Designed and traced quickmedian.

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)