/algorithms/treap_insert.tex

  1% treap_insert.tex
  2% 11-21-2001
  3\nopagenumbers
  4\parskip=2pt
  5\parindent=0pt
  6
  7{\sl An Iterative Algorithm for Treap Insertion
  8	\hfill \rm Farooq Mela $<$farooq.mela@gmail.com$>$}
  9
 10\medskip
 11
 12\noindent {\bf Algorithm I} ({\it Treap Insertion}).
 13Given a set of nodes which form a treap {\tt T}, and a key to insert
 14{\tt K}, this algorithm will insert the node into the treap while maintaining
 15it's heap properties. Each node is assumed to contain {\tt KEY}, {\tt PRIO},
 16{\tt LLINK}, {\tt RLINK}, and {\tt PARENT} fields. For any given node {\tt N},
 17{\tt KEY(N)} gives the key field of {\tt N}, {\tt PRIO(N)} gives the priority
 18field of {\tt N}, {\tt LLINK(N)} and {\tt RLINK(N)} are pointers to {\tt N}'s
 19left and right subtrees, respectively, and {\tt PARENT(N)} is a pointer to the
 20node of which {\tt N} is a subtree. Any or all of these three link fields may
 21be $\Lambda$, which for {\tt LLINK(N)} and {\tt RLINK(N)} indicates that {\tt
 22N} has no left or right subtree, respectively, and for {\tt PARENT(N)}
 23indicates that {\tt N} is the root of the treap. The treap has a field
 24{\tt ROOT} which is a pointer to the root node of the treap.
 25
 26\medskip
 27You can find an implementation of this algorithm, as well as many others, in
 28{\bf libdict}, which is available on the web at
 29{\tt http://github.com/fmela/libdict}.
 30\medskip
 31
 32\parindent=36pt
 33
 34\item{\bf I1.} [Initialize.]
 35Set ${\tt N} \gets {\tt ROOT(T)}$, ${\tt P} \gets \Lambda$.
 36
 37\item{\bf I2.} [Find insertion point.]
 38If ${\tt N} = \Lambda$, go to step I3.
 39If ${\tt K} = {\tt KEY(N)}$,
 40the key is already in the treap and the algorithm terminates with an error.
 41Set ${\tt P} \gets {\tt N}$;
 42if ${\tt K} < {\tt KEY(N)}$,
 43then set ${\tt N} \gets {\tt LLINK(N)}$,
 44otherwise set ${\tt N} \gets {\tt RLINK(N)}$. Repeat this step.
 45
 46\item{\bf I3.} [Insert.]
 47Set ${\tt N} \gets {\tt AVAIL}$.
 48If ${\tt N} = \Lambda$, the algorithm terminates with an out of memory error.
 49Set ${\tt KEY(N)} \gets {\tt K}$,
 50${\tt LLINK(N)} \gets {\tt RLINK(N)} \gets \Lambda$, and
 51${\tt PARENT(N)} \gets {\tt P}$.
 52Set {\tt PRIO(N)} equal to a random integer.
 53If ${\tt P} = \Lambda$, set ${\tt ROOT(T)} \gets {\tt N}$, and go to step I5.
 54If ${\tt K} < {\tt KEY(P)}$,
 55set ${\tt LLINK(P)} \gets {\tt N}$; otherwise,
 56set ${\tt RLINK(P)} \gets {\tt N}$.
 57
 58\item{\bf I4.} [Sift up.]
 59If ${\tt P} = \Lambda$ or ${\tt PRIO(P)} \leq {\tt PRIO(N)}$, go to step I5.
 60If ${\tt LLINK(P)} = {\tt N}$, rotate {\tt P} right; otherwise,
 61rotate {\tt P} left. Then set ${\tt P} \gets {\tt PARENT(N)}$, and repeat this
 62step.
 63
 64\item{\bf I5.} [All done.] The algorithm terminates successfully.
 65
 66\medskip
 67\parindent=0pt
 68{\bf Rotations}
 69
 70\noindent {\bf Algorithm R} ({\it Right Rotation}).
 71Given a treap {\tt T} and a node in the treap {\tt N}, this routine will rotate
 72{\tt N} right.
 73
 74\parindent=36pt
 75\item{\bf R1.} [Do the rotation.]
 76Set
 77${\tt L} \gets {\tt LLINK(N)}$ and
 78${\tt LLINK(N)} \gets {\tt RLINK(L)}$.
 79If ${\tt RLINK(L)} \neq \Lambda$,
 80then set ${\tt PARENT(RLINK(L))} \gets {\tt N}$.
 81Set ${\tt P} \gets {\tt PARENT(N)}$, ${\tt PARENT(L)} \gets {\tt P}$.
 82If ${\tt P} = \Lambda$, then set ${\tt ROOT(T)} \gets {\tt L}$;
 83if ${\tt P} \neq \Lambda$ and ${\tt LLINK(P)} = {\tt N}$,
 84set ${\tt LLINK(P)} \gets {\tt L}$, otherwise
 85set ${\tt RLINK(P)} \gets {\tt L}$.
 86Finally, set ${\tt RLINK(L)} \gets {\tt N}$,
 87and ${\tt PARENT(N)} \gets {\tt L}$.
 88
 89\parindent=0pt
 90
 91\medskip
 92The code for a left rotation is symmetric. At the risk of being repetitive, it
 93appears below.
 94\medskip
 95
 96\noindent {\bf Algorithm L} ({\it Left Rotation}).
 97Given a treap {\tt T} and a node in the treap {\tt N}, this routine will rotate
 98{\tt N} left.
 99
100\parindent=36pt
101\item{\bf L1.} [Do the rotation.]
102Set
103${\tt R} \gets {\tt RLINK(N)}$ and
104${\tt RLINK(N)} \gets {\tt LLINK(R)}$.
105If ${\tt LLINK(R)} \neq \Lambda$,
106then set ${\tt PARENT(LLINK(R))} \gets {\tt N}$.
107Set ${\tt P} \gets {\tt PARENT(N)}$, ${\tt PARENT(R)} \gets {\tt P}$.
108If ${\tt P} = \Lambda$, then set ${\tt ROOT(T)} \gets {\tt R}$;
109if ${\tt P} \neq \Lambda$ and ${\tt LLINK(P)} = {\tt N}$,
110set ${\tt LLINK(P)} \gets {\tt R}$, otherwise
111set ${\tt RLINK(P)} \gets {\tt R}$.
112Finally, set ${\tt LLINK(R)} \gets {\tt N}$,
113and ${\tt PARENT(N)} \gets {\tt R}$.
114
115\parindent=0pt
116
117\bye
118% EOF