Chapter 16. Greedy Algorithms
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1 Chapter 16. Greedy Algorithms Algorithms for optimization problems (minimization or maximization problems) typically go through a sequence of steps, with a set of choices at each step. A greedy algorithm always makes the choice that is the best at the current step. That is, it makes a locally optimal choice in the hope this choice will lead to a globally optimal solution. For some optimization problems, the greedy algorithm does not yield optimal solutions, but for many problems, it does. An example of greedy algorithms does not work: Consider finding a shortest path through a layered network: Construct a path by always adding an edge of shortest length. COMP3600/6466: Lecture
2 16.1 Greedy Algorithms s a b c d e f The shortest path is s c t. COMP3600/6466: Lecture t
3 16.1. Incompatible task scheduling Suppose that there are n tasks T 1,T 2,...,T n, where task T i must start at time s i and finish at time f i with s i f i. No two tasks can be performed at the same time (as there is only one CPU). We say that two tasks T i and T j are incompatible if their time intervals [s i, f i ] and [s j, f j ] are overlapping with each other i j, i.e., [s i, f i ] [s j, f j ] /0. The incompatible task scheduling problem is to admit as many tasks as possible while the admitted tasks are compatible with each other. First attempt at greedy solution: Repeatedly choose the earliest starting task that is compatible with previously chosen tasks. This doesn t work. Second attempt at greedy solution: Repeatedly choose the earliest finishing task that is compatible with previously chosen tasks. This works! Notice that this schedule algorithm usually is also refereed as the EDF algorithm (the Earliest Deadline First algorithm). COMP3600/6466: Lecture
4 16.1. Incompatible task scheduling (continued) Theorem. The solution delivered by the EDF algorithm is optimal: Repeatedly choose the earliest-finishing task that is compatible with previously chosen tasks. Proof. Let S 1,S 2,S 3,...,S k be any solution (including the optimal solution) to the incompatible task scheduling problem, where S i is the choice of a task at the ith step. Let G 1,G 2,G 3,... G k be the greedy solution. According to the greedy rule, G 1 finishes no later than S 1. Therefore, S 2 is compatible with G 1, so G 2 finishes no later than S 2, S 3 is compatible with G 2, and so on. In general, for any i with 1 i < k, G i finishes no later than S i, and so S i+1 is compatible with G 1,...,G i. Therefore, the greedy solution can be continued for another task G i+1, which finishes no later than S i+1. Thus, the solution consisting of G 1,...,G k is at least as good as the solution consisting of S 1,...,S k. As the solution S 1,S 2,S 3,...,S k is any solution (including the optimal solution), the greedy solution is at least as good as the solution, i.e., it is an optimal solution to the problem. COMP3600/6466: Lecture
5 16.1. Incompatible task scheduling (continued) The greedy solution has at least as many tasks as any other solution. It thus must have the maximum possible number of tasks admitted. We have also proved that the greedy solution finishes no later than any optimal solution. If all tasks are sorted by their finishing time, the following pseudocode is for the problem. Greedy CPU Scheduling(s, f ) 1 A {T 1 }; /* A is the solution */ 2 j 1; 3 for i 2 to n 4 do if s i f j 5 then A A {T i }; 6 j i; 7 return A. To implement the algorithm, sorting all tasks in order of their finishing time takes O(nlogn) time, the rest takes O(n) time. COMP3600/6466: Lecture
6 COMP3600/6466: Lecture
7 16.x Load balancing problem Given a set of m machines M 1,M 2,...,M m and a set of n jobs, each job j has a processing time t j > 0 with 1 j n. We seek to assign each job to one of the m machines so that the loads placed on all machines are as balanced as possible, where the load at a machine is the sum of processing times of all jobs allocated to the machine. Unfortunately this problem is NP-hard even if m = 2, i.e., it is very unlikely that this problem can be solved in polynomial time unless P=NP. We instead find a feasible solution to it. We also want to know how far this approximate solution from the optimal solution of the problem. If we are able to provide a provable approximation guarantee between a feasible solution delivered by a polynomial algorithm and the optimal solution of the problem, we then call this algorithm is an approximation algorithm for the load-balancing problem. COMP3600/6466: Lecture
8 16.x Load balancing problem (cont.) Let A(i) denote the set of jobs assigned to machine M i. Under an assignment, machine M i needs to work for a total time of T i = j A(i) t j, which is the load at machine M i for all i, 1 i m. Let T be the maximum load among all machines, i.e., T = max{t i 1 i m}. We aim to minimize a quantity known as the makespan, i.e., the minimum value of T. In other words, our objective is to minimize max {T i 1 i m}, The greedy strategy: examine the jobs in the job sequence one by one, and assign the current examining job j to a machine M i with the minimum load at the moment. COMP3600/6466: Lecture
9 16.x Load balancing problem (cont.) Greedy Balance(n,m) 1 for i 1 to m do 2 T i 0; /* the work load at machine i */ 3 A(i) /0; /* the set of jobs assigned to machine i */ 4 endfor; 5 for j 1 to n do 6 Let M i be a machine achieving the minimum load, i.e., T i = min {T i 1 i m}; 7 Assign job j to machine M i ; 8 A(i) A(i) { j}; 9 T i T i +t j ; 10 endfor Exercise: What s the running time of Algorithm Greedy Balance? COMP3600/6466: Lecture
10 16.x Load balancing problem (cont.) Lemma: Let T be the optimal makespan (load), then (i) T 1 m n j=1t j ; (ii) T max{ t j 1 j n}, as each job is not allowed to be distributed to more than one machine for its processing. COMP3600/6466: Lecture
11 Load balancing problem (cont.) Theorem: Algorithm Greedy-Balance produces an assignment of jobs to machines with makespan T 2T, where T and T are the loads delivered by the greedy algorithm and an optimal load of the problem, i.e., the proposed algorithm is 2-approximation algorithm for the load balancing problem. Proof: We assume that machine M i attains the maximum load T in our assignment and job j is the last job assigned to machine M i, respectively. The load of M i is the smallest prior to the addition of job j, which is T i t j, and every other machine i with 1 i m and i i has a load at least T i t j. Thus, we have m k=1 T k m(t i t j ). (1) COMP3600/6466: Lecture
12 As m k=1 T k m(t i t j ), we have T i t j 1 m = 1 m m T k k=1 n t j j=1 16.x Load balancing problem (cont.) T, by the lemma (2) Following the assumption that the makespan T is equal to T i, we have T = T i = (T i t j ) +t j T + T = 2T. (3) Exercise: Devise a 3 -approximation algorithm for the load balancing problem. 2 COMP3600/6466: Lecture
13 16.y Design of greedy algorithms Check the problem is an optimization problem (minimization or maximization problem) identify which greedy strategy is applied to the problem devise the greedy algorithm show the correctness of the proposed algorithm analyze the time complexity of the proposed algorithm COMP3600/6466: Lecture
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