Trees 1: Theory, Models, Generic Heap Algorithms, Priority Queues

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Trees 1 Theory, Models, Generic Heap
Algorithms, Priority Queues

• Andy Wang
• Data Structures, Algorithms, and Generic
  Programming

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Trees 1 Overview

• General treestheory and terminology
• Tree traversals
• Binary trees
• Vector implementation of binary trees
• Partially ordered trees (POT)
• Heap algorithms
• Heapsort
• Priority queue

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Theory and Terminology

• Definition A tree is a connected graph with no
  cycles
• Consequences
• Between any two vertices, there is exactly one
  unique path

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A Tree?
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A Tree? Nope
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A Tree?
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A Tree? Yup
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Theory and Terminology

• Definition A rooted tree is a graph G such
  that
• G is connected
• G has no cycles
• G has exactly one vertex called the root of the
  tree

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Theory and Terminology

• Consequences
• The depth of a vertex v is the length of the
  unique path from root to v
• G can be arranged so that the root is at the top,
  its neighboring vertices are vertices of depth 1,
  and so on
• The set of all vertices of depth k is called
  level k of the tree

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A Rooted Tree
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Theory and Terminology

• Definition A descending path in a rooted tree
  is a path, whose edges go from a vertex to a
  deeper vertex

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Theory and Terminology

• Consequences
• A unique path from the root to any vertex is a
  descending path
• The length of this path is the depth of the vertex

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depth 0
root
depth 1
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depth 2
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depth 3
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Theory and Terminology

• Definition If there is a descending path from
  v1 to v2, v1 is an ancestor of v2, and v2 is a
  descendant of v1.

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Theory and Terminology

• Suppose v is a vertex of depth k
• Any vertex that is adjacent to v must have depth
  k - 1 or k 1.

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Theory and Terminology

• Suppose v is a vertex of depth k
• Vertices adjacent to v of depth k 1 are called
  children of v.

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depth 0
root
depth 1
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depth 2
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depth 3
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Theory and Terminology

• Suppose v is a vertex of depth k
• If k gt 0, there is exactly one vertex of depth k
  1 that is adjacent to v in the graph. This
  vertex is called the parent of v.

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depth 0
root
depth 1
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Theory and Terminology

• Definitions
• A vertex with no children is called a leaf

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Theory and Terminology

• Definitions
• The height of a tree is the maximum depth of its
  vertices

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depth 0
root
depth 1
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height
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depth 2
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Theory and Terminology

• Definitions
• The root is the only vertex of depth 0. The root
  has no parent.

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Tree Traversals

• Definition A traversal is the process for
  visiting all of the vertices in a tree
• Often defined recursively
• Each kind corresponds to an iterator type
• Iterators are implemented non-recursively

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Preorder Traversal

• Visit vertex, then visit child vertices
  (recursive definition)
• Depth-first search
• Begin at root
• Visit vertex on arrival
• Implementation may be recursive, stack-based, or
  nested loop

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Preorder Traversal
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Preorder Traversal
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Postorder Traversal

• Visit child vertices, then visit vertex
  (recursive definition)
• Depth-first search
• Begin at root
• Visit vertex on departure
• Implementation may be recursive, stack-based, or
  nested loop

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Postorder Traversal
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Postorder Traversal
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Postorder Traversal
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Postorder Traversal
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Levelorder Traversal

• Visit all vertices in level, starting with level
  0 and increasing
• Breadth-first search
• Begin at root
• Visit vertex on departure
• Only practical implementation is queue-based

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Levelorder Traversal
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Levelorder Traversal
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Tree Traversals

• Preoder depth-first search (possibly
  stack-based), visit on arrival
• Postorder depth-first search (possibly
  stack-based), visit on departure
• Levelorder breadth-first search (queue-based),
  visit on departure

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Binary Trees

• Definition A binary tree is a rooted tree in
  which no vertex has more than two children

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root
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Binary Trees

• Definition A binary tree is complete iff every
  layer but the bottom is fully populated with
  vertices.

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Binary Trees

• A complete binary tree with n vertices and high H
  satisfies
• 2H lt n lt 2H 1
• 22 lt 7 lt 22 1

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root
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Binary Trees

• A complete binary tree with n vertices and high H
  satisfies
• 2H lt n lt 2H 1
• H lt lg n lt H 1
• H floor(lg n)

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Binary Trees

• Theorem In a complete binary tree with n
  vertices and height H
• 2H lt n lt 2H 1

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Binary Trees

• Proof
• For level k, there are 2k vertices
• Total number of vertices
• n 20 21 2k
• n 1 21 22 2k
• n 1 2(1 21 2k-1)
• n 1 2(n - 2k)
• n 1 2n 2k 1
• n 2k 1 - 1

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Binary Trees

• Let k H
• number of vertices for height H 2H 1 1
• number of vertices for height H lt 2H 1
• Let k H 1
• number of vertices for height H - 1 2H 1
• number of vertices for height H gt 2H
• 2H lt n lt 2H 1

