d087: Collecting Treasures

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d087: Collecting Treasures

Po-Lung Chen

Team Dont Block Me, National Taiwan University

Po-Lung Chen (Dont block me) d087: Collecting Treasures 1 / 31

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Problem Description

Given M open intervals as platforms.

One can move from platform A to platform B if A and B overlap.

Travel from one starting platform toevery other platformsexactly onceand back.

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Problem Description

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Problem Description

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Problem Description

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Notes

We may construct aninterval graph for the problem.

The problem is equivalent to find aHamilton cycleon this interval graph.

However, finding a Hamilton cycle on the general graph is NP-complete.

But in this “special” graph we can solve it by agreedy algorithm.

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Notes

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Notes

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The Algorithm

Keeping two paths starting from the leftmost interval.

Each time we consider the path with smallerendpoint.

Among all intervals which overlap with this path, we pick the interval with smallestendpoint.

If there is no interval overlap with the path, the answer is “no”.

Finally we will get two paths, check if the last interval at both paths overlap.

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The Algorithm

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The Algorithm

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The Algorithm

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Illustration (Finished!)

1-2-3-5 1-9-8-4-7-6

Now we have to check whether 5 and 6 overlap or not.

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Illustration (Finished!)

1-2-3-5 1-9-8-4-7-6

Now we have to check whether 5 and 6 overlap or not.

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Naïve Implementation

For each time, we scan onall intervals and find the interval we want.

This cost O(n) per iteration.

Total time complexity is O(n²).

Can we solve this problem faster?

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Naïve Implementation

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Naïve Implementation

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Naïve Implementation

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Carefully Implementation

From the view of thecandidateintervals,

each interval will goes into the set once, and be picked out once, too.

There are a total of O(n) add/remove operations to the set of candidate intervals.

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Carefully Implementation

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Carefully Implementation

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Carefully Implementation (Contd.)

The interval we want in the candidate set is the one withsmallest endpoint.

If we use a heaporsegment tree, we can update the candidate set in O(log n) time per operation.

The overall time complexity is O(n log n).

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Carefully Implementation (Contd.)

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Carefully Implementation (Contd.)

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Thank you for your listening!