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_posts/2024-02-25-cs-6006.md

@@ -257,7 +257,7 @@ title: CS 6.006 笔记
 ### Master Theorem
 
 The Master Theorem provides a way to solve recurrence relations in which recursive calls decrease problem size by a constant factor. Given a recurrence relation of the form $T(n) = aT(n/b)+ f(n)$ and $T(1) = Θ(1)$, with branching factor $a ≥ 1$, problem size reduction factor $b > 1$, and
-asymptotically non-negative function $f(n)$, the Master Theorem gives the solution to the recurrence by comparing $f(n)$ to $a^{log_bn} = n^{log_b a}$, the number of leaves at the bottom of the recursion
+asymptotically non-negative function $f(n)$, the Master Theorem gives the solution to the recurrence by comparing $f(n)$ to $a^{log_bn} = n^{log_b a}$, the number of leaves at the bottom of the recursion
 tree. When $f(n)$ grows asymptotically faster than $n^{log_b a}$, the work done at each level decreases
 geometrically so the work at the root dominates; alternatively, when $f(n)$ grows slower, the work
 done at each level increases geometrically and the work at the leaves dominates. When their growth
@@ -398,7 +398,7 @@ def direct_access_sort(A):
 - Idea! Represent each key $k$ by tuple $(a, b)$ where $k = an + b$ and $0 ≤ b < n$
 - Specifically $a = bk/nc < n$ and $b = (k mod n)$ (just a 2-digit base-n number!)
 - This is a built-in Python operation (a, b) = divmod(k, n)
-- Example: $[17, 3, 24, 22, 12] \rightarrow [(3,2), (0,3), (4,4), (4,2), (2,2)] \rightarrow \[32, 03, 44, 42, 22\](n = 5)$
+- Example: [17, 3, 24, 22, 12] ⇒ [(3,2), (0,3), (4,4), (4,2), (2,2)] ⇒ [32, 03, 44, 42, 22\](n=5)
 - How can we sort tuples?
 
 ### Tuple Sort
@@ -408,7 +408,7 @@ def direct_access_sort(A):
 - How to sort? **Idea**! Use other **auxiliary sorting algorithms** to separately sort each key
 - (Like sorting rows in a spreadsheet by multiple columns)
 - What order to sort them in? Least significant to most significant!
-- Exercise: $[32, 03, 44, 42, 22] =⇒ [42, 22, 32, 03, 44] =⇒ \[03, 22, 32, 42, 44\](n=5)$
+- Exercise: [32, 03, 44, 42, 22] =⇒ [42, 22, 32, 03, 44] =⇒ [03, 22, 32, 42, 44\](n=5)
 - **Idea**! Use tuple sort with auxiliary direct access array sort to sort tuples (a, b).
 - **Problem**! Many integers could have the same a or b value, even if input keys distinct
 - Need sort allowing **repeated keys** which preserves input order