4
votes
Accepted
How to convert a decision tree to an automaton?
A decision tree is a model of computation which makes sense for instance of constant size. In contrast, a language is usually a collection of instances of unbounded size. An automaton (in this context)...
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3
votes
Accepted
how does the shap algorithm work in polynomial time?
First, some notation.
A decision tree is a binary tree in which each internal node is labelled by one of $x_1,\ldots,x_n$, and has two outgoing edges labelled $0$ and $1$. Leaves are labelled with ...
- 278k
3
votes
Accepted
Lower bound and worst case scenario
When analyzing algorithms it makes little sense to consider the best-case scenario as it is very often trivial and not very informative.
You can convince yourself that almost every algorithm can be ...
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3
votes
In information theory, why is the entropy measured in units of bits?
Here is a simple intuition. Suppose we consider a random value that takes the values 0 or 1 with equal probability. Then this value can be represented in a single bit. Moreover, its Shannon entropy ...
D.W.♦
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3
votes
Accepted
What is the main difference between binary decision tree and binary decision diagram(BDD)?
One is a tree; one is not. A BDD is a directed acyclic graph (dag), but not necessarily a tree. This allows a BDD to be more compact, in some cases.
That's the only essential difference. Of course,...
D.W.♦
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3
votes
Accepted
What do Arora and Barak mean by $x|_S$ in their definition of certificate complexity?
It is a typo. $S$ indeed should be a subset of $[n] = \{1,2,\ldots,n\}$. On page 263, in the second sentence of their proof of Theorem 12.5, they list a 0-certificate (or 1-certificate) as a subset of ...
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2
votes
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What machine learning training algorithm to use for this kind of string dataset?
You want to perform supervised learning, i.e., you have set of training examples (in this case, a set of 4-vectors with a label that corresponds to one of the 100 songs). So in particular, you want to ...
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2
votes
Accepted
Decision Tree Learning Deviation - Russell and Norvig
TL;DR: They are basically doing a chi-squared test to compare the distributions.
In more detail: Let's break this down into two parts, $\Delta = \Delta_1 + \Delta_2$ where
$$\begin{align*}
\Delta_1 &...
D.W.♦
- 164k
2
votes
Connected Components - Linear Decision Trees
Connected components are a concept of topology. Under the usual topology on $\mathbb{R}^n$, we say that two points $x,y \in S$ belong to the same connected component if there is a path from $x$ to $y$ ...
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2
votes
Accepted
Longest palindrome substring in logarithmic runtime complexity
If a you are given a palindrome $p$ of size $N$ (as you say in the beginning of the question), then the longest palindrome is $p$ itself and you don't need to do any computation.
If your input is not ...
- 29.6k
2
votes
Decision tree and information-theoretic lower bound
No. The outcome is the median, not the sorted list. Thus, there are only 3 possible outcomes, not 3! outcomes. The information-theoretic lower bound says that if there are N possible outcomes, then ...
D.W.♦
- 164k
2
votes
Accepted
Finding maximum takes at least $\lceil n/2 \rceil$ comparisons
In fact, at least $n-1$ comparisons are needed. Indeed, consider the graph on the vertex set $A$ whose edges correspond to pairs of elements which were compared. If less than $n-1$ comparisons were ...
- 278k
2
votes
Accepted
decision trees and numeric attributes
Each vertex in a decision tree is associated with a question of the form "$x_i < c$?" (or "$x_i \leq c$?"), where $x_i$ is one of the input and $c$ is a constant. The decision whether to go left or ...
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2
votes
Expressivity of Polysize Decision Trees
This is not an exact characterization, but the class of languages recognizable by poly-size DT is included in $\mathrm{AC}^0$, specifically within its second level: that is, any DT can be converted to ...
- 1,506
2
votes
Accepted
Basic exercises on decision trees
ID3 is a heuristic. It is not guaranteed to generate an optimal decision tree.
Here is an algorithm for Exercise 1:
If $X_3 = 1$, then answer $2X_1-1$.
