CS 130a DISCUSSION SESSION 3

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CS 130a - DISCUSSION SESSION 3
Shravan Samindla
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Binomial Heap

• Set of binomial trees
• Each tree follows heap-order property (min-heap
  or max-heap)?
• A binomial tree of order k
• has root which has k children
• children are binomial trees of order 0, 1, ...,
  k-1
• has 2k nodes
• Can think of as binary representation of N
• 1 represents existence of corresponding binomial
  tree

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Binomial Heap (contd...)?

• Each binomial tree obeys heap property
• there can be at-most one binomial tree for each
  order

Why binomial?
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Binomial Heap Operations

• Insert
• Find-Min
• Delete-Min
• Decrease-key
• Delete
• Merge

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Binomial Heap - Implementation

• array/linked-list of root nodes
• 'parent' 'left-child' 'right-sibling'
• for root node, 'parent' points to...
• In project, for each node we also have a pointer
  to the largest element in the sub-tree rooted at
  that node.

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structure
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Example
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Binomial Heap - properties
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Binomial Heap - Merge

• 2 cases
• both are binomial trees of order k
• each of the two inputs are heaps with gt 1 root
  each.

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Binomial Heap - Merge
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Binomial Heap - Merge
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Binomial Heap - Merge
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Binomial Heap - Merge
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Binomial Heap - Merge
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Binomial Heap - Insert

• Create a heap with one node of element to be
  inserted.
• Merge this new heap with the heap that the
  element is to be inserted into
• Whats the running time of insert, merge?

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Binomial Heap - Merge

• A sequence of N inserts takes linear time
• Why?
• Note that
• if N ....0 gt takes 1 step
• if N ...01 gt takes 2 step
• and so on...until log(N) steps

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Binomial Heap - DeleteMin
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Binomial Heap - DeleteMin
so whats the running time of deleteMin?
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Binomial Heap - decreaseKey
will it affect other binomial trees in the heap?
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Binomial Heaps - Delete

• Change the value to -?
• percolate up the tree
• deleteMin
• So, whats the characteristic of binomial heaps
  w.r.t running times?

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In the project...

• When deleting an element
• incoming pointers can be
• 'parent' of children
• 'largest' of other nodes in the subtree
• make sure to handle these correctly

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Red-Black Trees

• A search tree with basically
• black node count to the leaves being the same
• children of red node must be black

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Red-Black Trees - Insertion

• Basic Idea
• Preserve rank property
• preserve search tree property
• Three cases
• parent is black
• parent is red
• parent's sibling is black
• parent's sibling is red

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Red-Black Trees - Insertion
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Red-Black Trees - Insertion
what about case 3?
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Red Black Trees - Insertion

• Why insert a node as red?
• Why perform rotations, not just color change?
• Can the grand-parent be red?
• What if the great-grand-parent is red?
• What's the difference in the result in case of
  rotation and double-rotation?

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Red-Black Trees - Insertion
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Red-Black Trees - Insertion
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Red Black Trees - Deletion

• Reduce all to the following form
• root is red both children are black
• use color flip,totation if necessary
• If one of the children is red, then root has to
  be black
• by appropriate rotation make sure a red node is
  on the path we are interested in

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THANK YOU