Add comparison of segment tree to simple array and prefix sums

jxu · web-flow · commit 577b49ceb128 · 2022-07-11T14:42:51.000-04:00
diff --git a/src/data_structures/segment_tree.md b/src/data_structures/segment_tree.md
@@ -24,7 +24,10 @@ To start easy, we consider the simplest form of a Segment Tree.
 We want to answer sum queries efficiently. 
 The formal definition of our task is:
 We have an array $a[0 \dots n-1]$, and the Segment Tree must be able to find the sum of elements between the indices $l$ and $r$ (i.e. computing the sum $\sum_{i=l}^r a[i]$), and also handle changing values of the elements in the array (i.e. perform assignments of the form $a[i] = x$). 
-If we computed each query naively, each query would take $O(n)$ time. The Segment Tree should be able to process both queries in $O(\log n)$ time.
+The Segment Tree should be able to process **both** queries in $O(\log n)$ time.
+
+A naive array implementation can update elements in $O(1)$, but requires $O(n)$ to recompute each sum query.
+Precomputed prefix sums can compute sum queries in $O(1)$, but updating an array element requires $O(n)$ changes to the prefix sums. 
 
 ### Structure of the Segment Tree
 
