Greedy Algorithm



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Bog'liq
greedy (1)

Theorem 5.6. The schedule created by selecting the earliest-ending activity that doesn't conflict with those already selected is optimal and feasible.
Proof. Feasibility follows as we select the earliest activity that doesn't conflict. Let E = (( and i 1 , b i 1 ) , . . . , ( a i Æ , b i Æ )) b y the output created b y selecting the earliest-ending activi t y and S = (( a j 1 , b j 1 ) , . . . , ( a j k , b j k )) a n y other feasible s c hedule.


W e claim that f or all m k , ( a i m , b i m ) exists and b i m b j m (or in l a yman's terms the m th activity in schedule E always ends before then m th activity in schedule S ). Assume the claim is false and m is the smallest counterexample. Necessarily m = 1 cannot be a counterexample as the schedule E by construction takes the first-ending event and must end before an event in any other schedule.

If m is a cou n terexample, then b i m - 1 b j m - 1 a j m and b i m > b j m . This means that the event j m ends prior to the event i m and is feasible with the other events of E. But then event j m would have has been chosen over event i m , a contradiction. So the claim is true.




Therefore, the number of events in E is at least that of any other feasible schedule S , proving the optimality of E. _

Let's take a moment to reflect on the general strategy employed in this proof: We considered the greedy solution E and any solution S and then proved that E is better than S by arguing that if it weren't, then it would contradict the construction of E. _





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