Update 8-2.md

docs/Chap08/Problems/8-2.md

@@ -22,20 +22,31 @@
 
 **d.** (a) Yes. (b) No. (c) No.
 
-**e.** Using $O(k)$ outside the input-arr.
+**e.**
+
+Thanks [@Gutdub](https://github.com/Gutdub) for providing the solution in this [issue](https://github.com/walkccc/CLRS/issues/150).
 
 ```cpp
-COUNTING-SORT(A, k)
+MODIFIED-COUNTING-SORT(A, k)
     let C[0..k] be a new array
-    for i = 0 to k
+    for i = 1 to k
         C[i] = 0
-    for i = 1 to A.length
-        C[A[i]] = C[A[i]] + 1    // C[i] now contains the number of elements equal to i
-    p = 0
-    for i = 0 to k
-        for j = 1 to C[i]
-            p = p + 1
-            A[p] = i
+    for j = 1 to A.length
+        C[A[j]] = C[A[j]] + 1
+    for i = 2 to k
+        C[i] = C[i] + C[i - 1]
+    insert sentinel element NIL at the start of A
+    B = C[0..k - 1]
+    insert number 1 at the start of B
+    // B now contains the "endpoints" for C
+    for i = 2 to A.length
+        while C[A[i]] != B[A[i]]
+            key = A[i]
+            exchange A[C[A[i]]] with A[i]
+            while A[C[key]] == key // make sure that elements with the same keys will not be swapped
+                C[key] = C[key] - 1
+    remove the sentinel element
+    return A
 ```
 
-Not stable, in place, in $O(n + k)$.
+In place (storage space is $\Theta(k)$) but not stable.