Trees%201:%20Theory,%20Models,%20Generic%20Heap%20Algorithms,%20Priority%20Queues

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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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Review Question

• Given the following information, can I uniquely
  identify a tree?
• Nodes listed based on an inorder traversal
• D, B, F, E, A, C
• Nodes listed based on a preorder traversal
• A, B, D, E, F, C

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Review Question

• D B F E A C
• A
• B
• D
• E
• F
• C

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Review Question

• D B F E A C
• A x
• B x
• D x
• E x
• F x
• C x

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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
3
2
1
0
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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

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 5 4 3
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The Push Heap Algorithm

• Add new data at next leaf

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 5 4 3 8
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The Push Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 5 4 3 8
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The Push Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 5 4 3 8
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The Push Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 8 4 3 5
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The Push Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 8 4 3 5
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The Push Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 8 4 3 5
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The Push Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
8 6 7 4 3 5
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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

0 1 2 3 4 5 6
0 l r ll lr rl rr
8 6 7 4 3 5
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The Pop Heap Algorithm

• Copy last leaf to root

0 1 2 3 4 5 6
0 l r ll lr rl rr
5 6 7 4 3 5
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The Pop Heap Algorithm

• Remove last leaf

0 1 2 3 4 5 6
0 l r ll lr rl rr
5 6 7 4 3
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The Pop Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
5 6 7 4 3
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The Pop Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
5 6 7 4 3
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The Pop Heap Algorithm

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

0 1 2 3 4 5 6
0 l r ll lr rl rr
7 6 5 4 3
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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()