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Binary Tree Traversals

• Inorder traversal
• Definition left child, visit, right child
  (recursive)
• Algorithm depth-first search (visit between
  children)

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Inorder Traversal
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Inorder Traversal
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Inorder Traversal
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Inorder Traversal
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Binary Tree Traversals

• Other traversal apply to binary case
• Preorder traversal
• vertex, left subtree, right subtree
• Inorder traversal
• left subtree, vertex, right subtree
• Postorder traversal
• left subtree, right subtree, vertex
• Levelorder traversal
• vertex, left children, right children

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Vector Representation of Complete Binary Tree

• Tree data
• Vector elements carry data
• Tree structure
• Vector indices carry tree structure
• Index order levelorder
• Tree structure is implicit
• Uses integer arithmetic for tree navigation

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Vector Representation of Complete Binary Tree

• Tree navigation
• Parent of vk v(k 1)/2
• Left child of vk v2k 1
• Right child of vk v2k 2

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Partially Ordered Trees

• Definition A partially ordered tree is a tree T
  such that
• There is an order relation gt defined for the
  vertices of T
• For any vertex p and any child c of p, p gt c

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Partially Ordered Trees

• Consequences
• The largest element in a partially ordered tree
  (POT) is the root
• No conclusion can be drawn about the order of
  children

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Heaps

• Definition A heap is a partially ordered
  complete (almost) binary tree. The tree is
  completely filled on all levels except possibly
  the lowest.

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root
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Heaps

• Consequences
• The largest element in a heap is the root
• A heap can be stored using the vector
  implementation of binary tree
• Heap algorithms
• Push Heap
• Pop Heap

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The Push Heap Algorithm

• Add new data at next leaf
• Repair upward
• Repeat
• Locate parent
• if POT not satisfied
• swap
• else
• stop
• Until POT

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The Push Heap Algorithm

• Add new data at next leaf

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The Push Heap Algorithm

• Add new data at next leaf

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The Push Heap Algorithm

• Repeat
• Locate parent of vk v(k 1)/2
• if POT not satisfied
• swap
• else
• stop

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The Push Heap Algorithm

• Repeat
• Locate parent of vk v(k 1)/2
• if POT not satisfied
• swap
• else
• stop

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The Push Heap Algorithm

• Repeat
• Locate parent of vk v(k 1)/2
• if POT not satisfied
• swap
• else
• stop

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The Push Heap Algorithm

• Repeat
• Locate parent of vk v(k 1)/2
• if POT not satisfied
• swap
• else
• stop

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The Push Heap Algorithm

• Repeat
• Locate parent of vk v(k 1)/2
• if POT not satisfied
• swap
• else
• stop

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The Push Heap Algorithm

• Repeat
• Locate parent of vk v(k 1)/2
• if POT not satisfied
• swap
• else
• stop

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The Pop Heap Algorithm

• Copy last leaf to root
• Remove last leaf
• Repeat
• find the larger child
• if POT not satisfied
• swap
• else
• stop
• Until POT

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The Pop Heap Algorithm

• Copy last leaf to root

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The Pop Heap Algorithm

• Copy last leaf to root

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The Pop Heap Algorithm

• Remove last leaf

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The Pop Heap Algorithm

• Repeat
• find the larger child
• if POT not satisfied
• swap
• else
• stop

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The Pop Heap Algorithm

• Repeat
• find the larger child
• if POT not satisfied
• swap
• else
• stop

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The Pop Heap Algorithm

• Repeat
• find the larger child
• if POT not satisfied
• swap
• else
• stop

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Generic Heap Algorithms

• Apply to ranges
• Specified by random access iterators
• Current support
• Arrays
• TVectorltTgt
• TDequeltTgt
• Source code file gheap.h
• Test code file fgss.cpp

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Priority Queues

• Element type with priority
• typename T t
• Predicate class P p
• Associative queue operations
• void Push(t)
• void Pop()
• T Front()

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Priority Queues

• Associative property
• Priority value determined by p
• Push(t) inserts t, increases size by 1
• Pop() removes element with highest priority
  value, decreases size by 1
• Front() returns element with highest priority
  value, no state change

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The Priority Queue Generic Adaptor

• template lttypename T, class C, class Pgt
• class CPriorityQueue
• C c
• P LessThan
• public
• typedef typename Cvalue_type value_type
• int Empty() const return c.Empty()
• unsigned int Size() const return c.Size()
• void Clear() c.Clear()
• CPriorityQueue operator(const CPriorityQueue
  q)
• if (this ! q)
• c q.c
• LessThan q.LessThan
• return this

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The Priority Queue Generic Adaptor

• void Display(ostream os, char ofc \0) const
  
• c.Display(os, ofc)
• void Push(const value_type t)
• c.PushBack(t)
• g_push_heap(c.Begin(), c.End(), LessThan)
• void Pop()
• if (Empty())
• cerr ltlt error ltlt endl
• exit(EXIT_FALIURE)
• g_pop_heap(c.Begin(), c.End(), LessThan)
• c.PopBack()