Simplified the introduction to models.

diff --git a/TODO b/TODO
index 8f8561e2e3e20c53034d28259a9bab5a09450892..24a643841eafd89c871fdebbf0211effd7671e14 100644 (file)
--- a/TODO
+++ b/TODO
@@ -14,7 +14,3 @@ Ranking:
 Typography:
 
 - formatting of multi-line \algin, \algout
-
-Diaz:
-
-> 2: simplify, remove proof sketches

diff --git a/ram.tex b/ram.tex
index 87bed0cadefff21e635063ce28a23230a61f2f68..74a651cf6bad05ed5a61cbbd76699116376cfea0 100644 (file)
--- a/ram.tex
+++ b/ram.tex
@@ -196,37 +196,12 @@ course trivial as the adjacency matrix has only~$\Theta(n^2)$ bits.}
 
 \para
 Compared to the RAM, the PM lacks two important capabilities: indexing of arrays
-and arithmetic instructions. We can emulate both with poly-logarithmic slowdown,
-but it will turn out that they are rarely needed in graph algorithms. We are
-also going to prove that the RAM is strictly stronger, so we will prefer to
-formulate our algorithms for the PM and use the RAM only when necessary.
-
-\thm
-Every program for the Word-RAM with word size~$W$ can be translated to a~PM program
-computing the same with $\O(W^2)$ slowdown (given a~suitable encoding of inputs and
-outputs, of course). If the RAM program does not use multiplication, division
-and remainder operations, $\O(W)$~slowdown is sufficient.
-
-\proofsketch
-Represent the memory of the RAM by a~balanced binary search tree or by a~radix
-trie of depth~$\O(W)$. Values are encoded as~linked lists of symbols pointed
-to by the nodes of the tree. Both direct and indirect accesses to the memory
-can therefore be done in~$\O(W)$ time. Use standard algorithms for arithmetic
-on big numbers: $\O(W)$ per operation except for multiplication, division and
-remainders which take $\O(W^2)$.\foot{We could use more efficient arithmetic
-algorithms, but the quadratic bound is good enough for our purposes.}
-\qed
-
-\thm
-Every program for the PM running in polynomial time can be translated to a~program
-computing the same on the Word-RAM with only $\O(1)$ slowdown.
-
-\proofsketch
-Encode each cell of the PM's memory to $\O(1)$ integers. Store the encoded cells to
-the memory of the RAM sequentially and use memory addresses as pointers. As the symbols
-are finite and there is only a~polynomial number of cells allocated during execution
-of the program, $\O(\log N)$-bit integers suffice ($N$~is the size of the program's input).
-\qed
+and arithmetic instructions. We can emulate both with slowdown $\O(W^2)$, where
+$W$~is the word size, but it will turn out that they are rarely needed in graph
+algorithms. On the other hand, a~PM program can be translated to a~RAM program
+with only constant slowdown, so the RAM is strictly stronger. We will therefore
+prefer to formulate our algorithms for the PM and use the RAM only when
+necessary.
 
 \para
 There are also \df{randomized} versions of both machines. These are equipped