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Goomba structure: Difference between revisions
→Goomba Cluster: rewritten with a full explanation
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==Goomba Cluster== | ==Goomba Cluster== | ||
A '''Goomba Cluster''' is | A '''Goomba Cluster''' is created by releasing cloned Goombas in different angles at the same HOLP. Because the Goombas are facing different directions, they have slightly different hitboxes, allowing them to be individually stomped. | ||
Goomba Clusters are typically used to build Goomba Bridges and Goomba Staircases, both of which require a staircase of length <math>n</math> to set the HOLP to where Goomba <math>n+1</math> must be placed. Without a Goomba Cluster, an entire bridge of length <math>n</math> must be built twice to build a bridge of length <math>n+1</math>, causing both time and the number of slots to grow with order <math>2^N</math>. | |||
However, with a Goomba Cluster, all of the Goombas at position <math>n</math> can be placed once, removing the need to keep setting the HOLP. Each Goomba position on the bridge still requires a HOLP update, however, the HOLP must only be set there once. Therefore, at position <math>n</math>, the number of Goombas required is the number of Goomba positions after that position, which is <math>N+1-n</math>. | |||
Since there is a position containing <math>n</math> Goombas for every <math>n\in [1,N]</math>, the total number of Goombas required is <math>\frac {N(N+1)}2=\frac {N^2}2+\frac N2</math>, something that grows much more efficiently on the order of <math>N^2</math>. | |||
==Goomba Bridge== | ==Goomba Bridge== | ||