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95 lines
2.2 KiB
Markdown
95 lines
2.2 KiB
Markdown
---
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title: "heaps-and-heapsort"
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aliases:
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tags:
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- cosc201
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---
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## 0.1 Overview
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[[notes/heap]]
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## 0.2 Operations
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### 0.2.1 Add element
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Assumptions
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- access first vacant position
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- set (or find) the value $H.q$ stored in any (occupied) position $q$
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- access parent of any given position
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- identify when we're at the root
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(all in constant time)
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Outcome: Change $H$ by adding x to it, while maintaining the heap conditions
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```
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p <- first vacancy, H.p <- x
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while p is not the root and H.parent(p) < H.p do
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swap H.parent(p) and H.p
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p <- parent(p)
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end while
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```
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### 0.2.2 Remove the maximum
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Outcome: Change H by removing its maximum (i.e., root) value wile maintaining the heap conditions
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```
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v <- H.root
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set H.root to be the value stored in the last occupied position
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p <- root
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while p has children
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if the largest value, H.c of a child of p is greater than H.p then
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swap H.c and H.p, p <-c
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else
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Break
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end if
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end while
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return v
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```
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### 0.2.3 Complexity
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In addition, we move along a branch from an added element up to the root, fixing violations as we go
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In removal, we move from the root down through some branch until all violations are fixed (can only occur if node has children)
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So both loops do most Ο(lg n)
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### 0.2.4 Storage
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- Array
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- root at position 0 and children at 1 and 2
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- children of 1 to in 3 and 4, children of 2 go in 5 and 6
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- first vacant pos --> heap[n]
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- value at pos q --> heap[q]
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- get parent of q --> parent(q) = (q-1)/2
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- get children of q --> (2 * q) ± 2
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- identify if q is root --> q == 0
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### 0.2.5 Implementation
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Use java.util.PriorityQueue
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## 0.3 Heap Sort
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In place and ϴ(n lg n)
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- start with array
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- using itself as a heap, add the elements one at a time until all been added
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- Then remove them one at a time - the largest elements gets removed first and the place where is needs to be put gets freed from the map
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## 0.4 Heap vs Merge
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heap --> in place, ϴ(n lg n)
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merge --> not in place, Ο(n lg n)
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Merge is preferred because
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- MS can take advantage of partially sorted data (hence ϴ() vs Ο())
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- MS memory accesses are good fast
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- overwrites allow for optimizations that swaps do not
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extra memory cost of merge sort is negligible
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∴ Merge sort is faster |