Red Black Trees.

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Presentation transcript:

Red Black Trees

Properties

Properties the root is black

Properties the root is black every external node is black

Properties the root is black every external node is black children of a red node are black

Properties the root is black every external node is black children of a red node are black all external nodes have the same black depth

Properties the root is black every external node is black children of a red node are black all external nodes have the same black depth Black Depth: the number of black ancestors minus one

Properties the root is black every external node is black children of a red node are black all external nodes have the same black depth Black Depth: the number of black ancestors minus one It is assumed external nodes are empty place holders (null nodes)

The height of a red-black tree with n nodes is O(log n) Properties the root is black every external node is black children of a red node are black all external nodes have the same black depth The height of a red-black tree with n nodes is O(log n) Black Depth: the number of black ancestors minus one It is assumed external nodes are empty place holders (null nodes)

We can convert a red black tree to a (2,4) tree

We can convert a red black tree to a (2,4) tree Merge every red node into its parent

We can convert a red black tree to a (2,4) tree Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 03,09 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 03,09 25,26 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 19,27 03,09 25,26 28 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 19,27 03,09 25,26 28 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 19,27 03,09 25,26 28 61,81 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 19,27 03,09 25,26 28 61,81,87 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 19,27 03,09 25,26 28 61,81,87 93,94 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 19,27 03,09 25,26 28 61,81,87 93,94,97 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 41,91 19,27 03,09 25,26 28 61,81,87 93,94,97 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 41,91 19,27 03,09 25,26 28 36 61,81,87 93,94,97 Merge every red node into its parent

We can convert a red black tree to a (2,4) tree 28 41,91 19,27 03,09 25,26 28 36 61,81,87 93,94,97 Merge every red node into its parent

03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 We can convert a (2,4) tree into a red black tree

03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z make w and z the two children of v We can convert a (2,4) tree into a red black tree

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z make w and z the two children of v

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z make w and z the two children of v

if v is a 2-node keep the (black) children of v as is 03 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z make w and z the two children of v 03 09

03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 OR if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z make w and z the two children of v 09 03

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z make w and z the two children of v

if v is a 2-node keep the (black) children of v as is 03,09 25,26 19,27 28 61,81,87 93,94,97 41,91 36 if v is a 2-node keep the (black) children of v as is if v is a 3-node create a new red node w give v’s first two (black) children to w w and v’s third child be the two children of v if v is a 4-node create red nodes w and z give v’s last two (black) children to z make w and z the two children of v 94 93 97

Insertion and deletion involve rotations, somewhat of an AVL-flavour

See G&T 10.5 and demo

fin