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imperfect_notes/Sorting/2.md

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+Radix Sort 
+----------
+- sort elements from lowest significant to most significant values
+- explain: basically counting sort on each bit / digit
+- **Stability:** inherently stable - won't work if unstable
+- **complexity:** $O(n \log\max a[i])$
+
+
+
+
+
+Sex-Tuples
+----------
+> Given A[n], all distinct
+> find the count of sex-tuples such that
+> $$\frac{a b + c}{d} - e = f$$
+> Note: numbers can repeat in the sextuple
+
+- Naive: ${n \choose 6} = O(n^6)$
+- Optimization. Rewrite the equation as $ab + c = d(e + f)$
+    - Now, we only need ${n \choose 3} = O(n^3)$
+    - Caution: $d \neq 0$
+    - Once you have array of RHS, sort it in $O(\log n^3)$ time.
+    - Then for each value of LHS, count using binary search in the sorted array in $\log n$ time.
+    - Total: $O(n^3 \log n)$
+