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Algoritmen & Datastructuren2014 2015

Priority Queues & Search Trees

Philip Dutr

Dept. of Computer Science, K.U.Leuven

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Overview Lecture

Copyright Ph.Dutr, Spring 20152

Priority Queues Binary Trees

Binary Heap

Promotion, Insertion, Demotion, Delete Max

Heapsort Binary Search Tree

Search, Insert,Analysis, Delete

2-3 Trees

Search, Insert,Analysis Red-Black Trees

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Priority Queues


Collections: insert and delete items Most recent Stack

Least recent Queue

Largest (or smallest) Priority Queues

Applications of priority queues

Operating systems [load balancing, interrupt handling]

Statistics [maintain largest M values in a sequence]

Graph searching [Dijkstra's algorithm, Prim's algorithm]

Streams of Data [Monetary transactions]

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Priority Queues


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Priority Queues


Insert / Remove operations

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Binary Trees


Binary tree Empty or node with links to left and right binary trees

Complete tree

Perfectly balanced, except for bottom level

Property: Height of tree with N nodes is 1+log2N

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Complete binary tree in nature


Hyphaene Compressa

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Binary heap


Heap-ordered binary tree Keys in nodes

No smaller than childrens keys.

Array representation

Take nodes in level order No explicit links needed

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Binary heap


Largest key is a[1] Use array indices to move

through tree:

Parent of node kis at k/2

Children of node kare at2kand 2k+1

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Promotion in a heap


Scenario: Node's key is replaced by a larger key than its parent's key

Exchange key in node with key in parent

Repeat until heap order restored

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Insertion in a heap


Add node at end, then swim it up At most log2N compares

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Demotion in a heap


Scenario: Node's key becomes smaller than one (or both) of its

children's keys

Exchange key in node with key in larger child

Repeat until heap order restored

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Delete maximum in a heap


Exchange root with node at end, then sink it down At most 2log2N compares

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Overview


Heapsort:

Keep all elements in priority queue Remove largest, rearrange queue

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Binary Search Trees (BST)


A BST is a binary tree in symmetric order A binary tree is either:

Empty

Two disjoint binary trees (left and right)

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Binary Search Trees (BST)


Symmetric order:Each node has a key, and every nodes key is:

Larger than all keys in its left subtree.

Smaller than all keys in its right subtree

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BST in Java


Often also size of the (sub)tree:

size(x) = size(x.left) + size(x.right) + 1

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BST Search


Return value corresponding to given key,or nullif no such key.

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BST Search


Implementation: Iterative (while-loop)

Recursion

# compares = depth of node

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BST Insert


Search for key, thentwo cases:

Key in tree reset value

(if no duplicate keys allowed)

Key not in tree add new node.

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BST Analysis


Many different BSTs are possible for same set of keys #compares = depth of node

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BST Analysis


Experiment: random keys, trees stay flat

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BST Analysis


Search hits in a BST tree from N random keysrequire ~1.39log2N compares, on average.

Cfr. quicksort

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BST Ordered Operations


Find minimum key? Move left as far as possible

Find maximum key?

Move right as far as possible

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BST Ordered Operations


Floor: find largest key root: floor couldbe in right subtree, or the root

Ceiling: find smallest key >= given key Cfr. supra

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BST Delete


Lay approach

Make node empty, but leave node in tree

Tombstone overhead

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BST Delete: minimum


To delete the minimum

key:

Go left until finding a node

with a null left link

Replace that node by its

right link

(Update subtree counts)

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BST Delete: Hibbard


Search for node with key

Case 0: 0 children

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BST Delete: Hibbard


Case 1: 1 child

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BST Delete: Hibbard


Case 2: 2 children

Find successor xof node with key k (node t)

Delete the minimum in t's right subtree

Put xin t's spot

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BST Summary


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Problem with BSTs


BST has poor worst case performance

Ideal: perfectly balanced BST

Too expensive

Theoretical solution to keep tree balanced:

2-3 tree

Practical implementation: red-black tree

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2-3 Trees


Allow 1 or 2 keys per node

2-node: one key, two children

3-node: two keys, three children

Perfect balance

Every path from root to null link has same length.

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2-3 Trees Search


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2-3 Trees Insert


Case 1: insert into a 2-node

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2-3 Trees Insert


Case 2: insert into a 3-node, whose parent is a 2-node

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2-3 Trees Insert


Case 3: insert into a 3-node, whose parent is a 3-node

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2-3 Trees Insert


If you reach the root and it's a 4-node:

split the root into three 2-nodes

Splitting the root increases height by 1

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Insertion Example


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Insertion Example


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Local transformations in a 2-3 tree


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Global properties


Each transformation maintains perfect balance and

symmetric order

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2-3 Trees Analysis


Every path from root to null-link has same length

Worst case: log2N [all 2-nodes]

Best case: log3N 0.631 log2N [all 3-nodes]

Between 12 and 20 for a million nodes

Between 18 and 30 for a billion nodes

Guaranteed logarithmic performance for search and insert

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2-3 Trees Analysis


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Red-Black (RB) Trees


Represent 2-3 tree as BST

Use internal left-leaning links as glue for 3-nodes

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Red-Black (RB) Trees


Invariants:

No node has two red

links connected to it

Every path from root

to null link has the

same number of blacklinks (black-balance)

Red links lean left

Implementation

1 field in node

indicates whether left

link is red

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RB Trees Rotations


Left rotation: Orient a (temp) right red link to the left

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RB Trees Rotations


Right rotation: orient a left red link (temp) to the right

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RB Trees Color Flip


Recolor to split a (temp) 4-node

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RB Trees Insertion


Basic strategy: Maintain 1-1 correspondence with 2-3

trees by applying elementary red-black tree operations

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RB Trees Insertion


Simple case: Insert in tree with exactly 1 node

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RB Trees Insertion


Case 1: insert into 2-node at bottom

Do standard BST insert; color new link red

If new red link is a right link, rotate left

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RB Trees Insertion


Simple case: insert in tree with exactly 2 nodes

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RB Trees Insertion


Case 2: insert in 3-node Do standard BST insert; color new link red.

Rotate to balance the 4-node (if needed).

Flip colors to pass red link up one level.

Rotate to make lean left (if needed).

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Insert Example


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Insert Example


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Insert Example


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RB Trees Analysis


Height with N nodes is

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Real Story

On-line business wanted customer database with

real-time updates Database provider decided to use RB tree

Max height 80 (2^40 records), error triggered otherwise

Rebalance only after insertions, not deletions

extended service outage

Court case

Legal testimony: If implemented properly, the height of a red-black

BST with N keys is at most 2log2N.

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