Question

You are consulting for a trucking company that does a large amount of business shipping packages between New York and Boston. The volume is high enough that they have to send a number of trucks each day between the two locations. Trucks have a fixed limit W on the maximum amount of weight they are allowed to carry. Boxes arrive at the New York station one by one, and each package i has a weight w_i. The trucking station is quite small, so at most one truck can be at the station at any time. Company policy requires that boxes are shipped in the order they arrive; otherwise, a customer might get upset upon seeing a box that arrived after his make it to Boston faster. At the moment, the company is using a simple greedy algorithm for packing: they pack boxes in the order they arrive, and whenever the next box does not fit, they send the truck on its way. Prove that, for a given set of boxes with specified weights, the greedy algorithm currently in use actually minimizes the number of trucks that are needed. Hint: Use the stay ahead method.

          You are consulting for a trucking company that does a large amount of business shipping packages
between New York and Boston. The volume is high enough that they have to send a number of trucks
each day between the two locations. Trucks have a fixed limit W on the maximum amount of weight
they are allowed to carry. Boxes arrive at the New York station one by one, and each package i has a
weight w_i. The trucking station is quite small, so at most one truck can be at the station at any time.
Company policy requires that boxes are shipped in the order they arrive; otherwise, a customer might
get upset upon seeing a box that arrived after his make it to Boston faster. At the moment, the company
is using a simple greedy algorithm for packing: they pack boxes in the order they arrive, and whenever
the next box does not fit, they send the truck on its way.
Prove that, for a given set of boxes with specified weights, the greedy algorithm currently in use actually
minimizes the number of trucks that are needed. Hint: Use the stay ahead method.

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You are consulting for a trucking company that does a large amount of business shipping packages
between New York and Boston. The volume is high enough that they have to send a number of trucks
each day between the two locations. Trucks have a fixed limit W on the maximum amount of weight
they are allowed to carry. Boxes arrive at the New York station one by one, and each package i has a
weight wi. The trucking station is quite small, so at most one truck can be at the station at any time.
Company policy requires that boxes are shipped in the order they arrive; otherwise, a customer might
get upset upon seeing a box that arrived after his make it to Boston faster. At the moment, the company
is using a simple greedy algorithm for packing: they pack boxes in the order they arrive, and whenever
the next box does not fit, they send the truck on its way.
Prove that, for a given set of boxes with specified weights, the greedy algorithm currently in use actually
minimizes the number of trucks that are needed. Hint: Use the stay ahead method.

Added by Peter W.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Aarya B

06/05/2023

Step 1

.., Bn. Let's also denote the sequence of trucks used by the greedy algorithm as T1, T2, ..., Tm, and the sequence of trucks used by an optimal algorithm as O1, O2, ..., Ok. We will prove by induction that for each i, the total weight of boxes in trucks T1 Show more…

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Transcript

00:02 In this question, we have two situations.

00:03 First one is what? so, slash for all we have what? all i is less than k and we have ti that is equals to what? that is equals to t dash i and tk this is equals to what? this is equals to t dash k.

00:28 This is factorial.

00:29 Tk factorial equals to what? t dash k.

00:31 Second we have what? second we have tk factorial this is equal to t dash k is equals to that is either.

00:44 Now, from the first situation here the greedy solution is what? the first one is what we have? we have tk is equals to 1 is equals to 1 plus t dash k that is what? that is the greedy solution switched trucks before what? trucks before the optimal solution.

01:33 Ok.

01:33 So, here but the greedy solution only switched to what? only switched trucks when the current truck is full.

01:41 When what? when current truck is full.

01:51 Here since what we have? since we have ti this is equals to what? this is equals to t dash i.

02:03 Ok and i is what? i is less than k.

02:07 So, the content of the current truck after adding what? after adding the k minus 1th place 1th package.

02:30 Ok.

02:32 Are identical for greedy and the optimal solution.

02:35 So, the greedy solution switched truck that means the truck can't fit the kth package to the optimal solution switched to what? switched truck as well.

02:45 So, here the situation cannot arise.

02:48 So, situation cannot arise here.

02:58 Now, here for the second step what we have? for the second step we have t dash k is equals to 1 plus t k that is what? that is optimal solution switches trucks before the greedy solution.

03:35 Before the greedy solution.

03:43 So, here construct a sequence what? so, we will construct a sequence that is t double dash i...

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