If $X_3 = 0$, then answer $2X_4-1$.
For ...
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2
votes
Accepted
Can inputs in the decision tree model be computed?
A decision tree is a special kind of "program" which computes a function, usual from $\{0,1\}^n$ to $\{0,1\}$. Let's take an example from Wikipedia:
A decision tree is a binary tree. ...
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2
votes
Accepted
Decision tree for searching element in sorted-array
You haven't really defined your computation model, so here is a suggestion.
The input to the algorithm is a sorted array $A$ of length $n$ and an element $x$.
The output is either a position $i$ such ...
- 278k
1
vote
Accepted
'Address Function' example of decision tree complexity is not clear
The decision tree that Arora and Barak mention gets as input $x,y$ and is only supposed to compute the function $f(x,y) = y_x$. It is not supposed to recover the entire input.
The decision tree ...
- 278k
1
vote
Can we create a decision tree for any comparison sorting algorithm even if it is very complicated?
In theory, yes you should be able to create a decision tree for every comparison based sorting algorithm. You can write a program that will do so (for a fixed comparison-based sorting algorithm of ...
- 612
1
vote
Accepted
Can we create a decision tree for any comparison sorting algorithm even if it is very complicated?
In general, a decision tree would enumerate all possible permutations $(N!)$ for given input of $N$ elements, out of which only $1$ would be desirable.
A sorting algorithm doesn't generate all ...
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1
vote
How to come up with combination a short-circuit evaluation table?
Looking at your question again, I think you really may only need two simple rules. After reviewing these rules, then we can see how to build a table.
Note that I will write
$$(a || b) = (a \lor b)$$
$...
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vote
how does the shap algorithm work in polynomial time?
In this paper: https://arxiv.org/pdf/2007.14045.pdf, a polynomial-time algorithm is given to compute the SHAP score for deterministic and decomposable Boolean circuits. This algorithm can also be used ...
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how does the shap algorithm work in polynomial time?
slight optimization over @yuval-filmus excellent answer.
In his comment here: https://github.com/slundberg/shap/issues/24 @sh1ng suggests an improvement. Rephrasing a bit, he's saying that we can use ...
- 369
1
vote
Why decision tree method for lower bound on finding a minimum doesn't work
In the comments I said that $n$ is the number of different possible outcomes of a minimum algorithm and that this is only a lower bound for the number of leaves of the decision trees of the comparison-...
- 549
1
vote
Why decision tree method for lower bound on finding a minimum doesn't work
Interesting question!
The way I understand this is that on sorting it happens that one comparison allows you to discard approximately a half of the possible answers. However on the case of the minimum,...
- 1,248
1
vote
Lower bound on comparison-based sorting
If there is a comparison sort whose running time is $f(n)$ on $g(n)$ of the inputs, then there are at least $g(n)$ leaves at depth at most $f(n)$, and so $g(n) \leq 2^{f(n)}$.
- 278k
1
vote
In information theory, why is the entropy measured in units of bits?
The entropy of a random variable $X$ can be described in terms of prefix-free binary encodings. Let $T(X)$ be the minimal average codeword length of a binary prefix code for $X$, and let $X^{\otimes n}...
- 278k
1
vote
Accepted
Lower bound for merging $m$ sorted arrays (decision tree leaves count - permutations)
Stirling's formula shows that
$$
\binom{N}{pN} \sim \frac{2^{H(p)N}}{\sqrt{2\pi p(1-p)N}},
$$
where $H(p)$ is the binary entropy function:
$$
H(p) = p\log \frac{1}{p} + (1-p) \log \frac{1}{1-p}.
$$
In ...
- 278k
1
vote
Decision tree complexity of deciding whether array is "zigzag"
You can prove a lower bound of $n-3$ comparisons, almost matching the trivial upper bound of $n-1$ comparisons (surely the gap can be closed with a bit more work). The proof uses an adversary argument....
- 278k
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