CSE 326: Data StructuresLecture #18

Fistful of Data Structures

Steve WolfmanWinter Quarter 2000

Today’s Outline

• What Steve Didn’t Get To On Monday• Warm-up: augmenting leftist heaps• Binomial Queues• Treaps• Randomized Skip Lists• What Steve Won’t Get To (Ever?)

Thinking about DecreaseKey in Leftist Heaps

Why not just percolate up?

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decreaseKey( , 3)

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node

DecreaseKey in Leftist Heaps

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decreaseKey(15, 3)

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Now just merge the two?

Fixing DecreaseKey in Leftist Heaps

decreaseKey(15, 3)

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2220This is

still leftist

This may not be leftist

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7So, fix it!

Now, merge!

DecreaseKey runtime

How many nodes could possibly have the wrong Null Path Length up the line?

runtime:

Delete in Leftist Heaps

decreaseKey(15, -)deleteMin()

runtime:

Binomial Trees

A binomial tree of rank 0 is a one node tree.

A binomial tree of rank k is a binomial tree of rank k-1 with another binomial tree of rank k-1 hanging from its root.

rank 0

rank 1

First Five Binomial Treesrank 0 rank 1 rank 2 rank 3

rank 4

How many nodes does a binomial tree of rank k have?

Binomial Queue Heap Data Structure

• Composed of a forest of binomial trees, no two of the same rank

• Heap-order enforced within each tree

• Ranks present can be computed using the binary representation of the tree’s size

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size = 10 = 10102

rank 1 rank 3

rank 0rank 1rank 2rank 3

Insertion in Binomial Queues

If there’s no rank 0 tree, just putthe new node in as a rank 0 tree.

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rank 1 rank 2insert(10)

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rank 1 rank 2

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rank 0

Insertion in Binomial Queues

insert(10)

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rank 0 rank 1

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rank 2

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rank 0 rank 1

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It’s like addition of binary numbers!1+1 = 0 plus a carry tree1+1+1 = 1 plus a carry tree runtime:

Merge in Binomial Queues

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rank 1 rank 2

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rank 0 rank 1

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rank 0

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rank 3

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0110 + 0011 = 1001

runtime:

DeleteMin in Binomial Queues

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These are one Binomial Queue

These are another

Just merge the two:8

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Binomial Queue Summary

• Implements priority queue ADT– Insert in amortized O(1) time– FindMin (with some tricks) in O(1) time– DeleteMin in O(log n) time– Merge in O(log n) time

• Memory use – O(1) per node– about the cost of skew heaps

• Complexity?

Treap Dictionary Data Structure

• Treaps have the binary search tree– binary tree property– search tree property

• Treaps also have the heap-order property!– randomly assigned

priorities

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heap in yellow; search tree in blue

prioritykey

Legend:

Tree + Heap… Why Bother?

Insert data in sorted order into a treap; what shape tree comes out?

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insert(7)

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insert(8)

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insert(9)

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insert(12)

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prioritykey

Legend:

Treap Insert• Choose a random priority• Insert as in normal BST• Rotate up until heap order is restored

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insert(15)

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Treap Delete• Find the key• Increase its value to • Rotate it to the fringe• Snip it off

delete(9)

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Treap Summary

• Implements Dictionary ADT– insert in expected O(log n) time– delete in expected O(log n) time – find in expected O(log n) time

• Memory use– O(1) per node– about the cost of AVL trees

• Complexity?

Perfect Binary Skip List

• Sorted linked list• # of links of a node is its height• The height i link of each node (that has one) links

to the next node of height i or greater

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Find in a Perfect Binary Skip List

• Start i at the maximum height• Until the node is found or i is one and the next

node is too large:– If the next node along the i link is less than the target,

traverse to the next node– Otherwise, decrease i by one

runtime:

Randomized Skip List Intuition

It’s far too hard to insert into a perfect skip list, but is perfection necessary?

What matters in a skip list?

Randomized Skip List

• Sorted linked list• # of links of a node is its height• The height i link of each node (that has one) links

to the next node of height i or greater• There should be about 1/2 as many nodes of

height i+1 as there are of height i

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Find in a RSL

• Start i at the maximum height• Until the node is found or i is one and the next

node is too large:– If the next node along the i link is less than the target,

traverse to the next node– Otherwise, decrease i by one

Same as for a perfect skip list!

runtime:

Insertion in a RSL

• Flip a coin until it comes up heads; that takes i flips. Make the new node’s height i.

• Do a find, remembering nodes where we go down • Put the node at the spot where the find ends• Point all the nodes where we went down (up to the

new node’s height) at the new node• Point the new node’s links where those redirected

pointers were pointing

Insertion Example in RSL

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insert(22)with 3 flips

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runtime:

Range Queries and Iteration

• Range query: search for everything that falls between two values

• Iteration: successively return (in order) each element in the structure

How do we do them?

How fast are they?

Randomized Skip List Summary

• Implements Dictionary ADT– insert in expected O(log n)– find in expected O(log n)– delete?

• Memory use– expected constant memory per node– about double a linked list

• Complexity?

What We Won’t DiscussPairing heaps - practically, the fastest and best implementation of heaps for

decreaseKey and merge; they use the leftist cut and merge technique

Red-Black Trees - a balanced tree that uses just a one-bit color flag and some invariants to maintain balance: see www/homes/sds/rb.html

AA-Trees - a cross between Red-Black trees and B-Trees that is relatively simple to code and gives worst case O(log n) running time

Deterministic skip lists - a version of skip lists that gives worst case O(log n) running time

To Do

• Finish Project III• Browse chapters 10 & 12 in the book• Form Project IV teams!

– groups of 4-6– 2 1/2 week project– demos at the end

Coming Up

• Quad Trees• k-D Trees• Quiz (February 17th)• Project III due (February 17th by 5PM!)• Project IV distributed (February 18th)