
More informationTools for parsimonious edgecolouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR201010
Tools for parsimonious edgecolouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR201010 Tools for parsimonious edgecolouring of graphs
More information