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Sunday, June 30, 2013

Generalized Tower of Hanoi - K Pegs, N Disks

I recently found online that Facebook is giving a fast track job opportunity for those who would solve a specific problem which may be a little more complex than a simple interview question. The process happens online - you login with your Facebook id and have a specific amount of time to write a solution in the language you desire. Here's the url in case you want to try it. They also provide a sample problem and an allotted 45 minutes to solve it. The problem is the well known Tower of Hanoi puzzle but generalized to k pegs and n disks. Given an initial and final configuration of the pegs you have to output the minimal number of steps needed to reach the final configuration. To make it easier they bound the k and n to 1 <= n <= 8, 3 <= k <= 5 and let you assume that there's a solution that takes at least 7 steps.

One basic approach here is brute forcing all possible configurations which results in an undirected graph. And how do you find something in a graph ? Using a search strategy - in my case I chose Breath-First Search.

Before going into the details of this approach let's see what other approaches we can find. Brute forcing will generate in the worst case 5^8 (390625) different configurations and then we'll have to search through them. Is there any other way where we wouldn't do a brute force, but go through the minimal path the first time? It seems like we could use different strategies of navigating through the pegs of a Hanoi Tower. There's basically 3 configuration cases that will take different approaches in navigation. Those are n < k, n=k and n > k. The point is there's a specific move pattern for each of these cases, that you could do to achieve the desired configuration. This article goes into much detail on that. There's also the way of multi-threading - we could parallelize this in some way - for example we could start from both ends.

I chose the brute force this time. While coding this approach a few problems may arise; mainly the problem of how would you represent the pegs. I chose to use a byte array, where each cell represents a peg and each bit of a peg represents a disk (if set).


Code looks like this:

Note:
The findLowestBitSet method basically returns the number of trailing zeros. In Java there's a util method for that:

The method could be optimized to look like this:

113 comments:

  1. Could you please explain the algorithm with some example , i am not able to follow it.In which order you are moving disks?
    With what values searchPath() , start[] and end[] array are filled with while calling it?

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  40. Holders Real Madrid will reach the UEFA Champions League semi-final provided they
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  41. A week after Canelo vs Chavez went toe-to-toe in an instant classic, boxing fans get another star-studded fight when Saul “Canelo” Alvarez (48-1-1, 34 KOs) and Julio Cesar Chavez Jr. (50-2-1, 32 KOs) meet at the T-Mobile Arena in Las Vegas.
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  42. After a quick explosion in Game 4 where Isaiah Thomas knocked down his first five three-point attempts,
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  46. The 101st Indianapolis 500 is set for Sunday, May 28, but first the lineup and the starting grid, including the pole kangaroo, have to be determined. This is done at Pole Day, where the top nine drivers battle for the top position to lead the race.

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  48. The United States Golf Association (USGA) has announced sectional qualifying sites for the 117th US Open 2017 Championship, which will be held at Erin Hills, in Erin, Wis., on June 15-18. Erin Hills is hosting its first U.S. Open and third USGA championship.



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  49. The United States Golf Association (USGA) has announced sectional qualifying sites for the 117th US Open 2017 Championship, which will be held at Erin Hills, in Erin, Wis., on June 15-18. Erin Hills is hosting its first U.S. Open and third USGA championship.



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  52. The Wallabies must successfully target dynamic All Blacks second rowers Brodie Retallick and
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  53. The Wallabies must successfully target dynamic All Blacks second rowers Brodie Retallick and
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    ReplyDelete

  54. Kelley sees Canelo as having to fight a very smart Golovkin vs Canelo Live Stream fight, use his hand speed, and make him miss with his shots for him to win.
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