Commit 66e3e642 by Filip Stedronsky

### 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. However, in this chapter, we shall use a much more fine-grained notion of space efficiency and measure space requirements in bits. Imagine we have a universe $X$ and we want a data structure to hold a single element $x \in X$. For example, $X$ can be the universe of all length-$n$ sequences of integers from the range $[m]$ and we want a data structure to hold such a sequence (we shall limit ourselves to finite universes only). Imagine we have a data structure whose size is parametrized by some parameter $n$ (e.g. number of elements). Let us define $X(n)$ as the universe of all possible values that a size-$n$ data structure (as a whole) can hold. For example if we 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 $OPT := \lceil\log |X|\rceil$ bits (which is essentilly the entropy of a uniform probability distribution over $X$). Let us denote $s(n)$ the number of bits needed to store a size-$n$ data structure. The information-theoretical optimum is $OPT(n) := \lceil\log |X(n)|\rceil$ (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 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. 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 compact data structure} is one that uses at most $\O(OPT)$ 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 $\O(OPT(n))$ bits of space.} Note that some linear-space data structures are not even compact -- because we are counting bits now, not words. For example, a binary search tree representing a length-$n$ sequence of numbers from range $[m]$ needs $\O(n (\log n + \log m))$ bits, whereas $OPT$ is $n \log m$. For $n >> m$, this does not satisfy the requirements for a compact data structure. Note that some linear-space data structures are not even compact -- because we are counting bits now, not words. For example, a linked list representing a length-$n$ sequence of numbers from range $[m]$ needs $\O(n (\log n + \log m))$ bits ($\log n$ bits are used to represent a next-pointer), whereas $OPT$ is $n \log m$. 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 fast operations on these space-efficient data structures. ... ...
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