From a5cd69394d19afb854a537283fa447570da12007 Mon Sep 17 00:00:00 2001
From: "Brady J. Garvin" <bgarvin@cse.unl.edu>
Date: Wed, 11 Oct 2023 14:05:51 -0500
Subject: [PATCH] Recorded work from Tuesday.
---
divide-and-conquer/notes.md | 45 ++++++++++++-------
.../src/features/median/kthSmallest.js | 12 ++++-
2 files changed, 41 insertions(+), 16 deletions(-)
diff --git a/divide-and-conquer/notes.md b/divide-and-conquer/notes.md
index ce6f832..d817da9 100644
--- a/divide-and-conquer/notes.md
+++ b/divide-and-conquer/notes.md
@@ -112,7 +112,7 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
```
In: [8, 7, 6, 9, 5], k = 2
- Out: 7
+ Out: 7 (but how?)
/ \
/ \
In: [8, 7], k = 2 In: [6, 9, 5], k = 0
@@ -131,24 +131,39 @@ 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
+ j (crossed)
+ 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
+ j (crossed)
+ i
+ In: [5, 7, 6], k = 2
+ Split: [], 5, [7, 6]
+ Out: 7
|
|
|
- In: […], k = …
- In: […], k = …
- Split: […], …, […]
- Out: …
+ In: [7, 6], k = 2 - 1 (because we threw away one element, 5) = 1
+ *
+ i
+ j (crossed)
+ i
+ In: [6, 7], k = 1
+ Split: [6], 7, []
+ Out: 7
```
## Analysis
@@ -156,9 +171,9 @@ 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)
diff --git a/divide-and-conquer/src/features/median/kthSmallest.js b/divide-and-conquer/src/features/median/kthSmallest.js
index 7125d1f..0378a9b 100644
--- a/divide-and-conquer/src/features/median/kthSmallest.js
+++ b/divide-and-conquer/src/features/median/kthSmallest.js
@@ -31,5 +31,15 @@ export function kthSmallest(sequence, k) {
k >= 0 && k < sequence.length,
'Tried to find the kth smallest element when k is out of bounds.',
);
- return sequence[0]; // TODO: stub
+ if (sequence.length < 2) {
+ return sequence[0];
+ }
+ const [left, pivot, right] = partition(sequence);
+ if (k === left.length) {
+ return pivot;
+ }
+ if (k < left.length) {
+ return kthSmallest(left, k);
+ }
+ return kthSmallest(right, k - 1 - left.length);
}
--
GitLab