Commit 66e3e642 authored by Filip Stedronsky's avatar Filip Stedronsky
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Succinct: parametrize everything by $n$ to make asymptotics make sense;...

Succinct: parametrize everything by $n$ to make asymptotics make sense; simpler example of non-compact linear-space structure
parent f74bb19c
...@@ -11,31 +11,33 @@ sctructures have linear space complexity, which is asymptotically optimal. ...@@ -11,31 +11,33 @@ sctructures have linear space complexity, which is asymptotically optimal.
However, in this chapter, we shall use a much more fine-grained notion of space However, in this chapter, we shall use a much more fine-grained notion of space
efficiency and measure space requirements in bits. efficiency and measure space requirements in bits.
Imagine we have a universe $X$ and we want a data structure to hold a single Imagine we have a data structure whose size is parametrized by some parameter
element $x \in X$. For example, $X$ can be the universe of all length-$n$ sequences $n$ (e.g. number of elements). Let us define $X(n)$ as the universe of all possible
of integers from the range $[m]$ and we want a data structure to hold such values that a size-$n$ data structure (as a whole) can hold. For example if we
a sequence (we shall limit ourselves to finite universes only). have a data structure for storing strings from a fixed alphabet, $X(n)$ may be the
universe of all length-$n$ strings from this alphabet.
The information-theoretically optimal size of such a data structure is Let us denote $s(n)$ the number of bits needed to store a size-$n$ data structure.
$OPT := \lceil\log |X|\rceil$ bits (which is essentilly the entropy of The information-theoretical optimum is $OPT(n) := \lceil\log |X(n)|\rceil$
a uniform probability distribution over $X$). (which is essentially the entropy of a uniform distribution over $X(n)$).
Now we can define three classes of data structures based on their fine-grained space Now we can define three classes of data structures based on their fine-grained space
efficiency: efficiency:
\defn{An {\I implicit data structure} is one that uses at most $OPT + \O(1)$ bits of space.} \defn{An {\I implicit data structure} is one that uses at most $OPT(n) + \O(1)$ bits of space.}
A typical implicit data structure contains just its elements in some order and nothing more. A typical implicit data structure contains just its elements in some order and nothing more.
Examples include sorted arrays and heaps. Examples include sorted arrays and heaps.
\defn{A {\I succinct data structure} is one that uses at most $OPT + {\rm o}(OPT)$ bits of space.} \defn{A {\I succinct data structure} is one that uses at most $OPT(n) + {\rm o}(OPT(n))$ bits of space.}
\defn{A {\I compact data structure} is one that uses at most $\O(OPT)$ bits of space.} \defn{A {\I compact data structure} is one that uses at most $\O(OPT(n))$ bits of space.}
Note that some linear-space data structures are not even compact -- because we are counting Note that some linear-space data structures are not even compact -- because we
bits now, not words. For example, a binary search tree representing a length-$n$ sequence are counting bits now, not words. For example, a linked list representing a
of numbers from range $[m]$ needs $\O(n (\log n + \log m))$ bits, whereas $OPT$ is length-$n$ sequence of numbers from range $[m]$ needs $\O(n (\log n + \log m))$
$n \log m$. For $n >> m$, this does not satisfy the requirements for a compact data bits ($\log n$ bits are used to represent a next-pointer), whereas $OPT$ is $n
structure. \log m$. For $n \gg m$, this does not satisfy the requirements for a compact
data structure.
And of course, as with any data structure, we want to be able to perform reasonably And of course, as with any data structure, we want to be able to perform reasonably
fast operations on these space-efficient data structures. fast operations on these space-efficient data structures.
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