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Core Condor

Red black trees are binary search trees where each node is colored red or black. Properties of red black trees include: (1) the root and external nodes are black, (2) no two consecutive nodes on any path from the root to an external node are red, and (3) all paths from the root to an external node contain the same number of black nodes. Insertion and deletion of nodes may violate these properties and require rebalancing through rotations and recoloring of nodes.

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Red Black Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Red Black Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example Red-Black Tree 10 7 8 1 5 30 40 20 25 35 45 60 3
Properties ,[object Object]
Properties ,[object Object]
Properties 10 7 8 1 5 30 40 20 25 35 45 60 3
Properties ,[object Object],[object Object],[object Object],[object Object],[object Object]
Properties ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Properties ,[object Object],[object Object],[object Object]
Insert ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Classification Of 2 Red Nodes/Pointers ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],LLb a b c d gp pp p
XYr ,[object Object],[object Object],[object Object],a b c d gp pp p a b c d gp pp p
LLb ,[object Object],[object Object],[object Object],y x a b z c d a b c d gp pp p x y z
LRb ,[object Object],[object Object],[object Object],[object Object],y x a b z c d b c a d gp pp p y x z
Delete ,[object Object],[object Object],[object Object]
Delete A Black Leaf ,[object Object],10 7 8 1 5 30 40 20 25 35 45 60 3
Delete A Black Leaf y ,[object Object],[object Object],py 10 7 1 5 30 40 20 25 35 45 60 3
Delete A Black Degree 1 Node ,[object Object],y ,[object Object],py 10 7 8 1 5 30 40 20 25 35 45 60 3
Delete A Black Degree 2 Node ,[object Object],10 7 8 1 5 30 40 20 25 35 45 60 3
Rebalancing Strategy ,[object Object],10 7 8 1 5 30 40 20 25 35 45 60 3 y py
Rebalancing Strategy ,[object Object],60 10 7 8 1 5 30 40 20 25 35 45 3 y py
Rebalancing Strategy ,[object Object],[object Object],60 10 7 8 1 5 30 40 20 25 35 45 3 y
Rebalancing Strategy ,[object Object],[object Object],[object Object],[object Object],[object Object],a b y py v
Rb0 (case 1) ,[object Object],[object Object],[object Object],a b y py v y a b py v
Rb0 (case 2) ,[object Object],[object Object],[object Object],a b y py v y a b py v
Rb1 (case 1) ,[object Object],[object Object],[object Object],a b y py v a b y v py
Rb1 (case 2) ,[object Object],[object Object],[object Object],a y py v b c w c y w py a b v
Rb2 ,[object Object],[object Object],[object Object],a y py v b c w c y w py a b v
Rr(n) ,[object Object],Rr(2) a y py v b c w
Rr(0) ,[object Object],[object Object],a b y py v a b y v py
Rr(1) (case 1) ,[object Object],[object Object],[object Object],a y py v b w c y c w py a v b
Rr(1) (case 2) ,[object Object],[object Object],[object Object],a y py v b w c d x y d x py a v b c w
Rr(2) ,[object Object],[object Object],[object Object],a y py v b w c d x d y x py a v b c w

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Red black 1

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Editor's Notes

  1. Every red node has a black parent.
  2. Height = 5
  3. Collapsed height = 3.
  4. Color flip to be defined later. Binary search trees in which each node contains a binary search tree are used, for example, in IP router tables. Each rotation of the outer binary search tree requires changes in the binary search trees of the involved nodes resulting in an O(log n) cost per rotation.
  5. Initially, p is newly inserted node. Pp and gp may be null. In each round, p, pp, and gp are moved up by 2 levels. If p is a red root, make it black. If p or pp black done! If pp is a red root, make it black. O.w., gp exists
  6. If py doesn’t exist, the whole tree is one black node deficient. This is OK.
  7. y represents both a subtree and its root. This includes the case when y is null. Broken lines indicate pointers whose colors are unspecified. v must exist because y is one black node/pointer deficient.