Binary search tree with minimum potential
How to construct a n-node binary serach tree such that its potential is the minimum? The size, rank and potential are defined as follows
The size $s(v)$ is the number of nodes in a subtree (include v) with the root $v$.
The rank $r(v) = log_2(s(v))$.
The potential of a tree $Phi(T) = sum_{v in T} r(v)$.
We are trying to minimize $Phi(T)$ with $n$ given nodes. I think the answer is a binary search tree. And through calculation I caluclate the potential is $leq 2n$. But I don't know how to prove this is indeed the minimum.
trees
asked Nov 25 at 7:44
Lawerance
1012
add a comment |
How to construct a n-node binary serach tree such that its potential is the minimum? The size, rank and potential are defined as follows
The size $s(v)$ is the number of nodes in a subtree (include v) with the root $v$.
The rank $r(v) = log_2(s(v))$.
The potential of a tree $Phi(T) = sum_{v in T} r(v)$.
We are trying to minimize $Phi(T)$ with $n$ given nodes. I think the answer is a binary search tree. And through calculation I caluclate the potential is $leq 2n$. But I don't know how to prove this is indeed the minimum.
trees
asked Nov 25 at 7:44
Lawerance
1012
add a comment |
How to construct a n-node binary serach tree such that its potential is the minimum? The size, rank and potential are defined as follows
The size $s(v)$ is the number of nodes in a subtree (include v) with the root $v$.
The rank $r(v) = log_2(s(v))$.
The potential of a tree $Phi(T) = sum_{v in T} r(v)$.
We are trying to minimize $Phi(T)$ with $n$ given nodes. I think the answer is a binary search tree. And through calculation I caluclate the potential is $leq 2n$. But I don't know how to prove this is indeed the minimum.
trees
asked Nov 25 at 7:44
Lawerance
1012
How to construct a n-node binary serach tree such that its potential is the minimum? The size, rank and potential are defined as follows
The size $s(v)$ is the number of nodes in a subtree (include v) with the root $v$.
The rank $r(v) = log_2(s(v))$.
The potential of a tree $Phi(T) = sum_{v in T} r(v)$.
We are trying to minimize $Phi(T)$ with $n$ given nodes. I think the answer is a binary search tree. And through calculation I caluclate the potential is $leq 2n$. But I don't know how to prove this is indeed the minimum.
trees
trees
asked Nov 25 at 7:44
Lawerance
1012
asked Nov 25 at 7:44
Lawerance
1012
asked Nov 25 at 7:44
Lawerance
1012
asked Nov 25 at 7:44
Lawerance
1012
asked Nov 25 at 7:44
Lawerance
1012
1012
add a comment |
add a comment |
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