feat: improve algorithms
Signed-off-by: Matej Focko <mfocko@redhat.com>
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1 changed files with 41 additions and 25 deletions
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@ -114,6 +114,19 @@
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\SetKwProg{Fn}{function}{ is}{end}
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\SetKwProg{Proc}{procedure}{ is}{end}
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\newcommand{\avlDeleteRebalance}{\hyperref[algorithm:avl:deleteRebalance]{\texttt{deleteRebalance}}}
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\newcommand{\avlDeleteFixNode}{\hyperref[algorithm:avl:deleteFixNode]{\texttt{deleteFixNode}}}
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\newcommand{\avlDeleteRotate}{\hyperref[algorithm:avl:deleteRotate]{\texttt{deleteRotate}}}
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\newcommand{\findParentNode}{\hyperref[algorithm:findParentNode]{\texttt{findParentNode}}}
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\newcommand{\wavlInsertRebalance}{\hyperref[algorithm:wavl:insertRebalance]{\texttt{insertRebalance}}}
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\newcommand{\wavlFixZeroChild}{\hyperref[algorithm:wavl:fix0Child]{\texttt{fix0Child}}}
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\newcommand{\wavlDeleteRebalance}{\hyperref[algorithm:wavl:deleteRebalance]{\texttt{deleteRebalance}}}
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\newcommand{\wavlBottomUpDelete}{\hyperref[algorithm:wavl:bottomUpDelete]{\texttt{bottomUpDelete}}}
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\newcommand{\wavlFixDelete}{\hyperref[algorithm:wavl:fixDelete]{\texttt{fixDelete}}}
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\begin{document}
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\chapter*{Introduction}
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@ -354,9 +367,9 @@ When propagating the error, we can encounter 3 cases (we explain them with respe
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We have implemented the deletion rebalance by implementing following functions:
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\begin{enumerate}
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\item \texttt{deleteRebalance} that handles updating the current node and its parent and iteratively calls subroutine handling previously described \textit{one step of a rebalancing}
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\item \texttt{deleteFixNode} that handles one adjustment of rebalancing as described above
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\item \texttt{deleteRotate} that handles rotation and updating of ranks, if necessary
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\item \avlDeleteRebalance{} that handles updating the current node and its parent and iteratively calls subroutine handling previously described \textit{one step of a rebalancing}
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\item \avlDeleteFixNode{} that handles one adjustment of rebalancing as described above
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\item \avlDeleteRotate{} that handles rotation and updating of ranks, if necessary
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\end{enumerate}
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\begin{algorithm}
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@ -369,7 +382,7 @@ We have implemented the deletion rebalance by implementing following functions:
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$(y, parent) \gets (parent, parent.parent)$\;
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}
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\BlankLine
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\While{$y \neq nil \land \texttt{deleteFixNode}(T, y, parent)$}{
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\While{$y \neq nil \land \avlDeleteFixNode(T, y, parent)$}{
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$y \gets parent$\;
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\eIf{$parent \neq nil$}{
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$parent \gets parent.parent$\;
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@ -385,22 +398,22 @@ We have implemented the deletion rebalance by implementing following functions:
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\begin{algorithm}
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\Proc{$\texttt{deleteFixNode}(T, x, parent)$}{
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\uIf(\tcp*[h]{Handles rule 1}){balance-factor of $x$ is $0$}{
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\uIf(\tcp*[h]{Handles \hyperref[avl:rules:delete:1]{rule 1}}){balance-factor of $x$ is $0$}{
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update rank of $x$\;
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\Return{$true$}\;
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}
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\ElseIf(\tcp*[h]{Handles rule 2}){balance-factor of $x$ is $-1$ or $1$}{
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\ElseIf(\tcp*[h]{Handles \hyperref[avl:rules:delete:2]{rule 2}}){balance-factor of $x$ is $-1$ or $1$}{
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\Return{$false$}\;
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}
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\BlankLine
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$(l, r) \gets (x.left, x.right)$\;
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$(rotateL, rotateR) \gets (\texttt{rotateLeft}, \texttt{rotateRight})$\;
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\BlankLine
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\tcp{Handles rule 3}
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\tcp{Handles \hyperref[avl:rules:delete:3]{rule 3}}
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\eIf{balance-factor of $x$ is $2$}{
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\Return{$\texttt{deleteRotate}(T, x, r, 1, rotateL, rotateR)$}\;
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\Return{$\avlDeleteRotate(T, x, r, 1, rotateL, rotateR)$}\;
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}{
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\Return{$\texttt{deleteRotate}(T, x, l, -1, rotateR, rotateL)$}\;
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\Return{$\avlDeleteRotate(T, x, l, -1, rotateR, rotateL)$}\;
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}
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}
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\caption{\texttt{deleteFixNode} algorithm for the AVL tree}\label{algorithm:avl:deleteFixNode}
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@ -417,19 +430,19 @@ There are two operations that are not described using helper functions and they
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\begin{algorithm}
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\Proc{$\texttt{deleteRotate}(T, x, y, leaning, rotateL, rotateR)$}{
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$factor \gets $ balance-factor of $y$\;
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$f \gets $ balance-factor of $y$\;
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\BlankLine
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\uIf(\tcp*[h]{Handles rule 3ab}){$factor = 0 \lor factor = leaning$}{
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\uIf(\tcp*[h]{Handles rules \hyperref[avl:rules:delete:3a]{3a} \& \hyperref[avl:rules:delete:3b]{3b}}){$f = 0 \lor f = leaning$}{
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$rotateL(T, x)$\;
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}
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\Else(\tcp*[h]{Handles rule 3c}){
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\Else(\tcp*[h]{Handles \hyperref[avl:rules:delete:3c]{rule 3c}}){
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$rotateR(T, y)$\;
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$rotateL(T, x)$\;
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}
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\BlankLine
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update ranks of $x$, $y$ and new root of the subtree\;
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\BlankLine
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\Return{$factor \neq 0$}\;
