From 369965d5b6031ef5b2bf3e69ef2a88f57919cd04 Mon Sep 17 00:00:00 2001
From: "Brady J. Garvin" <bgarvin@cse.unl.edu>
Date: Tue, 25 Oct 2022 16:46:20 -0500
Subject: [PATCH] Designed and traced quickmedian.
---
divide-and-conquer/notes.md | 55 +++++++++++++++++++++++--------------
1 file changed, 35 insertions(+), 20 deletions(-)
diff --git a/divide-and-conquer/notes.md b/divide-and-conquer/notes.md
index f4a53c4..df4fe2a 100644
--- a/divide-and-conquer/notes.md
+++ b/divide-and-conquer/notes.md
@@ -69,11 +69,11 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
```
In: [8, 7, 6, 9, 5]
- Out: …
+ Out: ????
/ \
/ \
- In: […] In: […]
- Out: … Out: …
+ In: [8, 7] In: [6, 9, 5]
+ Out: 7.5 Out: 6
```
* Option A: We are missing information from the recursive calls. Add output(s) so that they can return that information.
@@ -81,15 +81,15 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
## Design (second attempt)
-* More generally, we want …, so we add an input `k`:
+* More generally, we want kth smallest element, so we add an input `k`:
```
In: [8, 7, 6, 9, 5], k = 2
Out: …
/ \
/ \
- In: […], k = 2 In: […], k = 0
- Out: … Out: …
+ In: [8, 7], k = 2 In: [6, 9, 5], k = 0
+ Out: ???? Out: 5
```
## Partitioning
@@ -104,24 +104,39 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
```
In: [8, 7, 6, 9, 5], k = 2
- In: […], k = 2
- In: […], k = 2
- Split: […], …, […]
- Out: …
+ *
+ i
+ j
+ In: [8, 7, 6, 5, 9], k = 2
+ *
+ i (crossed)
+ j
+ i
+ In: [5, 7, 6, 8, 9], k = 2
+ Split: [5, 7, 6], 8, [9]
+ Out: 7
|
|
|
- In: […], k = …
- In: […], k = …
- Split: […], …, […]
- Out: …
+ In: [5, 7, 6], k = 2
+ *
+ i (crossed)
+ j
+ i
+ In: [5, 7, 6], k = 2
+ Split: [], 5, [7, 6]
+ Out: 7
|
|
|
- In: […], k = …
- In: […], k = …
- Split: […], …, […]
- Out: …
+ In: [7, 6], k = 1
+ *
+ i (crossed)
+ j
+ i
+ In: [6, 7], k = 1
+ Split: [6], 7, []
+ Out: 7
```
## Analysis
@@ -129,9 +144,9 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
* Uneven split:
T(n) = T(n - 1) + Θ(n) for n ≫ 0
- T(n) = Θ(⋯)
+ T(n) = Θ(n²)
* Even split:
T(n) ≈ T(n/2) + Θ(n) for n ≫ 0
- T(n) = Θ(⋯)
+ T(n) = Θ(n)
--
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