How to avoid initializing memory [in theory]
Consider an algorithm that uses a large memory area. If the running time of the algorithm is smaller than the size of the memory, initializing the memory will take longer than running the algorithm. However, using a shrewd trick, it’s possible to refrain from initializing the memory.
This mysterious trick is used quite frequently in research articles, often without explanation and a reference to Exercise 2.12 in The Design and Analysis of Computer Algorithms by Aho, Hopcroft, and Ullman, 1974.
Develop a technique to initialize an entry of a matrix to zero the first time it is accessed, thereby eliminating the O(||V||2) time to initialize an adjacency matrix.
To add insult to injury, this exercise is marked with a single star, and it’s widely believed that the number of stars indicates the number of authors who couldn’t solve the problem either.
Can your figure it out?
The idea is to keep track of which memory positions have been used so far. Using this information, you may postpone the initialization until the first time a memory cell is encountered.
One solution would be to use an array to keep track of active memory positions. Every time a memory position is accessed, check whether the address is in this array of active memory positions. If not, initialize this memory cell and add the address to the array.
For this scheme to work without increasing the overall running time, it’s crucial that the lookups in the array can be performed in constant time. To achieve this, use double bookkeeping. Not only do you use an array of active memory positions, but each memory position is also extended to contain a pointer into this array. In this figure, for instance, the memory has three active memory cells: cell numbers 0, 4, and 6.
Memory Active ------------------------- ------- Address Contents Pointer 0 134431 X----------> 0 1 938434 --------> 4 2 432754 | -----> 6 3 292343 | | 4 874944 X--- | 5 002345 | 6 654243 X------ 7 112903
Checking whether a memory cell is active is a two-step procedure. First, you check whether the pointer attached to this memory position is valid: Does it point to a position in the active array? If not, the memory position contains garbage and has not been initialized. If the pointer is valid, you also need to check whether the corresponding pointe in the active array actually points to this memory address. If it does, the memory cell is active.
This check only requires two memory accesses. Hence, it can be performed in constant time. Adding a new address to the active list is also done in constant time: Store the memory address in the next free position in the active array and set up the pointer associated with the memory location accordingly.
See The fastest sorting algorithm? for an O(n log log n) time sorting algorithm that uses this trick, and quite a few others.
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