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\Return{$f \neq 0$}\;
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}
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\caption{\texttt{deleteRotate} algorithm for the AVL tree}\label{algorithm:avl:deleteRotate}
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\end{algorithm}
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@ -489,7 +502,7 @@ Inserting values into WAVL tree is equivalent to inserting values into regular b
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\Return\;
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}
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\BlankLine
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$parent \gets \texttt{findParentNode}(key, T.root)$\;
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$parent \gets \findParentNode(key, T.root)$\;
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\If{$parent = nil$}{
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\Return\;
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}
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@ -502,7 +515,7 @@ Inserting values into WAVL tree is equivalent to inserting values into regular b
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$parent.right \gets insertedNode$\;
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}
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\BlankLine
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$\texttt{bottomUpInsertRebalance}(T, insertedNode)$\;
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$\wavlInsertRebalance(T, insertedNode)$\;
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}
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\caption{Insert operation on binary search tree}\label{algorithm:wavl:insert}
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\end{algorithm}
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@ -555,18 +568,21 @@ Once the node does not have a parent (is root node) or is not (0, 1) or (1, 0) n
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Bottom-up rebalancing after the insertion into WAVL tree is identical to AVL tree insertion:
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\begin{algorithm}
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\Proc{$\texttt{bottomUpInsertRebalance}(T, node)$}{
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\Proc{$\texttt{insertRebalance}(T, node)$}{
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\tcp{Handles \hyperref[avl:rules:insert:2]{rule 2}}
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\While{$node.parent \neq nil \land (node.parent\, is\, (0, 1)\, or\, (1, 0))$}{
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$\texttt{promote}(node.parent)$\;
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$node \gets node.parent$\;
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}
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\BlankLine
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\tcp{Handles \hyperref[avl:rules:insert:1]{rule 1}}
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\lIf{$node.parent = nil \lor node\, is\, not\, \text{0-child}$}{\Return}
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\BlankLine
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\tcp{Handles \hyperref[avl:rules:insert:3]{rule 3}}
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\eIf{$node = node.parent.left$}{
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$\texttt{fix0Child}(T, node, node.right, \texttt{rotateLeft}, \texttt{rotateRight})$\;
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$\wavlFixZeroChild(T, node, node.right, \texttt{rotateLeft}, \texttt{rotateRight})$\;
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}{
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$\texttt{fix0Child}(T, node, node.left, \texttt{rotateRight}, \texttt{rotateLeft})$\;
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$\wavlFixZeroChild(T, node, node.left, \texttt{rotateRight}, \texttt{rotateLeft})$\;
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}
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\BlankLine
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}
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@ -577,12 +593,12 @@ Bottom-up rebalancing after the insertion into WAVL tree is identical to AVL tre
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\Proc{$\texttt{fix0Child}(T, x, y, rotateToLeft, rotateToRight)$}{
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$z \gets x.parent$\;
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\BlankLine
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\uIf{$y = nil \lor y\, is\, \text{2-child}$}{
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$newRoot \gets rotateToRight(T, z)$\;
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\uIf(\tcp*[h]{Handles \hyperref[avl:rules:insert:3a]{rule 3a}}){$y = nil \lor y\, is\, \text{2-child}$}{
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$rotateToRight(T, z)$\;
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\BlankLine
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$\texttt{demote}(z)$\;
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}
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\ElseIf{$y\, is\, \text{1-child}$}{
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\ElseIf(\tcp*[h]{Handles \hyperref[avl:rules:insert:3b]{rule 3b}}){$y\, is\, \text{1-child}$}{
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$rotateToLeft(T, x)$\;
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$rotateToRight(T, z)$\;
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\BlankLine
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@ -597,7 +613,7 @@ Bottom-up rebalancing after the insertion into WAVL tree is identical to AVL tre
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\section{\texttt{delete}}
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\begin{algorithm}
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\Proc{$\texttt{bottomUpDeleteRebalance}(T, y, parent)$}{
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\Proc{$\texttt{deleteRebalance}(T, y, parent)$}{
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\uIf{$y \text{ is (2, 2)}$}{
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$\texttt{demote}(y)$\;
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$parent \gets y.parent$\;
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@ -613,7 +629,7 @@ Bottom-up rebalancing after the insertion into WAVL tree is identical to AVL tre
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$z \gets \text{3-child of } parent$\;
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\If{$z \neq nil$}{
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$\texttt{bottomUpDelete}(T, z, parent)$\;
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$\wavlBottomUpDelete(T, z, parent)$\;
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}
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}
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\caption{Initial phase of algorithm for the rebalance after deletion from the WAVL tree}\label{algorithm:wavl:bottomUpDeleteRebalance}
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@ -656,9 +672,9 @@ Bottom-up rebalancing after the insertion into WAVL tree is identical to AVL tre
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}
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$p \gets parent$\;
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\eIf{$parent.left = x$}{
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$\texttt{fixDelete}(T, x, p.right, p, false, \texttt{rotateLeft}, \texttt{rotateRight})$\;
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$\wavlFixDelete(T, x, p.right, p, false, \texttt{rotateLeft}, \texttt{rotateRight})$\;
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}{
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$\texttt{fixDelete}(T, x, p.left, p, true, \texttt{rotateRight}, \texttt{rotateLeft})$\;
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$\wavlFixDelete(T, x, p.left, p, true, \texttt{rotateRight}, \texttt{rotateLeft})$\;
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}
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}
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\caption{Propagation of the broken rank rule after deletion from the WAVL tree}\label{algorithm:wavl:bottomUpDelete}
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