Floyd’s buildHeap, Sorting · -Heap review and Array implementation of Heap -Floyd’s buildHeap...

Floyd’s buildHeap, Sorting

CSE 373: Data Structures and Algorithms

Thanks to Kasey Champion, Ben Jones, Adam Blank, Michael Lee, Evan McCarty, Robbie Weber, Whitaker Brand, Zora Fung, Stuart Reges, Justin Hsia, Ruth Anderson, and many others for sample slides and materials ...

Autumn 2018

Shrirang (Shri) Mare shri@cs.washington.edu

-Heap review and Array implementation of Heap -Floyd’s buildHeap algorithm - Intro to Sorting - Insertion sort -Heap sort

Heap is a tree-based data structure that satisfies - (a) structure property: it’s a complete tree - (b) heap property, which states: - for min-heap: - for max-heap:

- Operations (for min-heap): - removeMin()- peekMin()- insert()

- Applications: priority queue, sorting, ..

Heap review

paren t ≤ ch ildrenparen t ≥ ch ildren

removeMin()

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min-heap

function removeMinlast = last node in the treeminvalue = rootswap root with lastpercolateDown(root)return minvalue

end function<latexit sha1_base64="0oZpHWbxCd26cpWZMvf2MmWbEhw=">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</latexit><latexit sha1_base64="0oZpHWbxCd26cpWZMvf2MmWbEhw=">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</latexit><latexit sha1_base64="0oZpHWbxCd26cpWZMvf2MmWbEhw=">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</latexit><latexit sha1_base64="0oZpHWbxCd26cpWZMvf2MmWbEhw=">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</latexit>

Calls: removeMin()

removeMin()

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Broken heap

Calls: removeMin() 1

removeMin()

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1: function percolateDown(node)2: l = left child of node3: r = right child of node4: smallest = smallest in {l, r, node}5: if smallest 6= node then6: exchange node with smallest7: percolate(smallest)8: end if9: end function

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82 43 50 36

Broken heap

nodepercolateDown(root)

smallest

removeMin()

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Broken heap

smallestpercolateDown(smallest)

removeMin()

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Broken heap

smallest

removeMin()

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Broken heap

smallestpercolateDown(smallest)

removeMin()

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Broken heap

smallest

removeMin()

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smallest

min-heappercolateDown(smallest)

removeMin()

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smallest

min-heappercolateDown(smallest)

O(n ) + c1 + O(n log n ) = O(n )

Runtime of removeMin():

How can we do better?

Binary heap: Array implementation

!14Indices 0 1 2 3 4 5 6 7 8 9 10 11 12

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13 14 15

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!15Indices 0 1 2 3 4 5 6 7 8 9 10 11 12

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13 14 15

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!16Indices 0 1 2 3 4 5 6 7 8 9 10 11 12

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13 14 15

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!17Indices 0 1 2 3 4 5 6 7 8 9 10 11 12

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13 14 15

1 354 6294 1139

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!18Indices 0 1 2 3 4 5 6 7 8 9 10 11 12

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13 14 15

1 354 6294 1139 96 95 75 82 43 50 36 49

!19Indices 0 1 2 3 4 5 6 7 8 9 10 11 12

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13 14 15

1 354 6294 1139 96 95 75 82 43 50 36 49

With array starting at index 1

leftCh ild(i) = 2irig h tCh ild(i) = 2i + 1

paren t(i) = ⌊ i2 ⌋

!20Indices 0 1 2 3 4 5 6 7 8 9 10 11 12

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13 14 15

With array starting at index 0

leftCh ild(i) = 2i + 1rig h tCh ild(i) = 2i + 2

paren t(i) = ⌊ i − 12 ⌋

1 354 6294 1139 96 95 75 82 43 50 36 49

Sorting

- - Problem: Arrange items in a collection in a specified order. - - Lots of applications:

- lookup / search-merging sequences-data processing-Lots of sorting algorithms out there -Why study sorting?

Sorting

1. Comparison Sorts - Order of elements determined by comparing them- Fastest comparison sort: - Elements should support compareTo()

2. Non-comparison Sorts - Order of elements determined by leveraging properties of input- Typical runtime: - Also called as Niche Sorts aka “linear sorts”

Types of sorting algorithms

O(n log n )

In this class we’ll focus on comparison sorts

Insertion sort

0 1 2 3 4 5 6 7 8 9

2 3 6 7 5 1 4 10 2 8

https://visualgo.net/en/sorting

-Runtime:

-Stable:

- In-place:

Insertion sort

0 1 2 3 4 5 6 7 8 9

2 3 6 7 5 1 4 10 2 8

for i = 0 to n docurrent = A[i]j = findNewIndex(current, i)shift elements from j to i� 1 by 1A[j] = current

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-1. Run Floyd’s buildHeap -2. removeMin() n times

Heap sort

Floyd’s buildHeap, Sorting · -Heap review and Array implementation of Heap -Floyd’s buildHeap...

Documents

SHORTEST PATHS BY DIJKSTRA’S AND FLOYD’S ALGORITHMkundu/dstr/3-floyd.pdf · DIJKSTRA’S AND FLOYD’S ALGORITHM Dijkstra’sAlgorithm: •Finds shortest path from a givenstartNode

Binary heap

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Crossing the Line to Floyd’s

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Binomial Heap

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Four different data structures, each one best in a different setting. Simple Heap Balanced Heap Fibonacci Heap Incremental Heap Our results

Discussion #34 Warshall’s and Floyd’s Algorithms

Floyd’s Algorithm (shortest-path problem)

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Discussion #34 1/17 Discussion #34 Warshall’s and Floyd’s Algorithms

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Ch11: Heap and Heap Sort - Naresuan University · Ch11: Heap and Heap Sort 305233, 305234 Algorithm Analysis and Design JirapornPooksook NaresuanUniversity. What is the (binary) heap?

1 Chapter 6 Heapsort. 2 About this lecture Introduce Heap Shape Property and Heap Property Heap Operations Heapsort: Use Heap to Sort Fixing heap

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