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SPACE COMPLEXITY using visual studio .net toencode 3 of 9 on web,windows application GS1 Barcode Types computed by a stand barcode 39 for .NET ard oracle machine that makes calls to f 1 and uses space t 1 L + i=1 (li + log2 li ). We shall apply this lemma with f i = gi and t = O(log .

V1 . ) = O(log Vt ), using the bounds L = log2 (d 2 Vt ) and li = O(1) (as Visual Studio .NET 3 of 9 guaranteed by Claim 5.9).

Indeed, in this application L equals the length of the input to f t = gt . Proof Sketch: We compute f t by allocating space for the emulation of the globaltape and the local-tapes of each level in the recursion. We emulate the recursive computation by capitalizing on the fact that all recursive levels use the same global-tape (for making queries and receiving answers).

Recall that in the actual recursion, each level may use the global-tape arbitrarily so long as when it returns control to the invoking machine the global-tape contains the right answer. Thus, the emulation may do the same, and emulate each recursive call by using the space allocated for the global-tape as well as the space designated for the localtape of this level. The emulation should also store the locations of the other levels of the recursion on the corresponding local-tapes, but the space needed for this t 1 (i.

e., i=1 log2 li ) is clearly smaller than the length of the various local-tapes (i.e.

, t 1 i=1 li ). Conclusion. Combining Claim 5.

9 and Lemma 5.10, we conclude that the evaluation of g O(log . V1 . ) can be reduced to the evaluation of g1 in space O(log V1 . ); that is, g O(log V1 . ) can be computed by a standard oracle machine that makes calls to g1 and uses space O(log V1 . ). Recalling that G .net vs 2010 barcode 3 of 9 1 can be constructed in log-space (based on the input graph G 0 ), we infer that G = G O(log .

V1 . ) can be constructe Code 3/9 for .NET d in log-space. Theorem 5.

6 follows by recalling that G (which has constant degree and logarithmic diameter) can be tested for connectivity in log-space (see Exercise 5.13). Using a similar argument, we can test whether a given pair of vertices are connected in the input graph (see Exercise 5.

15). Furthermore, a corresponding path can be found within the same complexity (see Exercise 5.17).

. 5.3. Non-deterministic Space Complexity The difference betw Visual Studio .NET Code 3 of 9 een space complexity and time complexity is quite striking in the context of non-deterministic computations. One phenomenon is the huge gap between the power of two formulation of non-deterministic space complexity (see Section 5.

3.1), which stands in contrast to the fact that the analogous formulations are equivalent in the context of time complexity. We also highlight the contrast between various results regarding (the standard model of) non-deterministic space-bounded computation (see Section 5.

3.2) and the analogous questions in the context of time complexity; one good example is the question of complementation (cf. 5.


. 5.3.1. Two Models Recall that non-det VS .NET 3 of 9 erministic time-bounded computations were de ned via two equivalent models. In the off-line model (underlying the de nition of NP as a proof system (see De nition 2.

5)), non-determinism is captured by reference to the existential choice of an auxiliary ( non-deterministic ) input. Thus, in this model, non-determinism refers. 5.3. NON-DETERMINISTIC SPACE COMPLEXITY to choices that are .NET barcode code39 transcendental to the machine itself (i.e.

, choices that are performed off-line ). In contrast, in the on-line model (underlying the traditional de nition of NP (see De nition 2.7)) non-determinism is captured by reference to the non-deterministic choices of the machine itself.

In the context of time complexity, these models are equivalent because the latter on-line choices can be recorded (almost) for free (see the proof of Theorem 2.8). However, such a recording is not free of charge in the context of space complexity.

Let us take a closer look at the relation between the off-line and on-line models. The issue at hand is the cost of emulating off-line choices by on-line choices and vice versa. We stress the fact that in the off-line model the non-deterministic choices are recorded for free on an adequate auxiliary input device, whereas such a free record is not available in the on-line model.

The fact that the on-line model can be ef ciently emulated by the off-line model is almost generic; that is, it holds for any natural notion of complexity, because on-line non-deterministic choices can be emulated by using consecutive bits of the (off-line) non-deterministic input (and without signi cantly affecting any natural complexity measure). In contrast, the ef cient emulation of the off-line model by the online model relies on the ability to ef ciently maintain (in the on-line model) a record of nondeterministic choices, which eliminates the advantage of the off-line non-deterministic input (which is recorded for free in the off-line model). This ef cient emulation is possible in the context of time complexity, because in that context a machine may store a sequence of non-deterministic choices (performed on-line) and retrieve bits of it without signi cantly affecting the running-time (i.

e., almost free of charge ). This naive emulation of the off-line choices by on-line choices is not free of charge in the context of space-bounded computation, because (in the on-line model) each on-line choice that we record (i.

e., store) is charged in the space complexity. Furthermore, typically the number of non-deterministic choices is much larger than the space bound, and thus the naive emulation is not possible in the context of space complexity (because it is prohibitively expensive in terms of space complexity).

Let us recapitulate the two models and consider the relation between them in the context of space complexity. In the standard model, called the on-line model, the machine makes non-deterministic choices on the y (as in De nition 2.7).

20 Thus, if the machine may need to refer to such a non-deterministic choice at a latter stage in its computation, then it must store this choice on its storage device (and be charged for it). In contrast, in the so-called off-line model the non-deterministic choices are provided from the outside as the bits of a special non-deterministic input. This non-deterministic input is presented on a special read-only device (or tape) that can be scanned in both directions like the main input.

We denote by NSPACEon-line (s) (resp., NSPACEoff-line (s)) the class of sets that are acceptable by an on-line (resp., off-line) non-deterministic machine having space complexity s.

We stress that, as in De nition 2.7, the set accepted by a non-deterministic machine M is the set of strings x such that there exists a computation of M on input x that is accepting. (In the case of an on-line model this existential statement refers to possible non-deterministic choices of the machine itself, whereas in the case of an off-line model we refer to a possible choice of a corresponding non-deterministic input.

). An alternative but equivalent de nition is obtained by considering machines that read a non-deterministic input from a special read-only tape that can be read only in one direction. This stands in contrast to the off-line model, where the non-deterministic input is presented on a read-only tape that can be scanned freely..

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